The speed of light in a gravitational field

[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?

 

[NobodySpecial]

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.

 

[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.

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:

<br /> 3^{-1/2}<br />

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

<br /> 2^{-1/2}<br />

 

[e2m2a]

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?

 

[Mentz114]

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.

 

[pervect]

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.

 

[PAllen]

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:

<br /> 3^{-1/2}<br />

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

<br /> 2^{-1/2}<br />

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.

 

[Passionflower]

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.

 

[Jonathan Scott]

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:

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

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.

 

[PAllen]

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.

 

[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. 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.

 

[Passionflower]

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.

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.

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.

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.

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.

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.

 

[pervect]

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.

 

[pervect]

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.

And Pallen is asking the right questions about the problem setup.

 

[PAllen]

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?

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.

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.

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).

 

[PAllen]

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.

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.

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.

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.

 

[Passionflower]

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:

<br /> \rho = R \left( \sqrt {3}\sqrt {2}-\sqrt {2}+\ln \left( -\sqrt {3}+\sqrt {3}<br /> \sqrt {2}-\sqrt {2}+2 \right) \right) <br />

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

<br /> T = R+R\ln \left( 2 \right)<br />

The radar distance in proper time for R2 is:

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

And for R3:

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

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.

 

[PAllen]

Ok, so we have a Schwarzschild radius of R and two stationary observers R2 and R3.

Then the ruler distance between them is:

<br /> \rho = R \left( \sqrt {3}\sqrt {2}-\sqrt {2}+\ln \left( -\sqrt {3}+\sqrt {3}<br /> \sqrt {2}-\sqrt {2}+2 \right) \right) <br />

This is the key point. Which events are 2R and 3R are used to calculate this? That I what I am asking about (as was Pervect). Given an answer to this, a calculation is easy, but what criterion is used? If one rules out using a light based definition of simultaneity (else circularity for this problem) and Born rigidity (because you don't have approximate flatness), then what do you use? I hope there is a good answer, but you haven't given it.

[EDIT] Actually, given a simultaneity criterion, proper distance between the events at 2R and 3R becomes easy. However, there is still a further question of whether this is what an observer fixed (by rocket thrust) at 2R would measure using some procedure. The follow up question becomes what is induced 3-metric on some spatial hypersurface for the non-inertial observer at 2R.

 

[Passionflower]

This is the key point. Which events are 2R and 3R are used to calculate this? That I what I am asking about (as was Pervect). Given an answer to this, a calculation is easy, but what criterion is used? If one rules out using a light based definition of simultaneity (else circularity for this problem) and Born rigidity (because you don't have approximate flatness), then what do you use? I hope there is a good answer, but you haven't given it.

Events? What events?

I provided the calculations of the ruler distance and the radar distance both for coordinate and proper time between two stationary observers.

 

[PAllen]

Events? What events?

I provided the calculations of the ruler distance and the radar distance both for coordinate and proper time between two stationary observers.

I am asking what your definition of ruler distance is? To me it must mean:

There a set of events from 2R to 3R that the observer at 2R considers to be simultaneous. If the observer at 2R is non-inertial, this spacetime path is not generally a geodesic. Given this path, you can then easily compute its proper length. The problem I have is how define, for a realistic observe at 2R, what is the path of simultaneity between 2R and 3R?

[EDIT] What I am concerned about is, for example, the distance between 2R and 3R measured by a free falling observer would (I think) be quite different from the distance measured by an observer held in fixed position at 2R. Thus the idea of "the ruler distance" without further definition, does not make sense to me.

 

[Passionflower]

I gave you the answers to the questions you were asking.

I do not believe there is anything I can add as clearly you do not think too highly of my understanding of GR.

 

[bcrowell]

Reading Passionflower's #3, it seems to me that the following statement is clearly well defined and relates directly to quantities that can be measured in fairly obvious ways:

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.

For example, the Schwarzschild coordinate r can in principle be determined by measuring the Kretschmann invariant, and the Schwarzschild coordinate t can then in principle be determined by transporting a clock to the appropriate location along a radial path from some reference point and using the Schwarzschild metric to connect the clock time to the Schwarzschild t coordinate, based on the clock's known trajectory (measured in the form of r as a function of proper time).

Since the statement is well defined in relation to actual observables, the next question is whether it's true. The answer is yes, as can be determined by setting the line element in the Schwarzschild metric equal to zero and solving for dr/dt's dependence on r.

