Confused about DeSitter---again
Serenus Zeitblom
I think it is pretty well-known by now that DeSitter space can be
sliced in lots of different ways, eg by flat slices or by
spherical ones. What I don't understand is this: why does
anyone take the flat slicing seriously? I mean, it is obviously
going to lead to a geodesically incomplete spacetime---you can
have timelike worldlines coming in from "beyond the universe"!
That is pretty ridiculous---if you are going to let me do that,
I can declare that the world-tube of George Bush is a
"spacetime", simply by cutting away the rest of the universe.
When George eats a pretzel, that pretzel comes from beyond
the Georgiverse---but if that is ok for DeSitter, it should be
ok for George.
We don't normally do that kind of thing; geodesic
incompleteness is tolerated only when the curvature diverges or
something like that. So why should we tolerate it in the
case of DeSitter space? The only special thing there is that,
being empty of matter, de Sitter is extremely symmetric, so
you can hide the incompleteness with a nice choice of
coordinates. This is just like the Milne Universe, a
so-called cosmology which is really just a chunk of
Minkowski space. The fact is, however, that the full DeSitter
has spheres as slices; the flat slicing is just a
mathematical trick of no real importance, obtained by
cutting away half of the Penrose diagram, and hence just
like the George Bush Universe constructed above.
In short, DeSitter is REALLY a cosmology with finite
spherical space.
Right?
Stephen Blake
serenusze...@yahoo.com (Serenus Zeitblom) wrote in message
news:<c7fd6c7a.03080...@posting.google.com>...
> I think it is pretty well-known by now that DeSitter space can be
> sliced in lots of different ways, eg by flat slices or by
> spherical ones. What I don't understand is this: why does
> anyone take the flat slicing seriously?
>...
> In short, DeSitter is REALLY a cosmology with finite
> spherical space.
> Right?
I think I'm in agreement with the main point of your post. The
space-like slices of de Sitter space $t=const$ for the planar metric
(eqn (14) of [1])
ds^{2}=dt^2-e^{2t}(dx^{2}+dy^{2}+dz^{2})
intersect the cosmological horizon of an observer with worldline
x=y=z=0. Therefore, the planar coords are not physically meaningful
to an observer who is a positivist. The same goes for the global
metric (eqn (9) of [1]) (and its conformal cousin eqn (11) of [1])
ds^{2}=dt^{2}-\cosh^{2}t(d\chi^{2}+\sin^{2}\chi(d\theta^{2}+\sin^{2}\theta
d\phi^{2})
because the spacelike hypersurfaces t=const intersect the cosmological
horizon of the observer with worldline \chi=0.
The only coords which make sense physically are the static coords (eqn
(16) of [1])
ds^{2}=(1-r^{2})dt^{2}-(dr^{2}/(1-r^{2})+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}))
and here the space-like slices t=const are hyperspheres. In fact, it
turns out that an observer in de Sitter space with a radar set who
defined time t=(t2+t1)/2 and range as R=(t2-t1)/2 where t1 is the time
of the transmitted pulse and t2 is the time of the received pulse
would find the static metric by using r instead of R where
\cosh^{2}R=1/(1-r^{2}). In this case the observer's cosmological
horizon is the hypersurface R=\infty.
References
[1] 'Les Houches Lectures on de Sitter Space', M. Spradlin, A.
Strominger and A. Volovich, arXiv:hep-th/0110007
Steve Carlip
> I think it is pretty well-known by now that DeSitter space
> can be sliced in lots of different ways, eg by flat slices or by
> spherical ones. What I don't understand is this: why does
> anyone take the flat slicing seriously?
There's a lot of work going on these days on what quantum
gravity means in an asymptotically de Sitter space. I'm no
expert on this, but one serious proposal (see, e.g., Bousso,
hep-th/0010252) is that the quantum theory should be
formulated on a ``causal diamond'' -- the points both in
the past and the future of an observer's world line -- and
that the rest of the space is either irrelevant or, in stronger
versions of this conjecture, is gauge-equivalent and thus
contains no additional physics.
If this is right, then the relationship of a slicing to the global
structure is irrelevant, because the global structure is just
a classical artifact.
Another very different argument for looking at the flat slicing
is given by Aguirre and Gratton in astro-ph/0111191. They
recognize that the resulting spacetime is incomplete, but
argue for boundary conditions at a null surface that lead
to a nice picture of eternal inflation.
I don't want to argue that either of these considerations is
``right'' -- who knows? -- but they both mean that the flat
slicing is not just trivially wrong.
Steve Carlip
Steve Carlip
> I think it is pretty well-known by now that DeSitter space
> can be sliced in lots of different ways, eg by flat slices or by
> spherical ones. What I don't understand is this: why does
> anyone take the flat slicing seriously?
There's a lot of work going on these days on what quantum
Serenus Zeitblom
ste...@ntlworld.com (Stephen Blake) wrote in message news:<d8a8f2ec.03081...@posting.google.com>...
