The Galilean Limit of GR and 2 Forms of ADM
Rock Brentwood
by De Pietri et al. whose series [1], [2], [3] uncovered the
surprisingly subtle nature of the problem. The problem of continuously
linking the infrastructure of the two theories WAS the original
problem Einstein had been working on before c. 1910 before basically
dropping it and going full tilt with the Minkowski idea of 4-D
geometry. It was only in the 1930's that we even saw a curved
spacetime formulation of non-relativistic gravity (Cartan's
treatment).
So, we come to 1992-1994 and to quote from [2], "As far as we know, a
manifestly-covariant formulation of Newton's theory of gravitation has
never been proposed until now." (note in particular that Cartan's
formulation does NOT come from a Lagrangian formulation; the De Pietri
version does).
So, the following are basically just some crib notes I've been putting
together alongside my reading these references ... something I started
out back earlier this year but put on hold for a short while.
So, first the problems.
There are several issues I have with the whole treatment. First of
all, because the coupling suppresses two orders in the signature
parameter alpha = 1/c^2, this means that in the Galilean limit alpha -
> 0, you have to retain TWO orders of terms, not just one.
So, in addition to the 11 degrees of freedom corresponding to the
centrally extended Galilean group they have 16 more to handle the
second order.
I don't think all of this is necessary. A closer examination of what's
minimally required to get a theory powerful enough to allow the one
body solution to pass over continuously to its non-relativistic form
seems to show that you only need maybe 4 or 5 more degrees of freedom
over the 11.
But the second problem I have with the treatment is, by far, the
greater one and is also a problem that reflects on the ADM formulation
itself. Both suffer the same deficit. And the problem is this.
If you're going to fully embody any kind of Galilean limit to the
fully extended Galilean group, 11 parameters and all, then not only
should be starting off with an ansatz that has 11 degrees of freedom,
instead of the 10 of the Poincare' group (and De Pietri et al. are
okay on that), but you should also be going over to the U(1) bundle
representation of the group itself.
That means, you're working within the 5-dimensional projective
representation.
De Pietri et al. do not. And the gap shows!
The most conspicuous way this oversight is the handling of the
Lagrangian for the test body; the second biggest way it shows up is
the awkwardness of handling the Lagrangian for the field, itself.
So, now some answers and starting points for answers...
There is a lot of simplification to be had, overall, when fully
incorporating the 5-D representation -- not the least is the ability
to more consistently account for formulations, such as the Papapetrou
formulation, where you have 5 equations for continuity instead of 4 (4
momentum + 1 for mass).
In a quantum setting, the 5th degree of freedom (at least in the non-
relativistic theory) is the coordinate conjugate to mass. That's
represented as the phase coordinate u in the expression exp(imu) that
you would see, for instance, in the Foldy-Wouthuysen (sp?) transform.
The coordinate, u, itself, is basically the action, itself, for the
test body, divided by its mass.
First, we fix up the problem with the test body Lagrangian; and we
start out by busting a myth.
Myth #1: the Lagrangian for a test body in non-relativistic dynamics
is necessarily quasi-invariant.
Fact: once you incorporate the extra coordinate, then the mass shell
condition for the particle in coordinate space splits into two forms:
(1) the 5-D light cone condition:
dr^2 + 2dtdu + a du^2 = 0
(2) the "soldering relation":
ds = dt + a du.
This gives you the proper time s for the particle in LINEAR form. It
works just as well when the signature parameter a = 1/c^2 for
relativistic theory; and when a = 0 for non-relativistic theory. The
action is (modulo the constraints):
S = integral m ds.
In momentum space, the mass shell condition becomes
(1) the 5-D "restricted mass shell":
P^2 - 2MH + a H^2 = 0
(2) the linear invariant
M - aH = m
where M is the relativistic mass, H the kinetic energy.
Interestingly, I worked out the exercise of how this deforms in a
curved setting and the result is that H will end up picking up a
second term that gives you the gravitational potential.
Second, there is the notion that
Myth #2: Galilean gravity does not allow for the gauging of anything
analogous to the "Poincare' shift" term -vx/c^2 seen in the Lorentz
transformation and that
Myth #3: "absolute time" is the signature feature uniquely specific to
the Galilean signature.
Fact: once you incorporate the 5-D representation, the coordinates r =
(x,y,z) and t transform infinitesimally under rotations w = (w1,w2,w3)
and boosts v = (v1,v2,v3) the usual way:
delta(dr) = w x dr - u dt, delta(dt) = -au.dr
BUT ... the extra coordinates give you a vestige of the Poincare'
shift term:
delta(du) = u.dr
still present in the Galilean setting AND a vestige of absolute time
delta(ds) = 0
still present in the Lorentzian setting.
Since u does not (directly) appear in the De Pietri treatments
(although they kept coming back to it indirectly by way of other
related issues and essentially tried to work around it), they were
still in the game of "Galilei gravity ONLY allows for absolute time,
and ONLY Galilei gravity allows for absolute time".
Finally, there is the treatment involving ADM. As I was looking at
this, I noticed something interesting that links ADM to the above-
mentioned "soldering" relation and that by the magic of some unusual
prestidigitation, one can come up with REGULAR metrics for the
Galilean setting that transform continuously to/from the Lorentzian
signature (and even allow you to pass over to the Euclidean domain
where a < 0).
Let's start out with ADM in a somewhat standard form. The metric may
be represented by the following line element
Lambda = -(N dt)^2 + q_{ij} (dx^i - N^i dt) (dx^j - N^j dt)
where i, j indices range over 1,2,3
The dual metric is given by the quadratic form
lambda = -(1/N (d_t - N^i d_i))^2 + q^{ij} d_i d_j
where d_t = partial with respect to t, d_i partial with respect to
x^i (i=1,2,3).
