causality in 2d regge calculus

[David Hillman]

I'm playing with 2d Lorentzian regge calculus to try to see how it works
and in the process learn something about GR. I understand how the edge
lengths determine a Lorentz metric, how to glue together two triangles
at an edge by laying them out in Minkowski space. I understand the
conditions to ensure that the light cone structure works correctly at
edges and at vertices. I know how to compute a deficit angle. I
understand why this setup forces the Euler characteristic to be zero.

So, now I'm trying to understand how causality works. Consider a
hexagonal lattice with horizontal edges having squared-length 5 and
other edges having squared-length 1. All edges spacelike. This is a
triangulation of Minkowski space. So there is seemingly no problem
telling how geodesics travel through the vertex. A horizontal line is a
spacelike geodesic and looks like a Cauchy surface.

Now, modify the above by changing arbitrary squared-length-5 edges to
squared-length-6. (All edges still spacelike.) This is still seemingly a
valid Lorentzian regge thing. Now there is nonzero curvature wherever
you have a vertex that hits triangles containing at least one 5-edge and
one 6-edge.

Suppose all horizontal lattice lines on or below a given one (call it
line 1) have all 5-edges on them. On line 2 suppose some edges are 5's
and some are 6's. Line 0 (the line below line 1) is still a spacelike
geodesic. Propagate this seemingly Cauchy surface upwards. You approach
line 1. Some of the vertices on line 1 have nonzero curvature and some
don't. But this seemingly has nothing to do with what came before.
Everything looks the same everywhere on line 0 and below. So how are
these vertices with curvature caused?

Maybe I am not understanding what causality means in the context of GR?
I thought the metric at/near a Cauchy surface determined the metric
everywhere (in the globally hyperbolic case). Or, am I doing something
wrong regge-wise? Or, is the regge thing simply not analogous to the
continuous case? Or what?

[Chris Hillman]

On Thu, 27 Jun 2002, David Hillman wrote:

> I'm playing with 2d Lorentzian regge calculus

BTW, have you seen this?

http://xxx.lanl.gov/abs/gr-qc/0206077

I haven't had a chance to even skim it, alas, but perhaps Steve Carlip can
comment on whether this paper appears to have any obvious bearing on the
conjecture that the general solution of the EFE (or at least a very large
class of solutions) can be expressed in terms of elliptic modular curves?

Chris Hillman

Home page: http://www.math.washington.edu/~hillman/personal.html

[Chris Hillman]

On Thu, 27 Jun 2002, David Hillman wrote:

> I'm playing with 2d Lorentzian regge calculus

As in E^(1,1) or E^(1,2) tangent spaces? (I'll assume the latter.)

> to try to see how it works and in the process learn something about GR.

That's a reasonable idea in principle, but unfortunately it turns out that
gtr is quite different in 2 versus 3 dimensions: in 2 dimensions there are
no long range gravitational forces! Reason: the Weyl tensor vanishes
identically. This is discussed in most gtr textbooks, including MTW.
You might also be interested in this book:

Steve Carlip,
Quantum Gravity in 2+1 Dimensions,
Cambridge University Press, 1998

> I understand how the edge lengths determine a Lorentz metric, how to
> glue together two triangles at an edge by laying them out in Minkowski
> space.

OK, I take that back: I'll assume you mean E^(1,1) :-/

> I understand the conditions to ensure that the light cone structure
> works correctly at edges and at vertices. I know how to compute a
> deficit angle. I understand why this setup forces the Euler
> characteristic to be zero.

Whoa! That doesn't sound right!

Are you somehow secretly assuming that the spacetime is topologically
spherical? If so, you should look for old posts by Robert Low citing
papers explaining why some topologies are inconsistent with extra
Lorentzian structure imposed on a topological manifold.

> So, now I'm trying to understand how causality works. Consider a
> hexagonal lattice with horizontal edges having squared-length 5 and
> other edges having squared-length 1. All edges spacelike. This is a
> triangulation of Minkowski space.

Whoa--- I'm now -completely- confused! I have no idea what you might
mean, but I hope Steve Carlip or John Baez or Robert Low or Charles Torre
(or someone else who has published papers/books on gravitation) are
reading this and can help.

> I thought the metric at/near a Cauchy surface determined the metric
> everywhere (in the globally hyperbolic case).

That sounds OK.

> Or, am I doing something wrong regge-wise?

Hmm... I guess you -might- somehow be confusing levels of structure and
misunderstanding some essential notion behind the Regge calculus.

Have you read the chapter in MTW on the Regge calculus yet? (But note
that ideally you should study all of the "Track I" material first,
supplemented by a more recent textbook like Wald. I can understand the
temptation to cut corners but when it comes to curved spacetime I think
this seems to have been a bad idea for almost everyone who has tried it.)

Note too that IIRC there is at least one preprint on the LANL server
concerning Regge calculus, and other papers can be found in the print
literature.

Sorry I (at least) cannot be more helpful.

[David Hillman]

Chris Hillman wrote:

> On Thu, 27 Jun 2002, David Hillman wrote:
>
> > I'm playing with 2d Lorentzian regge calculus

...

> > I understand the conditions to ensure that the light cone structure
> > works correctly at edges and at vertices. I know how to compute a
> > deficit angle. I understand why this setup forces the Euler
> > characteristic to be zero.
>
> Whoa! That doesn't sound right!
>
> Are you somehow secretly assuming that the spacetime is topologically
> spherical? If so, you should look for old posts by Robert Low citing
> papers explaining why some topologies are inconsistent with extra
> Lorentzian structure imposed on a topological manifold.

No assumptions. Euler characteristic zero in 2-d means it's a torus or klein
bottle...and I didn't assume this; it just popped out of my calculations.
Which made me happy since in Hawking & Ellis pp. 39-40 it says that a compact
manifold admits a Lorentz metric iff its Euler invariant is zero. (I meant
that Euler = 0 in finite-triangulation (compact) case. Don't know how to
define Euler in other cases. According to Hawking/Ellis any noncompact
manifold admits a Lorentz metric.)

> > So, now I'm trying to understand how causality works. Consider a
> > hexagonal lattice with horizontal edges having squared-length 5 and
> > other edges having squared-length 1. All edges spacelike. This is a
> > triangulation of Minkowski space.
>
> Whoa--- I'm now -completely- confused! I have no idea what you might
> mean, but I hope Steve Carlip or John Baez or Robert Low or Charles Torre
> (or someone else who has published papers/books on gravitation) are
> reading this and can help.

Probably I confused you with the word hexagonal; was thinking about the dual.
A lattice like this:

--5--*--5--*--5--*--5--
/ \ / \ / \ /
/ \ / \ / \ /
--*--5--*--5--*--5--*--
/ \ / \ / \ / \
/ \ / \ / \ / \
--5--*--5--*--5--*--5--

I suppose that's called a triangular lattice? Anyway, there it is. The *'s are
vertices. Every edge is spacelike and has squared length 1, except for the
horizontal ones, which have squared length 5. This triangulates Minkowski
space with null lines running at an angle with the horizontal that is greater
than the angle of those lattice lines.

My point was that you can change any of those 5's to 6's and still have a
valid-seeming Regge 2-d thing. What does that say about causality?

This is related to the following question. Can one modify Minkowski space by
putting a little bump of curvature in the middle of it, leaving the rest flat?
If so, then I don't understand how causality works, since a Cauchy surface on
one side of the bump could not tell the difference between this space and
Minkowski space. But maybe such bumps cannot be constructed when the metric is
sufficiently continuous?

[Chris Hillman]

On Wed, 3 Jul 2002, David Hillman wrote:

> > > I understand the conditions to ensure that the light cone structure
> > > works correctly at edges and at vertices. I know how to compute a
> > > deficit angle. I understand why this setup forces the Euler
> > > characteristic to be zero.
> >
> > Whoa! That doesn't sound right!
> >
> > Are you somehow secretly assuming that the spacetime is topologically
> > spherical? If so, you should look for old posts by Robert Low citing
> > papers explaining why some topologies are inconsistent with extra
> > Lorentzian structure imposed on a topological manifold.
>
> No assumptions. Euler characteristic zero in 2-d means it's a torus or klein
> bottle...

Yes. By the way, I was trying to express two thoughts at once (never a
good idea!), and only noticed later that it probably looked like I think
that S^2 has Euler characteristic -zero-. Fear not; I know better than
that.

> and I didn't assume this; it just popped out of my calculations.

I'm still confused, but never mind.

> Which made me happy since in Hawking & Ellis pp. 39-40 it says that a
> compact manifold admits a Lorentz metric iff its Euler invariant is
> zero. (I meant that Euler = 0 in finite-triangulation (compact) case.
> Don't know how to define Euler in other cases.

How about Euler-Poincare characteristic? (From the homology groups.)
See e.g. the algebraic topology text by Greenberg & Harper.

> > > So, now I'm trying to understand how causality works. Consider a
> > > hexagonal lattice with horizontal edges having squared-length 5 and
> > > other edges having squared-length 1. All edges spacelike. This is a
> > > triangulation of Minkowski space.
> >
> > Whoa--- I'm now -completely- confused!
>

> Probably I confused you with the word hexagonal; was thinking about the
> dual. A lattice like this:
>
> --5--*--5--*--5--*--5--
> / \ / \ / \ /
> / \ / \ / \ /
> --*--5--*--5--*--5--*--
> / \ / \ / \ / \
> / \ / \ / \ / \
> --5--*--5--*--5--*--5--

(Re the ongoing threads concerning Coxeter groups, root lattices, regular
polytopes, and suchlike: if 5 equalled 1, that would the root lattice A_2,
heh. Someone just asked whether the Weyl groups W(D_n) are the dihedral
groups (no) and I should have said, even though John explained this in a
recent week, that W(A_n) are not the alternating groups Alt(n) but the
-symmetric- groups Sym(n). Again, I plan to discuss all this in detail in
a forthcoming post in the thread "Computational Invariant Theory,
Anyone?".)

> I suppose that's called a triangular lattice? Anyway, there it is. The
> *'s are vertices. Every edge is spacelike and has squared length 1,
> except for the horizontal ones, which have squared length 5.

The horizontal edges are spacelike, right?

> This triangulates Minkowski space with null lines running at an angle
> with the horizontal that is greater than the angle of those lattice
> lines.

Hmm... if you are only studying compact spacetimes with characteristic
zero, i.e. a two topological torus equipped with a "flat" Lorentzian
metric, you must mean you are triangulating the "flat torus", right?
E.g. cut a square (two edges spacelike and two timelike) from E^(1,1) and
identify opposite edges in the usual way to make a topological torus,
keeping the Lorentz metric tensor in the interior and extending it the the
two circles in the obvious way (the metric is then smooth so we have a new
Lorentzian manifold with toroidal topology and vanishing Gaussian
curvature).

> My point was that you can change any of those 5's to 6's and still have a
> valid-seeming Regge 2-d thing. What does that say about causality?

I'm confused. Are you thinking of something like this?

Some E^2 integer sided right triangles:

/| /|
5 / | 4 13 / | 12
/__| /__|

3 5

Some E^(1,1) integer sided right triangles (horizontal edge spacelike;
other two timelike):

/| /|
3 / | 5 5 / | 13
/__| /__|

4 12

Some E^(1,1) isoceles triangles (all edges timelike):

/| /|
3 / | 5 / |
/ | 10 / |
\ | \ | 26
3 \ | 5 \ |
\| \|

Or, you can turn these on their side so that all edges are -spacelike-.

What you say "change any of those 5's to 6's", do you have in mind tending
to a limit in which two edges of each triangle become -null-, or do you
have in mind introducing some curvature at some vertices by altering the
edge lengths to the triangles "pop out of the plane"?

(Note that for triangulated manifolds using simplices whose interiors are
flat [vanishing Riemann tensor], curvature is concentrated in the
codimension two facets; in three dimensional triangulations these are
edges; in two dimensional triangulations they are vertices.)

I still don't see what this has to do with "causality". Maybe someone
else can explain what I'm missing?

> This is related to the following question. Can one modify Minkowski
> space by putting a little bump of curvature in the middle of it, leaving
> the rest flat?

Of course! Same way you'd do that for say E^3; take any spherically
symmetric bump function f(r) supported on the unit ball with f(0) = 1, or
more generally any bump function of compact support, and consider

ds^2 = f(r)^2 dr^2 + r^2 (du^2 + sin(u)^2 dv^2),

0 < r < infty, 0 < u < Pi, -Pi < v < Pi

> If so, then I don't understand how causality works, since a Cauchy
> surface on one side of the bump could not tell the difference between
> this space and Minkowski space.

Are you somehow asking about the ADM reformulation of "local gtr"? If so,
then clearly you cannot expect to take two solutions to the EFE and "mix
them" using a partition of unity (a finite set of smooth [continuous] bump
functions everywhere summing to zero with support on some finite open
cover of your compact smooth [topological] manifold). You might try
searching the LANL preprint server under "Superluminal Censorship" for a
"no go" theorem which might be relevant to what you may or may not be
asking.

> But maybe such bumps cannot be constructed when the metric is
> sufficiently continuous?

Are you talking about the Regge calculus for constructing approximate
solutions in gtr? If so, did you find the chapter in MTW helpful?

I'm more confused than ever about what you are trying to do--- to
understand and use the Regge calculus? To understand the ADM
reformulation of local gtr? To understand restrictions on modifying a
valid solution to the EFE using bump functions, or blending two valid
solutions using partitions of unity?

[David Hillman]

Chris Hillman wrote:

> I'm still confused, but never mind.

Thanks for still replying even though I don't seem to be making myself clear.

First of all, since you keep asking about my background: I have taken a graduate
course (with Newman at Pitt) in general relativity. I did fine in the course, in
that I could solve most of the assigned problems. But (through no fault of
Newman's) I never got a feel for the subject, the sort of intuition and
understanding that I like to have about things. I am trying to improve that now,
at least a little bit. It doesn't help that the course was about ten years ago.
Yes, I have MTW and Wald and Hawking/Ellis and O'Neill's "Semi-Reimannian
Geometry" and Pauli and Spivak volume 1. And I even try to read parts of them on
occasion, with limited success.

So, I am starting small, and looking at Regge calculus in two dimensions. Not even
Regge calculus: I am not yet thinking about the Einstein equations or the action.
I'm just trying to define, a la Regge, a piecewise-linear sort of Lorentzian
metric in 2-d. I'm thinking about 2-d manifolds made out of triangles with
"squared-lengths" (v^a g_ab v^b; these can be positive, negative or zero) assigned
to the edges.

Here is what I figured out. Consider the following triangle:

*
/ \
b c
/ \
*---a---*

Here a, b and c stand for the squared-lengths of the edges that they are
labelling. The first fact is: the induced metric is Lorentzian iff a^2 + b^2 + c^2
- 2ab - 2ac - 2bc > 0.

Fine. So any assignment of real numbers to edges gives you a valid Lorentzian
triangle if it satisfies this property. For instance, {a=1, b=1, c=5} works, since
1^2 + 1^2 + 5^2 - 2 1 1 - 2 1 5 - 2 1 5 = 5 > 0. It is easy to show that:

1) if two of the three numbers differ in sign, the triangle is valid;

2) if one of the three numbers is zero, the triangle is valid iff the other two
numbers are not equal;

3) if all three numbers are positive, then the triangle is valid iff one of them
(say c) satisfies the inequality c > a + b + 2 Sqrt[ab];

4) if all three numbers are negative, then the triangle is valid iff one of them
(say c) satisfies the inequality c < a + b - 2 Sqrt[ab].

