Rock Brentwood wrote:
> The Wheeler-DeWitt equation completely loses any sense of time and time flow...
> This is the modern formulation and embedding of the Zeno Paradox, itself ...
> This is the Problem of Time.
> ... the Problem of Time is not a flaw endemic to quantum theory or to any
> prospective theory of quantum gravity. It's a flaw contained in the very structure
> of Relativity, itself! [that hits hard the very question that this article was in reply to]
...
> [It entails a reformulation of the paradigm from bottom-up]
> So, let's spell out, in some detail, what this actually entails and
> what it looks like.
The reason I don't take this exercise seriously as a bona fide
alternative to (or extension of) Relativity is because there are
actually a few key obstacles standing in the way that have been noted
at various points in time over the last 30 years or so by those who
try to rework Newtonian gravity in modern notation and draw a
(continuous) connection with it in "theory-space" to GR. The simple
naive approach of gauging the 11-parameter Poincare' group and using
this as the basis for an extended gravitational dynamics falls flat on
its face because you *still* lose important terms under "contraction"
to the Galilean limit (alpha = 1/c^2 -> 0). That is: some kind of
account for the ghosts left behind by the O(alpha^2) terms has to be
provided for.
But there are some ideas I'd like to work out that also happen to
touch on some issues that may be regarded as kind of "loose
ends" (like: what happened to the 5th transport equation when going
over from non-relativistic to relativistic fluid dynamics?). Suddenly,
you can see them in a new, and hopefully clearer light.
So first, this is the issue that was brought up:
In representation theory, we encounter the Galilei group, which can be
defined as the subgroup of the general affine group over the space
{ (t,r): t in R, r in R^3 }
that preserves the two invariants:
dt^2, Del^2
where Del^2 is the Laplacian over R^3. This results in a 10 parameter
group, whose parameters are naturally identified as momentum (P),
kinetic energy (H), angular momentum (L) and "mass moment" (K).
The only representations of this group correspond to particles of mass
M=0 moving at infinite speed ("action-at-a-distance" particles or
"synchrons", for lack of a better name). This deficiency arises from
the fact -- nowadays well-known -- that Galilei admits an 11th
parameter ... M ... as a central charge.
In contrast, the theories of special and general relativity are based
on the Poincare' group, which has only 10 generators, and no central
charges. Both H and M are combined into a single generator, E, which
corresponds to total energy. The merger is the essence of Einstein's
"mass energy unification" result.
This creates a problem -- a serious mismatch. Historically, both
Relativity and its associated structure were devised by inductive
reasoning from the 10-parameter Galilei group, before anyone had a
clue as to what was going on with central charges. It's
"correspondence limit" is the 10-parameter group. That is: you can
devise a continuous 1-parameter family of groups that connects the 10-
parameter Poincare' and Galilei groups.
The same holds true of Minkowski geometry. It, too, was devised based
on the 10-parameter Poincare' group.
What, then, is it that gives us the 11-parameter group as its
"correspondence limit"? What is the structure of the theories based on
this symmetry group? And what are their relation to the theories of
Special and General Relativity?
Over what geometry does both the Galilei (11) and 11-parameter
extension of Poincare' naturally reside? How does it relate to
Minkowski space? (Strong hint: non-relativistic fluid dynamics has 5
transport equations, instead of 4: one for mass, 3 for momentum and 1
for energy).
So I want to expand on this a bit, broadening the scope of the general
idea. In a way, you can think of this as a kind of reading -- or
stealing away -- bits and pieces from a book written 100 years from
now. So, you'll forgive me a little if it doesn't all gel together.
The issues:
(1) Proper Time as the Lagrangian for both Relativistic *and* Non-
Relativistic Mechanics
(2) Schwarzschild & Newton Bridged Together
(3) The *Extra* Fluid Dynamic Equation and Stress Tensor Symmetry?
(4) T_{44} & the Mysterious Density "Z"; Deformation to Relativistic
Fluid Dynamics.
(5) The "General Relativistic" Newtonian Gravity Law.
(6) Generalization to Relativistic Form
The fluid dynamics example is an important one, because if you want to
write down any law of gravity (relativistic or non-relativistic), an
often-overlooked item on the to do list is that you need to first
shore up the fluid dynamics model you use, since the gravitational
dynamics formalisms in common use are essentially just extensions of
fluid dynamics with gravity incorporated.