The two reduced metrics q_{ij} and q^{ij} are inverses of one another.
When doing the Galilei limit, the signature parameter a = 1/c^2 is
pulled out of N by basically reducing this to a light-speed multiple,
N = cU. Then the timelike terms in the two quadratic forms are written
as
Lambda = -(U dt)^2/a + ...
lambda = -a (1/U (...))^2 + ...
Okay, so this is ADM. In the Galilei limit Lambda blows up, and lambda
drops down to a rank 3 representation of the Poisson operator. So, you
can't DIRECTLY take the Galilei limit of this.
BUT ... (and this is something that De Pietri seemed to be trying to
do in [1]):
Let's carry out trick #1: rewrite the metric and its dual in "gauge
form" as:
Lambda = -(1/a) (V (dt - a A_i dx^i))^2 + h_{ij} dx^i dx^j
lambda = -a (d_t/V)^2 + h^{ij} (d_i - a A_i d_t) (d_j - a A_j d_t).
In ADM form, the shift field N^i gives you a local velocity field
which essentially mirrors the Galilei term -vt in x -> x - vt. In
"gauge form", A_i gives you the "inertocity" field (see note [4])
which mirrors the term -av.x of the Poincare' shift in t -> t - av.x.
This too has an ill-defined Galilean limit.
Both forms of the metric and its dual are forms that De Pietri would
have had issues with because of their breakdown in the case where the
signature parameter a -> 0. The failing, as we're about to see, lies
directly in their not taking into account the 5-D nature of the
extended Galilei group.
So, this leads to trick #2. Start with the line element. Given our
discussion about the coordinate form of the mass shell condition, this
line element ought to be equated to
Lambda = -ds^2/a
where ds is the "absolute time" or "proper time" for the particle.
Second, there ought to be a "soldering relation". So, we'll
incorporate this idea by regarding the "gauge form" of ADM as actually
being a projection of a 5-dimensional metric. This is to be obtained
by imposing the soldering relation:
ds = V (dt - a A_i dx^i) + a du.
Now, substituting this into the line element, we get
g = 2 V (dt - a A_i dx^i) du + a du^2 + h_{ij} dx^i dx^j = 0.
So, here is a curved space representation of the coordinate form of
the mass shell condition. In it is contained the 5-D metric we're
looking form.
It has an inverse! Not just for non-zero signature parameters a, but
even for a = 0. That's trick #3. Inverting it, we find.
g^{-1} = -a (d_t/V)^2 + h^{ij} (d_i + a A_i d_t) (d_j + a A_j d_t)
+ 2 d_t d_u/V.
or just
g^{-1} = lambda + 2 d_t d_u/V.
Now, trick #4. Canonical coordinates can be defined through the
canonical 1-form:
theta = P_i dx^i - H ds + M du = p_i dx^i - T dt + m du.
The two forms of coordinates are related through the soldering
condition by
p_i = P_i + a V H A_i, T = HV, m = M - aH.
Under the operator correspondence, we have
d_t <-> -T, d_u <-> m, d_i <-> p_i (in t,u coordinates)
d_s <-> -H, d_u <-> M, d_i <-> P_i (in s,u coordinates).
The dual metric g^{-1} is then the quadratic mass shell invariant,
while m = M - aH is the linear invariant, as before. Imposing the
restricted mass shell condition
g^{-1} = 0,
we end up getting
lambda = -2 d_t d_u/V <-> 2 mT/V = 2mH.
And along the way, we've crossed over the roadblock a = 0 and have
inverted the Galilean metric to obtain lambda in its original form.
So, the two conditions required were
Lambda = -ds^2/a, lambda = -2 d_t d_u/V = -2 d_s d_u
along with the soldering relation
ds = V (dt - a A_i dx^i) + a du.
Now, as I mentioned at the outset, these are little more than crib
notes and I still don't fully understand what they mean (even though
I'm the one producing them). But it seems to be the ingredients
necessary to redo De Pietri's formulation into a form that explicitly
contains the 5-D representation.
I'm expecting, in part, that if their entire analysis is worked out
with the above notes, it may even be possible to get around the
horrible 27-degree-of-freedom blowup in complexity that their method
seemed to entail and maybe even keep this down to 11 degrees of
freedom; and a bona fide 11-degree-of-freedom gauge theory for gravity
which admits a smooth transition not only TO the Galilean limit, but
BEYOND, over to the other side a < 0, into the Euclidean domain.
References and Notes:
[1] De Pietri, Lusanna and Pauri, "Galilean Theories of Gravitation",
arXiv:gr-qc/9212002v2 13 Dec 1992.
[2] De Pietri, Lusanna and Pauri, "Newtonian Gravitation as a Gauge
Theory", I; arXiv:gr-qc/9505046v1 22 May 1994
[3] De Pietri, Lusanna and Pauri, ditto, II; arXiv:gr-qc/9405047v1 22
May 1994
[4] Inertocity suffers the same problem as does the Millennial Decade:
nobody ever got around to naming it, until I did, just now. While
physicists measure speed by how far you go in a given time, we in the
running sports and in track measure inertocity -- which is how long it
takes you get go a given distance. Inertocity has units, such as miles
per minute. Nobody ever got around to naming this concept. I just did.
Rock Brentwood
> Inertocity has units, such as miles per minute.
Rock Brentwood wrote:
> Myth #1: the Lagrangian for a test body in non-relativistic dynamics
> is necessarily quasi-invariant.
>
> Fact: once you incorporate the extra coordinate, then the mass shell
> condition for the particle in coordinate space splits into two forms:
> (1) the 5-D light cone condition:
> dr^2 + 2dtdu + a du^2 = 0
> (2) the "soldering relation":
> ds = dt + a du.