So now it is easy to write down Lorentzian triangles. Is that sufficient
information to write down a triangulated 2-d Regge-thing? The answer is seemingly
no, because just making the triangles be Lorentzian doesn't mean that the
light-cone structure will work properly at edges or vertices where the triangles
meet. Let's look at two triangles meeting at an edge.

Y
/ \
b c
/ \
V---a---W
\ /
d e
\ /
Z

If a > 0, then we can lay this out in Minkowski space by putting vertex V at the
origin and putting vertex W at the point {Sqrt[a],0} on the positive x axis. Now
it is straightforward to show that there are exactly two possible choices for
vertex Y: one is above the x axis and one is below. Similarly, there are two
choices for vertex Z. We choose one for Y, the other for Z, and we've successfully
laid out both triangles in Minkowski space. So: two triangles meeting at a
spacelike edge always join in a coherent manner. (The only thing left undetermined
is the orientation of space and time...but this is not determined in GR anyway.)

Same thing works, switched 90 Euclidean degrees, if a < 0. Now suppose a = 0.
Recall that we must have b != c. We can put V at the origin and W at {1,1}. Given
that choice, it turns out there is exactly one place to put Y. It is to the left
of the x = t line if c > b and to the right of that line otherwise. This implies
that there is a coherence condition if two triangles join at a null edge. The
condition (given the above triangles) is: if c > b then d < e, and vice versa: if
c < b then d > e. It is easy to see that this condition has to do with ensuring
that the light cone structure works properly: it ensures that, if we look at lines
emerging from vertex V, on one side of the null line they will be spacelike, and
on the other they will be timelike.

Next I considered vertices. Now we can't typically lay things out in Minkowski
space anymore, but one thing we can still do is try to make sure that the light
cone at a vertex looks normal. That is: the rays emerging from a vertex should be
spacelike, then timelike, then spacelike, then timelike as you go around the
vertex, with each region separated by a null ray. The fact that things will
alternate between spacelike and timelike is guaranteed by the coherence condition
described in the previous paragraph. So all we need to ensure is that there be
four null rays emerging from each vertex. Without a new coherence condition to
ensure this, there might be any even number of null rays (including none) emerging
from a vertex.

So, what is this condition? Consider triangle abc in the above diagram again, and
look at its angle at vertex v. We can give this angle a score: the number of null
rays emerging from it inside that triangle. Each null ray that goes through the
interior of the triangle will be given 2 points. Each null ray that is an edge of
the triangle will be given 1 point (since it will be in two triangles, and
therefore be counted twice). Our coherence condition will then be that the sum of
scores assigned to the angles meeting at a vertex must be 8. Here are the possible
angles and their scores:

1) a > 0, b > 0, a + b - c > 0 Score: 0
2) a > 0, b > 0, a + b - c < 0 Score: 4
3) a > 0, b < 0 Score: 2
4) a > 0, b = 0, a + b - c > 0 Score: 1
5) a > 0, b = 0, a + b - c < 0 Score: 3
6) a = 0, b < 0, a + b - c > 0 Score: 3
7) a = 0, b < 0, a + b - c < 0 Score: 1
8) a = 0, b = 0 Score: 2
9) a < 0, b < 0, a + b - c > 0 Score: 4
10) a < 0, b < 0, a + b - c < 0 Score: 0

(where I have omitted some cases that can be obtained from these by switching the
roles of a and b). The above table is the result of a calculation I did; it turned
out that a + b - c was the relevant quantity in all cases where there was a
question of whether there existed an interior null line or not. So, to repeat
myself, the condition is: you have a valid light-cone structure at your vertex iff
the scores of the angles surrounding that vertex, as given above, sum to 8.

Try finding any of this in MTW! Certain details are omitted. (Actually most
descriptions of Regge calculus seem to stick to the Euclidean case, which is much
simpler: it doesn't involve any coherence conditions. This focus of the literature
on the Euclidean case also made it difficult for me to understand the notion of
deficit angle etcetera in the Lorentzian case. Finally, thanks in part to a paper
by Sorkin and some stuff in Pauli, now I know such things as: the angles involved
in the Lorentzian case are complex, and some of them are infinite, and the deficit
angle is always finite and pure imaginary; and I know how to compute the deficit
angle...but never mind all that for now.)

It then occurred to me that, if the sum of scores of angles of a given triangle is
always 4, then I can prove that the Euler characteristic of a finite triangulation
satisfying the above conditions must be equal to zero. Why? Because it is a
triangulation we have 3F = 2E. If the sum of scores of angles for each triangle is
4, and the sum of scores at each vertex is 8, then we have 4F = 8V. Then V - E + F
= F/2 - 3F/2 + F = 0. I then enumerated all the different types of Lorentz
triangles and showed that, yes, the score of the angles always does sum to 4. I'll
omit the details, since no doubt there is an easier way to see this.

Next, I realized that the following lattice (infinite triangular lattice) is not
only a valid Regge 2-d thing (satisfying all of the above Lorentz and coherence
conditions), but is flat: it is a triangulation of Minkowski space. Here the edges
all are spacelike, with the horizontal ones having squared-length 5 and the others
having squared-length 1.

--5--*--5--*--5--*--5--
/ \ / \ / \ /
/ \ / \ / \ /
--*--5--*--5--*--5--*--
/ \ / \ / \ / \
/ \ / \ / \ / \
--5--*--5--*--5--*--5--

Next I realized that if I change any of these squared-length 5 edges to
squared-length 6, I also have a valid 2-d Regge thing, Lorentzian and satisfying
all coherence conditions. If I change all of the 5's to 6's, I again get Minkowski
space. If I change just some of the 5's to 6's then there are vertices which have
nonzero deficit angles.

This puzzled me to no end, and I seem to be having trouble explaining why. I will
now try again.

> I'm confused. Are you thinking of something like this?
>
> Some E^2 integer sided right triangles:
>
> /| /|
> 5 / | 4 13 / | 12
> /__| /__|
>
> 3 5
>
> Some E^(1,1) integer sided right triangles (horizontal edge spacelike;
> other two timelike):
>
> /| /|
> 3 / | 5 5 / | 13
> /__| /__|
>
> 4 12
>
> Some E^(1,1) isoceles triangles (all edges timelike):
>
> /| /|
> 3 / | 5 / |
> / | 10 / |
> \ | \ | 26
> 3 \ | 5 \ |
> \| \|
>
> Or, you can turn these on their side so that all edges are -spacelike-.

The triangles in my lattice are of the latter sort: all edges spacelike.

> What you say "change any of those 5's to 6's", do you have in mind tending
> to a limit in which two edges of each triangle become -null-, or do you
> have in mind introducing some curvature at some vertices by altering the
> edge lengths to the triangles "pop out of the plane"?

The latter: just change some 5's to 6's and leave the other edges as they were.
This introduces curvature at some vertices.

> > This is related to the following question. Can one modify Minkowski
> > space by putting a little bump of curvature in the middle of it, leaving
> > the rest flat?
>
> Of course! Same way you'd do that for say E^3; take any spherically
> symmetric bump function f(r) supported on the unit ball with f(0) = 1, or
> more generally any bump function of compact support, and consider
>
> ds^2 = f(r)^2 dr^2 + r^2 (du^2 + sin(u)^2 dv^2),
>
> 0 < r < infty, 0 < u < Pi, -Pi < v < Pi

Ah: the Euclidean case again. But it will be a good exercise for me to try to do
the analogous thing in 2-d Minkowski space. Though this construction may not be
the sort of example I am looking for.

Okay: let me try again to explain my problem. As I understand it, causality means
something like: given data on a Cauchy surface, you can compute the rest of
whatever-it-is you are looking at. In GR, I thought this meant, roughly: given
what the metric looks like on a Cauchy surface (and in its immediate
neighborhood), you can compute the rest of spacetime. (I understand that really
this only is supposed to work locally in GR.) Now in my example where I change a 5
to a 6, there seems to be curvature introduced only at a few points. Like this
(the x's mark vertices containing curvature):

t
|
|2 x
| x x
|1 x
|
-------------------------------------------x
|
|
|
|
|
|

Look at the Cauchy surface t = 0. Everything there is perfectly flat. Given all
the data about the manifold around that line, how can you predict that up ahead in
that very same manifold is some curvature? Obviously you can't.

Now the reason why I'm not sure that your example is quite like mine is this:
suppose you try to introduce a bump of curvature of radius 1 about the origin in
Minkowski space. A ball of radius 1 in Minkowski space is a huge thing that
includes both null lines passing through the origin. So it passes through all
Cauchy surfaces. So if you live on a Cauchy surface in this space, you have some
information about the bump of curvature. In my example you don't. So your example
does not puzzle me, but mine does.

So, I hope you or someone can tell me: am I all mixed up? Or does a Minkowski
space modified to have a few points with curvature and the rest not violate
causality? And if so, why is my Regge thing not capturing this important aspect of
GR? Is it because causality requires a metric that is C^n for n >= some number?

[eric alan forgy]

Hi,

I am no expert in Regge calculus either, but I'm interested in lattice
theories and have done quite a bit of meditating on the subject, so I'll
go ahead and give my 2 cents.

[snip of evidence that you have put quite a bit of effort into this :)]

> Okay: let me try again to explain my problem. As I understand it, causality means
> something like: given data on a Cauchy surface, you can compute the rest of
> whatever-it-is you are looking at. In GR, I thought this meant, roughly: given
> what the metric looks like on a Cauchy surface (and in its immediate
> neighborhood), you can compute the rest of spacetime. (I understand that really
> this only is supposed to work locally in GR.) Now in my example where I change a 5
> to a 6, there seems to be curvature introduced only at a few points. Like this
> (the x's mark vertices containing curvature):
>
>
> t
> |
> |2 x
> | x x
> |1 x
> |
> -------------------------------------------x
> |
> |
> |
> |
> |
> |
>
> Look at the Cauchy surface t = 0. Everything there is perfectly flat. Given all
> the data about the manifold around that line, how can you predict that up ahead in
> that very same manifold is some curvature? Obviously you can't.

My thought is that, sure you can pinch and squeeze a Lorentzian Regge
lattice all you like (within constraints that you have painfully worked
out in gory detail). However, there is no gaurantee that this playdough
manifold (I don't know why "Mr Bill" comes to mind) will have any relation
to anything "physical." If you lived in a universe where this manifold is
admissable, then there is no way to predict the crumples if you are
starting from t = 0. However, who says that this manifold is a solution
to GR? By my admitted naivety, I would say that your very arguments
"prove" that this playdough manifold is NOT a solution to GR and hence we
shouldn't lose sleep over it anyway.

Just think about it physically. I am sitting peacefully in a flat
Minkowski spacetime and all of a sudden I feel wild gravitational forces
pulling on me, tugging me left and right. Where is conservation of
momentum? That is what I imagine happening when I look at your playdough
manifold. It seems very unphysical to me.

> So, I hope you or someone can tell me: am I all mixed up? Or does a Minkowski
> space modified to have a few points with curvature and the rest not violate
> causality? And if so, why is my Regge thing not capturing this important aspect of
> GR? Is it because causality requires a metric that is C^n for n >= some number?

I could be way off, but I think the problem is that you haven't even
included the field equations yet and are trying to make physical
deductions about a fictitious simplicial manifold that is not even
physical in the first place.

What I would do is start out with a Riemannian triangulation and let this
evolve according to the (2+1)-d field equations and see what happens. I'm
not sure that you will see anything interesting until you go to (3+1)-d,
but it's a start :)

Good luck,
Eric

[Chris Hillman]

On Fri, 12 Jul 2002, David Hillman wrote:

> Thanks for still replying even though I don't seem to be making myself
> clear.

BTW, I haven't forgotten that we corresponded some years ago regarding
something (cellular automata?) and at one point Doug Lind (who was on my
own thesis commitee) lent me a copy of your Ph.D. thesis. Assuming I
haven't confused you with someone else, that is! Assuming my memory is
accurate, you proposed to study cellular automata in which the lattice is
itself allowed to dynamically evolve by some kind of combinatorial
operations like splitting a vertex

\ / \ /
* *
| |
| *
| / \

I thought that was intriguing, although I had seen similar ideas before
(but not developed very far).

> First of all, since you keep asking about my background: I have taken a
> graduate course (with Newman at Pitt) in general relativity.

Oh, wow--- Newman's recent papers are amazing! :-) In a very classical
sort of way, of course.

(GTR is in imminent danger of becoming passe in theoretical physics, since
current interest is rightly focused on replacing it with a quantum theory,
but as I recently remarked, GTR will no doubt if anything -increase- in
importance in astrophysics over the new century, and therefore this theory
is still well worth studying in great detail, even though it is now very
old. Just as Newtonian gravitation is still very well worth studying in
great detail--- I happen to be particularly fascinated by resonances and
mode-locking phenomena, for instance. There remain -fundamental-
phenomena in each theory which are -still- incompletely or indeed only
very poorly understood. So if your interest is more in mathematical
challenges or astrophysics than in -fundamental- physics, pursing
understanding of gtr may be worth some time and effort. If your interest
is in fundamental physics, e.g. quantum gravity, it makes sense to try for
only the bare minimal understanding of gtr which is needed to study the
main approaches to quantum gravity which have been introduced so far.)

> I did fine in the course, in that I could solve most of the assigned
> problems. But (through no fault of Newman's) I never got a feel for the
> subject, the sort of intuition and understanding that I like to have
> about things. I am trying to improve that now, at least a little bit.
> It doesn't help that the course was about ten years ago. Yes, I have MTW
> and Wald and Hawking/Ellis and O'Neill's "Semi-Reimannian Geometry"

Good, good. Assuming your main interest is not in quantum gravity or
fundamental physics, I'd urge you to buy two or three more from this list:

author = {Bernard F. Schutz},
title = {A First Course in General Relativity},
publisher = {Cambridge University Press},
year = 1985}

(very good for what it covers-- should help give general intuition for
important topics like gravitational radiation),

author = {Hans Stephani},
title = {General Relativity: An Introduction of the Theory of the
Gravitational Field},
publisher = {Cambridge University Press},
edition = {Second},
note = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},
year = 1990}

(nice discussion of a few exact solutions beyond Kerr and friends, well
organized and tasteful selection of material),

author = {Robert M. Wald},
title = {General Relativity},
publisher = {University of Chicago Press},
year = 1984}

(standard textbook, very readable; best text for connections with the rest
of physics/math, no pun intended),

author = {F. de Felice and C.J.S. Clarke},
title = {Relativity on Curved Manifolds},
publisher = {Cambridge University Press},
year = 1990}

(best for measurement theory and the real ONBs I employ as a matter of
habit, since the so-called "physical components" of tensors expanded wrt
such "anholonomic bases" have immediate physical interpretations),

author = {Ray D'Inverno},
title = {Introducing {E}instein's Relativity},
publisher = {Clarendon Press},
year = 1995}

which is mostly very clear and again has a excellent choice of topics, but
is perhaps a bit too vague in places.

These books are all in print and available in inexpensive paperback
editions.

