# The speed of light in a gravitational field

by e2m2a
Tags: field, gravitational, light, speed
 P: 155 Does a photon gain speed if it travels in "free-fall" toward the earth? A particle of mass gains speed as it free-falls to the earth, and yet, mass is just another form of energy. Light is energy. So, does light gain speed?
 P: 474 No the speed of light is constant (in a vacuum) Mass is not just another form of energy, energy can be converted into mass and v.v. but gravitational mass is not equivalent to energy.
P: 1,555
I suspect you will get probably 10 postings that will only give you information about the speed of light measured locally, which is always c. But of course people deserve more information.
 Quote by e2m2a Does a photon gain speed if it travels in "free-fall" toward the earth? A particle of mass gains speed as it free-falls to the earth, and yet, mass is just another form of energy. Light is energy. So, does light gain speed?
In the case of Earth we can use the Schwarzschild solution as an approximation, in this case, in Schwarzschild coordinates, the coordinate speed of light actually slows down the closer it gets to the Earth. This is not only true for light but for all objects that have a coordinate velocity above a critical speed of:

$$3^{-1/2}$$

This is also the case for the proper velocity (e.g. the velocity wrt local shell observers) but in this case the critical velocity is

$$2^{-1/2}$$

P: 155

## The speed of light in a gravitational field

I can understand that relative to an observer on the earth, clocks would tick at the same rate locally at any point above the earth. The problem appears to arise when time measurements are made at different points.

For example, if the observer synchronized two clocks, one at the earth's surface and one at a certain height above the earth, measured the distance between the two points, and released a photon by some mechanism when the clocks were at the same time relative to the earth-observer's frame, the observer would measure the photon takes less time to reach the earth for the pre-measured distance traveled by the photon because the clock at the earth's surface would tick slower non-locally relative to the clock at the higher point. (The observer would be "blind" to this.)

Now, of course, an observer sufficiently far away from the earth would correctly state that this apparent increase in speed is due to the earth-observer's two clocks are always out of synchronization, the distance traveled by the photon contracted, and the clocks run at different rates, but this is the observations of someone outside of the frame of the earth.

Therefore, wouldn't it be correct for the observer on the earth to conclude that the speed of the photon increased relative to the earth observer's frame?
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 Quote by e2m2a Does a photon gain speed if it travels in "free-fall" toward the earth? A particle of mass gains speed as it free-falls to the earth, and yet, mass is just another form of energy. Light is energy. So, does light gain speed?
the photon will gain energy, exactly equivalent to the amount it would have gained if it's speed had increased through falling. But the energy gained is in the increase in frequency, not speed, which is constant.
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P: 7,204
 Quote by e2m2a Therefore, wouldn't it be correct for the observer on the earth to conclude that the speed of the photon increased relative to the earth observer's frame?
Not really.

The solution to the problem of clocks ticking at different rates at different heights is easy.
If the accuracy of the experiment is limited by the change in clock rates with altitude, you only need to measure the velocity over a smaller altitude change. In the limit as the altitude change approaches zero, there's no effect on the velocity measurement - and in practice, there's very little effect even with relatively large altitude changes.
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 Quote by Passionflower I suspect you will get probably 10 postings that will only give you information about the speed of light measured locally, which is always c. But of course people deserve more information. In the case of Earth we can use the Schwarzschild solution as an approximation, in this case, in Schwarzschild coordinates, the coordinate speed of light actually slows down the closer it gets to the Earth. This is not only true for light but for all objects that have a coordinate velocity above a critical speed of: $$3^{-1/2}$$ This is also the case for the proper velocity (e.g. the velocity wrt local shell observers) but in this case the critical velocity is $$2^{-1/2}$$
Curious what you think these speeds mean. Personally, the only thing meaningful is to model a process for measurement in a given scenario. For inertial frames in SR, you can generally dispense with this only because all reasonable approaches agree with commonly chosen coordinate values. In all other scenarios (including non-inertial frames in SR), the scenario and measurement model are crucial, and different choices lead to different answers.