This I have to retract. The "phase" coordinate u offsets the mass, so
in a Lagrangian formulation the action would be S = -mu. This is best
seen by applying the soldering relation ds = du + a dt to the
relativistic Lagrangian:
Ldt = -mc^2 ds = -mc^2 (dt + a du) = -m/a dt - m du
Thus
(L + mc^2) dt = -mdu.
In the De Pietri treatment, the two orders of suppression coming from
the coupling constant 8 pi c^3/G mean that there are residual order -2
and order -1 equations in the combined action for test masses and
fields. The order -2 residual is cancelled out by redefining the
Lagrangian as L + mc^2. This is what reduces it from invariant to
quasi-invariant -- because now the action has a non-invariant term
(mc^2 dt) in it. The order -1 residual is -mdu and gives you the
dynamics for test masses.
Only, in their treatment, they are not using the projective 5-D
representation so there is no u coordinate. So, their discussion has
the added complication of trying to talk around this element.
The action integral is
S = -integral m du
with the constraint dr^2 + 2dtdu + a du^2 = 0. For a = 1/c^2, this
gives you
S = integral (mc^2 dt - mc^2 ds) ~~ -integral mc^2 ds
(where ds = dt + a du) and for a = 0 it gives you the non-relativistic
integral
S = integral m dr^2/(2dt) = integral (1/2 m (dr/dt)^2 dt).
Rock Brentwood
particularly III, "ADM Made Easy":
http://groups.google.com/group/sci.physics.research/msg/376858288e2793ba?hl=en&dmode=source
On Dec 16, 2:02�am, Rock Brentwood <markw...@yahoo.com> wrote:
> I'm trying to wade through the treatment of the subject header's issue
> by De Pietri et al. ...
with their quote setting the stage:
> "As far as we know, a
> manifestly-covariant formulation of Newton's theory of gravitation has
> never been proposed until now."
>
> There are several issues I have with the whole treatment.
[All centering on the fact that they don't make full use of the
geometry that is naturally associated with the extended Galilei
group].
> So, in addition to the 11 degrees of freedom corresponding to the
> centrally extended Galilean group they have 16 more to handle the
> second order.
>
> I don't think all of this is necessary.
It turns out ... (ta da!) it's not.
You CAN get away with 11 degrees of freedom, after all. The result
allows you to write down a family of models, parameterized by the
"signature parameter" a, with a = 1/c^2 > 0 for Lorentzian geometries,
a = 0 for Newton-Cartan geometries, and a < 0 for Euclidean
geometries.
But, in order to overcome the two orders of suppression of the
coupling in the parameter a -- the problem that ran central to the de
Pietri treatment, you have to exploit the full power of the 5-D
geometry and the 11-parameter extension of the Poincare' group that is
to be continuously linked to the centrally extended Galilei group.
The resulting models are powerful enough to embody the 1-body solution
(a biggie), while also apparently being strictly less powerful than
the 27-parameter model posed by de Pietri. In fact, it may even be
more restrictive than GR, in that not all solutions to the field
equations in GR necessarily carry over to solution-families
parametrized continuously by (a) (although, all the solutions to GR
for Lorentzian geometries will be included as particular solutions for
specific values of a > 0).
First, some ...
(1) Background
One of the reasons for writing down the 3+1 form-valued vector form
for the equations governing the underlying geometry (this was the body
of the first 2 parts of the "Maxwell Equations of Gravity") was to
make more clearly visible the appearance and effect of the signature
parameter a, as well as providing for a much more intuitive
formulation more closely connected to things familiar to those who
study fluid dynamics (plus the calculations are a lot easier, as I'll
begin to show, with the example of the spherical solution).
To recapitulate, in the beginning: everything is in the setting of a
Riemann-Cartan geometry, with a metrical connection (omega^a_b) and
frame field (theta^a:a=0,1,2,3). These objects are collected into the
following:
t = theta^0, x = (theta^1,theta^2,theta^3),
T = Theta^0, X = (Theta^1,Theta^2,Theta^3)
with Theta representing the torsion. The connection and its associated
curvature forms (Omega^a_b) may be written as two sets of vectors --
representing the gauge degrees of freedom corresponding respectively
to rotational symmetries:
sigma, Sigma
and boost symmetries
alpha, Alpha.
Each of these are 3-vector valued forms; sigma, alpha being 1-forms,
Sigma, Alpha being 2-forms.
Products are taken component-wise as wedge products. Thus, the vector
identity a*b = -b*a for the cross product ()*() switches sign of both
a and b are odd-degree forms, and becomes a*b = b*a. Similarly, the
dot product ().() becomes anti-commutative a.b = -b.a for odd-degree
forms a, b.
The Cartan equations rendered in this form become:
dt + a alpha.x = T,
dx - sigma * x + alpha t = X
d(sigma) - (sigma * sigma)/2 + a (alpha * alpha)/2 = Sigma
d(alpha) - sigma*alpha = Alpha
The Ricci identities for the frame field are
dT + a alpha.X = a Alpha.x
dX - sigma * X + alpha T = -Sigma * x + Alpha t
and the Bianchi identities for the curvature 2-forms are
d(Sigma) - sigma * Sigma + a alpha * Alpha = 0
d(Alpha) - sigma * Alpha - alpha * Sigma = 0.
(2) Example: Spherical Geometry
Thus, for a spherically symmetric solution, in isotropic form, we
would have
t = (1 + aU) d(tau), x = e^B dr
where dr = (dx, dy, dz) and U, B are functions only of |r|.
>>From this, you can almost directly determine the connections when the
torsion is 0:
alpha = C r d(tau)
sigma = D r * dr
Upon substitution, you end up with
0 = T = a dU d(tau) + a C r d(tau) e^B . dr.