FWIW, what worked for me was solving the EFE a gazillion times and
systematically trying to apply all the nifty trickery I read about in
textbooks (later, research papers) to specific solutions. If you
interested in trying a similar route to intuitive understanding, I'd
suggest that you start by reading three survey papers:

author = {Jiri Bic\'ak},
title = {Selected solutions of {E}instein's field equations:
their role in general relativity and astrophysics},
booktitle = {{E}instein Field Equations and Their Physical Implications
(Selected essays in honour of {J}uergen {E}hlers)},
editor = {Bernd G. Schmidt},
publisher = {Springer-Verlag},
year = 2000,
series = {Lecture Notes in Physics},
volume = 540,
note = {gr-qc/0004016}}

author = {W. B. Bonnor},
title = {Physical interpretation of vacuum solutions of {E}instein's
equations. {P}art {I}: {T}ime independent solutions},
journal = {Gen. Rel. Grav.},
volume = 24,
pages = {551--574},
year = 1992}

author = {W. B. Bonnor and J. B. Griffiths and M. A. H. MacCallum},
title = {Physical interpretation of vacuum solutions of {E}instein's
equations. {P}art {II}: {T}ime dependent solutions},
journal = {Gen. Rel. Grav.},
volume = 26,
pages = {687--729},
year = 1994}

author = {J\"urgen Ehlers and Wolfgang Kundt},
title = {Exact Solutions of the Gravitational Field Equations},
booktitle = {Gravitation: an Introduction to Current Research},
editor = {Louis Witten},
publisher = {Wiley},
year = 1962,
pages = {49--101}}

The paper by Elhers and Kundt is of course now very old but still well
worth reading. Bonnor's review of stationary vacuums is strangely worded
in places but also very good. The recent review by Bicak is superb.

And coming July 15 is the long-awaited second edition of the monograph:

author = {D. Kramer and H. Stephani and E. Herlt and M. MacCallum},
title = {Exact Solutions of {E}instein's Field Equations},
publisher = {Cambridge University Press},
series = {Cambridge monographs on mathematical physics},
volume = 6,
year = 1980}

I should however point out that there are specific reasons why studying
exact solutions can be very misleading if you are not aware of certain
issues. Indeed, classic work of Marsden et al. shows that roughly
speaking for spacetimes with compact hyperslices, the solutions which
exhibit a certain kind of "instability" under "linearized perturbations"
are precisely the ones which have at least one Killing vector field
(symmetry). Even more roughly speaking: the solutions with at least one
dimensional symmetry group are precisely the "atypical" ones. The point
is of course that not very many exact solutions are known which have no
symmetries at all! See the textbook by Wald for a brief discussion of
Marsden et al. (Robert Low knows more about this than I do, and if I've
gotten anything seriously wrong no doubt he'll correct what I just said.)

I should also add that of course every time I read another book/paper (at
least, a good paper) or solve the EFE for the n-th time and then try to
figure out what the "new" solution might be good for, I learn more.

> Pauli

The 1921 book on relativity?!! I'd -strongly- advise against trying to
learn gtr first time through from any pre-MTW textbook! Once you have the
modern tools from differential geometry, you can go back and read pre-1965
classics with the inestimable advantage of highsight. Be aware that gtr
was very poorly understood before 1965 or so, and since then, one could
argue, the story has consisted more of the discovery of completely
unexpected subtleties than of "positive" breakthroughs in understanding of
the nature of the space of solutions of the EFE!

(Even widely heralded results like the proof by Klainerman and
Christodolou that the Minkowski vacuum is stable under sufficiently small
nonlinear perturbations, which is the first theorem to say something
general about some small neighborhood of one [very special!] point in the
solution space, turns out to give less physical insight than is required,
because there is apparently no way to say in advance how small is
"sufficiently small". Roughly speaking. This is related to issues which
appear to be troubling you, as you can probably see even from this brief
description.)

> and Spivak volume 1.

A fine if awesome textbook, but I find the notation a bit too fussy and
many think its a bit wordy. I like this textbook better:

author = {William M. Boothby},
title = {An Introduction to Differentiable Manifolds and {R}iemannian
Geometry},
edition = {Second},
series = {Pure and Applied Mathematics},
volume = 120,
publisher = {Academic Press},
year = 1986}

> So, I am starting small, and looking at Regge calculus in two
> dimensions. Not even Regge calculus: I am not yet thinking about the
> Einstein equations or the action. I'm just trying to define, a la Regge,
> a piecewise-linear sort of Lorentzian metric in 2-d. I'm thinking about
> 2-d manifolds made out of triangles with "squared-lengths" (v^a g_ab
> v^b; these can be positive, negative or zero) assigned to the edges.

OK, fine, that makes sense, but just be aware that four dimensional gtr is
completely different from its lower dimensional cousins.

> Here is what I figured out. Consider the following triangle:
>
> *
> / \
> b c
> / \
> *---a---*
>
> Here a, b and c stand for the squared-lengths of the edges that they are

^^^^^^^^^^^^^^^

> labelling.

OK

> The first fact is: the induced metric is Lorentzian iff
> a^2 + b^2 + c^2 - 2ab - 2ac - 2bc > 0.

Hmm... how precisely are you inducing a metric in the interior of the
triangle, given the squared edge lengths?

I skipped over a lot of stuff for now while I await your explanation.

> Try finding any of this in MTW! Certain details are omitted. (Actually
> most descriptions of Regge calculus seem to stick to the Euclidean case,
> which is much simpler: it doesn't involve any coherence conditions.

Just checking: you have studied the ADM reformulation and understand how
in that chapter of MTW one begins by "triangulating" spacelike
hyperslices, right? Here, in each triangulated hyperslice, the curvature
is concentrated in the edges of the (irregualar) tetrahedral net. One can
choose any triangle and "fold down" the pair of tetrahedra meeting at that
triangle; doing this systematically for triangles meeting at a given edge
one finds a deficit or excess associated with that edge.

> > /| /|
> > 3 / | 5 / |
> > / | 10 / |
> > \ | \ | 26
> > 3 \ | 5 \ |
> > \| \|
> >
> > Or, you can turn these on their side so that all edges are -spacelike-.
>
> The triangles in my lattice are of the latter sort: all edges spacelike.

OK, good, at least I know what kind of "triangulation" of your
two-manifolds (with tangent spaces having E^(1,1) structure) you have in
mind.

> Ah: the Euclidean case again.

Absent imposition of field equations, you can do same thing in any
Lorentzian manifold. Of course, one should expect that imposing the EFE
does prevent one from cooking up completely arbitrary geometries, and of
this turns out to be true.

In future threads, I may discuss in detail stationary axisymmetric vacuums
in gtr, where there is a beautiful and extensive theory of the master PDE,
which turns out to admit things analogous to the Backlund transformations
which give rise to soliton solutions of the KdV equation. Actually, there
are several mathematically similar classes with two Killing vector fields,
including Ernst vacuums (stationary axisymmetric), Ehlers vacuums,
polarized Gowdy vacuums, and polarized aligned colliding plane wave
vacuums. The upshot is that from 1965 to the present we can follow steady
progress toward a fairly adequate understanding of these large classes of
solutions (but none of them are of course anything close to an open
neighborhood in the solution space, of course!). It is interesting that
physical/geometric insight is however lagging far behind mathematical
insight into "symmetries of the field equation itself": at present we have
a better understanding of these subspaces of vacuum solutions than we do
of what any one solution in these subspaces "means"! See for example the
review by Bonnor et al. for some discussion of this point.

> Okay: let me try again to explain my problem. As I understand it,
> causality means something like: given data on a Cauchy surface, you can
> compute the rest of whatever-it-is you are looking at. In GR, I thought
> this meant, roughly: given what the metric looks like on a Cauchy
> surface (and in its immediate neighborhood), you can compute the rest of
> spacetime. (I understand that really this only is supposed to work
> locally in GR.)

Right, unless you have a globally hyperbolic spacetime.

> Now in my example where I change a 5 to a 6, there seems to be curvature
> introduced only at a few points. Like this (the x's mark vertices
> containing curvature):
>
>
> t
> |
> |2 x
> | x x
> |1 x
> |
> -------------------------------------------x
> |
> |
> |
> |
> |
> |
>
> Look at the Cauchy surface t = 0. Everything there is perfectly flat.
> Given all the data about the manifold around that line, how can you
> predict that up ahead in that very same manifold is some curvature?
> Obviously you can't.

I think you are saying that imposing a quasihyperbolic field equation on
the curvature, such as the EFE, should (and does) prevent curvature from
"spontaneously arising"; physically, in order to curve a region of
spacetime which is "at present flat", we need to bring in some
gravitational radiation from somewhere, or move some nongravitational
mass-energy around to directly change the curvature. (These are not of
course independent possibilities, in general, but see my detailed
discussion in past posts of the Vaidya null dust--- since EM field energy
can move at the speed of light, EM energy itself can arrive in a given
location at the same time that any gravitational radiation does, or even,
if appropriate symmetry conditions are met, without accompanying
gravitational radiation). Intuitively, this is fine, but in trying to
make the discussion precise -great care is required- in order to
distinguish between features of the coordinate chart and features of the
geometry. One of the most important distinctions between Riemann and
semi-Riemannian geometry is that this task is much more straightforward in
the former case. If you only are familiar with the former you are sure to
come to grief when you first start playing around with the latter.

> Now the reason why I'm not sure that your example is quite like mine is
> this: suppose you try to introduce a bump of curvature of radius 1 about
> the origin in Minkowski space.

When I replied previously it still wasn't clear if you planned to
eventually impose the EFE on your Lorentzian manifolds or not. As I said,
clearly you are right to expect that curvature (which is associated in
gtr, modulo some indeterminacy in "location", with gravitational field
energy) cannot be spontaneously created in gtr without a physical cause.

> A ball of radius 1 in Minkowski space is a huge thing that includes both
> null lines passing through the origin.

Oh! I see one misunderstanding right here--- the topology of Lorentzian
manifolds is induced by the usual euclidean balls, if you like. Levels of
structure: we can dress up any "naked topological four-manifold" with
either a Riemannian or Lorentzian metric tensor.

> So, I hope you or someone can tell me: am I all mixed up?

What I just said should help a lot! :-)

> Or does a Minkowski space modified to have a few points with curvature
> and the rest not violate causality?

I think you are saying that you expect perturbations of Minkowski vacuum
to a new spacetime which is still required to satisfy the vacuum EFE
cannot be made arbitrarily. If so, of course you are correct. Indeed, I
would highly recommend that you study the chapter in Schutz's textbook on
gravitational radiation (weak fields!) and then study the excellent
discussion of PP waves (strong fields!) in the review paper by Ehlers and
Kundt. See also past posts by myself in which I discussed in detail
various PP wave solutions and also various Beck vacuums (which can be
interpreted as "nonrotating" cylindrically symmetric gravitational waves)
and their generalization to Ehlers vacuums (cylindrically symmetric but
"rotating"). As a matter of fact, I may discuss in detail some slow
rotation perturbations of Weyl and Beck vacuums in the near future.
These exhibit many interesting features including analogues of Faraday
rotation of gravitational waves, energy loss due to emission of
gravitational waves, gravitomagnetically dominated regions, and the
ability using standard methods for solving boundary value problems of
producing large families of solutions with interesting behavior. And this
is -before- you start messing about with Backlund transformations and the
like! :-/

> And if so, why is my Regge thing not capturing this important aspect of
> GR?

I will await your explanation of how you are inducing a Lorentzian metric
in the interior of each triangle and defining your deficit "angles".
Right now it is not clear to me that you are in fact doing 2-dim Regge at
all.

> Is it because causality requires a metric that is C^n for n >= some
> number?

Your last question has a simple enough partial answer: you certainly need
to impose -some- smoothness conditions, if for no other reason than that
we need to define the EFE and this requires some degree of smoothness in
the metric. In the literature it is common to assume C^2 or C^3 depending
upon how you count. The natural naive choice is C^infty, and in reading
textbooks it is often best to think at this level. In discussions of
maximal analytic extensions, one assumes C^omega (real analytic) and of
course this is -far- more stringent than C^infty and indeed probably
unreasonable. But a typical surprise from gtr: even C^infty may be
unreasonable: there are some solutions (Robinson-Trautman vacuums and null
dusts) with a horizon which is not a geometric singularity, but which do
not permit smooth extensions through the horizion. (See the review paper
by Bicak.) OTH, while the RT solutions are things of great beauty, like
any other exact solution they are easily misinterpreted (see the two part
review paper by Bonnor et al.). I've discussed this issue at length in
the past--- I'd recommend in particular this paper:

author = {Dieter R. Brill},
title = {Aspects of Analyticity},
note = {gr-qc/9507019}

[Dan Christensen]

David Hillman <d...@cablespeed.com> writes:

> So, I am starting small, and looking at Regge calculus in two dimensions. Not even
> Regge calculus: I am not yet thinking about the Einstein equations or the action.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

...

>> > This is related to the following question. Can one modify Minkowski
>> > space by putting a little bump of curvature in the middle of it, leaving
>> > the rest flat?
>>
>> Of course! Same way you'd do that for say E^3; take any spherically
>> symmetric bump function f(r) supported on the unit ball with f(0) = 1, or
>> more generally any bump function of compact support, and consider
>>
>> ds^2 = f(r)^2 dr^2 + r^2 (du^2 + sin(u)^2 dv^2),
>>
>> 0 < r < infty, 0 < u < Pi, -Pi < v < Pi
>
> Ah: the Euclidean case again. But it will be a good exercise for me to try to do
> the analogous thing in 2-d Minkowski space. Though this construction may not be
> the sort of example I am looking for.
>
> Okay: let me try again to explain my problem. As I understand it, causality means
> something like: given data on a Cauchy surface, you can compute the rest of
> whatever-it-is you are looking at. In GR, I thought this meant, roughly: given
> what the metric looks like on a Cauchy surface (and in its immediate
> neighborhood), you can compute the rest of spacetime. (I understand that really
> this only is supposed to work locally in GR.)

This is true *if* the metric satisfies the Einstein Field equations.
So, one can conclude that the metric above, with a localized bump
perturbation, does *not*.

> Now in my example where I change a 5
> to a 6, there seems to be curvature introduced only at a few points. Like this
> (the x's mark vertices containing curvature):
>
>
> t
> |
> |2 x
> | x x
> |1 x
> |
> -------------------------------------------x
> |
> |
> |
> |
> |
> |
>
> Look at the Cauchy surface t = 0. Everything there is perfectly flat. Given all
> the data about the manifold around that line, how can you predict that up ahead in
> that very same manifold is some curvature? Obviously you can't.

Right. So your piecewise linear spacetime must not satisfy the Regge
analog of the Einstein equations.

> Now the reason why I'm not sure that your example is quite like mine is this:
> suppose you try to introduce a bump of curvature of radius 1 about the origin in
> Minkowski space. A ball of radius 1 in Minkowski space is a huge thing that
> includes both null lines passing through the origin. So it passes through all
> Cauchy surfaces. So if you live on a Cauchy surface in this space, you have some
> information about the bump of curvature. In my example you don't. So your example
> does not puzzle me, but mine does.

I don't understand this. Both seem to violate causality in the same
way to me. A Cauchy surface far from the origin in Chris's example
has no information about the bump.

Dan

PS: Please keep your lines shorter than 72 characters.

--
Dan Christensen
jdc+...@uwo.ca

[Jeremy Henty]

In article
<Pine.GSO.4.31.020711...@ux13.cso.uiuc.edu>, eric
alan forgy wrote:

> Just think about it physically. I am sitting peacefully in a flat
> Minkowski spacetime and all of a sudden I feel wild gravitational
> forces pulling on me, tugging me left and right. Where is
> conservation of momentum?

But that is not what you would feel. You would actually feel wild
tidal forces *stretching* you top-to-bottom, left-to-right and
front-to-back. Momentum would be conserved.