> serenusze...@yahoo.com (Serenus Zeitblom) wrote in message
> news:<c7fd6c7a.03080...@posting.google.com>...
>
> > spherical space.
> > Right?
>
> I think I'm in agreement with the main point of your post. The
> space-like slices of de Sitter space $t=const$ for the planar metric
> (eqn (14) of [1])
>
> ds^{2}=dt^2-e^{2t}(dx^{2}+dy^{2}+dz^{2})
>
> intersect the cosmological horizon of an observer with worldline
> x=y=z=0. Therefore, the planar coords are not physically meaningful
> to an observer who is a positivist. The same goes for the global
> metric (eqn (9) of [1]) (and its conformal cousin eqn (11) of [1])
>
> ds^{2}=dt^{2}-\cosh^{2}t(d\chi^{2}+\sin^{2}\chi(d\theta^{2}+\sin^{2}\theta
> d\phi^{2})
>
> because the spacelike hypersurfaces t=const intersect the cosmological
> horizon of the observer with worldline \chi=0.
You are right that the spatial sections of these slicings intersect
the horizon of the observer at the origin. But if that observer
insists that the only part of the universe that really counts is the
part with which HE can interact, then he is not even a positivist ---
he's a solipsist! Or a cosmic egomaniac. Still, I see where you are
coming from, because a lot of people [notably L Susskind and co] talk
as if they believe that, to understand DeSitter, you only really need
to care about "the" static patch --- presumably the one centered on
Stanford, CA. The excuse for this is supposedly "black hole
complementarity", which allegedly ought to make us frightened of
global descriptions of spacetimes. To which I say, "phooey!"
>
> The only coords which make sense physically are the static coords (eqn
> (16) of [1])
>
> ds^{2}=(1-r^{2})dt^{2}-(dr^{2}/(1-r^{2})+r^{2}(d\theta^{2}+\sin^{2}\theta
> d\phi^{2}))
>
> and here the space-like slices t=const are hyperspheres.
I think you mean "balls" here, not spheres---ie you mean the inside as
well.
In fact, it
> turns out that an observer in de Sitter space with a radar set who
> defined time t=(t2+t1)/2 and range as R=(t2-t1)/2 where t1 is the time
> of the transmitted pulse and t2 is the time of the received pulse
> would find the static metric by using r instead of R where
> \cosh^{2}R=1/(1-r^{2}). In this case the observer's cosmological
> horizon is the hypersurface R=\infty.
Well, that certainly isn't what we normally do in cosmology. So why
should we do it here?
Chris Hillman
from s.a.r. to this newsgroup.
Stephen: you wrote:
> The only coords which make sense physically are the static coords
???!!!!
Since this claim appears to contradict the diffeomorphism covariance of
gtr, I think you need to explain this more fully!
> I think I'm in agreement with the main point of your post.
Heh, I wish you'd tell me what the point -was-, then! :-/
(I already replied to Serenus in sci.astro.research; you can tell from
what I wrote there that I had a lot of trouble understanding what he was
trying to say. Initially I thought he was confusing notions such as
coordinate charts, manifolds with boundaries, and maximal extensions of a
local solution to the EFE, but from your post it seems you appear to think
that he was claiming that there is no -operational- way of defining
slicings other than the S^3 slicings. If so, you could be right about
what he meant, but this claim currently makes absolutely no sense to me,
and I hope you will try again to explain what you meant.)
Serenus: -were- you trying to claim that there is no operational way of
defining slicings other than the S^3 slicings? If so, I still have no
idea why anyone would think that--- can you try again to explain what you
have in mind?
> The space-like slices of de Sitter space $t=const$ for the planar metric
> (eqn (14) of [1])
>
> ds^{2}=dt^2-e^{2t}(dx^{2}+dy^{2}+dz^{2})
-infty < t,x,y,z < infty
Grrrr!
(Omitting the intended range of the coordinates is a very bad habit of
physicists which in my experience causes endless and entirely needless
confusion.)
> intersect the cosmological horizon of an observer with worldline
^^^
???
> x=y=z=0.
Just to make sure we're all on the same page: the coordinate planes t = t0
in this chart, which are all isometric to E^3, are the orthogonal
hyperslices defined by a certain expanding but irrotational timelike
geodesic vector field X, which can be written X = @/@t in the above chart.
IOW, this chart is comoving with the family of inertial observers whose
world lines are represented by the integral curves of the vector field X.