Or, factoring out the a d(tau):
dU = C r e^B.dr = C e^B |r|.d|r|,
thus
C = e^{-B} U'/|r|.
Similarly, upon substitution in the equation for X, you obtain the
result
0 = X = e^B dB dr - D (r * dr) * e^B dr + a C r d(tau) (1+aU) d
(tau)
which reduces, via the identities
d(tau) d(tau) = 0
(r * dr) * dr = -r (dr.dr) - (r.dr) dr = -|r| d|r| dr
(note the sign flip on the second term in the expansion of the cross
product).
Thus
e^B dB dr = -D e^B |r| d|r| dr
or
D = -B'/|r|.
Thus, you find the following expressions for the connections:
alpha = e^{-B} U' r/|r| d(tau)
sigma = -B' r/|r| * dr.
Pay close attention to the fact that only alpha has the expression
involving the gravitational potential U. Whatever field equations are
to be posed, this must survive in some fashion when going over to the
Galilean limit. We'll shortly see what problem de Pietri ran up
against.
(3) Invariants and Lagrangians
Now, at this point, all that we stated was a geometry and what might
be regarded as the "kinematics" for the field whose configuration
variables are t, x, T, X, sigma, alpha, Sigma and Alpha. The first and
most natural approach to adopt in writing down Lagrangians is to find
all the invariant combinations that can be formed of the configuration
variables (we will not consider the possibility of quasi-invariant
Lagrangians, such as the Chern-Simons form, here).
The configuration variables are particularly suited to embody a local
version of Lorentz symmetry. Thus, if the 3-vectors w and u
respectively represent infinitesimal rotations and boosts, then the
various objects will transform infinitesimally as:
delta(t) = -a u.x, delta(x) = w * x - u t,
delta(T) = -a u.X, delta(X) = w * X - u T.
This, combined with the equations for T and X, determines what
transformation is required on the connection (and thus the curvature),
and it comes out to the following:
delta(sigma) = w * sigma - a u * alpha + dw,
delta(alpha) = w * alpha + u * sigma + du,
showing that these respectively gauge rotation and boost symmetries.
For the curvature 2-forms, the derived transformations are
delta(Sigma) = w * Sigma - a u * Alpha
delta(Alpha) = w * Alpha + u * Sigma.
>>From these are also derived transformations for the frame 2-forms S =
(x*x)/2 and (xt):
delta(S) = w * S + u * xt, delta(xt) = w * (xt) - a u * S
as well as for their covariant derivatives (the spin 3-forms)
delta(X*x) = w * (X*x) + u * (Xt-Tx),
delta(Xt-Tx) = w * (Xt-Tx) - a u * (X*x).
For the volume 3-form V = x.S/3 and the 3-form St we get
delta(V) = -u.St
delta(St) = w * S - a u V.
So ... now we have enough to ask what's required for an invariant
Lagrangian. If we add the further restrictive condition that only
algebraic combinations of the configuration variables be permitted,
then since we're excluding quasi invariance, then only combinations of
the frame, torsion and curvature will be permitted.
La: Sigma * xt - a Alpha * S
Lb: Sigma * S + Alpha * xt
Lc: tV
Ld: 1/2 (Sigma^2 - a Alpha^2)
Le: Sigma.Alpha
Lf: 1/2 (T^2 - a X^2)
These are the 6 invariant 4-forms that can be formed algebraically of
these variables. The terms Lb, Lf combined to yield a co-boundary term
(the Nieh-Yan invariant), Le and Ld are co-boundary terms. So, for
field dynamics, this leaves only La, Lb and Lc.
The term La is the Einstein-Hilbert Lagrangian up to proportion, while
Lc (when added) gives you a cosmological constant term. The term Lb is
parity-violating and gives you the term associated with the Immirzi
coefficient. We'll ignore that one.
(4) Field Equations and The "Stiffness" Problem
So ... when it comes time to formulate field equations, as outlined in
the previous articles where we developed the field equations from a
heuristic consideration of transport equations for fluid dynamics, the
energy current and momentum 3-currents came by way of the derivatives
d/dt, d/dx
respectively; while the angular momentum and boost currents came by
way of the respective derivatives
d/d(sigma), d/d(alpha).
Here, we will consider only the energy and boost currents (since the
other two currents yield X = 0, T = 0 for field equation in the
absence of spin sources ... in a Lorentzian setting, that is). We
obtain the following expressions:
dLa/dt --> -Sigma * x; dLa/dx -> t Sigma - a x * Alpha
dLc/dt --> V; dLc/dx --> -St.
The Einstein equations have, on the left-hand sides,
-Sigma * x = (... energy current ...)
t Sigma - a x * Alpha = (... momentum current ...).
So, now we return to the example for spherical geometries. Recall the
point made there -- ONLY alpha contains the potential U in it. When we
work out the expressions for Sigma and Alpha, Sigma gets only the
contribution
Sigma = ... + a alpha * alpha
which drops out in the Galilean limit a -> 0. But, then, so does the
term involving Alpha in the field equation for the momentum current.
Thus, we lose control over the gravitational potential U. That was the
dilemma facing Cartan, and those who came after; and it was the
problem that ran central to de Pietri's 1992-1994 treatment.
This is the "Stiffness Problem". It has 2 parts: (1) the equations
controlling the gravitational potential U actually come from the
SPATIAL part of the geometry, not the temporal part; but (2) the
spatial geometry becomes too stiff too fast, when going over to the
Galilean limit.
(5) Tenderizing Space. The 5-D Representation
The reason for this problem is that the Poincare' Shift term delta(t)
= -a u.x = -u.x/c^2 drops out, leaving behind no semblance of
"simultaneity relativity". So, there's no room left for the spatial
sections to wobble around.