==========
(1) Proper Time as the Lagrangian for both Relativistic *and* Non-
Relativistic Mechanics
If you take literally the idea that the 5th coordinate encapsulates
proper time with particles now subject to two constraints:
(Linear) "Soldering" constraint: dt = ds - alpha du
(Quadratic) "Metric" constraint: dr^2 + 2dtdu + alpha du^2 = 0
and that their conjugate momentum components are subject to
constraints of similar form
(Linear) "Mass-Energy" constraint: mu = M - alpha H = constant
(Quadratic) "Generalized Mass Shell" constraint: P^2 - 2MH + alpha
H^2 = 0
then -- incredibly enough -- you actually have enough to use the
*same* action, across the board, to both the Galilean and Relativistic
cases -- the proper time.
The action for a point-mass reads, simply,
S = integral ds,
subject to the constraints
dt = ds - alpha du, dr^2 + 2dtdu + alpha du^2 = 0.
The constraints are incorporated as Lagrangian multipliers, everything
written as a function of time, t:
S = integral (s' + A(s' - alpha u' - 1) + B(r'.r' + 2u' + alpha
u'^2)) dt,
where ()' = d()/dt. Varying s, u and r leads, respectively, to the
equations:
(1+A)' = 0, (2B(1 + alpha u') - alpha A)' = 0, (Br')' = 0.
Varying A and B leads to the constraints
s' - alpha u' = 1, r'.r' + 2u' + alpha u'^2 = 0.
For the Galilean case, alpha = 0, this gives us
s' = 1, r' = v, u' = -v^2/2
A' = 0, (Bv)' = 0, B' = 0.
Consequently, we find that A, B and v are constants of motion and that
the time-dependence of the coordinates is given up to integration
constants by
r = vt, s = t, u = -v^2t/2.
For the remaining cases, where alpha is not 0, we have
r' = v, (1 + alpha u')^2 = 1 - alpha v^2, s' = 1 + alpha u'.
A' = 0, (Bv)' = 0, (Bs')' = 0.
Again, we find that A, B and v are constants of motion and that the
time-dependence of the coordinates is given up to integration
constants by
r = vt, s = +/- root(1 - alpha v^2) t, u = (s-t)/alpha = -v^2t/(1 +/-
root(1 - alpha v^2)).
The coordinate s, in both cases, is (up to sign) the proper time and
the on-shell Lagrangian is the proper time.
==========
(2) Schwarzschild & Newton Bridged Together
The Schwarzschild solution is given by the line element
ds^2 = (1 + 2 alpha U) dt^2 - alpha (dr^2/(1 + 2 alpha U) + r^2 (dq^2
+ (sin q df)^2)),
where U = -GM/r is the gravitational potential of the force center.
If we adopt the soldering relation dt = ds - alpha du, and treat the
entire line element as a 5-metric line element set to 0, then the
quadratic constraint reduces to the form
dr^2/(1 + 2 alpha U) + r^2 (dq^2 + (sin q df)^2) + 2dt (du + U dt) +
alpha du^2 = 0.
The proper time is then given by the relation
ds = dt + alpha du.
This has a non-trivial Galilean limit,
dr^2 + r^2 (dq^2 + (sin q df)^2) + 2 dt (du + U dt) = 0.
The proper time is just the time, itself: ds = dt. Reverting the
spherical coordinates (r,q,f) to Cartesian form (x,y,z), we then get:
dx^2 + dy^2 + dz^2 + 2 dt (du + U dt) = 0.
The conjugate coordinates are given by the canonical 1-form
P_x dx + P_y dy + P_z dz - H ds + M du.
Associated with this is the inverse metric expressed by the quadratic
form:
lambda = P^2 - 2 M H - 2U M^2.
If we now subject the test particle to the quadratic constraint
lambda = 0,
and linear constaint
mu = M = constant
the net result is that for non-trivial states (non-zero mu), we get
the reduction:
H = P^2/(2 mu) + mu U.
The momentum coordinate, H, conjugate to the "absolute time", s,
assumes the familiar form; the one and the same as the Hamiltonian for
the test-particle.
When subject to canonical quantization
-i h-bar d = (dx P_x + dy P_y + dz P_z - ds H + du M),
we obtain the equations of motion
i h-bar @(psi)/@s = (-i h-bar del)^2 psi/(2 mu) + mu U psi,
-i h-bar @(psi)/@u = mu psi.