For example, let's say you have two observers on opposite sides of the center of Swarzchild solution. Well, let's immediately choose not to; your results are strongly affected by lensing, and one observer may see the other as a ring, and have a hard time defining simple measuring schemes.

Ok, choose observers on near opposite sides, so a light path between thme passes within e.g. 1.5 times the event horizon (or any other scenario of you choice; but it must be specified). Now propose how to measure speed of light. No single, simple, method can be used. Radar ranging gives you a proper time interval along one observer's world line. To get distance, you typically assume c. That won't do if you're trying to measures it. Ok, define some separate procedure (e.g. idealized parallax distance) to get an apparent distance. Now finally you have a distance and a time. I would be surprised if any such measurement procedure gives results matching any of the commonly used coordinate values.
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 Quote by PAllen Curious what you think these speeds mean.
You seem to have the impression that nothing physical can be calculated from using the Schwarzschild solution. If that is so then you are completely wrong.

If you pay attention I and a few others constantly use the Schwarzschild solution to make physical meaningful calculations on this forum. But I have not seen one single calculation from you using the Schwarzschild solution, or any other solution for that matter, all you seem to do is criticize those who actually do make the effort here.

Curious why you are at all interested in GR if you never want to do any calculations.

I certainly could make physically meaningful calculations about what I wrote above or quote several papers that discuss this situation. If you cannot, then if you want to learn about GR, I strongly suggest you are going to start making an effort. Looking at GR from a 30,000ft height and thinking you know it all without being able to do even simple calculations using the Schwarzschild solution is in my opinion being in a state of delusion.
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 Quote by Mentz114 the photon will gain energy, exactly equivalent to the amount it would have gained if it's speed had increased through falling. But the energy gained is in the increase in frequency, not speed, which is constant.
Relative to any fixed observer, the frequency is constant. Relative to a series of observers at different potentials, the frequency increases as the photon falls, because their clocks are increasingly time dilated.

An easier point of view is to look at momentum. Relative to an isotropic coordinate system, the rate of change of momentum of a test particle of energy E (which could be a photon), in a central weak field is given by the following expression:

$$\frac{d\mathbf{p}}{dt} = \frac{E}{c^2} \, \mathbf{g} \, \left ( 1 + \frac{v^2}{c^2}} \right )$$

where g is the Newtonian gravitational acceleration of the field and all of the quantities including c, the coordinate speed of light, are measured in the coordinate system rather than in local space.

Note that as for Newtonian gravity, this expression does not depend on the direction of motion, although unlike for Newtonian gravity, it does depend on the speed.

For a vertical photon, the change in momentum is entirely due to the change in the coordinate value of c.
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 Quote by Passionflower You seem to have the impression that nothing physical can be calculated from using the Schwarzschild solution. If that is so then you are completely wrong. If you pay attention I and a few others constantly use the Schwarzschild solution to make physical meaningful calculations on this forum. But I have not seen one single calculation from you using the Schwarzschild solution, or any other solution for that matter, all you seem to do is criticize those who actually do make the effort here. Curious why you are at all interested in GR if you never want to do any calculations. I certainly could make physically meaningful calculations about what I wrote above or quote several papers that discuss this situation. If you cannot, then if you want to learn about GR, I strongly suggest you are going to start making an effort. Looking at GR from a 30,000ft height and thinking you know it all without being able to do even simple calculations using the Schwarzschild solution is in my opinion being in a state of delusion.
I assume anything can be calculated from given coordinates, and to understand what coordinates mean, I would do calculations. My question, and it is purely a question is, as someone who has investigated these coordinates, what measurements do they correspond to? That is a perfectly fair and reasonable question, irrespective of whether I can answer it. At the present time, I am not interested in doing calculations (I last did such calculation over 30 years ago, am professionally involved in an unrelated field, and am not motivated to refresh and modernize my skills enough to readily do calcuations. I remain conceptually interested in GR and SR, and recognize the severe limitation of not doing calculations on my own. I will continue to ask questions I cannot answer myself).