This gets to the main issue I had with the de Pietri treatment. The
extended Galilei group DOES continue to have the Poincare' shift term
in it. In fact, it is absolutely critical to have it to consistently
formulate the transformation properties for the FIVE components of the
Galilei momentum (H = Kinetic energy, p = momentum 3-vector, M =
mass):
delta(H) = -u.P
delta(P) = w x P - u M
delta(M) = -a u.P.
In the Galilei setting, a = 0, and M = mu is the (invariant) mass.
What are the coordinates conjugate to these components, if all you
have are 4 coordinates? You need the 5th, u and then you can write the
canonical form
P.dr - H ds + M du = P.dr - H dt + mu du
where the invariant mass mu is given in terms of M and H by
mu = M - a H
and the coordinates s, t and u are related by
dt = ds - a du.
The transformation properties for the coordinates reflect those of the
momentum components,
delta(dr) = w x dr - u dt
delta(dt) = -a u.dr
delta(du) = u.dr
with delta(ds) = 0 -- the coordinate s is an invariant.
In the 5-dimensional representation, two things happen:
(a) the Poincare' shift term delta(dt) = -a u.dr survives in the
Galilean limit as delta(du) = u.dr,
(b) the Absolute Time invariant ds survives in the Lorentzian
signature as the "proper time" invariant.
So ... now this leads to the following modification of the Cartan
equations: we add in the frame 1-form u and torsion 2-form U, and
write the equations
du - a alpha.x = U, dU - a alpha.X = -a Alpha.x.
Defining s = t + a u, and S = T + a U, this leads to the following
equations:
ds = S, dS = 0.
The Absolute Time is "gauged" as an extra scalar field.
(5) 5-D Invariants and Lagrangians
Now, if we were to take this formulation in a 4-D setting, we would
seek to find 4-form invariants out of the expanded set of
configuration variables. However, we would run immediately into a
wall. Nothing substantial would end up being added to the set of 6
invariants previously derived. All we would accomplish is to split the
torsion term
Lf: (T^2 - a X^2)/2
into two invariants
S^2/2, (X^2 + 2TU + a U^2)/2.
But, something interesting happens that is a foreshadowing of more to
come. The second of these invariants has lost a power of the signature
parameter (a). So, we removed an order of suppression.
There is an 8th invariant formed of the torsion and frame field, but
it won't add much to our discussion, so I'll leave it to for you to
find.
But this misses an important point. When we write down transport
equations for the five components (H, M, P), we are replacing these by
their corresponding transport currents. The result is that we obtain a
set of equations (5 in all) governing the transport of mass (i.e. the
continuity equation), of momentum and kinetic energy.
But what's the stress tensor for the whole set? Since there are 5
equations, if we require the stress tensor to be symmetric, then there
needs to be a FIFTH COORDINATE in these equations -- i..e, the
coordinate u.
It is an interesting point, for instance, that the Schroedinger
equation can actually be recast as effective fluid dynamic equations
that involve 5 variables and not just 4 ... once we rewrite the
equation in hyperbolic form as
(P^2 - 2MH + a H^2) psi = 0, (M - a H) psi = mu psi;
under the operator correspondence
P = -i h-bar del, M = -i h-bar d_u, H = i h-bar d_s.
For the signature a = 0, this reduces to the Schroedinger equation
with the inclusion of the "mass phase" term
psi_0 = exp(imu/h-bar).
The extra coordinate u -- in this setting -- is just the Foldy-
Wouthuysen phase.
So, if we are to take seriously to heart the notion that the transport
equations are for a FIVE x 5 stress tensor, and not just a FOUR x 5
tensor, then we should be looking for 5-invariants, not 5-invariants.
In fact, we can immediately write down 6 corresponding to the
algebraic invariants La, ..., Lf by suffixing the invariant 1-form s
(and there are 3 more invariants besides these).
Look at what happens when we do it for the Einstein-Hilbert term and
the cosmological constant term:
La s = (Sigma * xt - a Alpha * S) s = Sigma *xts - a Alpha *Ss
Lc s = tVs
But
ts = (s - a u)s = ss - aus = -aus
since the wedge product ss = s ^ s of the 1-form vanishes. Thus, we
find
La s = -a (Sigma * xu + Alpha * S) s
Lc s = -a (uV) s
So ... instead of considering the invariants La s and Lc s, we'll
replace them with their order-of-magnitude reduced forms
La' = (Sigma * xu + Alpha * S) s
Lc' = (uV) s.
We've just shed an order of magnitude and removed the "stiffness"
problem.
The energy and momentum currents that come out of this are given by
the derivatives (d_s)_{u} = (d_t)_{u} and d/dx. For the mass, we can
take the derivative (d_u)_{s} for the current associated with
"relativistic mass" M, and (d_u)_{t} for the current associated with
the invariant mass mu.
These are 5 transport equations. Working out the expressions, we get
the following expressions as the left-hand side of these equations:
For P:
d(La')'/dx = (u Sigma + x * Alpha) s
d(Lc')/dx = -(Su) s
For H:
-d(La')/ds = -(Sigma * xu + Alpha * S)
-d(Lc')/ds = -uV = V u
For M:
d(La')/du = -Sigma * x s
d(Lc')/du = V s
For mu = M - a H:
d(La')/du_{t} = -Sigma * x t + a Alpha * S
d(Lc')/du_{t} = V t
The "stiffness problem" is now gone, and these retain the key terms
(particularly, x * Alpha) in the Galilean limit a -> 0.
(6) The "Langoliers Transform" as Return of Absolute Time
Notice how in these equations we find an additional factor s. In the
absence of torsion, ds = S = 0, and we can (at least locally) express
the 1-form s as the differential of Absolute Time s = d(tau).