In the conformal diagram
______
|\ /|
| \ / |
| \/ |
| /\ |
| / \ |
|/____\|
we can take x = y = z = 0 to correspond to the left vertical segment
(agreed?), and then this "expanding exponential" chart covers the region
______
|*****/|
|****/ |
|***/ |
|**/ |
|*/ |
|/_____|
Note that in the static chart
ds^2 = -(1-r^2/a^2) dt'^2 + dr^2/(1-r^2/a^2)
+ r^2 (du^2 + sin(u)^2 dv^2),
-infty < t < infty, 0 < r < a, 0 < u < pi, -pi < v < pi
the vector field X can be written
X = 1/(1-r^2/a^2) @/@t' + r/a @/@r
But this representation is of course only valid on the domain of our
static chart, which as you know only covers the region
______
|\ /|
|*\ / |
|**\/ |
|**/\ |
|*/ \ |
|/____\|
Another convenient chart covering the same region as your expanding
exponential chart is the outgoing Painleve chart
ds^2 = -dT^2 + (dr - r/a dT)^2 + r^2 (du^2 + sin(u)^2 dv^2),
-infty < T < infty, 0 < r < infty, 0 < u < pi, -pi < v < pi
Exercise: read off the obvious coframe and find the dual frame. Use this
to plot light cones in the Painleve chart (suppress the angular
coordinates). What is the significance of the surface r = a? Is this a
coordinate singularity? Show that in this chart X can be written
X = @/@T -r/a @/@r
and plot some of the integral curves. Also plot some radial null
geodesics.
Note that in this chart r is the same Schwarzschild type radial coordinate
as in the static chart. The cosmological horizon r = a (spoiler: this is
-not- a coordinate singularity!) corresponds to
______
|\ |
| \ |
| \ |
| \ |
| \ |
|_____\|
in the conformal diagram. This horizon is the boundary of the absolute
past of the world line x=y=z=0, i.e. the set of events in H^(1,3) which
can influence some event on our world line, and where the coordinate
planes T = T0 are all isometric to E^3 and are in fact the same slices as
above. In the conformal diagram, as you know, they look like curves which
all run from some event on the world line x=y=z=0 toward the upper right
corner.
Thus, all these slices -intersect- the boundary of the absolute past (the
set of events which can influence some event on the world line x=y=z=0),
but they all miss the -other- cosmological horizon
______
| /|
| / |
| / |
| / |
| / |
|/_____|
which is the boundary of the absolute future, i.e. the set of events which
can be influenced by some event on our world line. (As you no doubt know,
if you consider instead "the antipodal observer", the roles of these two
horizons are interchanged.)
Assuming the latter is the horizon you had in mind above, I agree with
your claim about the relationship of this horizon with the slices t = t0
from your expanding exponential chart.
But then you said:
> Therefore, the planar coords are not physically meaningful to an
> observer who is a positivist.
Whoa!!! I think you need to explain what you mean by "physically
meaningful" here!
Are you saying that only half of H^(1,3) lies in the absolute future of
some event on the world line x=y=z=0? If so, agreed. Are you also saying
that this implies that only the region
______
|\ /|
|*\ / |
|**\/ |
|**/\ |
|*/ \ |
|/____\|
makes sense to a strict logical positivist whose world line is represented
by x=y=z=0? If so, I am not sure that I understand what you have in
mind--- can you explain?
> [1] 'Les Houches Lectures on de Sitter Space', M. Spradlin, A.
> Strominger and A. Volovich, arXiv:hep-th/0110007
I have that paper in front of me, and as far as I can see, they don't make
any claim resembling what you or Serenus might be saying! Of course, I
am not sure if you meant to imply that they -do- make any such claim...
Be this as it may, I note that the first few sections of this paper
involve no QFT. The authors offer a nice review of some charts on
H^(1,3), but they entirely fail to emphasize the main point of my reply to
Serenus in s.a.r., namely that H^(1,3) has zillions of symmetries and
zillions of slicings into any of S^3, H^3, or E^3 slices. I didn't
explicitly say it there, but by the same token there are zillions of
"cosmological horizons"--- this is an observer dependent concept!
(Compare the event horizon in say Schwarzschild vacuum.)
Similarly, "the" conformal diagram really depicts one of a family of
charts; to see this note that there are zillions of "equatorial
three-spheres" and zillions of pairs of antipodal points on each such S^3;
each choice of an equator and pair of "poles" gives a chart which can
pictured in the usual conformal diagram, with the horizons (two diagonals)
corresponding to the boundaries of the absolute past and future of the
world line of either "pole".
I still think overlooking such points may be at the root of at least some
of the confusion I sense in this discussion.
By the way, Serenus, in the outgoing Eddington chart
ds^2 = -(1-r^2/a^2) dp^2 + 2 dp dr + r^2 (du^2 + sin(u)^2 dv^2),
-infty < p < infty, 0 < r < infty, 0 < u < pi, -pi < v < pi,
(same Schwarzschild radial coordinate!), which is one useful chart I
didn't mention in my previous reply (I could have mentioned a dozen
more!), it is convenient to adopt the frame
e_1 = @/@p + r^2/a^2/2 @/@r
e_2 = @/@r - @/@p - r^2/a^2/2 @/@r
e_3 = 1/r @/@u
e_4 = 1/r/sin(u) @/@v
rather than the static frame read off from the static chart. The reason
is that this frame is defined on the entire region
______
|*****/|
|****/ |
|***/ |
|**/ |
|*/ |
|/_____|
covered by this chart. Here Y = e_1 is an expanding but irrotational
timelike vector field with nonzero acceleration
D_Y Y = -r^2/a^2 e_2
Compare the static congruence Z = @/@t' (see the static chart above),
which has acceleration
D_Z Z = -r^2/a^2 1/(1-r^2/a^2) f_2
Here, Z is an expansionless irrotational timelike vector field whose
orthogonal hyperslices are all isometric to S^3 with uniform curvature
1/a^2, as I mentioned in my previous reply. But the orthogonal
hyperslices of Y are isometric to Riemannian-three manifolds having
three-dimensional Riemann tensor
r_(2323) = r_(2424) = 1/a^2 (1-r^2/a^2/2)
r_(3434) = 1/a^2 (1-r^2/a^2/4)
Thus, this is a simple example of a foliation of (half of) H^(1,3) into
hyperslices which are neither homogeneous nor isotropic.