Also, I have actually done a few calculations in and for my posts. In some cases, I don't put them in the post because I hate latex. In other cases, I have put results in using crude notations. But generally, I have not and don't plan (for now) to do any systematic calculations.
 PF Patron Sci Advisor P: 4,463 Getting back to something like the original question on this post, the way I would set up the problem would be: Assumming R is the event horizon radius, and a worldline of an observer maintaining fixed position 2R, and another world line at e.g. 3R, what speed of light would be measured between them? For me, this would immediately raise the question of what points of the world lines to consider simultaneous, because I can't conceive of measuring distance without this, and can't see how to measure speed without measuring distance. This would immediately put me in a quandary. I would attach no meaning to coordinate time. For one thing, both of these world lines are *extremely* non-inertial observers (think of the rocket g forces required to mainain constant position; barring tidal effects, they would correspond to Rindler observers with very high G, different for each world line; time would run very differently for each world line.). Normally, I would say a simple, light based simultaneity definition is good for many purposes. However, here the goal is to measure the radial speed of light, so I would consider any simultaneity definition involving light to be circular for this purpose and any using the coordinate time of the overall solution to be meaningless for such observers. I would be stuck, but at least I would feel I've asked relevant initial questions. I don't know whether Passionflower has gone through this exercise or not, but I don't see such issues even being mentioned. For me, I wouldn't see how to use Born rigidity either. The simple definitions I've seen are in terms of a comoving inertial frame. Over a span like 2R to 3R, I would be stumped by the fact that there is no remotely inertial frame that can cover this range distance. So I would be completely stumped by how to compute what GR predicts for this measurement, but I would at least feel I have asked relevant starting questions.
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 Quote by PAllen Assumming R is the event horizon radius, and a worldline of an observer maintaining fixed position 2R, and another world line at e.g. 3R, what speed of light would be measured between them? For me, this would immediately raise the question of what points of the world lines to consider simultaneous, because I can't conceive of measuring distance without this, and can't see how to measure speed without measuring distance.
From the information you give it is very easy to calculate the physical distance and roundtrip time (both in proper time for R2 and R3's clock) in terms of R.

Both yuiop and I have done similar calculations in various topics. This time I really leave it as an exercise to calculate it, it is straightforward and in no way ambiguous.

 Quote by PAllen This would immediately put me in a quandary. I would attach no meaning to coordinate time.
It is very easy to calculate the coordinate time in terms of R and from this you can calculate the proper time both for R2 and R3.

 Quote by PAllen For one thing, both of these world lines are *extremely* non-inertial observers (think of the rocket g forces required to mainain constant position; barring tidal effects, they would correspond to Rindler observers with very high G, different for each world line; time would run very differently for each world line.).
Another good reason to actually make the calculations and plot them for various actual values of R. If you do that you will find out that you are wrong calling those observers *extremely* non-inertial observers. How 'non inertial' really depends on the actual value of R. For instance if R is very large the proper acceleration of the stationary observers could be much smaller than 1g. Also in another calculation I made last week on this forum I demonstrated that those observers would not exactly correspond to Rindler observers. Also this calculation is trivial, we can immediately calculate the difference from Rindler acceleration in terms of R.

 Quote by PAllen I don't know whether Passionflower has gone through this exercise or not, but I don't see such issues even being mentioned.
Both yuiop and I have made similar calculations and by using prior postings you can obtain the correct formulas on how to calculate this situation. Basically all you need is the integrated distance between R2 and R3, the integrated coordinate time light takes to go from R2 and R3 and vice versa and on how to convert this time to proper time for both R2 and R3. Again, this is another good example why it is very important to do exercises, both yuiop and I went through these issues but you do not seem to realize it, and the reason I expect is that you have not tried the calculations yourself.

 Quote by PAllen For me, I wouldn't see how to use Born rigidity either.
I do not see how Born rigidity is relevant when we want to measure the roundtrip speed of light between two stationary points.