The next question is whether the statement is of any physical interest. I don't see any reason to think that it is. It refers to a coordinate velocity, and in general coordinate velocities are of no physical interest.

There are lots of things that are of physical interest that one can calculate from the Schwarzschild metric. Examples include the result of the Hafele-Keating experiment, the precession of Mercury's perihelion, the Shapiro time delay, the deflection of light rays, the geodetic effect, and round-trip propagation times in the GPS system. What all these things have in common is that (1) they relate directly to practical experiments, and (2) they don't depend for their interest on any special coordinate system.

 

[Passionflower]

The next question is whether the statement is of any physical interest. I don't see any reason to think that it is. It refers to a coordinate velocity, and in general coordinate velocities are of no physical interest.

One stationary space station, a known distance from the surface of a planet, could calculate the ruler distance to another stationary space station by measuring the round trip time of light, from the elapsed proper time he could derive the ruler distance by calculating the coordinate speed of light using Schwarzschild coordinates.

 

[bcrowell]

One stationary space station, a known distance from the surface of a planet, could calculate the ruler distance to another stationary space station by measuring the round trip time of light, from the elapsed proper time he could derive the ruler distance by calculating the coordinate speed of light.

I gave one impractical method by which the statement could be connected in a well-defined way to experimental observables. This is another impractical method by which the statement could be connected in a well-defined way to experimental observables. This has nothing to do with whether or not the statement is of any physical interest.

Another thing to consider is whether or not the statement answers the OP's question. It doesn't, because the OP's question was not asked in terms of Schwarzschild coordinates. The OP naively imagined that the question had a definite answer without regard to any specific coordinate system. The helpful way to answer the OP's question was to point out that the question was ill-defined.

 

[PAllen]

One stationary space station, a known distance from the surface of a planet, could calculate the ruler distance to another stationary space station by measuring the round trip time of light, from the elapsed proper time he could derive the ruler distance by calculating the coordinate speed of light using Schwarzschild coordinates.

The difficulty I have with this is the coordinate speed of light. Whose? Consider 2 cases: two space stations stationary relative to each other by being in different positions on the same orbiot; two stations under power, held stationary above the Earth as if it weren't rotating. Each of these observer would be quite different frames of reference, and the Swarzchild coordinate speed of light would not seem to have relevance to either of these scenarios.

 

[Passionflower]

The difficulty I have with this is the coordinate speed of light. Whose? Consider 2 cases: two space stations stationary relative to each other by being in different positions on the same orbiot; two stations under power, held stationary above the Earth as if it weren't rotating. Each of these observer would be quite different frames of reference, and the Swarzchild coordinate speed of light would not seem to have relevance to either of these scenarios.

Different positions in the same orbit?

Let me cut this short: do you think my calculations are right or wrong?

If they are wrong please say what you think is wrong and provide the right answer.

If they are right we are done.

There is no point in arguing with people who keep on going on about 'it is not defined', 'it is too hard', 'define it', 'it depends' and who never bother to write a single formula or do one single calculation, then when you confront them with hard formulas and answer they neither confirm nor deny it is right or wrong, they simply ignore the work and go on as they did before. Perhaps some of those 'high level' folks should take the famous 'shut up and calculate' saying to heart. Implying one knows it all but never showing any work does not impress me at all, on the contrary I fear many of those think they know more than they actually do.

 

[PAllen]

Let me cut this short: do you think my calculations are right or wrong?

If they are wrong please say what you think is wrong and provide the right answer.

If they are right we are done.

There is no point in arguing with people who keep on going on about 'it is not defined', 'it is too hard', 'define it', 'it depends' and who never bother to write a single formula or do one single calculation, then when you confront them with hard formulas and answer they neither confirm nor deny it is right or wrong, they simply ignore the work and go on as they did before. Perhaps some of those 'high level' folks should take the famous 'shut up and calculate' saying to heart.

Definition comes before calculation. I have given a set of requirements for a definition of distance from 2R to 3R, without using light (since we want to independently measure lightspeed), and in reference to specific world lines at 2R and 3R. I freely admit I have no idea how meet those requirements. Until that is achieved, there is nothing to calculate.

You give a calculation, but no description of how it relates to the problem (at least the specific problem I've posed, which seems like a good way to make the original post speculation concrete). I assume your calculation is a correct calculation of 'something'. I have seen no evidence that 'something' has anything to do with the problem I've posed.

 

[member 11137]

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?