Regards,

Jeremy Henty

[eric alan forgy]

Hi,

On Sun, 14 Jul 2002, Jeremy Henty wrote:
> eric alan forgy wrote:
>
> > Just think about it physically. I am sitting peacefully in a flat
> > Minkowski spacetime and all of a sudden I feel wild gravitational
> > forces pulling on me, tugging me left and right. Where is
> > conservation of momentum?
>
> But that is not what you would feel. You would actually feel wild
> tidal forces *stretching* you top-to-bottom, left-to-right and
> front-to-back. Momentum would be conserved.

Hmm... this is what I would expect from something that satisfied the EFE
or any other physically acceptable rules, but I was referring to this
unphysical geometry that David was concocting. If you arbitrarily pull and
tug on spacetime with no rules at all (other than that it remain
Lorentzian), there is no reason that I can see that you should have
physically meaningful rules pop out. That is the only point I was trying
to make. Then again, I could be mistaken, which wouldn't be the first time
:)

The reason I thought of conservation of momentum is that there was no
other mass present that was inducing the curvature in David's manifold
and it would be somewhat of a miracle if those lumps satsified the EFE so that
they qualified as pure gravitational waves too. That curvature would seem
to whip me around (or so I thought) unless they happened to be pure
gravitational waves in which case I would just be pulled and stretched as
you described. I'm sure there would be other troublesome consequences of
living a life in David's manifold, but that is the obvious one that came
to mind :)

Eric

[David Hillman]

Dan Christensen wrote:

What does it mean to say that a given Lorentz metric satisfies
the Einstein field equation? Doesn't this equation just say that
the stress/energy tensor equals some particular tensor involving
g's?

I know that it is thought that in "realistic" cases there are
constraints on the stress/energy tensor (various energy
conditions, for instance), and hence, via the Einstein equation,
on the metric. I'm not interested in realism here, except insofar
as it involves causality. Are constraints of this sort required
in order to have local causality? My impression was that they
were not, but perhaps this is wrong. What sort of constraints are
required?

Now while I'd love to know the answer to that last question, if
it does not imply that the dimension of spacetime must be bigger
than two then I don't see how it could possibly apply to the 2d
case. For in that case (signature {1,-1}, which is what I've been
talking about) we seem to have R_{a b} = 1/2 R g_{a b}. (I had
heard that someplace and I just now computed it myself and got
the same thing.) So these terms cancel in the Einstein equation,
which therefore simply says that gamma g_{a b} = T_{a b}, where
gamma is the cosmological constant. So even if we require the
stress/energy tensor to be zero, if we also set gamma to zero we
get absolutely no constraints on the metric. This is why I
thought it was not unreasonable to expect to see causality in my
example, even though I hadn't applied any facts about the
Einstein equation. Any Lorentz metric in 2-d satisfies the vacuum
Einstein equation with zero cosmological constant. Doesn't it?

[David Hillman]

Chris Hillman wrote:

> On Wed, 3 Jul 2002, David Hillman wrote:

> > This is related to the following question. Can one modify Minkowski
> > space by putting a little bump of curvature in the middle of it, leaving
> > the rest flat?
>
> Of course! Same way you'd do that for say E^3; take any spherically
> symmetric bump function f(r) supported on the unit ball with f(0) = 1, or
> more generally any bump function of compact support, and consider
>
> ds^2 = f(r)^2 dr^2 + r^2 (du^2 + sin(u)^2 dv^2),
>
> 0 < r < infty, 0 < u < Pi, -Pi < v < Pi

Okay: I see now how to do this in signature {1,-1} (thanks for the example).

ds^2 = dx^2 - dt^2

change coordinates:

x = r cos u
t = r sin u

and I get

ds^2 = cos 2u dr^2 - 2 r sin 2u dr du - r^2 cos 2u du^2

The matrix of the metric has determinant -r^2 < 0 (except at
coordinate singularity at origin) so that's a check that maybe I
haven't totally screwed up so far. Now I want to modify this
using f(r), keeping the determinant negative. So: let f(r) be
c^infinity bump function that is zero for r>=1 and whose absolute
value is always less than 1, and let

ds^2 = (cos 2u + f(r)) dr^2 - 2 r sin 2u dr du
- r^2 cos 2u du^2

Since this adds a term of absolute value < r^2 to the
determinant, the determinant remains negative, so this is still a
valid Lorentz metric. For certain f's satisfying the above, it
seems that the metric would now have a bump of curvature inside a
ball of Euclidean radius 1 about the origin and be completely
flat everywhere else.

So, my confusion (described in previous posts in this thread) may
have nothing to do with Regge calculus. Since our f is very
weakly constrained, it seems plausible that there exists an f
such that: 1) the manifold contains nonzero curvature; 2) the
line t = -2 is a Cauchy surface. The metric seemingly satisfies
the vacuum Einstein equations with zero cosmological constant
(see my reply to Dan Christensen in this thread). And clearly
there is no information contained in the neighborhood of the line
t = -2 to allow one to predict the bump of curvature (since it
looks locally exactly like the line t = -2 in Minkowski space).
So: am I wrong that such an f exists? Or, am I wrong to expect
the neighborhood of a Cauchy surface in GR to contain information
allowing one to predict how the manifold looks elsewhere? Or what?

[Moderator's note: Paragraphs reformatted. Please keep your line
lengths below 80 characters, preferably below 75 or so. -MM]

[Jeremy Henty]

eric alan forgy wrote:

> On Sun, 14 Jul 2002, Jeremy Henty wrote:

>> eric alan forgy wrote:

>> > ... I am sitting peacefully in a flat Minkowski spacetime and

>> > all of a sudden I feel wild gravitational forces pulling on me,
>> > tugging me left and right. Where is conservation of momentum?

>> But that is not what you would feel. You would actually feel wild
>> tidal forces *stretching* you top-to-bottom, left-to-right and
>> front-to-back. Momentum would be conserved.

> Hmm... this is what I would expect from something that satisfied the
> EFE or any other physically acceptable rules, but I was referring to
> this unphysical geometry that David was concocting. If you
> arbitrarily pull and tug on spacetime with no rules at all (other
> than that it remain Lorentzian), there is no reason that I can see
> that you should have physically meaningful rules pop out. That is
> the only point I was trying to make. Then again, I could be
> mistaken, which wouldn't be the first time >:)

Well, I could be wrong too, but I am pretty sure that what I said
would be true for *any* geometry. If you accept that gravitation
arises because test particles follow geodesics in a curved spacetime,
then it follows that locally gravity can only manifest itself as
tides. Of course an arbitrary geometry might well be physically
meaningless in other ways, but the gravity == curvature == tides
principle will always hold.

IMHO, of course.

Regards,

Jeremy Henty

[David Hillman]

Chris Hillman wrote:

> On Fri, 12 Jul 2002, David Hillman wrote:
>
> BTW, I haven't forgotten that we corresponded some years ago regarding
> something (cellular automata?) and at one point Doug Lind (who was on my
> own thesis commitee) lent me a copy of your Ph.D. thesis.

Yes, that was me. I guess it's harder to forget someone with your own last
name.

Thanks for the annotated references (deleted).

> > So, I am starting small, and looking at Regge calculus in two
> > dimensions. Not even Regge calculus: I am not yet thinking about the
> > Einstein equations or the action. I'm just trying to define, a la Regge,
> > a piecewise-linear sort of Lorentzian metric in 2-d. I'm thinking about
> > 2-d manifolds made out of triangles with "squared-lengths" (v^a g_ab
> > v^b; these can be positive, negative or zero) assigned to the edges.

> OK, fine, that makes sense, but just be aware that four dimensional gtr is
> completely different from its lower dimensional cousins.

> > Here is what I figured out. Consider the following triangle:
> >
> > *
> > / \
> > b c
> > / \
> > *---a---*
> >
> > Here a, b and c stand for the squared-lengths of the edges that they are
> ^^^^^^^^^^^^^^^
> > labelling.

> OK

> > The first fact is: the induced metric is Lorentzian iff
> > a^2 + b^2 + c^2 - 2ab - 2ac - 2bc > 0.

> Hmm... how precisely are you inducing a metric in the interior of the
> triangle, given the squared edge lengths?

Let triangle with vertices A, B and C have squared lengths a, b, c where the
side opposite vertex A has squared length a, etcetera. If one of the sides is
spacelike (say side AB), then lay out the triangle in Minkowski space (using
coordinates {x,t} with squared length given by x^2 - t^2) with A at the origin
and B at {Sqrt[c],0}. Then C = {x,t} must satisfy

SquaredLength[C,A] = b
SquaredLength[C,B] = a

Solving these simultaneous equations gives

x = (b + c - a) / (2 Sqrt[c])

t = plus or minus Sqrt[(a^2 + b^2 + c^2 - 2ab - 2ac - 2bc)/4c)]

So there is a solution iff a^2 + b^2 + c^2 - 2ab - 2ac - 2bc >= 0. If
it equals zero, the triangle is degenerate. If it's > 0, there are two
solutions, corresponding to choosing a direction of time. (If it's <
0, I'm guessing a Euclidean metric would work instead.) So you can lay
out the triangle uniquely in Minkowski space, apart from the group
generated by translation, reflection in space and/or time, and Lorentz
transformation.

If no sides are spacelike, do the same sort of thing with a timelike
side. You get the same sort of result. (With the same condition on
there being a solution.) Every triangle has at least one spacelike or
timelike edge. So: done. The three squared lengths are exactly what is
needed to determine the metric in a unique way.

Of course, supposedly this works in n dimensions. I haven't tried it, but a
counting argument shows that it is plausible: a metric in n dimensions has
(n+1)n/2 independent entries; an n-dimensional simplex has (n+1)n/2 edges.

[Dan Christensen]

David Hillman <d...@cablespeed.com> writes:

> Dan Christensen wrote:

>> David Hillman <d...@cablespeed.com> writes:

>> > Okay: let me try again to explain my problem. As I understand it,
>> > causality means something like: given data on a Cauchy surface,
>> > you can compute the rest of whatever-it-is you are looking at. In
>> > GR, I thought this meant, roughly: given what the metric looks
>> > like on a Cauchy surface (and in its immediate neighborhood), you
>> > can compute the rest of spacetime. (I understand that really this
>> > only is supposed to work locally in GR.)

>> This is true *if* the metric satisfies the Einstein Field equations.
>> So, one can conclude that the metric above, with a localized bump
>> perturbation, does *not*.

> What does it mean to say that a given Lorentz metric satisfies
> the Einstein field equation? Doesn't this equation just say that
> the stress/energy tensor equals some particular tensor involving
> g's?

Since you didn't specify a distribution of matter on the initial
surface, I assumed that you were talking about the vacuum Einstein
equations. And these of course put a strong restriction on the
possible metrics, making causality work.

If you allow matter, then I believe that GR is only well-posed (in
particular, causal) if the matter fields satisfy certain types of
differential equations. So there is still a restriction on the
possible metrics.

I'm no expert on this, so maybe someone else can say more.

Dan

[John Baez]

In article <878z47h...@uwo.ca>, Dan Christensen <jdc+...@uwo.ca> wrote:

>Since you didn't specify a distribution of matter on the initial
>surface, I assumed that you were talking about the vacuum Einstein
>equations. And these of course put a strong restriction on the

>possible metrics, [...]

I'm a bit confused, since I thought *every* metric was a
solution of the vacuum Einstein equations in 2d spacetime:
in this particular dimension we have the identity

R_{uv} = 1/2 R g_{uv}

so we automatically get

G_{uv} := R_{uv} - 1/2 R g_{uv} = 0

Of course in *other* dimensions the vacuum Einstein equations
do put strong restrictions on the metric, especially in dimension
3, where they say it's flat.

> [...] making causality work.

I'm not sure what you mean by this - probably because I wasn't
been following this thread too carefully until you showed up.

There's been a huge amount of work on 2d gravity, including a
lot of Regge-like stuff, but a lot of it uses actions other
than the usual vacuum Einstein action. If David Hillman is
feeling adventurous he can look at this paper, for example:

hep-th/0001124
Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results
Authors: J. Ambjorn, J. Jurkiewicz, R. Loll

Abstract: We review some recent attempts to extract information about
the nature of quantum gravity, with and without matter, by quantum field
theoretical methods. More specifically, we work within a covariant
lattice approach where the individual space-time geometries are
constructed from fundamental simplicial building blocks, and the path
integral over geometries is approximated by summing over a class of
piece-wise linear geometries. This method of ``dynamical triangulations''
is very powerful in 2d, where the regularized theory can be solved
explicitly, and gives us more insights into the quantum nature of 2d
space-time than continuum methods are presently able to provide. It also
allows us to establish an explicit relation between the Lorentzian- and
Euclidean-signature quantum theories. Analogous regularized gravitational
models can be set up in higher dimensions. Some analytic tools exist to
study their state sums, but, unlike in 2d, no complete analytic solutions
have yet been constructed. However, a great advantage of our approach
is the fact that it is well-suited for numerical simulations. In the
second part of this review we describe the relevant Monte Carlo techniques,
as well as some of the physical results that have been obtained from the
simulations of Euclidean gravity. We also explain why the Lorentzian
version of dynamical triangulations is a promising candidate for a
non-perturbative theory of quantum gravity.

[John Baez]

In article <878z47h...@uwo.ca>, Dan Christensen <jdc+...@uwo.ca> wrote:

>Since you didn't specify a distribution of matter on the initial
>surface, I assumed that you were talking about the vacuum Einstein
>equations. And these of course put a strong restriction on the

[Chris Hillman]

On Fri, 19 Jul 2002, Jeremy Henty wrote:

> If you accept that gravitation arises because test particles follow
> geodesics in a curved spacetime, then it follows that locally gravity
> can only manifest itself as tides. Of course an arbitrary geometry
> might well be physically meaningless in other ways, but the gravity ==
> curvature == tides principle will always hold.

This is not quite correct; gtr provides the simplest counterexample, but
related classical relativistic field theories of gravitation will provide
additional counterexamples.

First of all, at least in -gtr-, the equations of motion for nonspinning
test particles having only a mass charge ("world lines are timelike
geodesics") actually -follow- from the field equations. This important
point is explained in many textbooks; more detail can be found in the very
old but still useful review

author = {Joshua N. Goldberg},
title = {The Equations of Motion},

booktitle = {Gravitation: an Introduction to Current Research},
editor = {Louis Witten},
publisher = {Wiley},

pages = {102--129},
year = 1962}

("Mass charge": this is shorthand for saying that the test particles are
electrically neutral and more generally will interact with any
nongravitational fields only indirectly, via the gravitational effect of
the field energy of these nongravitational fields, if any such are
present, gasp, gasp.)