 Quote by PAllen So I would be completely stumped by how to compute what GR predicts for this measurement, but I would at least feel I have asked relevant starting questions.
Again measuring the distance and roundtrip speed of light, in proper time for both endpoints R2 and R3 is trivial and unambiguous.
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 Quote by PAllen Getting back to something like the original question on this post, the way I would set up the problem would be: Assumming R is the event horizon radius, and a worldline of an observer maintaining fixed position 2R, and another world line at e.g. 3R, what speed of light would be measured between them? For me, this would immediately raise the question of what points of the world lines to consider simultaneous, because I can't conceive of measuring distance without this, and can't see how to measure speed without measuring distance.
What I would suggest is to measure the speed between points that are closer together, i.e between 2R and 2.0001R, or more generally between R' and R'+epsilon.

Then, you can use a co-located and instantaneously co-moving inertial frame to measure the speed of light, because you've limited yourself to a region of space-time that's small enough that it's essentially flat.

Though if you look at the accelerating elevator problem, there isn't that much of a problem with accelerating clocks, as long as you make the region of your measurement small enough.

The problem of not being able to measure the one-way speed of light without defining how to syncrhronize clocks is a problem left over from special relativity. It has various resolutions - the most common is to measure the round trip speed, and state that you are explicitly assuming isotropy, so that the time is equal forwards and backwards.

Technically, nowadays, the speed of light is defined as a constant. So if you're actually measuring the speed of light, you probably should note that you're using a physical standard meter rod as your reference.

This then gives the question of what mathematical model to use for your physical meter rod - since you are calculating the result rather than performing the experiment. Born rigidity immediately comes to mind, and would be my suggestion. Basically you can figure out the expected stretch in your actual physical meter rod due to the stresses on it, and either call them experimental error, or make a note of how big they are and say that you are compensating for them.

 This would immediately put me in a quandary. I would attach no meaning to coordinate time.
In general coordinate anything doesn't have any particular physical meaning, unless one chooses the coordinates correctly. In the Schwarzschild geometry it happens to have some significance as a killing vector.
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 Quote by Passionflower From the information you give it is very easy to calculate the physical distance and roundtrip time (both in proper time for R2 and R3's clock) in terms of R. Both yuiop and I have done similar calculations in various topics. This time I really leave it as an exercise to calculate it, it is straightforward and in no way ambiguous.
Well - if you've done the calculation, you should be able to answer Pallen's questions, of how you define simultaneity, and along what particular curve you integrate the distance.

Just coming up with an answer isn't very convicing, if you can't show your work and define what it is that you're calculating when asked for details.

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 Quote by Passionflower From the information you give it is very easy to calculate the physical distance and roundtrip time (both in proper time for R2 and R3's clock) in terms of R. Both yuiop and I have done similar calculations in various topics. This time I really leave it as an exercise to calculate it, it is straightforward and in no way ambiguous.
Round trip time is in R2's proper time, for example, is straightforward. For distance, a pair of events considered simultaneous is required to do it any invariant way. What points on the the two world lines should be considered simultaneous?

 Quote by Passionflower Another good reason to actually make the calculations and plot them for various actual values of R. If you do that you will find out that you are wrong calling those observers *extremely* non-inertial observers. How 'non inertial' really depends on the actual value of R. For instance if R is very large the proper acceleration of the stationary observers could be much smaller than 1g. Also in another calculation I made last week on this forum I demonstrated that those observers would not exactly correspond to Rindler observers. Also this calculation is trivial, we can immediately calculate the difference from Rindler acceleration in terms of R.
I was assuming small R to make any effect on speed of light larger.
I also explicitly said it wasn't exactly Rindler, just similar until tidal effects come into play.

 Quote by Passionflower Basically all you need is the integrated distance between R2 and R3, the integrated coordinate time light takes to go from R2 and R3 and vice versa and on how to convert this time to proper time for both R2 and R3.
Integrated distance between which pair of events at R2 and R3? Which coordinate times at each location would you consider simultaneous? I consider this not meaningful without some operations definition appropriate for, e.g., the non-inertial observer at R2. And given the purpose, this operational defintion should not involve light. I disagree that 'same coordinate time' in the Swarzchild coordinates has any meaning as a definition of which point in the R3 world line should be considered simultaneous to a given point on the R2 time line for an R2 observer. Once that is answered, it is trivial, but to me that is a non-trivial question.
 Quote by Passionflower I do not see how Born rigidity is relevant when we want to measure the roundtrip speed of light between two stationary points. Again measuring the distance and roundtrip speed of light, in proper time for both endpoints R2 and R3 is trivial and unambiguous.
Born rigidity would provide a possible answer, without relying on light, to define which events at R2 and R3 could be considered simultaneous by an observer at R2.