Within the generalized theory of relativity (GTR), the speed of light is depending on position and on speed of the observer. This has been proved with an accuracy from about 2 per cent in 1966-1968 (radar signals). At the surface of the Earth the variation is about 2 cm per second (to be compared with c = 30 000 000 000 cm persecond).

 

[Passionflower]

The OP naively imagined that the question had a definite answer without regard to any specific coordinate system. The helpful way to answer the OP's question was to point out that the question was ill-defined.

That is not naive at all, in fact the poster is totally correct.

Below is a plot of light speeds between pairs of static observers (o1, o2) separated a fixed ruler distance of 1 with the radar distance as measured by a clock at observer o1. In the plot you can see the ruler distance (which is 1 for each pair) divided by the radar distance, this ratio is larger for pairs closer to the EH. This is a coordinate free plot as only the ruler distance and proper time is used.

[PLAIN]http://img169.imageshack.us/img169/4331/slowdownoflight.gif

On the x-axis 'd' is the ruler distance to the EH while the y-axis is 1 divided by the radar distance from 'd' to 'd+1' as measured by a clock at a ruler distance 'd' to the EH.

As you can see it is crystal clear that light slows down the closer you get to the EH.

So you are wrong, but I suspect you will simply ignore this graph or this posting or claim I lack understanding.

 

[JesseM]

That is not naive at all, in fact the poster is totally correct.

Below is a plot of light speeds between pairs of static observers (o1, o2) separated a fixed ruler distance of 1 as measured by a clock at observer o1. In the plot you can see the ruler distance (which is 1 for each pair) divided by the radar distance, this ratio is larger for pairs closer to the EH. This is a coordinate free plot as only the ruler distance and proper time is used.

I don't think it's really right to say that using rulers and proper times makes something "coordinate free". After all, inertial coordinate systems are typically defined in terms of a set of rulers and synchronized clocks...anything that allows you to assign a position and a time to any arbitrary event can be treated as defining a coordinate system. (and if two people describe definitions of distances and times using the exact same system of rulers and clocks, but the first uses the words says that the readings on the rulers and clocks define the 'coordinates' of events while the second just talks about 'distances' and 'times', then surely this mere difference in wording does not mean the first is giving a coordinate-dependent description of events while the second is giving a coordinate-independent description)

 

[Passionflower]

I don't think it's really right to say that using rulers and proper times makes something "coordinate free". After all, inertial coordinate systems are typically defined in terms of a set of rulers and synchronized clocks...anything that allows you to assign a position and a time to any arbitrary event can be treated as defining a coordinate system. (and if two people describe definitions of distances and times using the exact same system of rulers and clocks, but the first uses the words says that the readings on the rulers and clocks define the 'coordinates' of events while the second just talks about 'distances' and 'times', then surely this mere difference in wording does not mean the first is giving a coordinate-dependent description of events while the second is giving a coordinate-independent description)

So you are claiming that ruler distance and radar distance as measured by a clock for static observers depend on the chosen coordinate chart in a Schwarzschild solution?

This is getting weirder by the minute. Anything goes to 'prove' me wrong I suppose.

 

[JesseM]

So you are claiming that ruler distance and radar distance as measured by a clock for static observers depend on the chosen coordinate chart in a Schwarzschild solution?

No, I just say that any definition of distances and times in terms of rulers and clocks can itself be taken as a definition of a coordinate system; in fact, the only physical way to define any coordinate system is in terms of some set of physical rulers (or other physical distance measures like radar distance) and physical clocks. Do you agree that textbooks (following the example of Einstein's original 1905 paper) typically define inertial coordinate systems in such a physical way too, by picturing a network of rigid rulers and synchronized clocks like the one illustrated here? If so, would you say this somehow means that statements about "velocity" in such a system of rulers and clocks are "coordinate independent"? I would say the only coordinate-independent statements about such a system are purely local ones like "object A was next to marking x=5 on the ruler when the clock at that marking read t=13", any attempt to turn such local facts into statements about distances or velocities means you are making coordinate-dependent statements.

Anything goes to 'prove' me wrong I suppose.

Why do you have to make things personal like this? I would have said the same if someone else was making your argument about coordinate-independence, and it's not like there haven't been plenty of times I've agreed with you about stuff. This is a discussion board, everyone's opinions get challenged periodically, no need to make bitter remarks about it and create bad feelings for no good reason.