BTW, the book in which this article appeared has several other old but
still useful reviews, including

author = {F. A. E. Pirani},
title = {Gravitational Radiation},

booktitle = {Gravitation: an Introduction to Current Research},
editor = {Louis Witten},
publisher = {Wiley},

pages = {199--226},
year = 1962}

author = {R. Arnowitt and S. Deser and C. W. Misner},
title = {The Dynamics of General Relativity},
booktitle = {Gravitation: An Introduction to Current Research},

editor = {Louis Witten},
publisher = {Wiley},

pages = {227--265},
year = 1962}

author = {Yvonne Bruhat},
title = {The {C}auchy Problem},
booktitle = {Gravitation: An Introduction to Current Research},

editor = {Louis Witten},
publisher = {Wiley},

pages = {130--168},
year = 1962}

author = {J\"urgen Ehlers and Wolfgang Kundt},
title = {Exact Solutions of the Gravitational Field Equations},
booktitle = {Gravitation: an Introduction to Current Research},
editor = {Louis Witten},
publisher = {Wiley},
year = 1962,
pages = {49--101}}

I think the editor (Louis Witten, father of Edward) must have made a real
effort to help the authors of the reviews write papers which would be
comprehensible to graduate students circa 1960, and suspect that modern
graduate students will still appreciate this effort :-/

Second of all, as the qualifier "nonspinning" above hints, in gtr there
are spin-spin forces which will force "off course" a -spinning- test
particle (with only mass charge) moving through a gravitational field with
a "rotating" source. This is very easy to see in the weak-field GEM
formalism (see for example the textbook by Wald) but remains true in
strong fields. You can look for previous posts by me explaining (with
citations) the Bel decomposition of the Riemann tensor, relative to an
arbitrary timelike congruence X, into three three-dimensional tensors:

1. Electrogravitic tensor E[X]_(ab); symmetric; 6 alg. indpt. cmpts.,

2. Magnetogravitic tensor B[X]_(ab); traceless; 8 alg. indpt. cmpts.,

3. Topothesiogravitic tensor, L[X]_(ab); symmetric; 6 alg. indpt. cmpts.

Note that this adds up to 20 alg. indpt. cmpts. as it should. Here, X
defines a family of observers (possibly noninertial, possibly not locally
nonrotating) and E[X]_(ab) completely describes the tidal forces they
measure. B[X]_(ab) completely describes the -initial- spin-spin forces
such an observer measures on a gryoscope (even less massive than himself)
which he suddenly drops. Indeed, if X is geodesic, these can be read
right off the components of B[X]_(ab) just as the tidal forces can be read
right off the components of E[X]_(ab) (components wrt an appropriate ON
basis, that is!). Finally, L[X]_(ab) completely describes the sectional
curvatures associated with hyperplane elements orthogonal to X; if X is
vorticity-free (hypersurface-orthogonal; this corresponds to locally
nonrotating observers) it describes the three-dimensional Riemann tensor
of hyperslices orthogonal to X.

("Topothesia": topographic description of a place, as in surveying.)

Here, (2) is closely related to GEM formalism, for which see for example
the review papers:

http://xxx.lanl.gov/abs/gr-qc/0207065

(I haven't had a chance to read this yet; the second author, a member of
the Gravity Probe B development team, has recently proposed an intriguing
quasi-heuristic argument regarding GEM which ignores a host of subtle but
fundamental issues and is probably wrong, but at a glance this review
looks fine)

http://xxx.lanl.gov/abs/gr-qc/0106043

http://xxx.lanl.gov/abs/gr-qc/0105096

In a vacuum, the Bel decomposition reduces to

1. E_(ab); traceless symmetric; 5 alg. indpt. cmpts.,

2. B_(ab); traceless symmetric; 5 alg. indpt. cmpts.

which adds up to 10 alg. indpt. cmpts., as it should, since then the
Riemann tensor agrees with the Weyl tensor.

Chris Hillman

[Dan Christensen]

ba...@galaxy.ucr.edu (John Baez) writes:

> In article <878z47h...@uwo.ca>, Dan Christensen <jdc+...@uwo.ca> wrote:
>
>>Since you didn't specify a distribution of matter on the initial
>>surface, I assumed that you were talking about the vacuum Einstein
>>equations. And these of course put a strong restriction on the
>>possible metrics, [...]
>
> I'm a bit confused, since I thought *every* metric was a
> solution of the vacuum Einstein equations in 2d spacetime:

John is right, of course. As an excuse for my confusion, let me just
say that in the past when I've worked with (1+1)-dimensional versions
of gravity, I've used field equations which reduce to R = 0 when there
is no matter or cosmological constant. So there was a restriction on
the possible spacetimes. But in 2d GR, there is no restriction.

So, now I'm also confused by David Hillman's questions. It does seem
to be the case that knowing a 1+1-dimensional Lorentzian metric on a
spacelike "surface" (and knowing extrinsic curvature type information,
which was trivial in his example) doesn't determine the metric at
points in the future all of whose past rays intersect the surface,
even though the metric will satisfy the vacuum Einstein equations.

Are we missing something, or is this just the way it is?

If that's just the way GR is, I wonder if the R = 0 theory is any
better? In general, it can be an interesting theory: it has black
holes, a Newtonian limit, gravitational waves and other features
analogous to GR, and the field equations can be obtained from those of
GR by a certain dimensional limit. Lots of papers by R.B. Mann from
the late 80's and early 90's discuss this theory.

Dan

[Chris Hillman]

On Wed, 17 Jul 2002, David Hillman wrote:

> Chris Hillman wrote:
>
> > On Wed, 3 Jul 2002, David Hillman wrote:
>
> > > Can one modify Minkowski
> > > space by putting a little bump of curvature in the middle of it, leaving
> > > the rest flat?
> >
> > Of course! Same way you'd do that for say E^3; take any spherically
> > symmetric bump function f(r) supported on the unit ball with f(0) = 1, or
> > more generally any bump function of compact support, and consider
> >
> > ds^2 = f(r)^2 dr^2 + r^2 (du^2 + sin(u)^2 dv^2),
> >
> > 0 < r < infty, 0 < u < Pi, -Pi < v < Pi

Oops, that should have been something like [1 + f(r)]^2. Sorry about
that.

Maybe some more detail would be helpful, although IIRC an almost identical
example is discussed in detail in MTW. Take the cobasis

o^1 = [1 + f(r)] dr

o^2 = r du

Taking exterior derivatives and using Cartan's equation

do^j = -w^j_k /\ o^k

we -guess- the connection one-forms, finding that up to algebraic
symmetries the only nonvanishing connection one-form is

-du
w^1_2 = --------
1 + f(r)

Using the second Cartan equation

O^j_k = dw^j_k + w^j_m /\ w^m_k

we find that up to algebraic symmetries the only nonvanishing curvature
two-form is

f''(r)
O^1_2 = -------------- o^1 /\ o^2
r [1 + f(r)]^3

Finally, using

O^j_k = R^j_k|mn| o^m /\ o^n

we read off

f''
R^1_(212) = ---------
r (1+f)^3

This procedure, which is due to Cartan, is explained in MTW in the section
on computing curvature. It is almost always the most efficient procedure
for computing the Riemann tensor, even in two dimensions.

BTW, in two dimensions, up to sign, R^1_(212) is just the Gaussian
curvature. Also, for a simple construction of explicit -smooth- (C^infty)
bump functions see

author = {Morris W. Hirsch},
title = {Differential Topology},
publisher = {Verlag},
series = {Graduate texts in mathematics},
volume = 33,
year = 1976}

> So, my confusion (described in previous posts in this thread) may have
> nothing to do with Regge calculus. Since our f is very weakly
> constrained, it seems plausible that there exists an f such that: 1) the
> manifold contains nonzero curvature; 2) the line t = -2 is a Cauchy
> surface.

Of course. I could easily produce a huge class of explicit examples. In
fact, what the heck, I will! Take the cobasis

o^1 = -[ 1 + f(t) g(x) ] dt

o^2 = [ 1 + f(t) g(x) ] dx

where

{ f = 1 on |t| < 1/2
{ 0 < f < 1 on 1/2 < |t| < 1
{ f = 0 on |t| > 1

{ g = 1 on |x| < 1/2
{ 0 < g < 1 on 1/2 < |x| < 1
{ g = 0 on |x| > 1

are -smooth- bump functions, for example constructed a la Hirsch. Up the
algebraic symmetries, the only nonvanishing component of the Riemann
tensor is, uhm,

(f g'' - g f'') (g f')^2 - (f g')^2
R_(1212) = ---------------- + -------------------
(1+fg)^3 (1+fg)^4

As you see, this vanishes outside the square |t| <1, |x| < 1.

> The metric seemingly satisfies the vacuum Einstein equations with zero
> cosmological constant (see my reply to Dan Christensen in this thread).

As I warned you in a previous post, in dimension two, the EFE is
essentially trivial (the spacetimes just discussed also satisfy "Ricci
vanishes"), and in dimension three, the EFE still does not even admit
gravitational radiation! One way to understand why is to ask yourself why
in two dimensions, the Riemann tensor R_(abcd) can be reconstructed from
the Ricci scalar R = R^a_a, and why in three dimensions the Riemann tensor
can be reconstructed from the Ricci tensor R_(ab). Thus, the EFE is only
associated with a nonabsurd gravitation theory in dimension four or
higher. This important point is discussed in MTW and other textbooks.

BTW, if you are reading math papers on differential geometry, be warned
that by an "Einstein manifold" they usually mean nothing but a
-riemannian- manifold for which "Ricci vanishes". Of course this makes
perfect sense mathematically, but it should be clear that it has nothing
to do with gtr, even in dimension four.

> And clearly there is no information contained in the neighborhood of the
> line t = -2 to allow one to predict the bump of curvature (since it
> looks locally exactly like the line t = -2 in Minkowski space).

Dunno about your example, but certainly that is true of mine.

> So: am I wrong that such an f exists?

You are correct. There are zillions of examples which are homeo to R^2
but have curvature concentrated in a compact region like a square.

> Or, am I wrong to expect the neighborhood of a Cauchy surface in GR to
> contain information allowing one to predict how the manifold looks
> elsewhere?

You are right about this too: given a hyperslice S in a globally
hyperbolic spacetime M satisfying the vacuum EFE, if suitable initial data
(e.g. extrinsic curvature and metric tensor) satisfies the constraint
equations, then the ADM evolution equations for a vacuum EFE solution will
yield M in some neighborhood of S. (Think of moving small topological
spheres over S to "thicken it" to a four dimensional region in M. Again,
do not become confused between the topology on M, which can be induced by
Riemannian "balls" [or by "cubes", etc.], and the Lorentzian "metric",
which is associated with noncompact "loci of constant distance" from a
given point. Do not become confused by the inconsistent terminology:
"metric topology" as in general topology or analysis textbooks and "metric
tensor" as that is used in semiriemannian geometry.)

> Or what?

As far as I can see, this confusion is due entirely to the triviality of
the EFE in low dimensions. You need to look at the EFE in four dimensions
or higher to see anything interesting like gravitational radiation. In
three dimensions you do at least get something like "gravitational
attraction" but since there is no radiation, "gravitity is not a long
range force in three dimensional gtr". IIRC, this is explained in MTW.

Chris Hillman

[Chris Hillman]

On Sun, 21 Jul 2002, David Hillman wrote:

> I guess it's harder to forget someone with your own last name.

It's not my last name, it's -Chris Hillman's- last name!

> > > The first fact is: the induced metric is Lorentzian iff
> > > a^2 + b^2 + c^2 - 2ab - 2ac - 2bc > 0.
>
> > Hmm... how precisely are you inducing a metric in the interior of the
> > triangle, given the squared edge lengths?
>
> Let triangle with vertices A, B and C have squared lengths a, b, c where
> the side opposite vertex A has squared length a, etcetera. If one of the
> sides is spacelike (say side AB), then lay out the triangle in Minkowski
> space (using coordinates {x,t} with squared length given by x^2 - t^2)
> with A at the origin and B at {Sqrt[c],0}.

So c > 0? Probably OK, since you can probably argue at least one edge
must be spacelike if the triangle is nontrivial. (The other two could
evidently be a null pair, a spacelike pair, a timelike pair, or some
combination.)

> Then C = {x,t} must satisfy
>
> SquaredLength[C,A] = b
> SquaredLength[C,B] = a

But how are you computing ||C-A||^2? Isn't this what you are trying to
define?

Chris Hillman (still confused)

[David Hillman]

> > Let triangle with vertices A, B and C have squared lengths a, b, c where
> > the side opposite vertex A has squared length a, etcetera. If one of the
> > sides is spacelike (say side AB), then lay out the triangle in Minkowski
> > space (using coordinates {x,t} with squared length given by x^2 - t^2)
> > with A at the origin and B at {Sqrt[c],0}.
>
> So c > 0? Probably OK, since you can probably argue at least one edge
> must be spacelike if the triangle is nontrivial. (The other two could
> evidently be a null pair, a spacelike pair, a timelike pair, or some
> combination.)

At least one edge is non-null. Since space and time are basically symmetrical
in signature {1,-1}, analogous arguments work for a triangle with a spacelike
edge and a triangle with a timelike edge.

> > Then C = {x,t} must satisfy
> >
> > SquaredLength[C,A] = b
> > SquaredLength[C,B] = a
>
> But how are you computing ||C-A||^2? Isn't this what you are trying to
> define?

Minkowski space. Metric used is ds^2 = dx^2 - dt^2. Squared-length of line
joining points {a,b} and {c,d} is (a-c)^2 - (b-d)^2. Right?

[eric alan forgy]

Hi -

On Sun, 21 Jul 2002, Dan Christensen wrote:

> ba...@galaxy.ucr.edu (John Baez) writes:

> > I'm a bit confused, since I thought *every* metric was a
> > solution of the vacuum Einstein equations in 2d spacetime:

You learn something new every day :)

This is pretty interesting. Is anyone else besides me bothered by this
fact?

> John is right, of course. As an excuse for my confusion, let me just
> say that in the past when I've worked with (1+1)-dimensional versions
> of gravity, I've used field equations which reduce to R = 0 when there
> is no matter or cosmological constant. So there was a restriction on
> the possible spacetimes. But in 2d GR, there is no restriction.

> So, now I'm also confused by David Hillman's questions.

Me too!

It seems that there really is no rule governing evolution in 2d spacetimes
according to GR. Is that right? It seems you can evolve into anything you
like. No uniqueness at all. I'm probably missing something, but it'd
certainly be nice if someone could explain what is going on :)

Eric

PS: If I weren't so far behind schedule in trying to graduate, I might try
to work it out myself so I'm hoping some merciful expert might say a word
or two :)

[Chris Hillman]

On Sun, 21 Jul 2002, Dan Christensen wrote:

> If that's just the way GR is, I wonder if the R = 0 theory is any
> better? In general, it can be an interesting theory: it has black
> holes, a Newtonian limit, gravitational waves and other features
> analogous to GR, and the field equations can be obtained from those of
> GR by a certain dimensional limit. Lots of papers by R.B. Mann from the
> late 80's and early 90's discuss this theory.

I think David was indeed ultimately interested in exploring the Regge
approximation scheme for the ADM reformulation of the vacuum EFE, but
hadn't yet realized that in 1+1 the vacuum EFE is trivial or that in 2+1
the vacuum EFE does not allow for gravitational radiation. Thus, as I said
in my other post, for purely mathematical reasons, the vacuum EFE does not
permit sensible "dimensional reductions" from 3+1.

-However-... I think David might settle at least in the short term for a
"toy theory" involving curved spacetimes which can perhaps be attacked
with a Regge type approach but which has fewer than four dimensions. It
occurs to me that one possibility for such a theory might be the 2+1 EFE
with source a (minimally coupled) massless scalar field, which is the
simplest imaginable relativistic field theory and admits causality
phenomena (also, of course, if one ignores the gravitational effect of the
field energy, it has nice general solutions in closed form on E^(1,2).)
Of course, one would need to figure out an ADM type constraint and
evolution equation reformulation and then to figure out the corresponding
Regge scheme.