You have still not given any definition of the simultaneity condition. The same Swarzchild t values at R2 and R3 I claim has no physical meaning for a real observer at R2 (at least until validated with some procedure for simultaneity that invovolves neither light nor coordinates).
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 Quote by pervect What I would suggest is to measure the speed between points that are closer together, i.e between 2R and 2.0001R, or more generally between R' and R'+epsilon.
Yes, that would be much easier. I was trying to accentuate a hypothetical possibility you would measure something different from c in trip up and down the gravity well. I was aiming to see if you could define a measurment over a span where flatness cannot be assumed.

 Quote by pervect The problem of not being able to measure the one-way speed of light without defining how to syncrhronize clocks is a problem left over from special relativity. It has various resolutions - the most common is to measure the round trip speed, and state that you are explicitly assuming isotropy, so that the time is equal forwards and backwards. Technically, nowadays, the speed of light is defined as a constant. So if you're actually measuring the speed of light, you probably should note that you're using a physical standard meter rod as your reference.
I am familiar with this issue and choosing to ignore it. But even for a two way measurement of lightspeed, you first have to come up with a distance, which seems to require simultaneity, which I couldn't solve, and still don't understant how to solve for a distance like 2R to 3R.

 Quote by pervect This then gives the question of what mathematical model to use for your physical meter rod - since you are calculating the result rather than performing the experiment. Born rigidity immediately comes to mind, and would be my suggestion. Basically you can figure out the expected stretch in your actual physical meter rod due to the stresses on it, and either call them experimental error, or make a note of how big they are and say that you are compensating for them.
Yes, Born rigidity is what I gave some thought to, but again, for my purpose of measuring lightspeed where flatness is not approximately correct, I didn't see how to apply definitions of Born rigidity I've seen.

 Quote by pervect In general coordinate anything doesn't have any particular physical meaning, unless one chooses the coordinates correctly. In the Schwarzschild geometry it happens to have some significance as a killing vector.
I expect it has significance, but not obviously for the purpose here, since it has nothing to do with how an (accelerated) observer held at 2R would measure simultaneity to 3R.
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 Quote by PAllen Assumming R is the event horizon radius, and a worldline of an observer maintaining fixed position 2R, and another world line at e.g. 3R, what speed of light would be measured between them?
Ok, so we have a Schwarzschild radius of R and two stationary observers R2 and R3.

Then the ruler distance between them is:

$$\rho = R \left( \sqrt {3}\sqrt {2}-\sqrt {2}+\ln \left( -\sqrt {3}+\sqrt {3} \sqrt {2}-\sqrt {2}+2 \right) \right)$$

Now the radar distance T in coordinate time between them is:

$$T = R+R\ln \left( 2 \right)$$

The radar distance in proper time for R2 is:

$$\tau_{R2} = 1/2\, \left( R+R\ln \left( 2 \right) \right) \sqrt {2}$$

And for R3:

$$\tau_{R3} = 1/3\, \left( R+R\ln \left( 2 \right) \right) \sqrt {6}$$

From this you can calculate the (average) speed of light, if you do this you will find that both the coordinate speed and the speed from r1 to r2 (r1 < r2) in proper time is always < c. Only the speed from r2 to r1 in proper time is > c.
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 Quote by Passionflower Ok, so we have a Schwarzschild radius of R and two stationary observers R2 and R3. Then the ruler distance between them is: $$\rho = R \left( \sqrt {3}\sqrt {2}-\sqrt {2}+\ln \left( -\sqrt {3}+\sqrt {3} \sqrt {2}-\sqrt {2}+2 \right) \right)$$