The way that s, t and u combine clearly has something to do with the
technical fix "imaginary time" that lies behind the speculation of
Hawking and his people regarding signature change. However, we have
NOT carried out a purely formal and technical expedient of
"complexifying" the metric or even "complexifying" the time-like
direction, for the sake of being able to (say) do path integrals, or
other kinds of Euclideanization.
Rather, they have emerged automatically. Rather than being written in
by hand, they emerge as a consequence of the (highly visual and
meaningful) 5-D representation commonly used in the representation
theory for the Galilei group.
Only now, this representation is extended to the Poincare' group and
-- by virtue of the fact that we can take the signature parameter a <
0 -- to the 4-D EUCLIDEAN group.
It is a little-known (and scandalous) secret that the famous author
Steven King is actually the secret alter ego of Steven Hawking (I
mean, come on Steve. Didn't you think your name would be a dead
giveaway?!) He disguises his true form by faking the ability to speak
and walk on his own while in disguise as the author (pretty effective
disguise, one might add!)
But the obsession with time travel (e.g. the Shining) and other
spacetime anomalies is telling.
In the Langoliers, a group of travellers get stuck in a spacetime
anomaly. This anomaly has the unusual property that things are
"winding down" -- that is, their time-like direction is acting more
and more like "imaginary time", with propagation turning into decay.
The real world is still moving in time -- but at an extremely slow
pace. The travellers' time, meanwhile, seems to be going on a sideways
slant. And what they find out is that each instant of real time is a
slice in spacetime, and that spacetime, itself, is flowing in a SECOND
timelike direction.
In this direction, everything decays and eventually a bunch of giant
(and cute) Mouth-like creatures come rampaging to gobble everything up
into nothingness.
The physical account for this anomaly is that the travellers were
stuck on an anomalous projection of the "absolute time" (s), so that
their (t) coordinate was at a slant from the real worlds' (t)
coordinate. This occurred by virtue of their (t) being defined by a
different "soldering relation" t_{Traveller} = t(s, u), while the real-
world t continued to be defined by the soldering relation t = s - a u.
So, they picked up a little bit of the (u) coordinate.
The signature that results from the quadratic form
P^2 - 2MH + a H^2
is, in fact, independent of the signature parameter a. It's 4+1, de
Sitter. So, even though there are two time-like directions (t, s) it
is NOT a 3+2 signature, we're talking about here. When this
orthogonalization is done, the result is a kind of COMPLEX time -- the
time-like signature is replaced by a (1+1) signature, which is added
to the (3+0) signature of the spatial dimensions to yield a (4+1)
signature.
This is best seen by using the canonical correspondence to
P <-> del, M <-> d_u, H <-> d_s
to revert to the "Klein-Gordon" operator
del^2 + 2 d_s d_u + a d_s^2 = del^2 + 2 d_t d_u - a d_t^2
since
(d_t)_{u} = (d_s)_{u}, (d_u)_{t} = (d_u)_{s} - a (d_t)_{u}.
Inverting this, we get the 5-metric
dr^2 + 2 dt du + a du^2 = dr^2 + 2 ds du - a du^2.
For non-zero signature parameters a, we can eliminate du = (ds - dt)/a
and write this as
dr^2 + ds^2/a - dt^2/a.
For a > 0, the real-time (t) is Lorentzian and absolute time (s) is
Euclidean. For a < 0 they swap roles. Combined, they give you a kind
of *imaginary* time s + it.
Rock Brentwood
where I'm going to discuss (a) the transformation of the equations and
notation into indexed invariant 5-D notation, (b) the CORRECT form of
the Cartan equation for gravity (i.e. that the right-hand side is not
T_{00} but T_{55}!), and (c) a brief discussion of the ramifications
for providing the foundation of an extension of the path integral
formalism to curved spaces AND to non-relativistic theory.
To recap:
> > I'm trying to wade through the treatment of the subject header's issue
> > by De Pietri et al. ...
(Their quote)
> > "As far as we know, a
> > manifestly-covariant formulation of Newton's theory of gravitation has
> > never been proposed until now."
>
> > So, in addition to the 11 degrees of freedom corresponding to the
> > centrally extended Galilean group they have 16 more to handle the
> > second order.
>
> > I don't think all of this is necessary.
>
> It turns out ... (ta da!) it's not.
>
> You CAN get away with 11 degrees of freedom, after all. The result
> allows you to write down a family of models, parameterized by the
> "signature parameter" a, with a = 1/c^2 > 0 for Lorentzian geometries,
> a = 0 for Newton-Cartan geometries, and a < 0 for Euclidean
> geometries.
(1) Index Notation
(1.1) 4-D
First, we'll embody the previous notation used for the formulation of
4-D Lorentzian Riemann-Cartan geometries.
The indexing convention used, in flat space, leads to coordinatization
of the following form:
t = x^0, r = (x, y, z) = (x^1, x^2, x^3)
The metric components are
g^{00} = -a, g^{i0} = 0 = g^{0j}, g^{ij} = delta^{ij}
g_{00} = -1/a, g_{i0} = 0 = g_{0j}, g_{ij} = delta_{ij}
and the anti-symmetric Levi-Civita
e_{0123} = 1 = e^{0123}.
The connection 1-forms reduce to the following:
sigma = (omega^2_3, omega^3_1, omega^1_2)
alpha = (omega^1_0, omega^2_0, omega^3_0)
with
omega^0_0 = = 0
omega^i_j = -omega^j_i (i,j = 1, 2, 3)
omega^0_j = a omega^j_0 (j = 1, 2, 3)
and similar equations for the curvature 2-forms Sigma and Alpha,
relative to the 2-forms (Omega^a_b).
These properties make the connection metrical (i.e. 0-covariant
derivative for the metric); thus the space is Riemann-Cartan.