The reason I suggest this as a "toy theory" is that it is really not a toy
at all! Indeed, it can be used to study some of the most important
questions in classical gtr, because it is well known that the 2+1 EFE for
a minimally coupled massless scalar field is equivalent to the study of
the Beck family of solutions to the 3+1 vacuum EFE! See for example

http://xxx.lanl.gov/abs/gr-qc/9608042

http://xxx.lanl.gov/abs/gr-qc/9608041

Note that the authors call the Beck family the "Einstein-Rosen
cylindrically symmetric gravitational wave", but the solution was first
found by Beck and it is -not- the family of all cylindrically symmetric
vacuums but rather the family of all -nonrotating- cylindrically symmetric
vacuums.

As I recently mentioned in another thread, one of the many reasons the
Beck vacuums are interesting is that they allow one to study the energy
carried away by outgoing gravitational radiation, or the stresses caused
in a thin shell by incoming radiation. In particular, I might discuss a
Beck vacuum interior containing a cylindrically symmetric gravitational
sine wave (exact vacuum solution) matched across a massive thin shell to a
Levi-Civita vacuum exterior (a -static- cylindrically symmetric vacuum).
This exhibits readily computable and interesting stresses on the shell due
to the presence of the radiation inside the shell. One attractive feature
of the interior solution is that it arises very simply from some well
known standard material involving the application of Fourier-Bessel series
to classical boundary value problems encountered by every undergraduate
math/physics student.

As it happens, I am feeling somewhat inspired for a variety of reasons to
yak about the Weyl, Beck, and Kramer vacuums and their generalizations to
minimally coupled massless scalar field solutions, non-null
electrovacuums, and to allowing one Killing vector to acquire vorticity
(in the case of the Weyl family, this is a fancy way of saying that the
Ernst family is the family of all stationary axisymmetric vacuum
solutions). One of the points, emphasizing something which Bonnor
stresses in his review (recently cited by me in another post) and
elsewhere, is that a single chart can (after changing coordinates or at
least renaming them) often be interpreted in several ways. As usual, this
is the old local versus global phenomenon. As usual, it can be quite
taxing to distinguish between valid and naive attempted interpretations.
Again, I could have thrown in above two more classes: the "nonlinear
interaction zone" of two colliding plane waves with aligned polarization,
and an "interior region" of a class of black hole solutions. All these
things are pretty much equivalent mathematically, and as I said, there are
now a great variety of rather sophisticated techniques for attacking these
various special cases of the EFE, including ideas imported from the theory
of solitons. The key idea uniting all the classes of spacetimes I
mentioned is of course the presence of two Killing vector fields, one of
which is vorticity free. These may be a pair of timelike and spacelike
fields, or a pair of spacelike fields. So again, studying the proposed
toy theory by a Regge scheme would probably really be worthwhile (assuming
such a scheme exists!).

A very nice example of the problem of interpretation, which is always
crucially important, but never easy to resolve (OTH, multiple distinct
correct interpretations may be possible), was discussed in a long extinct
post by me and, as I recently found out, much earlier and much more
extensively by Bonnor in an old paper: the so-called "static plane vacuum"
(sometimes called the "Taub vacuum") has several possible interpretations,
but its interpretation as the exterior field of a static thin plane is not
one of the reasonable ones! I independently noticed the problems with
this; in his paper, Bonnor argues that this vacuum is better interpreted
as a "funny chart" for the exterior field of a static "ray" mass. The
paper in question is cited in the review paper already cited:

author = {W. B. Bonnor},
title = {Physical interpretation of vacuum solutions of {E}instein's
equations. {P}art {I}: {T}ime independent solutions},
journal = {Gen. Rel. Grav.},
volume = 24,
pages = {551--574},
year = 1992}

author = {W. B. Bonnor and J. B. Griffiths and M. A. H. MacCallum},
title = {Physical interpretation of vacuum solutions of {E}instein's
equations. {P}art {II}: {T}ime dependent solutions},
journal = {Gen. Rel. Grav.},
volume = 26,
pages = {687--729},
year = 1994}

Another very nice example, already noted by Kramer, is the Kramer vacuum
corresponding to the "sine wave" Beck vacuum just mentioned. This turns
out to be a wave propagating along an axis, with Gaussian falloff, which
generalizes a corresponding solution in Maxwell's theory of EM which was
only discovered in 1987! (I won't give the reference yet to Kramer's
paper because it has some annoying incorrect numerical factors, arrrghgh,
so I'll need to write out the solutions in question correctly when I find
time.)

Chris Hillman

[Urs Schreiber]

"eric alan forgy" <fo...@students.uiuc.edu> schrieb im Newsbeitrag
news:Pine.GSO.5.31.02072...@ux11.cso.uiuc.edu...

[...]

> It seems that there really is no rule governing evolution in 2d spacetimes
> according to GR. Is that right? It seems you can evolve into anything you
> like. No uniqueness at all. I'm probably missing something, but it'd
> certainly be nice if someone could explain what is going on :)

I've been reading some string theory, so I'll give this one a try. I hope I get
it right. Otherwise someone please correct me.

The short answer to "what is going on?" should be: Gauge invariance. All
configurations of ordinary (Einstein-Hilbert) GR in d=2 are locally gauge
equivalent.

Here is why: The metric in d=2, being a symmetric tensor, has 3 degrees of
freedom. The reparametrization invariance of GR allows us to freely specify 2
coordinate functions, which removes 2 of those 3 degrees of freedom. Hence one
can always locally find a coordinate system in which the metric reads

g = e^f(t,x) diag(-1,1) ,

where e^f is the conformal factor. But precisely in 2 spacetime dimensions GR
has another gauge invariance besides diff invariance, namely *conformal
invariance*, i.e. it is invariant under a conformal transformation of the
metric tensor. To see this it is helpful to consider the "conformal weight" of
a couple of objects. If a quantity "A" transforms as

A -> e^(n f) A

under the conformal transformation

g -> e^f g

then it is said to have conformal weight n. (Maybe I am confusing n with -n,
but that is just convention.)

A simple calculation reveals the conformal weights of the following objects

g: 1
g^-1 : -1
det(g): d
sqrt(det(g)): d/2
Gamma: 0
Riemann: 0
Ricci: 0
R: -1
sqrt(det(g)) R: d/2 - 1

(Here "Gamma" is the Levi-Civita connection with one contravariant index (in
physicist convention), "Riemann" the Riemann tensor with one contravariant
index, "Ricci" the Ricci tensor with two covariant indices and "R" is the
curvature scalar.)

Therefore under a conformal transformation the Einstein-Hilbert Lagrangian
transforms as

sqrt(det(g)) R -> e^(d/2-1) sqrt(det(g)) R

and thus it is invariant precisely if

d/2-1 = 0 <=> d= 2 .

This means that in d=2 there are no local degrees of freedom in the vacuum
Einstein-Hilbert theory after the gauge has been fixed. Hence the EFE vanish
identically.

--
Urs.Sc...@uni-essen.de

[Chris Hillman]

On Tue, 23 Jul 2002, I wrote:

> it is well known that the 2+1 EFE for a minimally coupled massless
> scalar field is equivalent to the study of the Beck family of solutions
> to the 3+1 vacuum EFE!

Uh oh, that might have been a brain blip on my part; the paper I cited
does -not- contain the claim I thought I recalled, and now I suspect my
memory may have been erroneous. Of course, it would be -very- easy to
check the quoted claim, but I'm too lazy :-/

(There -is- a very close connection between the Beck family of exact
vacuum solutions and the usual 3+1 wave equation, which I have mentioned
in past posts, but that's something else.)

A hint for anyone less lazy than I: there are two obvious approaches, and
one should check both give the same result. First, one can figure out the
Einstein tensor in 1+2 dimensions, and let this stand as the RHS with the
1+2 stress-energy tensor from a massless scalar field on the RHS. Here,
recall that in 1+2 dimensions, given a symmetric tensor P_(ab), its
traceless part is

P_(ab) - 1/3 g_(ab) P

where P = P^a_a is the trace, and the tracedual is

Q_(ab) = P_(ab) - 2/3 g_(ab) P

where by construction Q = -P. Second, check that in 3+1 the field
equations for a massless scalar field are equivalent to

R_(ab) = 8 Pi (D_a phi) (D_b phi) (3+1 EFE)

D^a D_a phi = 0 (curved spacetime wave equation!)

This evidently has an immediate dimensional reduction (don't forget to
modify the constant 8 factor on the RHS of the first equation). This
should probably take a few minutes at most.

BTW, John Baez mentioned the fact that every 1+1 spacetime is a solution
of the 1+1 vacuum Einstein equation. Eric Alan Forgy asked:

> Is anyone else besides me bothered by this fact?

Once you understand how to reconstruct the Riemann tensor from the
curvature scalar R in 1+1 (or from the Ricci tensor in 2+1), I think it is
obvious why things could not be otherwise. (This reconstruction problem
is given as an exercise in most gtr textbooks.) The point is that this
has -nothing to do- with the EFE itself, but is simply one more instance
of a well-known class of "dimensional" phenomena: lotsa things which can
exist in three (spatial) dimensions can't exist in two (like knots!), and
lotsa things which can happen in two dimensions can't happen in one (like
moving around a compact obstacle!).

Chris Hillman

[David Hillman]

Chris Hillman wrote:

> On Sun, 21 Jul 2002, Dan Christensen wrote:
>
> > If that's just the way GR is, I wonder if the R = 0 theory is any
> > better? In general, it can be an interesting theory: it has black
> > holes, a Newtonian limit, gravitational waves and other features
> > analogous to GR, and the field equations can be obtained from those of
> > GR by a certain dimensional limit. Lots of papers by R.B. Mann from the
> > late 80's and early 90's discuss this theory.
>
> I think David was indeed ultimately interested in exploring the Regge
> approximation scheme for the ADM reformulation of the vacuum EFE, but
> hadn't yet realized that in 1+1 the vacuum EFE is trivial or that in 2+1
> the vacuum EFE does not allow for gravitational radiation. Thus, as I said
> in my other post, for purely mathematical reasons, the vacuum EFE does not
> permit sensible "dimensional reductions" from 3+1.

No: believe it or not, what I was interested in was an answer to my question.

Which, to restate simply (at the risk of repeating myself), is: in 2-d GR, you
can have a metric in which Minkowski space is modified by putting a C-infinity
bump of curvature within radius 1 of the origin, leaving the rest completely
flat. The metric satisfies the vacuum EFE since any 2d Lorentz metric does.
The line t = -2 is a Cauchy surface in Minkowski space. One might think that
it would also be a Cauchy surface in this bump space. But if so, then the
theorem that a Cauchy surface and its neighborhood contain enough info to
generate the entire space cannot be true, since both t = -2 lines look
identical. So, what is wrong with this picture? Does the theorem that a Cauchy
surface and its neighborhood contain enough info to generate the entire space
not hold in 2d? If that is what is going on, it would be nice to know. (And in
general I'd like to understand better when this theorem holds and when it
doesn't.) Or, is there some reason why the line t=-2 in a bump space turns out
never to satisfy the definition of a Cauchy surface? If so, what is the
reason?

[Urs Schreiber]

"Urs Schreiber" <Urs.Sc...@uni-essen.de> schrieb im Newsbeitrag
news:ahkl8l$1a76$1...@rs04.hrz.uni-essen.de...

[...]

> A simple calculation reveals the conformal weights of the following objects

Oops, sorry, this was a little too simple. What I wrote of course only holds
for global conformal transformations with _constant_ conformal factor e^f.
Otherwise, in d=2, the curvature scalar also picks up a term proportional to
the Laplacian of f.

--
Urs.Sc...@uni-essen.de

[Urs Schreiber]

"David Hillman" <d...@cablespeed.com> schrieb im Newsbeitrag
news:3D3E1380...@cablespeed.com...

[...]

> The line t = -2 is a Cauchy surface in Minkowski space. One might think that
> it would also be a Cauchy surface in this bump space. But if so, then the
> theorem that a Cauchy surface and its neighborhood contain enough info to
> generate the entire space cannot be true, since both t = -2 lines look
> identical. So, what is wrong with this picture? Does the theorem that a
> Cauchy surface and its neighborhood contain enough info to generate the
> entire space not hold in 2d?

I think the answer is this: A Cauchy surface should contain enough information
to generate the entire space *up to symmetries/gauge transformations*. The
vacuum Einstein-Hilbert action in d=2 is invariant under diffeos and under
conformal transformations. By means of these it is possible to transform every
d=2 metric tensor locally to the flat Minkowski metric diag(-1,1). Hence
*every* spacelike line should be a Cauchy surface in d=2, since one does not
need any information at all to locally generate the entire spacetime up to
gauge transformations.

(However, since I am not an expert on this I may be wrong. Corrections are
welcome.)

--
Urs.Sc...@uni-essen.de

[Chris Hillman]

On Wed, 24 Jul 2002, David Hillman wrote:

> you can have a metric in which Minkowski space is modified by putting a
> C-infinity bump of curvature within radius 1 of the origin, leaving the
> rest completely flat.

I think that what may be going on here is that you are somehow confusing
yourself by what might at first seem to be a minor misemphasis. The above
makes good sense only if you delete "in 2-d GR".

Let me restate what you just said in a way which I hope will help clear up
the confusion: certainly there are 1+1 Lorentzian manifolds which agree
with E^(1+1) outside some compact region R and inside have metric tensor
modified by a bump function, e.g. perhaps something like a Kerr-Schild
Ansatz for a double null chart:

ds^2 = 2 dp dq + f(p,q) dq^2

where p = (t+z)/sqrt(2), q = (t-z)/sqrt(2); here @/@p and @/@q are null
vector fields and f(p,q) is a smooth bump function. ("Smooth": C^infinity
but certainly -not- C^omega or "real analytic".) Similarly for n+1, of
course. This has nothing whatever to do with any physical theory; it is
simply an elementary observation about semiriemannian geometry.

John Baez has already answered the questions in your post (in his comment
addressed to Dan Christensen), and I thought I had too at least twice but
now I can't seem to find the post I thought I had submitted (maybe it
simply hasn't yet appeared).

> The metric satisfies the vacuum EFE since any 2d Lorentz metric does.

Again, I think you are confusing yourself by getting the logic the wrong
way around.

In 1+1 you can reconstruct the Riemann tensor (rank 4) from the Ricci
scalar. That is a purely mathematical fact about semiriemannian geometry
which is -entirely independent- of the EFE or indeed of any physical
theory. Likewise, in 2+1 you can reconstruct the Riemann tensor (rank
four) from the Ricci tensor (rank two), and this is entirely independent
of any physical theory.

These purely mathematical facts do imply however that the EFE has a chance
of modeling a long-range fundamental physical interaction only in n+1, n >
2. When n = 3 of course, we have gtr, which as we all know does model a
long range fundamental interaction (gravitation).

Put another way: the way to think about "every 1+1 Lorentzian manifold
trivially satisfies the EFE" is that the 1+1 EFE is trivial and therefore
entirely uninteresting, and therefore there is no point to trying to
construct and then worry over an ADM evolution/constraint reformulation.

Or again: the phrase "in 2-d GR" is objectionable because there isn't any
such thing, at least not in the sense of a nontrivial theory.

If you are willing to go to 2+1, you should be able to find an ADM
reformulation of the 2+1 EFE coupled to the 2+1 curved spacetime wave
equation, i.e. "the field equations for a 2+1 minimally coupled
massless scalar field solution". This models a massless scalar field
(field equation the wave equation) whose field energy is the sole
source of the gravitational field or spacetime curvature. For purely
mathematical reasons, as I said, in 2+1 the Riemann tensor is
completely determined by the Ricci tensor, so the 2+1 EFE does not
permit gravitational radiation. However, the massless scalar field
does of course permit radiative solutions, so I conjecture that
studying a Regge type reformulation of this toy theory might be
interesting.