(@ = partial derivative operator)
The u coordinate is the phase factor associated with the mass, mu, and
we obtain Schroedinger evolution with respect to the "absolute time"
s.
All of this remains intact when passing continuously over to the
Relativistic case, by continuously varying the parameter alpha. The
result becomes equivalent to the Klein-Gordon equation for the test
particle on a Schwarzschild background.
Only, here: the evolution is with respect to the "absolute time"
coordinate s, not t. The coordinates t and s are no longer equivalent.
==========
(3) The *Extra* Fluid Dynamic Equation and Stress Tensor Symmetry?
As you probably know, fluid dynamics has a serious misfit when going
over from non-relativistic to relativistic form: there's a 5th
equation that somehow gets lost. So ... where did it go?
If you want to formulate a gravitational dynamics, then the first
thing to note is that both Newtonian gravity and General Relativity
are built atop a fluid dynamics foundation. Whether it be the Poisson
equation or the Einstein field equation, what appears on the right-
hand side are quantities defined in fluid dynamics. Thereore, both
formulations of gravitational dynamics are simply extensions of fluid
dynamics, expanded to included the action under gravity.
Thereore, to get a better understanding on the *correct* form of the
correspndence limit to non-Relativistic theory from (the correct form
of) Relativity -- a limit that properly targets the 11-parameter
Galilei group, instead of the 10-parameter Galilei group as Relativity
currently does -- then you first have to address what's going on with
the FIFTH equation in fluid dynamics.
That's right: in non-relativistic fluid dynamics, there are FIVE
equations, not four. One governs the transport of kinetic energy,
three the transport of momentum, and the fifth ... that's the
conservation law, which governs the transport of mass.
This is yet another place, where you see the misfit between the five
components (P = (P_1,P_2,P_3), H = -P_4, M = P_5) of momentum, versus
the four (E = -P_0, P = (P_1,P_2,P_3)) of relativistic dynamics:
you've lost the equations for H and M and replaced it by only one
equation: that for E.
Moreover, if there are 5 equations, and not 4, then in order to get
symmetry for the corresponding stress tensor, you need a 5th
coordinate in each equation! So, even at the classical level there is
need for repair to complete what's left undone.
The non-relativistic equations are transport equations for local
densities (rho, pi, eta) which give us the currents, respectively, or
(M, P, H). They take the form
@rho/@t + del.(v rho) = 0
@pi/@t + del.(v pi) = F
@eta/@t + del.(v eta) = B + v.F,
where F is the force-density and B the power density. The momentum
density pi and energy density eta are subject to the relations
pi = rho v, eta = rho v^2/2.
The dot ().() denotes scalar product for 3-vectors. The vectors in the
equation above are v, pi, del and F. The scalars are t, rho, eta and
B.
First, this set of equations -- which is just the skeletal form of the
final complete set -- are required to be "scale invariant" in the
sense that if one chunks a bunch of subsystems together and defines
the "total" rho, pi, eta as averages; then the form of the resulting
averaged equations should remain the same. First, we can always assume
the same form for (rho v) by *defining* the velocity by the relation
<pi> = <rho> v.
This requires the assumption:
* if <pi> = 0, wherever <rho> = 0.
For other non-linear combinations, this requires adding extra terms to
account for the various covariances, e.g.,
<rho v^i v^j> = <rho v^i><v^j> + P^{ij},
<rho v^2/2> = <rho>(<v>^2/2 + U),
<rho v eta> = <rho v><eta> + q,
thus introducing extra terms corresponds to the "stress" (P),
"internal energy" (U), "heat transfer" (q).
The resulting system now looks like this:
D(rho) = @rho/@t + del.(v rho) = 0
D(pi) = @(rho v)/@t + del.(rho v v + P) = F
D(eta) = @(rho (v^2/2 + U))/@t + del.(rho (v^2/2 + U) v + v.P + q) =
B + v.F.
(@ = partial derivative operator)
We may then ask how this transforms under the (10-parameter) Galilei
group. By including the v.P term in the last equation we'll have
ensured the invariance of U, P and q under boosts.
Under the Galilei transform
r -> r - Vt, t -> t, v -> v - V, del -> del, @/@t -> @/@t + V.del,
we have the following transformation
D(rho) -> D(rho), D(pi) -> D(pi) - D(rho) V, D(eta) -> D(eta) - D
(pi).V + D(rho) V^2/2
This reflects the momentum space transformation
M -> M, P -> P - MV, H -> H - P.V + MV^2/2.
