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Apr 14, 2020, 3:17:51 AM4/14/20

to

[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 / dÂ² * 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?

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 / dÂ² * 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?

Nov 24, 2020, 1:04:17 AM11/24/20

to

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

> [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
> 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."

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.

Nov 26, 2020, 4:46:53 PM11/26/20

to

Ned Latham <nedl...@internode.on.net> 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

>

> "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
> [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.

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.

> 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.

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.]

> 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.

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

--

-- "Jonathan Thornburg [remove -animal to reply]" <jth...@astro.indiana-zebra.edu>

Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA

currently on the west coast of Canada

"There was of course no way of knowing whether you were being watched

at any given moment. How often, or on what system, the Thought Police

plugged in on any individual wire was guesswork. It was even conceivable

that they watched everybody all the time." -- George Orwell, "1984"

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