(I have no idea what if anything might go wrong if you try this exercise,
because I haven't tried it myself, but I would expect perhaps naively that
nothing would go wrong.)

I stress again that the fact that the nature of the EFE is
signficantly different in 1+1 or 2+1 from n+1 where n > 2 should not
be troubling. As I said in the other post, there are zillions of such
"dimensional phenomena" in mathematics, e.g. in E^n for n =/=3 you
can't have knotted curves, and in E^n for n < 2 you can't move around
compact objects. In particular, such dimensional phenomena are the
order of the day for many PDEs. For example: the nature of harmonic
functions in 1, 2, or n dimensions (n>2) are significantly different
from each other.

> The line t = -2 is a Cauchy surface in Minkowski space. One might think
> that it would also be a Cauchy surface in this bump space.

I think the same misemphasis is again causing trouble here.

When you refer to "Cauchy surface" you presumably are thinking of an
evolution/constraint equation approach to the formulation of a physical
theory, in particular, I suspect you are thinking of the ADM local
reformulation of the vacuum EFE. But, usually one only considers the
-vacuum- EFE in 3+1 or higher, for the reason above. As I just suggested,
it might make some sense to study the mcmsf EFE in 2+1.

> But if so, then the theorem that a Cauchy surface and its neighborhood
> contain enough info to generate the entire space cannot be true, since
> both t = -2 lines look identical. So, what is wrong with this picture?
> Does the theorem that a Cauchy surface and its neighborhood contain
> enough info to generate the entire space not hold in 2d?

The real problem is that the EFE is essentially trivial in 1+1, and the
-vacuum- EFE is essentially trivial in 2+1. Thus there is little or no
point in discussing an ADM type reformulation of a trivial field theory!

OTH, ADM is sensible for 3+1 and for globally hyperbolic spacetimes
one can indeed reconstruct the spacetime geometry by evolving, using
the ADM evolution equations, from initial data on some initial
hyperslice. This initial data cannot be chosen arbitrarily; it must
satisfy the ADM constraint equations. To carry out the evolution, one
must provide some kind of "gauge-fixing rule", which essentially says
how to evolve the -chart- as one evolves the geometry of the
hyperslices. Such rules usually lead to the hyperslice evolving a
"kink" which stops the evolution, but one can always obtain at least a
kind of "slab region" in the spacetime from any legal initial
hyperslice. Then one can choose another slice in that region (or
choose another gauge fixing condition!) and try to evolve a bit
further. Eventually you can reconstruct the entire spacetime---
assuming it is globally hyperbolic! If not, you will only get a
"conformal block" bounded by one or more Cauchy horizons, and gtr does
not allow a unique prediction for what happens after that. For
example, in the conformal diagram for the maximal analytic extension
of the usual exterior region of the Reissner-Nordstrom electrovacuum,
the diamonds are conformal blocks, and the hatched region represents
the history of a collapsing shell of EM radiation (say) which
illustrates why this is only to be expected:

8\ /8
8 \ / 8
8 \ / 8
8 \/ 8
8 /\ #8
8 / \##8
8 / #\#8
8/ ###\8 i^+
/\ ### /\
/ \### / \ scri^+
/ #\# / \
/ ###\/ \
\ ### /\ ext / i^0
\### / \ /
\# / \ / scri^-
\/ \/
8\ /8 i^-
8 \ / 8
8 \ / 8
8 \/ 8

(I'm not implying that this particular maximal extension is physically
reasonable, but it does illustrate the conceptual point.)

BTW, in previous posts I and others have discussed Cauchy horizons and the
ADM reformulation of the 1+3 vacuum EVE extensively. (There is also an
ADM reformulation valid for nonvacuum solutions but this is more
complicated.)

> If that is what is going on, it would be nice to know. (And in general
> I'd like to understand better when this theorem holds and when it
> doesn't.) Or, is there some reason why the line t=-2 in a bump space
> turns out never to satisfy the definition of a Cauchy surface? If so,
> what is the reason?

I hope what I just said completely clarifies the confusion. If not, I'll
have to ask someone else to try.

Chris Hillman

[Chris Hillman]

On Wed, 24 Jul 2002, I wrote:

> On Tue, 23 Jul 2002, I wrote:

> > it is well known that the 2+1 EFE for a minimally coupled massless
> > scalar field is equivalent to the study of the Beck family of solutions
> > to the 3+1 vacuum EFE!

> Uh oh, that might have been a brain blip on my part; the paper I cited
> does -not- contain the claim I thought I recalled,

True.

> and now I suspect my memory may have been erroneous.

False. I've seen the fact I mentioned most recently somewhere else, but
for a slightly different formulation see the superb review paper

http://xxx.lanl.gov/abs/gr-qc/0004016

where one will find citations but also a brief discussion which suffices
to establish my claims that

> it would be -very- easy to check the quoted claim, but I'm too lazy

:-/

Chris Hillman

[eric alan forgy]

Hey David,

I hope you don't mind me taking wild stabs at these questions. You
obviously know much more about this than I do, but I end up learning a lot
by trying to explain things, so here it goes...

On Wed, 24 Jul 2002, David Hillman wrote:
>
> No: believe it or not, what I was interested in was an answer to my question.
>
> Which, to restate simply (at the risk of repeating myself), is: in 2-d GR, you
> can have a metric in which Minkowski space is modified by putting a C-infinity
> bump of curvature within radius 1 of the origin, leaving the rest completely
> flat. The metric satisfies the vacuum EFE since any 2d Lorentz metric does.
> The line t = -2 is a Cauchy surface in Minkowski space. One might think that
> it would also be a Cauchy surface in this bump space. But if so, then the
> theorem that a Cauchy surface and its neighborhood contain enough info to
> generate the entire space cannot be true, since both t = -2 lines look
> identical. So, what is wrong with this picture? Does the theorem that a Cauchy
> surface and its neighborhood contain enough info to generate the entire space
> not hold in 2d? If that is what is going on, it would be nice to know. (And in
> general I'd like to understand better when this theorem holds and when it
> doesn't.) Or, is there some reason why the line t=-2 in a bump space turns out
> never to satisfy the definition of a Cauchy surface? If so, what is the
> reason?

After learning that ANY Lorentzian metric in 2D satisfies the EFE, then
the only possible answer to your question that I can tell is that
causality is fine and Cauchy does tell you that you can predict the
evolution uniquely. The catch is that your schrunched up Lorentzian
manifold must be precisely the same as an unschrunched up Lorentzian
manifold. The two must be equivalent in some since (related by a gauge
transformation?).

I just read Urs' explanation that seems perfectly clear to me. You can
transform away all of your schrunches so that the metric is conformally
equivalent to a flat Minkowski metric. Then due to the magic of 2d, the
EFEs are apparently conformally invariant. Voila! An unschrunched
Lorentzian manifold is equivalent to a schrunched Lorentzian manifold in
2d.

How does that sound?

Eric

PS: I am the inventor of the term "schrunched manifold" (c) :)

[David Hillman]

Chris Hillman wrote:

> I think that what may be going on here is that you are somehow confusing
> yourself by what might at first seem to be a minor misemphasis. The above
> makes good sense only if you delete "in 2-d GR".

Obviously we are having difficulty understanding one another (and
therefore we are repeating ourselves...such discussions might be
better held in private; I tried, but probably got sent to
dev/null). But I think I am beginning to see where you are coming
from. For you, the true EFE is in 4-d; other d's don't deserve the
name. Especially 2d, which is "trivial" since:

> John Baez has already answered the questions in your post (in his comment
> addressed to Dan Christensen), and I thought I had too at least twice but
> now I can't seem to find the post I thought I had submitted (maybe it
> simply hasn't yet appeared).

I puzzled for a while about why you think John answered my question
(especially since he simply stated that in 2d the Einstein tensor is
zero, a fact that I had also noted in a previous post in the thread),
but now I think I do see why. In higher d, there are exactly enough
equations in the suitably posed EFE for the Cauchy problem to have a
solution. In 2d, because the Einstein tensor is zero, the one equation
needed is degenerate, so there are too few equations. Hence (according
to you) the situation is "unphysical". Probably none of you talked
about the number of equations business because you thought it was
obvious, but unfortunately for me very little is obvious.

Is this correct?

Fine. On the other hand, I like to learn things by understanding how
they work in various dimensions. The EFE does not explicitly have a
dimension written down, and it is easy to plug any number in for the
dimension. So, I thought it would be nice to see how the thing worked
in lower dimensions. I played with Regge calculus in 2d and since I
wasn't understanding how causality worked when curvature was present I
thought I'd look at the flat case. That is: at triangulations of 2d
Minkowski space. And there I found formulations where causality worked
fine. In particular I was looking for combinatorial solutions: by
which I roughly mean (since I don't know exactly what I mean anyway)
triangulations of Minkowski space that use only finitely many types of
triangles. In fact such solutions exist. The ones I've found are not
extraordinarily interesting, but they aren't trivial either. They're
certain cellular automata in which the cells can be taken to be
rectangles in Minkowski space whose sides are null. The value of the
cell is the squared length of the spacelike diagonal of the rectangle
(this quantity is probably proportional to the area of the rectangle;
I don't know how to compute areas in Minkowski space, but this
quantity certainly acts like an area). This was interesting to me
because I'd long known that one could not have anything but the most
trivial cellular automata satisfying SR in 2d when one considered the
spaces of the CA to be horizontal lines in Minkowski space...but
hadn't realized that if you let the spaces of the CA be spacelike
lines that had bends in them, you could get more interesting
solutions.

Now, what does this have to do with GR? Perhaps nothing. But if Urs is
correct, perhaps it does have something to do with it: perhaps by
choosing to impose the flatness condition, I am fixing the conformal
gauge. And perhaps there are other ways to fix this gauge which would
give me different combinatorial solutions.

In any case: however trivial 2d is, I am learning something by
thinking about it! Thanks for your (and everyone's) help in this.

> If you are willing to go to 2+1, you should be able to find an ADM
> reformulation of the 2+1 EFE coupled to the 2+1 curved spacetime wave
> equation, i.e. "the field equations for a 2+1 minimally coupled
> massless scalar field solution". This models a massless scalar field
> (field equation the wave equation) whose field energy is the sole
> source of the gravitational field or spacetime curvature. For purely
> mathematical reasons, as I said, in 2+1 the Riemann tensor is
> completely determined by the Ricci tensor, so the 2+1 EFE does not
> permit gravitational radiation. However, the massless scalar field
> does of course permit radiative solutions, so I conjecture that
> studying a Regge type reformulation of this toy theory might be
> interesting.

As in 2d, the vacuum EFE in 3d wouldn't be entirely uninteresting for me,
since I would be looking at tilings.

But you're right that I'd be also interested in treating cases where
stress-energy is nonzero, since the real world seems to be like
that. I'm used to trying to treat the most general situation possible,
since it is often simpler. But in this case the general case seems
poorly understood. As I understand it, I have to impose some sort of
global condition on stress/energy (a condition that holds at every
point in spacetime) so that I get a solvable PDE for the initial value
problem, but nobody knows what the most general such condition
is. True?

Actually it amazes me that this is the case: it must be that whenever
anyone concludes something about the universe using GR, they are
imposing some such condition! And, what: hoping that in the real world
the condition approximately holds? I naively thought that GR was a
complete theory of matter and curvature, matter actually being just an
aspect of curvature, so that if you specified the state of matter and
curvature on a Cauchy surface, GR would tell you what happens. But:
no? you really also have to describe an additional condition satisfied
globally by the matter? Well, I'll have to do some work if I hope to
understand all this.

[Moderator's note: GR needs to be supplemented by a theory of
matter to make specific predictions in the presence of matter.
That's what the rest of physics is for. - jb]

[ba...@galaxy.ucr.edu]

Chris Hillman <hil...@math.washington.edu> wrote:

>The real problem is that the EFE is essentially trivial in 1+1, and the
>-vacuum- EFE is essentially trivial in 2+1. Thus there is little or no
>point in discussing an ADM type reformulation of a trivial field theory!

I hope Steve Carlip isn't listening! :-) He wrote a whole book
on the Einstein equations in 2+1 dimensions, including a lot about
these equations for a vacuum and their ADM formulation...
and how to quantize them.

Of course, you're completely right that the vacuum Einstein equations
in 2+1 dimensions say the metric is flat, and that this makes
the theory *locally* trivial - all solutions look *locally*
alike. However, they can be different *globally*, and this leads
to a lot of fascinating math.

For example, the ADM formulation of the 2+1 vacuum Einstein equations
is nice because the phase space is closely related to the
"Teichmueller space" of a Riemann surface.

And when you quantize the vacuum Einstein equations in 2+1 dimensions,
they get even *more* interesting... though only globally.

(Amusingly, Ruth Williams just walked by holding a copy of Steve's
book, which she bought today... she's an expert on the Regge calculus
at DAMTP here in Cambridge, but she's working on a (2+1)d gravity
project now.)

[John Baez]

In article <ahlo98$ug6$1...@rs04.hrz.uni-essen.de>,
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:

>"David Hillman" <d...@cablespeed.com> schrieb im Newsbeitrag
>news:3D3E1380...@cablespeed.com...

>> The line t = -2 is a Cauchy surface in Minkowski space. One might think that

>> it would also be a Cauchy surface in this bump space. But if so, then the
>> theorem that a Cauchy surface and its neighborhood contain enough info to
>> generate the entire space cannot be true, since both t = -2 lines look
>> identical. So, what is wrong with this picture? Does the theorem that a
>> Cauchy surface and its neighborhood contain enough info to generate the
>> entire space not hold in 2d?
>
>I think the answer is this: A Cauchy surface should contain enough information
>to generate the entire space *up to symmetries/gauge transformations*. The
>vacuum Einstein-Hilbert action in d=2 is invariant under diffeos and under
>conformal transformations. By means of these it is possible to transform every
>d=2 metric tensor locally to the flat Minkowski metric diag(-1,1). Hence
>*every* spacelike line should be a Cauchy surface in d=2, since one does not
>need any information at all to locally generate the entire spacetime up to
>gauge transformations.
>
>(However, since I am not an expert on this I may be wrong. Corrections are
>welcome.)

This sounds right to me!

I hope Hillman understands the utterly crucial phrase *up to
symmetries/gauge transformations*, because this is the crux
of the issue. Let me try to explain this idea in less technical
terms, since it's incredibly important. I'll be a bit imprecise....

In many field theories, there are ways of changing
your fields that don't change the actual physics of what's going
on. For example, in electrostatics you can add any constant to the
potential without changing the physics. Transformations like
this are called GAUGE SYMMETRIES. We only hope to predict the
future "up to gauge symmetries". By this I mean, two predictions
that differ by a gauge symmetry are equally good as far as physics
goes; there's no way to decide which is right - but no need to, either.

We say a slice of spacetime is a "Cauchy surface" if knowing what's
going on in a little neighborhood of that slice is enough to determine
what's going on everywhere... UP TO GAUGE SYMMETRIES.

In general relativity, changing coordinates doesn't change the physics,
so they count as gauge symmetries. This really freaked Einstein
out - he was set back for quite a while by the fact that his equations
only predicted the future UP TO A CHANGE OF COORDINATES. In fact,
he tried to change the equations to eliminate this "problem", until
he realized it wasn't a problem!

In 2d spacetime, general relativity is even weirder: it has a lot
more solutions but also a lot more gauge symmetries - you can get
from any solution to any other by a gauge symmetry!

(At least locally... I haven't thought about general relativity
in 2d spacetime very much.)

