I have noticed that Einstein stresses in various places that, whereas
the speed/velocity of light is constant in SR, it is not in GR.

The reason for my current source of confusion here is that Einstein
wrote in German, where there is no linguistic distinction between
vector (velocity) and scalar (speed).

It is obvious that the vector must vary since light rays bend in
gravitational fields.

However, the situation for the scalar seems rather less clear to me,
at present.

Does anyone know of any accelerating (linear or rotating) reference
frame experiments which unambiguously resolve this question one way or
the other?

(pleaseremove spam from address for emailed response)

Chalky wrote:
On Jun 16, 8:54 pm, Igor Khavkine <igor... at (no spam) gmail.com> wrote:
I think it is best (that is, to avoid as much confusion as possible)
to consider the speed of light to be constant everywhere.

I am inclined to agree. It certainly makes matters simpler.

It's not a question of opinion, or of making things "simpler" ("Keep
things as simple as possible, BUT NO SIMPLER." -- A. Einstein). It's a
question of how this is measured and modeled. In our current best model,
General Relativity, there's no question: over a non-local path the
vacuum speed of light need not be c (even though the local speed of
light is c at each point along the path). This is a basic instance of
the curvature of spacetime.

To see this there's no need to consider light bending around the sun
(though that's an experimental confirmation of the model). One need not
even worry about synchronization of distant clocks. In Schwarzschild
spacetime and Schw. coordinates, for a clock at A and a mirror at B
directly above A, the round-trip vacuum speed of light is not c, where
speed means twice the number of standard meter sticks that can be laid
between A and B divided by the elapsed time during the light's flight
measured by a standard clock at point A. Adding another clock and
mirror, the speed for B->A->B differs from A->B->A; adding a mirror C an
equal number of meter sticks below A, the speed for A->C->A is different
from either of them. Unfortunately, for the earth and sun these
differences are too small to measure directly, and interference methods
suffer from the impossibility of making a Michelson interferometer rigid
enough to rotate in a vertical plane.

        Hmmm. I wonder if it could be sensitive enough to rotate in
        a horizontal plane on earth, and measure the effect due to
        the sun? Probably not, as these interferometers are
        notoriously unstable.... And, of course, Kennedy and
        Thorndike (and similar experiments) could have seen
        at least some fraction of it.

I have often thought that, given  all our observations of the
constancy of c were made in the accelerating frames we actually
inhabit, it seems a bit peculiar to conclude from this that c is
constant in inertial frames but perhaps not in accelerating frames.

The accelerations involved are so small they are inconsequential. The
math of GR is quite clear: the vacuum speed of light is locally c in any
locally inertial frame (using standard clocks and rulers). In an
accelerated system the speed of light cannot be isotropic, whether
measured one-way or round-trip [#]. But for tiny accelerations the
difference is smaller than tiny, and in practice here on earth,
unmeasurable.

        The relevant scale is c^2/g, and for earth's gravity
        that is about a lightyear.

        [#] One-way measurements have great difficulties with
        clock synchronization.

Tom Roberts

On Jul 5, 2:44=A0pm, Tom Roberts <tjroberts... at (no spam) sbcglobal.net> wrote:
In our current best model,
General Relativity, there's no question: over a non-local path the
vacuum speed of light need not be c (even though the local speed of
light is c at each point along the path).
[... example from Schwarzschild spacetime]

I note from http://en.wikipedia.org/wiki/Schwarzschild_coordinates
that the Schwartzchild chart does not represent radial distances
accurately.

Given this, it is difficult to see how the use of such a
coordinate system can prove the reality of effects which are beyond
the limits of observation.

Consider points a and b in an accelerating
frame accelerating in the direction a to b, and generated by starting
the acceleration of a and b at the same point in time, with the same
separation.
Observers at both a and b can confirm observationally that clocks on b
run faster than clocks on a.

Now, are you saying that for b, c is < 1, for the round photon trip
bab, and for a, c is >1 for the round photon trip aba, so that they
can also agree on separation?

And is that difference effective on the
outward photon trip, the return photon trip, or both?

Now consider an accelerating frame where both a and b start
accelerating from the same point in spacetime (but in different
spatial directions). Both will then find observationally that clocks
on the other are running slower.
Is c then>or<1 for either / both round trips?

Finally, and potentially more subversively, since we cannot tell,
experimentally, whether c is constant in inertial or accelerating
frames,

Chalky wrote:

Finally, and potentially more subversively, since we cannot tell,
experimentally, whether c is constant in inertial or accelerating
frames,

This is not true. We CAN tell for round-trip speed. For measurements of
round-trip speed, c is constant in locally-inertial frames occupied by
labs on earth; this has been measured literally zillions of times.

Chalky wrote:

Consider points a =A0and b in an accelerating
frame accelerating in the direction a to b, and generated by starting
the acceleration of a and b at the same point in time, with the same
separation.
Observers at both a and b can confirm observationally that clocks on b
run faster than clocks on a.