 

[PAllen]

I've seen a number of derivations of the idea that an observer away from an event horizon 'sees light as going slower' closer to the event horizon. I would guess that, properly defined, there is little dispute about this (e.g. MTW and Sean Carroll both have such derivations). I think it is also accepted that any sufficiently local measurement of lightspeed will be c (almost all posters here, plus a couple of GR texts say this). The seemingly open question here is whether there is a physically meaningful, preferred, way to talk about a non-local measurement of lightspeed in the radial direction by a (non-inertial) static observer at some fixed position above the event horizon. If there is well defined answer to this, it would not surprise me that this comes out different from c.

I have another thought on this. Ich has described of a 'maximally minkowski frame' that can be defined for highly non-inertial observers. He said, for example, that for a uniformly accelerting observer in SR it ends up with the Rindler coordinates and metric. It was based on building coordinates out from a 'prime observer' world line. As shown by the Rindler case, 'paths of simultaneity' are not necessarily (spacelike) 4-geodesics of the spacetime; and a simultaneity hypersurface is not necessarily locally Euclidean. Just as with the SR case, the existence of Euclidean hypersurfaces in the Schwarzschild metric has nothing to do with the way this procdedure would choose hypersurfaces of simultaneity in reference to a chosen prime observer.

Perhaps such a frame provides a meaningful way to talk about a static observer near the event horizon making a non-local measurement c in the radial direction (away and back from the center). I recall Ich's description did involve light signals, but perhaps it isn't circular because he only used it define 'comoving' chains of observers that show no red/blue shift relative to each other. Hopefully, there is no hidden speed assumption.

(I have no idea how to perform these operations, or what the results would be).

 

[Passionflower]

Why do you have to make things personal like this? I would have said the same if someone else was making your argument about coordinate-independence, and it's not like there haven't been plenty of times I've agreed with you about stuff. This is a discussion board, everyone's opinions get challenged periodically, no need to make bitter remarks about it and create bad feelings for no good reason.

I apologize JesseM.

There are a few individuals (not you) who pretend to know everything about relativity and at the same time have the urge of telling others how little they understand. They never show a formula or do a calculation, when their statements are challenged and supported by mathematics and graphs they simply ignore those challenges. That is very frustrating at times.

The seemingly open question here is whether there is a physically meaningful, preferred, way to talk about a non-local measurement of lightspeed in the radial direction by a (non-inertial) static observer at some fixed position above the event horizon. If there is well defined answer to this, it would not surprise me that this comes out different from c.

So just one posting before you posted this I showed a graph plotting the ruler distances between two observers divided by the radar time as measured by a clock at the observer closest to the EH for different distance. Clearly you can see that light takes more time for pairs closer to the EH. But you simply seem to re-raise the same question as if this posting never happened? Am I wrong in this?

Out of curiosity do you understand the plot and did you make any conclusions yourself? Or perhaps you disagree with the result, perhaps you think I miscalculated something? Or did you decide to simply ignore what I wrote?

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.

Good points, so I showed my work here: https://www.physicsforums.com/showpost.php?p=2980825&postcount=17.

Now since you asked for them it would be very polite to comment on them right? And if you think that there is anything wrong with it back it up with mathematics? Does that sound reasonable to you?

 

[PAllen]

So just one posting before you posted this I showed a graph plotting the ruler distance between two observers divided by the radar time as measured by a clock at the observer closest to the EH for different distance. Clearly you can see that light takes more time for pairs closer to the EH. But you raise the question as if this posting never happened.

Out of curiosity do you understand the plot and did you make any conclusions yourself? Or perhaps you disagree with the result, perhaps you think I miscalculated something? Or did you decide to simply ignore what I wrote?

I see the plot but don't know it's meaning because I haven't seen definitions of key terms like ruler distance.

I am not the only one who seems unsure of the physical applicability of this graph to a reasonable measurement process that might be used by a static observer near an event horizon.
-

l
Good points, so I showed my work here: https://www.physicsforums.com/showpost.php?p=2980825&postcount=17.

Now since you asked for them it would be very polite to comment on them right?
So you think the calculations are they right or wrong?

Here you simply give a formula for ruler distance, without defining its basis. A simple question I've asked and never gotten any answer to is how this ruler distance is defined. Further, I have made a claim which you may or may not agree with (others here have agreed with it): to define a 'ruler' you must define a spacelike path considered to be 'simultaneous' by some observer; then the interval along this path may be taken to be its length. I have asked whether you agreed with this and, if so, what is the path and rationale for it in your ruler distance formula (I believe Pervect also asked this).

