Change of speed of light in a gravitational field

[notknowing]

From all the sources I have (and including wikipedia), it seems that the speed of light is lowered by the presence of a gravitational field.
Now since the speed of light is related to the electrical permittivity and the magnetic permeability of the vacuum, one could interpret this by saying that the gravitational field modifies the properties of the vacuum. Instead of a geometrical description og gravity (GR), one could therefore probably obtain an equivalent description in which "curvature" is replaced by "modification of the (aether) vacuum".
I appreciate your comments.

 

[George Jones]

I think you're confusing coordinate speed with speed with respect to an orthonormal frame.

At any vacuum event p in spacetime, electrical permittivity and the magnetic permeability of the vacuum are related to the speed of light with respect to an orthonormal frame at p. This speed is always c.

 

[notknowing]

I think you're confusing coordinate speed with speed with respect to an orthonormal frame.

At any vacuum event p in spacetime, electrical permittivity and the magnetic permeability of the vacuum are related to the speed of light with respect to an orthonormal frame at p. This speed is always c.

I do not agree : The speed is not always c. This only applies to inertial frames. I quote from (http://www.physlink.com/Education/AskExperts/ae13.cfm ) :

Most of us have heard of the result from _special_ relativity that the speed of light is the same for all observers in inertial frames.

The result is _not_ the same in general relativity. In general relativity, the statement becomes that the speed of light is the same (i.e., good old 'c') for all observers in _local_ inertial frames.

Local inertial frames in general relativity are just those frames of reference in which the observer is in gravitational free fall. A fancy way of looking at it is that the _local_ frame of reference of a free falling observer corresponds to a small patch of _flat_ spacetime tangent to the globally curved spacetime. As long as the observer confines measurements to a small enough local region, the approximation provided by the small tangent patch of flat spacetime can be made to be an arbitrarily good approximation to the true spacetime, which is actually curved in the main. The speed of light in flat spacetime is, of course, the usual value of c.

For example, if one had a closed laboratory in orbit (i.e., in free fall) around the Earth and one did an experiment inside that laboratory to measure the speed of light, one would get the usual published value of c. All such observers would get one and the same value for c.

If, however, the distance through which the light traveled in the course of measuring its speed was too great, the deviation of the reference frame from being 'flat' would become apparent. That is, gravitational effects would begin to become apparent.

So, it is absolutely true that the speed of light is _not_ constant in a gravitational field [which, by the equivalence principle, applies as well to accelerating (non-inertial) frames of reference]. If this were not so, there would be no bending of light by the gravitational field of stars. One can do a simple Huyghens reconstruction of a wave front, taking into account the different speed of advance of the wavefront at different distances from the star (variation of speed of light), to derive the deflection of the light by the star.

Indeed, this is exactly how Einstein did the calculation in:

'On the Influence of Gravitation on the Propagation of Light,' Annalen der Physik, 35, 1911.

which predated the full formal development of general relativity by about four years. This paper is widely available in English. You can find a copy beginning on page 99 of the Dover book 'The Principle of Relativity.' You will find in section 3 of that paper, Einstein's derivation of the (variable) speed of light in a gravitational potential, eqn (3). The result is,

c' = c0 ( 1 + V / c2 )

where V is the gravitational potential relative to the point where the speed of light c0 is measured.

So, the fact that the speed of light changes in a gravitational field was expressed by Einstein himself in 1911 (though he made in this paper an error in the derivation of the bending of light, which he later, luckily for him, corrected before the experiments were made).
Further, the Shapiro delay has been experimentally verified (see http://en.wikipedia.org/wiki/Shapiro_effect ).
Further, one can find books on relativity in which the change of the speed of light is given explicitly ; see Ruppel, F., Mechanik, Relativitat, Gravitation, Springer Verlag, Berlin, Heidelberg, 1973 (in German) and I could probably find a number of other sources as well.

 

[pervect]

George is right, you (notknowing) are misunderstanding the point. The speed of light is always 'c' as measured by a local clock and ruler. If you have light coming from Hawking-knows-where, and you measure that speed with a ruler and a clock, you will always measure the speed of that light to be equal to 'c'.

The references you quote do not disagree on this point. Einstein's quote was about coordinate speeds, not locally measured speeds, for one example.

 

[Chris Hillman]

Coordinate velocity and Sakharov

Hi, notknowing,

From all the sources I have (and including wikipedia), it seems that the speed of light is lowered by the presence of a gravitational field.

I agree with George Jones and Pervect (hi, Pervect!) that you must be confusing coordinate speed with physical velocity. You cited the Wikipedia and some website unfamiliar to me, but you appear to have misunderstood something someone said (perhaps taken out of context?), or someone may have misinformed you. If you really want to know anything about gtr, you should obtain a good textbook; gosh knows you have a virtual cornucopia to choose from! I'd recommend Misner, Thorne, and Wheeler, Gravitation, Freeman, 1973 (known as MTW and still widely available), because it probably contains the answer to any answer you might ask.