Stephen: hmmm, at first I assumed you were only referencing a convenient
discussion of a few charts on H^(1,3), but now I wonder if you were
thinking of QFT on H^(1,3) and that some consideration involving preferred
Green's functions or something like that motivated your claim that "only
the static chart makes physical sense". If so, can you explain?
> The same goes for the global metric (eqn (9) of [1]) (and its conformal
> cousin eqn (11) of [1])
>
> ds^{2}=dt^{2}-\cosh^{2}t(d\chi^{2}+\sin^{2}\chi(d\theta^{2}+\sin^{2}\theta
> d\phi^{2})
>
> because the spacelike hypersurfaces t=const intersect the cosmological
^^^
???
> horizon of the observer with worldline \chi=0.
AKA x=y=z=0, agreed? If so, I have the same comment as above: which of
the two cosmological horizons associated with this observer do you mean?
> The only coords which make sense physically are the static coords (eqn
> (16) of [1])
> ds^{2}=(1-r^{2})dt^{2}-(dr^{2}/(1-r^{2})+r^{2}(d\theta^{2}+\sin^{2}\theta
> d\phi^{2}))
I really hope you plan to either clarify/justify or retract this claim!
> and here the space-like slices t=const are hyperspheres.
Yup.
> In fact, it turns out that an observer in de Sitter space with a radar
> set who defined time t=(t2+t1)/2 and range as R=(t2-t1)/2 where t1 is
> the time of the transmitted pulse and t2 is the time of the received
> pulse would find the static metric by using r instead of R where
> \cosh^{2}R=1/(1-r^{2}). In this case the observer's cosmological horizon
> is the hypersurface R=\infty.
I think you are claiming that you can give some "operational
interpretation" of the static chart in terms of radar observations, but
your description is too mangled for me to understand! Can you explain
clearly and concisely what you have in mind? The only bit I think I get
here is that the boundary of the events which can be influenced by some
event on the world line x=y=z=0 lies "at infinity" in the expanding
exponential chart.
I certainly would agree that various congruences (e.g. X above) and other
geometrical objects defined on a Lorentzian manifold M can be given more
or less direct operational interpretations in terms of idealized
experiments carried out, in imagination, by ideal observers in M.
However it seems you might be claiming that only a particular family of
charts on H^(1,3) (the static charts) are "physically meaningful", which
is IMO absurd under any interpretation of the phrase "physically
meaningful" I can think of right now. Maybe you only meant to say that
some coordinates (functions on H^(1,3)) have a -more obvious- operational
interpretation than others--- in that case, of course, I'd agree.
Chris Hillman
Serenus Zeitblom
> Serenus Zeitblom <serenusze...@yahoo.com> wrote:
> What I don't understand is this: why does
> > anyone take the flat slicing seriously?
>
> hep-th/0010252) is that the quantum theory should be
> formulated on a ``causal diamond'' -- the points both in
> the past and the future of an observer's world line -- and
> that the rest of the space is either irrelevant or, in stronger
> versions of this conjecture, is gauge-equivalent and thus
> contains no additional physics.
>
> If this is right, then the relationship of a slicing to the global
> structure is irrelevant, because the global structure is just
> a classical artifact.
OK. But I think it is fair to say that this point
of view is *very* controversial. Think of what it
means --- all but a tiny bit of deSitter is
"unphysical", and you can completely ignore
conformal infinity. Furthermore, by means of a natural
choice of coordinates in the "important" part of deSitter,
you can actually claim that spacetime is static!
I have done a highly unscientific
survey of opinions about this, and several of those I
consulted replied in terms that I had better not repeat
here.....I should say that a lot of people are impressed
by Bousso's entropy bound stuff, but the idea that you
can just ignore everything outside a causal diamond is
really pushing it.
> Another very different argument for looking at the flat slicing
> is given by Aguirre and Gratton in astro-ph/0111191.
Again, while their theory is a lot more sensible than
Susskind-Solipsism, I'm sure you recognise that it is
far indeed from mainstream opinion. [I stress that this
is not a criticism, and I urge people to have a look at this
very interesting paper.]