Under this action, both the power density, B, and force density, F,
are also invariant.
But now this puts us in an awkward position of having a 5x4 stress
tensor:
T^0_5 = rho, T^i_5 = T^0_5 v^i
T^0_j = rho v_j, T^i_j = T^0_j v^i + P^i_j
-T^0_4 = rho (v^2/2 + U), -T^i_4 = T^0_4 v^i + sum(v^j P^i_j) + q^i,
for i = 1, 2, 3 and j = 1, 2, 3; summation taken over j = 1, 2, 3.
Plus, we're in the awkward position of having used the *11* parameter
Galilei transformation for the momentum components, but only the *10*
parameter transformation for the coordinates.
To fix this, we take the (s = x^4, u = x^5) coordinates in place of t
= x^0 and note that the "soldering relation" dt = ds - alpha du
reduces to t = s for the Galilei case, alpha = 0. Then we write
T^4_5 = T^0_5 = rho, T^4_j = T^0_j = rho v_j, -T^4_4 = -T^0_4 = rho
(v^2/2 + U),
for j = 1, 2, 3.
The momentum space metric is given by the quadratic form (summation
convention used):
lambda = P^2 - 2MH + alpha H^2 = g^{mu nu} P_{mu} P_{nu}
where P_4 = -H, P_5 = M. This gives us
g^{ij} = delta^{ij}; for i, j = 1, 2, 3
g^{i4} = g^{i5} = 0 = g^{4j} = g^{5j}; for i, j = 1, 2, 3
g^{44} = alpha, g^{45} = 1 = g^{54}, g^{55} = 0.
The inverse metric is given by the line element
dr^2 + 2dsdu - alpha du^2
with r = (x^1, x^2, x^3), s = x^4, u = x^5; and works out to
g_{ij} = delta_{ij}; for i, j = 1, 2, 3
g_{i4} = g_{i5} = 0 = g_{4j} = g_{5j}; for i, j = 1, 2, 3
g_{44} = 0; g_{45} = g_{54} = 1; g_{55} = -alpha.
If we lower the indices using this metric, defining T_{ij} = g_{ik}
T^k_j, then we have for the Galilei case (alpha = 0),
T_{ij} = rho v_i v_j + P_{ij}; for i, j = 1, 2, 3
T_{i5} = T_{5j} = rho v_j; for i, j = 1, 2, 3
T_{55} = rho
T_{54} = -rho (v^2/2 + U),
T_{i4} = -rho (v^2/2 + U) v_i - sum (v^j P_{ij}) - q_i.
The one necessary condition for this to all work is that the internal
stress tensor P also be symmetric:
P_{ij} = P_{ji}.
In order to complete this picture, we need to also include,
T^5_i = T_{4i} = T_{i4} = -(rho (v^2/2 + U) v_i + P_{ij} v^j + q_i)
T^5_4 = T_{44}
T^5_5 = T_{45} = T_{54} = -rho (v^2/2 + U).
This doesn't tell us what T^5_4 = T_{44} ought to be. But the
requirement that the full system now be invariant under Galilei(11),
and not just Galilei(10):
r -> r - Vt, s -> s, u -> u + V.r
del -> del - V @/@u, @/@s -> @/@s + V.del - V^2 @/@u, @/@u -> @/@u -
alpha V.del
will give us the correct form as
T^5_4 = T_{44} = rho (v^2/2 + U)^2 + (v.P + q).v + Z
where Z is Galilei-invariant.
Altogether, what we find are the following equations,
D(rho) = @rho/@t + del.(v rho) - @(rho (v^2/2 + U)/@u = 0
D(pi) = @(rho v)/@t + del.(rho v v + P) - @(rho (v^2/2 + U) v + P.v +
q)/@u = F
D(eta) = @(rho (v^2/2 + U))/@t + del.(rho (v^2/2 + U) v + v.P + q) - @
(rho (v^2/2 + U)^2 + (v.P + 2q).v + Z)/@u = B + v.F.
This extension still preserves the transformation behavior
D(rho) -> D(rho), D(pi) -> D(pi) - D(rho) V, D(eta) -> D(eta) - D
(pi).V + D(rho) V^2/2,
but now the picture is completed both by making the *full* 5X5 stress
tensor symmetric, and by incorporating the *full* 5-coordinate/Galilei
(11) of the Galilei transformation.