The frame 2-form and volume 3-form are, respectively,
S = (x*x)/2 = (theta^2 theta^3, theta^3 theta^1, theta^1 theta^2)
V = S.x/3 = theta^1 theta^2 theta^3
Out of this, we derive the invariant 4-form combinations that can be
constructed algebraically from the curvature 2-forms, torsion 2-forms
and frame 1-forms:
La = xt.Sigma - aS.A = -1/4 e_{mnrs} theta^m theta^n Omega^{rs}
Lb = S.Sigma + xt.A = 1/2 theta^m theta^n Omega_{mn}
Lc = Vt = -1/24 e_{mnrs} theta^m theta^n theta^r theta^s
Ld = 1/2 (Sigma^2 - a Alpha^2) = 1/4 Omega_{mn} Omega^{mn}
Le = Sigma.Alpha = 1/(4a) e_{mnrs} Omega^{mn} Omega^{rs}
Lf = 1/2 (T^2 - a X^2) = -1/(2a) Theta_m Theta^m
(2) 5-D Projective Representation
In the projective representation, the flat-space coordinates are
redefined replacing t by (s,u) = (x^4, x^5); so indices run from 1-5.
The 0 coordinate derived by the "soldering relation"
x^0 = x^4 - a x^5.
The frame 1-forms and torsion 2-forms acquire the following additions
s = theta^4, u = theta^5, S = Theta^4, U = Theta^5
(this S is a scalar, not to be confused with the 3-vector S above
which denotes the "area" frame 2-form).
In 4-D this gives us 5 frame 1-forms, effectively a 4x5 array of frame
components. In the 5-dimensional geometry, it becomes a 5x5 array.
The connection components are related by
omega^u_v = 0 (u, v = 4, 5)
alpha = (omega^1_4, omega^2_4, omega^3_4),
omega^i_5 = -a omega^i_4,
omega^5_j = omega^j_4
with similar equations for the curvature 2-forms. This yields a
connection that is metrical with respect to the 5-metric whose
components are given by
g_{ij} = delta_{ij},
g_{i4} = g_{i5} = 0 = g_{4j} = g_{5j},
g_{44} = 0, g_{45} = 1 = g_{54}, g_{55} = -a.
This metric is invertible for all values of the signature parameter
(a) and corresponds to the 5-D line elements
dr^2 + 2 ds du - a du^2 = dr^2 + 2 dt du + a du^2 = dr^2 + ds^2/a -
dt^2/a
the last equation only holding for non-zero values of (a).
The inverse metric is
g^{ij} = delta^{ij},
g^{i4} = g^{i5} = 0 = g^{4j} = g^{5j}
g^{44} = a, g^{45} = 1 = g^{54}, g^{55} = 0.
This yields the following as its Klein-Gordon operator
del^2 + 2 d_s d_u + a d_s^2 = del^2 + 2 d_t d_u - a d_t^2
where, in th second equation d_u = (d_u)_{t} is taken with t fixed,
while in the first d_u = (d_u)_{s} is taken with s fixed. The
operators are related by
(d_t)_{u} = (d_s)_{u}
(d_u)_{t} = (d_u)_{s} + a (d_s)_{u}
(3) Fluid Dynamics Equations and the Correct Form of the Field
Equations
In non-relativistic theory, one has 5 transport equations -- but in
only 4 variables. This complicates the idea of forming a stress
tensor. Closely linked to this problem is formulating a suitable
equation for gravity versus matter.
In fact, Cartan has the wrong equation! It's not T_{00} on the right
hand side of the field equation, but an INVARIANT.
The two invariants in the projective representation are
ds = theta^4
(d_u)_{t} = (d_u)_{s} + a (d_s)_{u} = e_5 + a e_4,
where (e_m: m = 1, 2, 3, 4, 5) denotes the frame dual to (theta^m: m =
1, 2, 3, 4, 5).
What Cartan ACTUALLY has is not T_{00} on the right, but the invariant
T((d_u)_{t}, (d_u)_{t}) = T_{55} + a (T_{54} + T_{45}) + a^2 T_{44}.
When written in (t,u) form, using the index range (0,1,2,3,5) in place
of (1,2,3,4,5), this becomes
T((d_u)_{t}, (d_u)_{t}} = T_{55}.
This is what I'm trying to connect to the Lagrangian invariants I
described in the previous articles.
The fluid dynamics equations in non-relativistic theory are
Continuity:
d_s (rho) + d_i (rho u^i) = 0
Momentum Transport:
d_s (pi_j) + d_i (pi_j u^i + P^i_j) = 0
Energy Transport:
d_s (eta) + d_i (eta u^i + P^i_j u^j + Q) = 0
(since s = t + au = t in the non-relativistic limit a = 0).
The constitutive relations are given by
pi_j = rho delta_{ij} u^i
eta = rho delta_{ij} u^i u^j + U
and with u = (u^1, u^2, u^3) representing the local velocity field, U
the internal energy density and P^i_j the 3-D stress tensor and Q the
heat flux.
This can be written as a 5-D conservation law for a SYMMETRIC stress
tensor, when symmetry is defined relative to the 5-metric. The
compopnents T^i_j of the 5 x 5 stress tensor are largely determined by
this requirement.
The continuity equation is d_m T^m_5 = 0. The momentum transport
equation is d_m T^m_j = 0 for each j = 1, 2, 3. And the energy
transport equation is d_m T^m_4 = 0. The index m ranges over 1,2,3,4,5
using the coordinates s = x^4, u = x^5.
>From the continuity equation, we extract the components
T^4_5 = rho, T^i_5 = rho u^i.