> I don't want to argue that either of these considerations is
> ``right'' -- who knows? -- but they both mean that the flat
> slicing is not just trivially wrong.
OK, I take your point. Let me therefore re-state what
I am driving at. Leaving aside highly unconventional or
controversial interpretations, the following statement
holds: if our spacetime is asymptotically deSitter, then
the curvature of its distinguished spatial sections is
*positive*, not zero. In other words the acceleration of
the universe is prima facie evidence that Omega is
[slightly] greater than unity. I just thought that this
point is not sufficiently emphasised.
Stephen Blake
for the static patch of an observer made sense physically. I shall be stepping
back from this claim to a certain extent. However, before doing this, my
original post used some sloppy language, so I shall just review more
precisely what I meant.
We are concerned with the de Sitter space considered as the hyperboloid,
X_{1}^{2}+X_{2}^{2}+X_{3}^{2}+X_{4}^{2}-X_{5}^{2}=1 (eqn 1)
embedded in 5-d Minkowski space with coords X_{1}...X_{5} as in
refs [1,2]. The Penrose diagram for this dS space is as follows.
I^{+}
_____
|\ |
|*\ |
N |**\ | S
|** \ |
|*___\|
I^{-}
The worldline of our observer is the line N. The diagonal line is the future
event horizon of the observer. If the observer carries a radar and sends a
pulse out at his proper time t1 and receives the signal reflected from a
space-time event at his proper time t2, then he defines the range to the
event as R=(t2-t1)/2 and the time of the event as t=(t2+t1)/2 and then we
get coordinates,
X_{1}=cosh(t)
X_{2}=sinh(R)sin(theta)cos(phi)
X_{3}=sinh(R)sin(theta)sin(phi)
X_{4}=sinh(R)cos(theta)
X_{5}=sinh(t) (eqn 2)
where -infty<t<infty, 0<R<infty, 0<theta<pi, 0<phi<2.pi
and the metric is,
ds^{2}=[dt^{2}-dR^{2}-sinh(R)^{2}(dtheta^2+sin(theta)^{2}dphi^{2})]/(cosh(R)^{2})
where theta and phi are the angles used to point the radar. The substitution
cosh^{2}R=1/(1-r^{2}) will change the metric into the static metric of
(eqn 16) of [1]. These coords cover the northern causal diamond shown
by the asterisks in the Penrose diagram.
I was overcome by too much exhuberance for an observer-centric attitude
to physics when I claimed that these coordinates were the only ones which
made sense physically. This was too strongly worded and I should have said that
the northern causal diamond shown is the only region for which the observer
is able to operationally assign coords to the events, and the radar
is one procedure that he can use to do this. Events outside the observer's
causal diamond exist although the observer cannot operationally assign
coordinates to these events. For example, if the observer uses a telescope,
he will be able to see the light from events in the triangle at the bottom
of the Penrose diagram, but he could not send a radar pulse to these events.
Having admitted these errors, a defence of the observer-centric approach
is in order, lest anyone think I am in complete retreat. I am struck by
the fact that objects which one normally thinks of as the stuff of physical
reality seem to be contingent. For example, in the Unruh effect, an observer
who accelerates through Minkowski space perceives himself to be in a bath of
quanta with a definite temperature. In contrast, an inertial observer
does not perceive these quanta and just perceives a vacuum. For another
example, the stuff we call matter is characterized by mass and spin, yet
mass and spin label the irreducible unitary reprepresentations of the
Poincare group. Suppose that two observers, Alice and Bob, inhabit de Sitter
space and Alice has been translated along Bob's z-axis so that she is on a
distant galaxy. If Alice carries a small piece of test matter with her,
then Alice and Bob will not agree on the rest mass and spin of the piece of
matter because the matter is presumably characterized by the labels of
unitary irreps of the de Sitter group which are not the rest mass and spin
which label the unitary irreps of the Poincare group. (I should point
out that, currently, my knowledge of group rep theory is not strong
enough to do anything other that make this sort of qualitative argument.)
So, guided by these examples, I think we should start with a collection
of equivalent observers. Being mutually equivalent, the observers all agree
that the world they inhabit can be described by some fixed set of
mathematical elements x1,x2,... . Then, if Alice perceives an element x1,
then this element of physical reality appears to Bob as f(x1) where
f is an element of the group that connects the viewpoints of the observers.
The set of mathematical elements is the carrier space for a representation
of the group connecting the observers.
In the current case, the collection of equivalent observers are the
inertial observers in de Sitter space (they are not subject to forces).
The mathematical elements are points, lines, planes and hyperplanes. The
group is the de Sitter group. Alice considers a point p. From Bob's point of
view, this element of physical reality is the point f(p) where (say) f
is the element of the de Sitter group that translates Alice along Bob's z
axis as in the earlier example. When, as in this example, the mathematical
elements form the carrier space for the fundamental rep of the group, we
just recover Klein's Erlanger Programm.
The point p considered by Alice need not be within her future event horizon.