Yes. Assuming Minkowski spacetime.

Now, are you saying that for b, =A0c is < 1, for the round photon trip
bab, and for a, c is >1 for the round photon trip aba, so that they
can also agree on separation?

Yes, as long as one uses meter sticks undergoing Born rigid motion to
measure the distance used in the computation of speed

Finally, and potentially more subversively, since we cannot tell,
experimentally, whether c is constant in inertial or accelerating
frames,

This is not true. We CAN tell for round-trip speed. For measurements of
round-trip speed, c is constant in locally-inertial frames occupied by
labs on earth; this has been measured literally zillions of times.

If you start two separate bodies (or two ends of
the same body) accelerating at the same rate, at the same time, it is
obvious that they will both have the same velocity at all subsequent
times, in that original inertial frame. Consequently, I can see no
reason for postulating that one must accelerate harder than the other,
for their separations to remain constant subsequently. Sure, their
separation in the original inertial frame may become relativistically
shrunk in due course, but that is irrelevant because the lengths of
co-moving rulers would shrink by the same amount.

On Jul 10, 3:20 am, Tom Roberts <tjroberts... at (no spam) sbcglobal.net> wrote:
For measurements of
round-trip speed, c is constant in locally-inertial frames occupied by
labs on earth; this has been measured literally zillions of times.

I don't see how you can describe as 'locally inertial', a frame where
inertial bodies achieve speeds of > 15 feet/sec in less than one
second, if you stop applying force to keep them 'stationary'.

Your argument appears to fly in the face of the general postulate (aka
principle of equivalence)

Well, I don't trust Born rigid rulers because they are totally
artificial constructs embodying too many logical contradictions.

If you start two separate bodies (or two ends of
the same body) accelerating at the same rate, at the same time, it is
obvious that they will both have the same velocity at all subsequent
times, in that original inertial frame.

Chalky wrote:
On Jul 10, 3:20 am, Tom Roberts <tjroberts... at (no spam) sbcglobal.net> wrote:
For measurements of
round-trip speed, c is constant in locally-inertial frames occupied by
labs on earth; this has been measured literally zillions of times.

I don't see how you can describe as 'locally inertial', a frame where
inertial bodies achieve speeds of > 15 feet/sec in less than one
second, if you stop applying force to keep them 'stationary'.

Remember that physics is a QUANTITATIVE science. The phrase "locally
inertial" is always an approximation. But in practice, for labs on earth
and for tabletop light-speed measurements (mentioned above), that
approximation is FAR better than the experimental resolutions. As the
apparatus is supported against gravity, one normally analyzes such
experiments in the locally-inertial frame that is initially at rest
relative to the apparatus.

Exercise for the reader: Consider a lab measurement of the
round-trip speed of light over a 10 meter path. To estimate
the error induced by ignoring the acceleration of gravity:
A) compute the time duration of the round-trip light path
B) compute how far an object would fall during that time,
given it is initially at rest (hint: L = 0.5 g t^2).
This is an overestimate, as there is a better inertial
frame to use (at rest when the light is in the middle).
C) now compute the difference between the assumed
(horizontal) path and the actual path due to that
distance fallen (hint: neglect the curvature of the path
and add the vertical fall to the horizontal distance;
this clearly overestimates the actual path length).
D) now relate that error in path length to an error in the
measured speed (hint: fractional errors are the same).
Compare to a typical experimental resolution of ~0.1 parts
per billion.

This is only an (over) ESTIMATE. For real experiments you
need to look them up and read their error analysis.

Your argument appears to fly in the face of the general postulate (aka
principle of equivalence)

No. It merely includes the fact that physics is a quantitative science,
and for this specific physical situation the acceleration due to gravity
is completely negligible.

To induce Born rigid motion in a solid object, you must either:
a) attach the acceleration to a single point and let any
startup transients damp out, making sure the acceleration
is small enough so strain is negligible.

or:
b) couple the acceleration to every individual atom, varying
it appropriately for each atom so the proper distances
remain constant.

Chalky wrote:

Well, I don't trust Born rigid rulers because they are totally
artificial constructs embodying too many logical contradictions.

Not at all! For a ruler, Born rigid motion means that in a steady state
the atoms of the ruler arrange themselves so that the inter-atomic
distances have their usual values.

On Jul 13, 7:53 pm, Tom Roberts <tjroberts... at (no spam) sbcglobal.net> wrote:
For a ruler, Born rigid motion means that in a steady state
the atoms of the ruler arrange themselves so that the inter-atomic
distances have their usual values.

Which is un-physical. If the rulers are laid out vertically, the force
inducing acceleration transfers through the rulers at the speed of
sound in that material. Consequently, rulers DO stretch if hung from
the ceiling, and compress if stacked up from the floor, when
acceleration is induced.

How
many carpenters use different rulers for vertical and horizontal
measurements, and adjust the former for altitude?