So, in 2d spacetime, if you know what's happening on a Cauchy
surface, you can still predict the future "up to gauge symmetries" -
but now that's not saying very much! Basically, there's nothing
left to predict, since all solutions are physically the same!

[Urs Schreiber]

David Hillman wrote:

[...]

> Now, what does this have to do with GR? Perhaps nothing. But if Urs is
> correct, perhaps it does have something to do with it: perhaps by
> choosing to impose the flatness condition, I am fixing the conformal
> gauge. And perhaps there are other ways to fix this gauge which would
> give me different combinatorial solutions.

Perhaps one should add a remark here: Since, as Chris Hillman
has emphasised, the theory described by the EH-action in 2d is
essentially empty (the Lagrangian is a total derivative), it
probably makes little sense to consider special types of
invariances of its solutions. Surely, if every metric is a
solution then any transformation whatsoever is a symmetry.
This makes little sense by itself. But conformal invariance
(plus the usual diffeos) is the crucial symmetry that remains
when one adds couplings to standard massless matter in 1+1
dimensions, like ordinary scalar or spinor fields, because the
additional matter action terms are also conformally invariant
in 2d. Furthermore, by adding these matter contributions to
the EH action, the theory becomes far from empty and is
apparently worth studying - it is essentially string theory
(possibly with Euclidean target space when one comes from the
gravitational perspective). And in this case conformal
invariance is indeed the reason why it is consistent to assume
that all these 2d matter fields propagate on flat Minkowski
space.

Chris Hillman wrote:

> > If you are willing to go to 2+1, you should be able to find an ADM
> > reformulation of the 2+1 EFE coupled to the 2+1 curved spacetime wave
> > equation, i.e. "the field equations for a 2+1 minimally coupled
> > massless scalar field solution".

BTW, most ingredients of the ADM reformulation of *2d* gravity
minimally coupled to massless scalar fields can be found in
sections 12.2 and 12.3 (starting on p. 114) of

M. Henneaux, Lectures on String Theory, with Emphasis on
Hamiltonian and BRST-Methods, in L. Brink and M. Henneaux,
Principles of String Theory, Plenum Press (1988)

--
Urs.Sc...@uni-essen.de

[Urs Schreiber]

Chris Hillman <hil...@math.washington.edu> wrote in message news:<Pine.OSF.4.44.020724...@goedel2.math.washington.edu>...

[...]

> Or again: the phrase "in 2-d GR" is objectionable because there isn't any
> such thing, at least not in the sense of a nontrivial theory.

[...]

I understand how this is meant, namely with respect to local degrees
of freedom.

But there is apparently a whole research field going under the
headline "2-d GR", or maybe rather "2-d quantum GR". Taking a random
example from the literature, I'll quote from the introduction of [1]:

>>>
At first sight, two-dimensional general relativity appears "trivial,"
at least as a physical theory, since for instance the Einstein-Hilbert
action

I = 1/2pi int sqrt(g) R

is a topological invariant, so that the Einstein field equations are
automatically obeyed.
Yet, actually, on further investigation, two-dimensional quantum
general relativity proves to be a strikingly rich theory. What is
loosely called "critical" two-dimensional gravity is an essential
ingredient in string theory. "Noncritical" two-dimensional gravity is
a much more difficult subject which has been intensively studied with
various motivations including possible applications to string theory
and to the large N limit of quantum gauge theories with gauge group
SU(N).
In the earliest approach to the subject, introduced by Polyakov
[...], noncritical two-dimensional gravity is related to a quantum
field theory with Liouville action. [...] In a very different
approach, two dimensional gravity has been studied by counting
triangulations of surfaces, which can be related to "matrix models"
[...]. This approach has been developed with spectacular success
[...]. Another approach [...] uses ideas of "topological quantum field
theory" and can be reduced to a description in terms of the cohomology
of the moduli space of Riemann surfaces. [...]
A variety of arguments indicate that the theories constructed by
these different approaches are equivalent. In addition to heuristic
arguments, Liuoville theory can be compared to the matrix models by
comparison of critical exponents [...]. Topological gravity is related
to Liouville theory by an elegant argument to to Distler [...] that
involves a variant [...] of the usual bose-fermi equivalence of
Riemannian surfaces. The topological field theory approach is related
to the matrix models by explicit comparison in genus <= 3, by the
"string equation" and another similar equation that can be derived in
both frameworks, and by formal analogies.
<<<
(from pp. 871-872 of [1])

But I have only just come across this topic, so I cannot say more.
Maybe some of the 2d-GR experts on this group can say more. (I assume
that the above paragraph refers to J. Distler's work. Also I have seen
a related article by C. Torre.)

[1]
E. Witten, Two-Dimensional Gravity and Intersection Theory on Moduli
Space, in E. Brezin and S. Wadita (eds.), The large N expansion in
Quantum Field Theory and Statistical Physics, World Scientific (1993),
p. 871

[Chris Hillman]

On 30 Jul 2002, Urs Schreiber wrote:

> But there is apparently a whole research field going under the headline
> "2-d GR", or maybe rather "2-d quantum GR".

Are you sure this means the 1+1 EFE rather than 2+1 EFE? Same for your
reference to "2d gravity". In my own posts I tried to consistently use
"1+1, 2+1, 3+1, 4+1", etc., when signature mattered and "2,3,4,5", etc.
when it did not. This might have caused confusion even though I did try
once before to point out the terminological conflict. I also tried to
distinguish between pure math (semiriemannian geometry) and specific
physical theories, such as classical gtr.

Chris Hillman

[John Baez]

Chris Hillman <hil...@math.washington.edu> wrote:

>On 30 Jul 2002, Urs Schreiber wrote:

>> But there is apparently a whole research field going under the headline
>> "2-d GR", or maybe rather "2-d quantum GR".

>Are you sure this means the 1+1 EFE rather than 2+1 EFE?

>In my own posts I tried to consistently use

>"1+1, 2+1, 3+1, 4+1", etc., when signature mattered and "2,3,4,5", etc.
>when it did not.

A lot of people in this game use, say, "1+1" for 2d manifolds
with Lorentzian signature, and "2" for 2d manifolds with Riemannian
signature. Anyway, there's a huge amount of work on gravity in both
1+1 dimensions and 2 dimensions, some of it using the plain old
Einstein action (which gives a 2d TQFT), but a lot using more fancy
actions (e.g. "Liouville theory"). There's also a huge amount of
work on gravity in both 2+1 dimensions and 3 dimensions, mostly
using the Einstein action, but often with cosmological constant.

Just because something is simple enough to be almost completely
understood, does not mean it's not worth bothering with. :-)

[David Hillman]

> [Moderator's note: GR needs to be supplemented by a theory of
> matter to make specific predictions in the presence of matter.
> That's what the rest of physics is for. - jb]

(It is extremely embarrassing that I don't understand this, but such is life.)
Okay, the source of my confusion is the following. In Newtonian physics my
impression is that we can specify a bunch of, say, point-masses at various
positions, and compute what happens, assuming that gravity is the only force.
(Or, if we can't compute it, it's still determined by the theory.) The only
theory involved here is the theory of gravity. No other "theory of matter" is
required. Plug the initial conditions in, apply the theory of gravity, and
you're done. Is there no analogue of this in GR? (Or, is "there are no other
forces besides gravity," suitably interepreted, a theory of matter? What would
that theory say about the stress-energy tensor?)

[Chris Hillman]

On Tue, 30 Jul 2002, David Hillman wrote:

> Obviously we are having difficulty understanding one another (and
> therefore we are repeating ourselves...such discussions might be better
> held in private; I tried, but probably got sent to dev/null).

If you sent it from a *.com addy, yeah.

> But I think I am beginning to see where you are coming from. For you,
> the true EFE is in 4-d; other d's don't deserve the name.

No, I said what I -said-: the n+1 dimensional EFE is trivial for n < 3,
for the reasons I explained, as did other posters, as do most modern
textbooks.

You still haven't said whether you've attempted to do the two standard
exercises I mentioned, BTW. I think that you'll -need- to do them before
you can hope to understand what I was -really- saying in the post to which
you are responding.

> I puzzled for a while about why you think John answered my question
> (especially since he simply stated that in 2d the Einstein tensor is
> zero, a fact that I had also noted in a previous post in the thread),
> but now I think I do see why. In higher d, there are exactly enough
> equations in the suitably posed EFE for the Cauchy problem to have a
> solution. In 2d, because the Einstein tensor is zero, the one equation
> needed is degenerate, so there are too few equations. Hence (according
> to you) the situation is "unphysical".

No, I said -"trivial"-, and for the 1+1 EFE I referred to the same fact as
John. You did suss part of what I meant correctly--- the theory of an
equation of some kind which is -either- "trivial" or "unphysical" is not
likely to be interesting in its own right. However, "trivial" and
"unphysical" are -quite different notions-. Furthermore, the latter
might "disqualify" a theory (from the status of "interesting in its own
right") only in the eyes of physicists, while the former will most likely
"disqualify" it in the eyes of either mathematicians or physicists!

> On the other hand, I like to learn things by understanding how they work
> in various dimensions. The EFE does not explicitly have a dimension
> written down, and it is easy to plug any number in for the dimension.
> So, I thought it would be nice to see how the thing worked in lower
> dimensions.

As I said, this is indeed a very natural idea. The textbooks often offer
the problems I mentioned (show that in dimension two the Riemann tensor
can be recovered from the Ricci scalar, and in dimension three from the
Ricci tensor) precisely in order to demonstrate that while this -does-
yield some insight in the case of the EFE--- in particular it can draw
attention to the crucial role played by the Weyl tensor in four or higher
dimensions--- it is fair to say that the main insight is understanding of
why unfortunately you can get only very limited information about the
theory of higher dimensional EFE from 2+1 EFE, and essentially none from
1+1 EFE. There are in fact zillions of other observations to the effect
that gravitation is inherently four (or higher) dimensionsal, e.g. well
known observations about PP waves which I have often mentioned in past
posts.

> I played with Regge calculus in 2d and since I wasn't understanding how
> causality worked when curvature was present I thought I'd look at the
> flat case.

Remember that I myself pointed out that the while I have mostly bad news
concerning 2+1 EFE, there is also some significant good news: some of the
most important large families of exact solutions in 3+1 EFE are in fact
very closely related to certain kind of 2+1 EFE solutions. So, as I said,
if you are willing to go up one dimension and look study minimally coupled
massless scalar field ("mcmsf") solutions to the 2+1 EFE, you might be
able to explore issues like Cauchy horizons in a significantly simpler
setting.

No guarantees--- I haven't thought about this myself, it is just something
which occured to me and which appears promising at first glance.

> That is: at triangulations of 2d Minkowski space.

^^

Noted for future reference. Presumably you mean the plane E^(1+1), not
E^(2+1). Above I used "two dimensional" to mean two dimensional
semiriemannian manifolds, i.e. any signature, and so forth, and used 1+1
to specify Lorentzian signature; similarly for other dimensions.

> And there I found formulations where causality worked fine.

I hope you realize that I doubt anyone here, including yourself, knows
what you mean by "causality works". Certainly I have no idea!

I realize that this is a reasonable excuse for not having so far explained
clearly what you mean, but I mention it to encourage you to keep asking
yourself what you mean. My experience also suggests that it would be
helpful to pause and study up on 3+1 ADM and Regge, while keeping in the
back of your mind the question: "would any of this work in some sense for
2+1 mcmsf?" And even before that I think you should do the exercises
about the 1+1 and 2+1 EFE, making sure you understand why I said the
conclusions have nothing to do with the EFE or any other physical theory
one might try to impose on two and three dimensional Lorentzian
spacetimes; rather, there are some purely mathematical facts about the
Riemann tensor in those dimension which limits the interest of the EFE
since the LHS of the EFE happens to employ the Ricci tensor--- or
equivalently, the Einstein tensor; see John Baez's gtr tutorial.

> But you're right that I'd be also interested in treating cases where
> stress-energy is nonzero, since the real world seems to be like that.
> I'm used to trying to treat the most general situation possible, since
> it is often simpler. But in this case the general case seems poorly
> understood.

I can confirm that the nature of the solution space to the 3+1 EFE is
poorly understood. As yet there is not even concensus on the "right"
-definition- of this space. OTH, the families of vacuum solutions (and
even evacs and mcmsf solutions) with two commuting Killing vector fields
are increasingly well understood, and while currently known solution
generating techiniques tend to give rather complicated exact solutions,
they can already be used to a considerable extent to cook up a solution in
a given class with preassigned physical characteristics, e.g. an
asympotically flat Ernst vacuum with preassigned Geroch mass/ang. momentum
multipople moments.

> As I understand it, I have to impose some sort of global condition on
> stress/energy (a condition that holds at every point in spacetime)

Huh?

> so that I get a solvable PDE for the initial value problem, but nobody
> knows what the most general such condition is. True?
>
> Actually it amazes me that this is the case:

I have no idea what you're talking about. Are you possibly saying you are
suprised that to define/find say an electrovacuum you need to have at hand
a relativisitic classical field theory of EM? (Fortunately, you do:
Maxwell's theory of EM.) I see that John Baez had the same question:

> [Moderator's note: GR needs to be supplemented by a theory of
> matter to make specific predictions in the presence of matter.
> That's what the rest of physics is for. - jb]

Or are you perhaps refering to the energy conditions?

> In particular I was looking for combinatorial solutions: by which I
> roughly mean (since I don't know exactly what I mean anyway)

OK, agreed :-/

> triangulations of Minkowski space that use only finitely many types of
> triangles. In fact such solutions exist.

^^^^^^^^^

Sigh... I'll have to bug out with this post because for some reason I just
cannot seem to get across an absolutely crucial insight, which is that it
doens't really make good sense to speak of "solutions" to a -trivial-
equation (the 1+1 EFE)! John is -much- better than I at explaining stuff;
I tend to quickly lose interest in trying to communicate when I sense
there is insufficient common background knowledge to facillitate
communication. So I hope he'll step in here and try to clear up the
confusion.

Chris Hillman

[Chris Hillman]

On Tue, 30 Jul 2002 ba...@galaxy.ucr.edu wrote:

> Chris Hillman <hil...@math.washington.edu> wrote:
>
> >The real problem is that the EFE is essentially trivial in 1+1, and the
> >-vacuum- EFE is essentially trivial in 2+1. Thus there is little or no
> >point in discussing an ADM type reformulation of a trivial field
> >theory!
>
> I hope Steve Carlip isn't listening! :-) He wrote a whole book
> on the Einstein equations in 2+1 dimensions, including a lot about
> these equations for a vacuum and their ADM formulation...
> and how to quantize them.

I know.

> Of course, you're completely right that the vacuum Einstein equations
> in 2+1 dimensions say the metric is flat, and that this makes
> the theory *locally* trivial - all solutions look *locally*
> alike. However, they can be different *globally*,

Actually, I did know -that- :-/ (Needless, perchance, to say?)

> For example, the ADM formulation of the 2+1 vacuum Einstein equations is
> nice because the phase space is closely related to the "Teichmueller
> space" of a Riemann surface.

Cool!--- I'd like to know more about this if you ever feel like saying
more. But only after you/jim explain quantales in more detail :-/
Speaking of which, I just came across this paper

http://xxx.lanl.gov/abs/math.LO/0106059

This is the only preprint I've found so far which mentions the word
"quantale" in the abstract or title, but so far my search has been very
casual (almost wrote :-/ "uses" and "causal" respectively; -that- would
have been confusing!).

Chris Hillman