She might not be able to receive a signal from p but she recognises that p
is physically real because she knows that p appears to Bob as f(p) and f(p)
can be within Bob's future event horizon for some choice of the f.
Finally, referring to the space-like slices t=const of the
static coords/radar coords (eqn 2), Serenus wrote:
> I think you mean "balls" here, not spheres---ie you mean the inside as
> well.
The space-like slices are 3-spheres. They are 3-d spaces with constant
positive curvature. They do not have a boundary and they are not balls.
References
[1] "Les Houches Lectures on de Sitter Space", Spradlin, Strominger and
Volovich, arXiv:hep-th/0110007
[2] "The large scale structure of space-time", Hawking and Ellis.
Chris Hillman
On Wed, 20 Aug 2003, Stephen Blake wrote:
> We are concerned with the de Sitter space considered as the hyperboloid,
>
> X_{1}^{2}+X_{2}^{2}+X_{3}^{2}+X_{4}^{2}-X_{5}^{2}=1 (eqn 1)
>
> embedded in 5-d Minkowski space with coords X_{1}...X_{5} as in
> refs [1,2].
I still don't understand whether you and Serenus intend this discussion to
occur only in the dS/CFT context. If so, are you saying that in this
context, one must think of H^(1,3) as this particular embedded manifold?
I don't see why, if so.
Note that I previously assumed you only meant that (as in Hawking & Ellis
and the first few sections of Bousso's paper) this embedding is regarded
merely as a convenient way to obtain some charts, which also makes the
huge isometry group of H^(1,3) easy to appreciate, but not as having
direct physical significance.
> The Penrose diagram for this dS space is as follows.
^^^
> I^{+}
> _____
> |\ |
> |*\ |
> N |**\ | S
> |** \ |
> |*___\|
> I^{-}
>
> The worldline of our observer is the line N.
Just to be 100% sure we're all on the same page:
You should have said that we can define a -particular- "Penrose chart" and
then supressing the angular variables gives a -particular- conformal
diagram. (See a comment below.)
To pick out a particular Penrose chart, we choose a particular equatorial
S^3 (with its global static timelike congruence), and then choose a
particular pair of antipodal points which we call the "N. Pole" and the
"S. Pole". Then we have defined two -particular- causal diamonds
associated with these two -particular- observers, but of course there are
infinitely many other pairs of diamonds defined for other pairs of
observers; they are all obtainable from the first pair by self-isometries
of H^(1,3).
> The diagonal line is the future event horizon of the observer. If the
> observer carries a radar and sends a pulse out at his proper time t1 and
> receives the signal reflected from a space-time event at his proper time
> t2,
Whoa, whoa, whoa!! If this event is not on his world line, you have yet
to explain how this event is related to the proper time of our observer,
agreed?
I take it you mean something like this:
Choose an equatorial S^3 and pair of antipodal points as above. Let O for
Oliver ;-P be the observer whose world line passes through "North Pole" on
this equatorial slice, and let X be the uniformly expanding congruence as
discussed above. Let P for Peter be another observer whose world line is
another integral curve of this congruence. Suppose P carries a radar
reflector, while O carries a radar set.
Diagram of a thought experiment:
| |
O2| |
|\ |
| \ |
| \| P1
| /|
| / |
O1|/ |
| |
(I've drawn their world lines parallel, but really they are diverging
exponentially.)
Here, at event O1, Oliver sends out a radar pulse; at event P1 it hits and
is reflected by P's reflector; at event O2 the echo is recieved by Oliver.
Now, as I understand this exercise, we are trying to construct the
coordinate chart (1) "operationally" in terms of this kind of experiment.
(Right?)
That is, recalling that a "coordinate" is nothing but a monotonic function
defined on some region of M, we want to describe a thought experiment in
which we can in principle label some or all of the events in a particular
region of H^(1,3), to wit a particular causal diamond on H^(1,3). Here,
in order to label all the events, clearly we must define the values of the
coordinate functions on this diamond.
Assuming I understand what you are trying to do, then obviously we cannot
use the slices t = t0 (presumably what you meant by refering to Oliver's
"proper time" wrt events on -Peter's- world line) -before- we define them
operationally! That is, at this point we haven't yet defined the
coordinate t; once we know how to label events in our diamond by their t
coordinate, we will in effect have "constructed operationally" the
hyperslices t = t0, which as we know are all locally isometric to E^3.
> The substitution cosh^{2}R=1/(1-r^{2}) will change the metric into the
> static metric of (eqn 16) of [1]. These coords cover the northern causal
> diamond shown by the asterisks in the Penrose diagram.
Agreed, but IMO you have -not- yet described an "operational definition"
of this chart.
Also, I emphasize again that this chart is one of infinitely many similar
but distinct charts, defined on an infinite family of "diamonds", all of
which can be obtained from one another by self-isometries of H^(1,3).
> I was overcome by too much exhuberance for an observer-centric attitude
> to physics when I claimed that these coordinates were the only ones
> which made sense physically.