>From the momentum transport equation, we extract the components
T^4_j = pi_j = rho u_j, T^i_j = rho u^i u_j + P^i_j
where u_j = delta_{ij} u^i. Finally, from the energy transport
equation, we extract the components
T^4_4 = eta = rho u^2/2 + U, T^i_4 = eta u^i + P^i_j u^j + Q^i
The symmetry requirement means we have the following:
T_{ij} = T_{ji} -> T^i_j = T^j_i -> P^i_j = P^j_i.
Thus, the 3-D stress tensor is symmetric with respect to the 3-metric
delta_{ij}. In addition, we have
T_{i4} = T_{4i} -> T^i_4 = T^5_i -> T^5_j = eta u_j + u_i P^i_j + Q_j
where Q_j = delta_{ij} Q^i.
T_{i5} = T_{5i} -> T^i_5 = T^4_i - a T^5_i
In the non-relativistic limit, a = 0, and this gives us the
constitutive relation
pi_j = rho u_j.
For the 4-5 components, we have
T_{45} = T_{54} -> T^5_5 = T^4_4 - a T^5_4.
In the non-relativistic limit, this leads to
T^5_5 = T^4_4 = eta.
The component T^5_4 remains undetermined by these requirements. It
only shows up when we go over to a relativistic dynamics (a non-zero)
as an extra degree of freedom not notamally accounted for by
relativstic fluid dynamics.
Thus, when the dependence on the 5th coordinate is added, the
equations become
d_s (rho) + d_i (rho u^i) + d_u (eta) = 0
d_s (pi_j) + d_i (pi_j u^i + P^i_j) + d_u (eta u_j + u_i P^i_j + Q_j)
= 0
d_s (eta) + d_i (eta u^i + P^i_j u^j + Q) + d_u (T^5_4) = 0
In the absence of any dependence on u and assuming T^5_4 = 0, this
leads to the usual equations.
Note, in particular, what is playing the role of mass density rho:
rho = T^4_5 = T_{55} = T((d_u)_{s}, (d_u)_{s}).
Since a = 0, this is the invariant
rho = T((d_u)_{t}, (d_u)_{t}).
THAT is what should be appearing on the right-hand side of the Cartan
equation, not T_{00}!
(4) The "Unification" as a Replacement for Osterwalder-Shrader (sp?)
(Sorry if I keep getting the spelling wrong. I'm not a stickler for
names).
Thus, it is the failure to take fully into account the 5-D geometry
naturally associated with the centrally extended Galilei group that
leads to a confusion (and dilemma) about the actual form of the field
equations for non-relativistic gravity. This is already what came out
in the previous articles, where we found that the De Pietri treatment
radically simplifies (from 27 degrees of freedom down to 11) when we
extend the geometry from 4 to 5 dimensions.
Because of that confusion, the close (and CONTINUOUS) connection
between those equations and those for relativistic theory has been
obscured all that time since -- even when we get to de Pietri et al.
in the mid 1990's; who were arguably the first to actually write down
a geometric formulation and Lagrangian that passes over smoothly from
non-zero signature parameter (a) to a = 0.
The formulation I outlined passes over smoothly into a = 0 and BEYOND
a = 0, to a < 0 -- the Euclidean sector.
Hence, we also have a ready-made replacement for the Osterwalder-
Shrader Theorem that should serve as an alternate basis for path
itntegrals, suitable for generalizing them to curved spacetimes AND to
non-relativistic field theory(!)
This was the problem I was also alluding to in an earlier response in
the Path Integral articles. Namely: that it is not the question of
generalizing the Fourier transform that is getting in the way of
generalizing the path integral approach to curved spaces, but actually
it's the question of linking up the Lorentzian and Euclidean forms of
gravity (so that one may continuously transform between a Lorentzian
and Euclidean solution). This is why you see discussion of the idea of
a "complex time" or even "complex metric" formulation in Hawking's
attempt to bridge the gap between the Lorentzian and Euclidean
signatures.
What you actually need, in place of Osterwalder-Shrader, is a way to
take a solution, like Schwarzschild, and continuously transform it
into its Euclidean form. Apparently, people forgot to ask about what
happens at the transition point; e.g. how you get past the degenerate
metric that lies between the Lorentz and Euclidean signatures.
The above discussion works for crossing over the Galilean boundary
case, which is suitable for the type of signature transitions
characteristic of a Big Bang singularity (hence the article posted on
this topic). The other kind of signature transformation discussed
there was through the "Archimedean" degenerate signature, which is
more characteristic of causal horizons and event horizons.
The approach just described, based on combining the symmetry groups
across the boundary case, is suitable for the transition Lorentz ->
Galilei -> Euclid; because of the involvement of the central charge in
the boundary case, and the emergence of the associated 5-D geometry.
But this does not directly work for the other type of transition
Lorentz -> Archimedean -> Euclid, since there is no central charge
here.
However, when we take the homogeneous symmetry group (i.e. the
generators J = (J1,J2,J3) and K = (K1,K2,K3) respectively for
rotations and boosts) and combine it with the CO-AFFINE generators (R
= (R1,R2,R3), T), then the result is a kind of dual form of the affine
group. This group develops a central charge when passing over into the
Archimedean phase. Then, all of what's been described can be used here
as well. The roles of the translation generator P and time translation
generator H are now played respectively by the momentum translation
generator R and energy translation generator T. The central charge is
associated with T and gives you a kind of "absolute time" invariant
(much like what the 5-D representation did when we combined Galilei
with Lorentz and Euclid).
Both the affine and co-affine groups are restrictions of the conformal
group (alternatively, they can also be regarded as restrictions of SL
(5)). So, across both signature-changing boundaries, it is
distinguished subgroups of the conformal group that are involved. I
don't know what the conformal group, itself, does when going through
the signature phase changes Lorentz -> Galilei -> Euclid or Lorentz ->
Archimedean -> Euclid. Maybe the conformal group gets a central charge
in both cases; in which case, that is the group that one should be
using here.