On Tuesday, April 14, 2020 at 2:17:51 AM UTC-5, Ned Latham wrote:
[moderator's note: one of the standard references conderning the
empirical status of GR is the open-access article
http://www.livingreviews.org/lrr-2014-4
]
From http://www.einstein-online.info/spotlights/redshift_white_dwarfs
"A combination of Newtonian
gravity, a particle theory of light, and the weak equivalence
principle (gravitating mass equals inertial mass) suffices. It is,
therefore, perhaps best regarded as a test of that principle rather
than as a test of general relativity."
It's an empty claim, unless it is posed as a solution to a
non-relativistic formulation of Maxwell's equations on the curved
space-time background geometry that embodies Newtonian gravity - which
would either be in a Newton-Cartan geometry or a Bargmann geometry.
Preferably, the two cases (relativistic and non-relativistic) should be
unified in a single parameter family of equations & theories that
contains both as special cases, then one can *directly* test for
relativity versus non-relativistic theory by deriving error bars for the parameter.
I'm not aware of anyone who's actually written the non-relativistic form
of Maxwell's equations on a curved non-relativistic background. Flat
space-time is easy (just take the non-relativistic limit of the Maxwell-Minkowski equations ... the non-relativistic limit is equivalent
to the system that Lorentz posed in his papers in 1895-1904). But curved Newtonian space-time is an entirely difference matter.
A brief search shows up some items that might have something related Newton-Cartan, Galileo-Maxwell and Kaluza-Klein
https://arxiv.org/pdf/1512.03799.pdf
Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of
time
https://arxiv.org/pdf/1402.0657.pdf (For reference, Carroll is the
c = 0 limit of Minkowski geometry and of the kinematics given by the
Poincare' group.)
Generalized Maxwellian exotic Bargmann gravity theory in three spacetime dimensions
https://www.sciencedirect.com/science/article/pii/S037026932030397X
(This might be useful, but it's restricted to 2+1 dimensional
spacetimes.)
If using Bargmann geometry - the simplest and most elegant approach -
this requires adding an extra coordinate (u) and going over to a 4+1 dimensional geometry. A metric that includes both Newtonian gravity and Schwarzschild can be written by the following line element / constraint:
dx^2 + dy^2 + dz^2 - 2 alpha U/(1 + 2 alpha U) dr^2 + 2 dt du + alpha du^2
- 2U dt^2 = 0
with proper time, s, given by s = t + alpha u
U = -GM/r = gravitational potential per unit mass
alpha > 0 for relativity (with light speed c = root(1/alpha)), alpha = 0
for non-relativistic theory
Maxwell's equations can be expressed in terms of the potential 1-form
A = *A*.d*r* - phi dt + b du = A_x dx + A_y dy + A_z dz - phi dt + b du
where
*A* = (A_x, A_y, A_z) is the "magnetic potential"
phi is the electric potential
d*r* = (dx, dy, dz)
b is the same "b" that appears in the "B-field formalism" in QED,
except, here, it's in classical field theory, not QFT; and would be set
to 0 for this problem.
The field-potential equations are dA = F, where F = *B*.d*S* + *E*.d*r*
^ dt [+ (...) ^ du which we ignore, by assuming that b = 0 and *A* and
phi are independent of u]. where d*S* = (dy^dz, dz^dx, dx^dy).
We still have to write down a Lagrangian density L(A, F) for the field,
to get its field equations. The response fields would be the densities
defined by the derivatives
*J* = @L/@*A*, rho = -@L/@(phi), *D* = @L/@*E*, *H* = -@L/@*B*
@ = partial derivative curly-d symbol Since the background geometry is
curved (both in the relativistic and non-relativistic versions), then
the Lagrangian density has non-trivial dependence on the metric, so it
is not a simple linear relation between *D* and *E*, or *B* and *H*.
There's extra stuff involving the metric.
Whatever is written down should
(a) reduce equivalently to the Maxwell equations on the Schwarzschild background, when alpha = 1/c^2
(b) produce the Maxwell-Minkowski equations for at least one frame of
reference when alpha = 1/c^2
(c) produce the non-relativistic limit of the Maxwell-Minkowski
equations when alpha = 0
The Maxwell-Minkowski equations are
*D* + alpha *G* x *H* = epsilon (*E* + *G* x *B*)
*B* - alpha *G* x *E* = mu (*H* - *G* x *D*)
For non-relativistic theory, alpha = 0 and the dependence on *G* and on
frame is essential and cannot be eliminated; while for relativistic
theory, alpha > 0, and the *G* dependence might be eliminated, if
epsilon mu = alpha; but (again) the constitutive relation is non-trivial because metric components are mixed up in this, and epsilon and mu will
be variable. Nonetheless, they may *still* multiply out to alpha, in
which case, the *G* dependence can be removed.
For the non-relativistic case, *G* = *0* would probably be the case in
the center of mass frame of the gravitating body. But for a rigorous
test, different choices of *G* may need to be included in the
comparison.
The corresponding 3-form is made from the response 2 form of the 4D
theory and du
G = (*D*.d*S* - *H*.d*r* ^ dt) ^ du [+ ... dV + ... d*S* ^ dt which we
ignore and treat as 0]
and the field law dG = Q, where Q is the source 4-form, made from the
source 3-current of the 3D theory and du:
Q = (rho dV - *J*.d*S* ^ dt) ^ du [+ ... dV ^ dt, which we also ignore and treat as 0]
[this G not to be confused with the vector *G* up above.]
For the source-free field, Q = 0, and you just have the free field
equations. But they are non-trivial, since the metric is mixed in there
with them.
Now ... with all of that, you can then write down the wave equations,
solve them and compare the solutions to observation and arrive at an
estimate for the parameter alpha; and that will be your test.
But there is no valid test that can be claimed, unless it is a test of
*actual* theories (not just hand-waved ad hoc solutions) - which means
actual equations on actual background geometries, with actual
Lagrangians, etc.; all that laid out in detail. Because it's not
solutions you're testing, nor ad hoc fixes, but entire *theories* and frameworks.
In the case at hand, we want to prove that alpha > 0 and that the (alpha
= 0) case lies outside of the error bars. That, and that alone, is what establishes the relativistic law of gravity, in favor of the Newtonian
law of gravity.
I'm not aware of anyone who's actually done this rigorously, as an
actual test of entire theories and frameworks, rather than as a test of solutions, from first principles, like this. So, any claim that
"Newtonian theory accounts for the observed red-shift" is dead on
arrival. Not without a formulation of the non-relativistic form of the
Maxwell equations on a curved Newtonian spacetime it doesn't.
Likewise, any claim of tests that actually *do* distinguish between the
two paradigms needs to be made rigorous in the above sense, before it
can be considered as fully established. I don't know if this exercise
has actually be done yet, so I don't know if a truly rigorous test (with
actual error bars for alpha) has been done.
--- SoupGate-Win32 v1.05
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