OK.
> This was too strongly worded and I should have said that the northern
> causal diamond shown is the only region for which the observer is able
> to operationally assign coords to the events, and the radar is one
> procedure that he can use to do this.
Agreed, except that you have -not- in fact described how he can do this!
I claim your procedure as described above is incomplete or even circular.
> Events outside the observer's causal diamond exist although the observer
> cannot operationally assign coordinates to these events.
Agreed.
> For example, if the observer uses a telescope, he will be able to see
> the light from events in the triangle at the bottom of the Penrose
> diagram, but he could not send a radar pulse to these events.
Just to make sure everyone is following this:
In the conformal diagram, the interior points represent two-spheres having
various areas 4 pi r^2, where r is a Schwarzschild radial coordinate (the
one appearing in the Schwarzschild type static chart associated with this
choice of equatorial S^3 and a pair of antipodal points on this equator).
Roughly speaking, this radius is larger for points near the "center" but
goes to zero as you approach the two vertical edges. Thus, these two
segments represent world lines. Curves running through the interior of
the digram represent world sheets; curves which run "generally
horizontally" represent spacelike hyperslices. The diagonal
______
| /|
| / |
O| / |
| / |
| / |
|/_____|
is a null slice marking the boundary of the region
______
|*****/|
|****/ |
O|***/ |
|**/ |
|*/ |
|/_____|
consisting of all events which can be causally influenced by O; that is
all events which can be reached by a null or timelike curve starting from
an event on the world line of O, i.e. "the absolute future of O".
Similarly, the diagonal
______
|\ |
| \ |
O| \ |
| \ |
| \ |
|_____\|
is a null slice marking the boundary of the region
______
|\ |
|*\ |
O|**\ |
|***\ |
|****\ |
|*****\|
consisting of all events which can causally influence O; that is all
events such that some null or timelike curve drawn from this event can
reach an event on the world line of O, i.e. "the absolute past of O" for
short.
> Having admitted these errors, a defence of the observer-centric approach
Approach to the alleged dS/CFT correspondence?
(Bear with me, I haven't been a regular reader of s.p.r. for some time---
I do recall that the OP seemed to refer to a previous discussion, but as
far as I know it was the first post in a new thread. I did read Steve
Carlip's post and I do realize that one of Stephen's references suggests
that the context of this discussion may be understood by everyone but me
to be an alleged dS/CFT correspondence.)
> is in order, lest anyone think I am in complete retreat. I am struck by
> the fact that objects which one normally thinks of as the stuff of
> physical reality seem to be contingent.
Would "observer dependent" be an acceptable synonym here?
> For example, in the Unruh effect, an observer who accelerates through
> Minkowski space perceives himself to be in a bath of quanta with a
> definite temperature. In contrast, an inertial observer does not
> perceive these quanta and just perceives a vacuum. For another example,
> the stuff we call matter is characterized by mass and spin, yet mass and
> spin label the irreducible unitary reprepresentations of the Poincare
> group. Suppose that two observers, Alice and Bob, inhabit de Sitter
> space and Alice has been translated along Bob's z-axis
Arghgh! "Bob's z axis" is of course ill-defined as written, agreed?
I think you are again behaving as if you have already defined a foliation
into hyperslices, but I thought the whole point was to give an operational
definition explaining some sense in which this particular foliation is
allegedly "natural" for some purpose still not clear to me. So it seems
to me that this discussion is somehow circular.
[snip rest of thought experiment until Steven explains his meaning]
> So, guided by these examples, I think we should start with a collection
> of equivalent observers.
If I guess correctly, you mean something like this: to try to set up QFT
on a curved spacetime, we should start with some timelike congruence X,
where X is presumably defined in some coordinate free manner, and using
thought experiments involving observers whose world lines are integral
curves of X we should "operationally define" a "preferred chart" in which
to perform quantum mechanical computations.
Indeed, from "equivalent" I guess you mean that every integral curve of X
should be carried onto another such by some self-isometry of M. Or maybe
only by a diffeomorphism on M? In the former case, from the standpoint of
classical gtr this is a -huge- (and objectionable) restriction on M. If X
is to be a -geodesic- congruence M is even more restricted.
> Being mutually equivalent,
In what sense?
Are you implicitly restricting your spacetime to be one which can be
modeled by a homogeneous/isotropic Lorentzian manifold M such as H^(1,3)?
> the observers all agree that the world they inhabit can be described by
> some fixed set of mathematical elements x1,x2,...
Do you mean that X must also admit an operational definition of a
coordinate chart defined by four coordinate functions, call them t,u,v,w?
(If so, I claim you have yet to describe this operational definition.)
> Then, if Alice perceives an element x1,
A coordinate spatial hyperslice t = t0?
> then this element of physical reality appears to Bob as f(x1) where f is
> an element of the group that connects the viewpoints of the observers.
> The set of mathematical elements is the carrier space for a
> representation of the group connecting the observers.