The basic problem is if you plot some light cones in the Schwarzschild chart outside the event horizon of a nonrotating black hole, they appear to become "radially flattened" and "taller", because the coordinate vectors @/@t, @/@r etc. do not have unit length. If you don't correct for this, you will be incorrectly computing all velocities, including the velocity of a passing laser pulse. If you do correct for this, then of course the speed of light is c=1 (in relativistic or "geometrized" units) everywhere and everywhen, more or less by construction of the theory. (Since our default theory of gravitation is gtr, I assume this is the theory you have in mind.)

As for the quotation from some website which you offered (do you know who wrote it and why they can be considered an "expert"?), it appears that this author may be referring to a distinct issue, that old bugaboo, "local versus global". The clue is that he mentions light bending, which is a curvature effect not seen in an infinitesimally small region.

In the above, I was discussing physics and geometry "at the level of tangent spaces". If you consider "paths" (say of a laser pulse) and "speeds" measured over larger regions of spacetime, you will need to deal with the fact that second order curvature effects will show up. In addition, at the operational level by which we connect our mathematical computations in gtr (our default theory) with what real observers can actually measure, it turns out that there are many distinct notions of "distance in the large" and thus of "velocity in the large". This is a fascinating topic, but not one we should get into unless we are all comfortable with at least the Track One concepts covered in MTW.

Now since the speed of light is related to the electrical permittivity and the magnetic permeability of the vacuum, one could interpret this by saying that the gravitational field modifies the properties of the vacuum. Instead of a geometrical description og gravity (GR), one could therefore probably obtain an equivalent description in which "curvature" is replaced by "modification of the (aether) vacuum".
I appreciate your comments.

You will be interested in an idea put forward long ago by the famous physicist Andrei Sakharov (the same who was a well known political dissident in the former Soviet Union). You can read more about it in Box 17.2 of Misner, Thorne, and Wheeler, Gravitation. I should add that no-one has ever been able to make this work, but it continues to intrigue many physicists.

Chris Hillman

 

[pervect]

Hi, Chris - welcome to PF, and glad you could make it!

 

[notknowing]

Hi, notknowing,



I agree with George Jones and Pervect (hi, Pervect!) that you must be confusing coordinate speed with physical velocity. You cited the Wikipedia and some website unfamiliar to me, but you appear to have misunderstood something someone said (perhaps taken out of context?), or someone may have misinformed you. If you really want to know anything about gtr, you should obtain a good textbook; gosh knows you have a virtual cornucopia to choose from! I'd recommend Misner, Thorne, and Wheeler, Gravitation, Freeman, 1973 (known as MTW and still widely available), because it probably contains the answer to any answer you might ask.

The basic problem is if you plot some light cones in the Schwarzschild chart outside the event horizon of a nonrotating black hole, they appear to become "radially flattened" and "taller", because the coordinate vectors @/@t, @/@r etc. do not have unit length. If you don't correct for this, you will be incorrectly computing all velocities, including the velocity of a passing laser pulse. If you do correct for this, then of course the speed of light is c=1 (in relativistic or "geometrized" units) everywhere and everywhen, more or less by construction of the theory. (Since our default theory of gravitation is gtr, I assume this is the theory you have in mind.)

As for the quotation from some website which you offered (do you know who wrote it and why they can be considered an "expert"?), it appears that this author may be referring to a distinct issue, that old bugaboo, "local versus global". The clue is that he mentions light bending, which is a curvature effect not seen in an infinitesimally small region.

In the above, I was discussing physics and geometry "at the level of tangent spaces". If you consider "paths" (say of a laser pulse) and "speeds" measured over larger regions of spacetime, you will need to deal with the fact that second order curvature effects will show up. In addition, at the operational level by which we connect our mathematical computations in gtr (our default theory) with what real observers can actually measure, it turns out that there are many distinct notions of "distance in the large" and thus of "velocity in the large". This is a fascinating topic, but not one we should get into unless we are all comfortable with at least the Track One concepts covered in MTW.



You will be interested in an idea put forward long ago by the famous physicist Andrei Sakharov (the same who was a well known political dissident in the former Soviet Union). You can read more about it in Box 17.2 of Misner, Thorne, and Wheeler, Gravitation. I should add that no-one has ever been able to make this work, but it continues to intrigue many physicists.

Chris Hillman

Hi, thanks for your interesting comments. That does not however unvalidate the whole reasoning (from Warren Davis ; see http://web.hcmut.edu.vn/~huynhqlinh/olympicvl/tailieu/physlink_askexpert/ae_warren_davis.cfm.htm ) which I quoted. I do have Misner, Thorne, and Wheeler (MTW), Gravitation and some other good books too. MTW does not really treat this subject (change of speed of light by gravitation) so therefore I used some other references. When looking at the velocity of light, as measured over an extended region of space, it IS lowered and this is also described in the Shapiro effect, which is real. This brings me back to my original question (on the properties of the vacuum) on which I liked some input. I know about the theory put forward by Andrei Sakharav (and the description in MTW) and I am still convinced that he was on the right track, but as you mentioned, no-one has ever been able to make this work (though I believe one day it will work).

