The three classical tests of General Relativity.

[Ned Latham]

[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

"One of the three classical tests for general relativity is the
gravitational redshift of light or other forms of electromagnetic
radiation. However, in contrast to the other two tests - the
gravitational deflection of light and the relativistic perihelion
shift - you do not need general relativity to derive the correct
prediction for the gravitational redshift. 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."

That last sentence must surely be a non-contentious way of saying
that the gravitational red shift is not a definitive test of general
relativity.

The writer has apparently not considered that the same combination of
factors applies also to the gravitational deflection of light, which
implies that it too is not a definitive test of general relativity.

With two of the three classical tests of GR thus seen as inconclusive,
the question arises as to whether a similar combination could provide
the correct prediction for the relativistic perihelion shift. At first
glance, the idea would seem preposterous: a particle theory of light
must surely eschew Lorentz transforms and Einstein's second postulate,
and in that case an alternative way to the relationships implied by
the gamma() factor must be found. But as it happens, there is one:
postulating that gravity propagates through a field the energy of
which varies as the gamma() factor gives us F = G M m / d² * gamma(v),
which does indeed correctly predict the relativistic perihelion shift.

And yes, the above *is* speculative, but if the math produces the
correct prediction, can it be regarded as fanciful, or in some way
illegitimate? Shouldn't we keep such alternatives in mind when theory
is being tested?

[Rock Brentwood]

Post by Ned Latham
[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.

[Jonathan Thornburg [remove -animal to reply]]

Post by Ned Latham
[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
"One of the three classical tests for general relativity is the
gravitational redshift of light or other forms of electromagnetic
radiation. However, in contrast to the other two tests - the
gravitational deflection of light and the relativistic perihelion
shift - you do not need general relativity to derive the correct
prediction for the gravitational redshift. 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."
That last sentence must surely be a non-contentious way of saying
that the gravitational red shift is not a definitive test of general
relativity.

That's correct: gravitational redshift is a test of the weak equivalence
principal (WEP). If the WEP holds, then we can accurately model gravity
as a curved-spacetime phenomenon, i.e., we have a "metric theory of gravity'.
All metric theories of gravity predict the same gravitational redshift.
The WEP does not address what physical laws determine the curvature of
spacetime; different metric theories of gravity differ on this.

Post by Ned Latham
The writer has apparently not considered that the same combination of
factors applies also to the gravitational deflection of light, which
implies that it too is not a definitive test of general relativity.

This is not true. Different metric theories of gravity make different
predictions for light deflection (& for the corresponding Shapiro delay
of light). Notably, in the "parameterized post-Newtonian (PPN) formalism"
(where we expand metric components in power series in the Newtonian
gravitational potential and keep the leading post-Newtonian terms),
gravitational light deflection is proportional to $(1 + \gamma)/2$,
where $\gamma$ is a coefficient in a the diagonal space-space metric
components.

IMPORTANT: The WEP alone doesn't predict a value for $\gamma$! That is,
different metric theories of gravity -- all compatible with the WEP --
have different values for $\gamma$. This means that test of gravitational
light deflection (which amount to measurements of $\gamma$) are NOT tests
of the WEP; rather, they are tests of various metric theories of gravity
(of which GR is one) *all* of which are consistent with the WEP.

[Everything I wrote above about gravitational light deflection also
applies to the "Shapiro" gravitational delay of photons when passing near
a massive body. This too is proportional to $(1 + \gamma)/2$ in the PPN
approximation, and tests of this are also true tests of different metric
theories of gravity (e.g., GR), not tests of the WEP.]

Post by Ned Latham
With two of the three classical tests of GR thus seen as inconclusive,
the question arises as to whether a similar combination could provide
the correct prediction for the relativistic perihelion shift.

No. The perihelion shift (of the orbit of a body with nonzero rest
mass, as opposed to the orbits of photons probed by light deflection)
depends on a different set of PPN parameters: it's proportional to
$(2 + 2\gamma - \beta)/3$ (+ a term depending on the solar quadrupole
moment, which is now known to be very small), where $\beta$ is a
different coefficient appearing in the time-time metric component.

Just like $\gamma$, the WEP alone doesn't predict a value for $\beta$.
That is, different metric theories of gravity -- all compatible with
the WEP -- have different values for $\beta$ as well as $\gamma$. Thus,
tests of perihelion advance (which amount to measurements of a linear
combination of $\beta$ and $\gamma$) are NOT tests of the WEP; rather,
they are tests of various metric theories of gravity, all of which are
consistent with the WEP.



As our moderator noted, <http://www.livingreviews.org/lrr-2014-4> is a
superb OPEN-ACCESS review article on the theoretical analysis of such
experimental tests of gravity. If you want to delve deeper, the same
author has written a classic book on this topic:
Clifford M Will
"Theory and Experiment in Gravitational Physics", 2nd Ed
Cambridge U.P., 2018, ISBN 978-1-107-11744-0