We can add an off-setting term, E, to the right of the mass transport
equation. To preserve the transformation properties of the equations,
this requires modifying the right-hand sides of the other two
equations as follows:
F becomes F + v E,
B + v.F becomes B + v.F + (v^2/2) E.
==========
(4) T_{44} & the Mysterious Density "Z"; Deformation to Relativistic
Fluid Dynamics.
There is an extra density, Z, that suddenly appears out of nowhere. In
fact, this is already mandated solely by the "system chunking" scale
invariance requirement descibed above. It soaks up the covariances of
all the non-linear terms in the remainder of T_{44}.
I have no idea what Z is. Its units are those of energy-density or
pressure multiplied by velocity squared. It's left largely
undetermined by all three conditions: "scale-invariance", symmetry and
Galilei invariance.
But now, with this final element in place, it's possible to directly
convert this to "Relativistic" form. The scare-quotes mean that this
is not Relativistic fluid dynamics in the usual sense -- which is
grounded in the Poincare'(10) group, but "extended" Relativistic,
grounded in the Poincare'(11) group and it's 5-coordinate
representation.
So, the 5th equation remains intact. In invariant form, the equations
read:
@(T^{mu}_{nu})/@x^{mu} = K_{nu},
K_5 = K(@/@u) = 0.
So, there are actually TWO senses of the conservation law -- as there
are in the Galilean case. One is the force-law "continuity equation"
for the stress tensor -- the first equation. The other equation is a
bona fide continuity equation: one for the invariant mass mu = M -
alpha H.
The specific forms
pi = rho v, eta = rho (v^2/2 + U)
come out of the requirements (1) symmetry of the stress tensor and (2)
invariance under the space-time symmetry group. These have to be
derived anew.
==========
(5) The "General Relativistic" Newtonian Gravity Law.
Returning to the question raised at the outset of (3), the Poisson
equation
del^2 U = 4 pi G rho
can now be seen in a new light. This equation, following Cartan, is
written in "General Relativistic" form as:
R_{00} = K T_{00}
for a suitable constant K. The *correct* form comes straight out of
the fluid dynamics equations. The density rho appears in the equation
for mass transport. Consequently, it is NOT the 0-anything component
(the component associated with "total energy" E or "time" t), but the
5-5 component!
T_{55} = rho.
Thus, the correct equation should ultimately arise from
del^2 U = 4 pi G T_{55}.
If generalizing to the metric for Galilei(11), the left-hand side
should be replaced with the corresponding quadratic invariant:
(del^2 + 2 (@/@u)(@/@s)) U = 4 pi G T_{55}.
In invariant form, this reads:
@_{mu} (root(-g) g^{mu nu} @_{nu} U) = 4 pi G root(-g) T(@/@u, @/@u),
where g is the determinant of the *5*-metric (g_{mu nu}); and
@_{mu} = @/@x^{mu}.
==========
(6) Generalization to Relativistic Form
The potential, itself, naturally fits in with the metric as follows.
The geodesic law in GR can be written in first-order form as:
m g_{mu nu} dx^{nu}/ds = p_{mu}
dp_{mu}/ds = m @(phi)/@x^{mu}
where
phi = 1/2 g_{nu rho} dx^{nu}/ds dx^{rho}/ds.
Relating this to the equation involving the gravitational potential U
dp/ds = -m del U
gives us the following correspondence: phi = -2U. In turn, this can be
expressed as the following differential equation
g_{mu nu} dx^{mu}/ds dx^{nu}/ds + 2U = 0
or as the quadratic constraint (the "geodesic constraint"),
g_{mu nu} dx^{mu} dx^{nu} + 2U ds^2 = 0.
This seems to suggest *combining* both the Newtonian and Relativistic
equations into a larger set as follows:
@_{mu} (root(-g) g^{mu nu} @_{nu} U) = 4 pi G root(-g) T(@/@u +
alpha @/@s, @/@u + alpha @/@s)
g_{mu nu} dx^{mu} dx^{nu} + 2U ds^2 = 0,
where -- in the extended 5-D geometry -- ds and @/@u + alpha @/@s are
the two linear vector invariants. The problem (as well-known from
Cartan onwards) is: combine it with *what*? The Einstein equation
reads
8 pi G T_{mu nu} = c^3 G_{mu nu},
and does not have a sensible Galilean limit because of the c^3 factor.
The c^3 factor has to be directly incorporated into G_{mu nu} before
taking the limit.