 

[notknowing]

The link to Warren Davis did not seem to work correctly: So here is a copy of a short biography :

Warren F. Davis
Ph.D. Physics
President, Davis Associates, Inc.
Newton, MA USA

Biography
Warren F. Davis obtained both his Ph.D. and undergraduate degrees in physics from M.I.T. For his graduate thesis he undertook a theoretical examination of gravitational radiation as predicted by the Einstein, Brans-Dicke and Rosen bi-metric theories of gravity.

He has worked in a variety of applied areas, including radar signature analysis for the U.S. Pacific Test Range, digital flight simulation and onboard guidance for the Apollo lunar landing mission, signal analysis in radio astronomy, optical image reconstruction and aperture synthesis, and the development of new multidimensional fast Fourier transform techniques and algorithms.

He is currently the President of Davis Associates, Inc..

 

[Chris Hillman]

Ambiguity of "speed of light": local vs. global, not choice of basis

Hi, notknowing,

Hi, thanks for your interesting comments. That does not however unvalidate the whole reasoning (from Warren Davis ; see http://web.hcmut.edu.vn/~huynhqlinh/olympicvl/tailieu/physlink_askexpert/ae_warren_davis.cfm.htm ) which I quoted.

I don't think I recognize his name, but thanks for the information.

I do have Misner, Thorne, and Wheeler (MTW), Gravitation and some other good books too. MTW does not really treat this subject (change of speed of light by gravitation) so therefore I used some other references. When looking at the velocity of light, as measured over an extended region of space, it IS lowered and this is also described in the Shapiro effect, which is real.

Agreed, the Shapiro effect is measureable, and HAS been measured (confirming the gtr prediction to the limits of accuracy)!

OK, I think I see what happened now. In your initial post, you failed to clarify what you meant by the ambiguous phrase "speed of light". I guessed that you were trying to compute the relative velocity of a laser pulse as measured over a region of spacetime so small that curvature effects can be neglected, but incorrectly using the coordinate basis rather than a frame field (bundle of "local Lorentz frames"). But I overlooked the mention of "lightbending" in the passage you quoted, which is a clue that you actually were thinking of computing a "light path distance" and a "light travel time" over a larger region of spacetime, and dividing to obtain a "lightspeed in the large".

So it looks like in fact everyone does know that, as Einstein knew at an early date, accelerating observers (in either flat or curved spacetime) will in general measure values for "lightspeed in the large" which are dependent on the path taken by the light wave, the method of measurement, the motion of the observer, and other factors, but which usually won't yield a value of unity (in relativistic units). The Shapiro light delay effect is a reflection of this principle.

There is very important point lurking here which I hesitate to try to explain, and which is very well known to researchers in this field, but which is unfortunately slurred over in many textbooks. Namely: even in flat spacetime, there are many competing notions of "distance in the large" which have more or less direct operational significance but give different results. Naturally, in appropriate limits these will agree, however. In particular, all these methods agree on sufficiently small scales.

This brings me back to my original question (on the properties of the vacuum) on which I liked some input.

I'll leave that to others.

Chris Hillman

 

[notknowing]

Hi, notknowing,



I don't think I recognize his name, but thanks for the information.



Agreed, the Shapiro effect is measureable, and HAS been measured (confirming the gtr prediction to the limits of accuracy)!

OK, I think I see what happened now. In your initial post, you failed to clarify what you meant by the ambiguous phrase "speed of light". I guessed that you were trying to compute the relative velocity of a laser pulse as measured over a region of spacetime so small that curvature effects can be neglected, but incorrectly using the coordinate basis rather than a frame field (bundle of "local Lorentz frames"). But I overlooked the mention of "lightbending" in the passage you quoted, which is a clue that you actually were thinking of computing a "light path distance" and a "light travel time" over a larger region of spacetime, and dividing to obtain a "lightspeed in the large".

So it looks like in fact everyone does know that, as Einstein knew at an early date, accelerating observers (in either flat or curved spacetime) will in general measure values for "lightspeed in the large" which are dependent on the path taken by the light wave, the method of measurement, the motion of the observer, and other factors, but which usually won't yield a value of unity (in relativistic units). The Shapiro light delay effect is a reflection of this principle.

There is very important point lurking here which I hesitate to try to explain, and which is very well known to researchers in this field, but which is unfortunately slurred over in many textbooks. Namely: even in flat spacetime, there are many competing notions of "distance in the large" which have more or less direct operational significance but give different results. Naturally, in appropriate limits these will agree, however. In particular, all these methods agree on sufficiently small scales.



I'll leave that to others.

Chris Hillman

Yes indeed, now we are at the same wavelength. I should have clarified it better and I was not talking about coordinate speed (a concept which is for a physicist of little value).