Calling John Baez! "Mathematical elements"--- is that a standard term?
> In the current case, the collection of equivalent observers are the
> inertial observers in de Sitter space (they are not subject to forces).
I agree that the uniform exponentially expanding congruence X discussed
above for the case H^(1,3) happens to be an -inertial- congruence, i.e.
D_X X = 0.
> The mathematical elements are points, lines, planes and hyperplanes.
Do you mean this? From spatial hyperslices t = t0, and 1+2 hyperslices x
= x0, we can obtain by intersection submanifolds t = t0, x = x0. (For
example, two spheres as discussed above in connection with the geometric
meaning of the Schwarzschild radial coordinate.) By intersecting these we
can pick out special curves (including things some would be wont to call
"coordinate axes"; we sophisticates of course discourage such customs) and
then events.
There is an excellent discussion of this way of understanding general
coordinate charts (if I understand correctly what you have in mind) in the
classic semipopular book by Hilbert and Cohn-Vossen, Geometry and the
Imagination.
(Hmm... maybe they only discuss -orthogonal- charts; this restriction
arises if one demands that the coordinate hypersurfaces t = t0, x = x0, y
= y0, z = z0 intersecting at a given event must always be mutually
orthogonal.)
> The group is the de Sitter group.
I.e. the ten dimensional Lie group which is the isometry group of H^(1,3).
(Sophisticates probably know its Lie algebra well, under another name!)
> Alice considers a point p. From Bob's point of view, this element of
> physical reality is the point f(p) where (say) f is the element of the
> de Sitter group that translates Alice along Bob's z axis as in the
> earlier example.
The -whole point- as I see it of the Hugeness of This Group is that there
are infinitely many unidimensional subgroups which could be used for this
purpose. So, unless I'm missing something, you have not yet defined the
procedure you have in mind, whatever this might be.
> When, as in this example, the mathematical elements form the carrier
> space for the fundamental rep of the group, we just recover Klein's
> Erlanger Programm.
Recover the -Programm-? In what sense?
(You are discussing a -particular- Lie group, yes? So how can you
"recover" a very general -Programm-?)
> The point p considered by Alice need not be within her future event
> horizon.
-Future- horizon, as in this diagram, right?
______
| /|
| / |
A | / |
| / |
| / P |
|/_____|
> She might not be able to receive a signal from p
^^^^^^^^^^^^^^^^^^^^^
send a signal to?
The diagram for an event P from which Alice might not be able to -recieve-
a signal would presumably be something like this:
______
|\ |
| \ |
A | \ P |
| \ |
| \ |
|_____\|
> but she recognises that p is physically real because she knows that p
> appears to Bob as f(p) and f(p) can be within Bob's future event horizon
> for some choice of the f.
Are you saying that because Bob's history overlaps Alice's causal diamond,
she can gain access to Bob's observations of events she can't see if Bob
relays his observations to her?
If so, agreed!
BTW, if anyone still doesn't understand why I keep insisting on the
multiplicity of conformal diagrams, let Bob's world line be some integral
curve of our uniform exponentially expanding congruence X other than the
antipodal world line, and try to draw Bob's causal diamond in the original
conformal diagram! The point is that each conformal diagram associated
with our congruence X, say the conformal diagram associated with Alice,
admits a simple representation of the causal diamonds of Alice and her
antipodal twin, but the diamond of Bob, while equally well-defined
geometrically, cannot be represented faithfully (after suppressing angular
coordinates in the original Penrose chart) unless we adopt the conformal
chart coming from another Penrose chart, this one adapted to Bob's world
line instead of Alice's world line.
The huge isometry group of H^(1,3) is our friend, but it does mean that
any observer dependent concept can be realized in lotsa ways.
> Finally, referring to the space-like slices t=const of the
> static coords/radar coords (eqn 2), Serenus wrote:
>
> > I think you mean "balls" here, not spheres---ie you mean the inside as
> > well.
>
> The space-like slices
t = t0, in a static Schwarzschild chart on H^(1,3)?
> are 3-spheres. They are 3-d spaces with constant positive curvature.
> They do not have a boundary and they are not balls.
Agreed, I think.
Chris Hillman
Chris Hillman
In two posts which will probably soon appear, I requested from Serenus and
Stephen respectively further clarification of their most recent posts in
this thread. Unfortunately, due to circumstances beyond my control which
became known to me only after I submitted these posts, I will be unable to
read/respond to their answers! I will also be unable to read/respond to
email.
However, while I regret that I won't be able to see the answers to the
questions I asked so recently, I think that my contribution has consisted
of pointing out various ambiguities (hopefully without become too
obnoxious!). Fortunately, from the most recent posts by Stephen and
Serenus I expect they and others will be able to remove any remaining
deficiencies in for example the procedure Stephen has been describing for
"operationally constructing" a static Schwarzschild chart in the "causal
diamond" for a particular observer whose world line is an integral curve
of one of the static congruences on (part of) H^(1,3) which we have been
discussing.
Chris Hillman