 

[Passionflower]

Here you simply give a formula for ruler distance, without defining its basis.

You wrote you are very familiar with GR but now you say you do not know how to obtain the ruler distance between two stationary observers at two different r coordinates in a Schwarzschild solution.

This is the formula:
<br /> \sqrt {r_{{2}} \left( r_{{2}}-2\,M \right) }-\sqrt {r_{{1}} \left( r_{<br /> {1}}-2\,M \right) }+2\,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{<br /> {2}}-2\,M}}{\sqrt {r_{{1}}}+\sqrt {r_{{1}}-2\,M}}} \right) <br />

You obtain this formula by integration of:

<br /> \left({1-{\frac {2M}{r}}\right)^{-1/2}<br />

I and others went over this many times in several postings on this forum, did you read any of these?

 

[PAllen]

You wrote you are very familiar with GR but now you say you do not know how to obtain the ruler distance between two stationary observers at two different r coordinates in a Schwarzschild solution.

This is the formula:
<br /> \sqrt {r_{{2}} \left( r_{{2}}-2\,M \right) }-\sqrt {r_{{1}} \left( r_{<br /> {1}}-2\,M \right) }+2\,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{<br /> {2}}-2\,M}}{\sqrt {r_{{1}}}+\sqrt {r_{{1}}-2\,M}}} \right) <br />

You obtain this formula by integration of:

<br /> \left({1-{\frac {2M}{r}}\right)^{-1/2}<br />

I and others went over this many times in several postings on this forum, did you read any of these?

I have never claimed to be any kind of expert in GR. I have, in a few posts, described my background (which has peaks and valley's of understanding, and rusty computational skills, and no access to math and graphing software). I see no problem in these forums posing questions that may be fuzzy and that I don't know how to solve. People may discuss/answer/ignore as they see fit.

This definition doesn't relate to what I am asking. It looks like it is just using simultaneity as defined by the Schwarzschild t coordinate (could be wrong here, tell me if so). I am asking for a spacelike simultaneity path from the world line of a static oserver at e.g. 2R to the world line of a static observer at 3R, that comes from application of some reasonable operational definition of simultaneity applied by the non-inertial observer at 2R . The complication being that since we want to independently measure light speed, it is at least dicey to use a light based definition of simultaneity. Anything to do with Schwartzschild t coordinate does not seem relevant to this.

 

[Passionflower]

Well I gave you the formula for the ruler distance, if you decide to ignore it then so be it.

Perhaps some of the people who you seem to think so highly of can give you the formula for whatever it is you are looking for but I would not hold my breath. Why don't you ask them to write down the formula? It would be fun for me to see what they will come up with.

They will either admit that the formula I gave is the correct one or circumvent admitting this by acknowledging your statement without backing it up with any mathematics or formulas instead they will tell you it is all very difficult or so. And if all that fails they perhaps will give you some vague formula in tensorial form, that will shut up 99.9% of the respondents here and then they can keep claiming they know it all and others simply don't understand.

Yes why don't you do that ask for this formula?

 

[Mentz114]

It seems to me that the ruler-distance formula given by PassionFlower
<br /> s = \int \sqrt{g_{rr}}dr<br />
can be interpreted in two ways

1. an observer using an infinitesimal ruler placed end-to-end ( following what path ?)
2. a number of peaks in a monchromatic light ray traveling between the two shell observers.
The ruler's proper length or the wavelength measured in the starting frame are used in calibration.

Are these the same thing ?

(PassionFlower, please don't exercised about this, I also use this formula, but it seems good time to analyse what it means operationally).

 

[PAllen]

Well I gave you the formula for the ruler distance, if you decide to ignore it then so be it.

Perhaps some of the people who you seem to think so highly of can give you the formula for whatever it is you are looking for but I would not hold my breath. I suspect they will acknowledge your statement without backing it up with any mathematics or formulas instead they will tell you it is all very difficult or so. And if all that fails they perhaps will give you some vague formula in tensorial form, that will shut up 99.9% of the respondents here and then they can keep claiming they know it all and others simply don't understand.

Let me ask you this:

Consider 3 observers momentarily adjacent (but with different relative velocities) at 2R. One is free falling, the other is orbiting (unstably, since we 2R is well inside of the closest stable orbit), the last is head static by a rocket. Do you think they will agree on simultaneity? Do you think there is any reason any of their perceptions of simultaneity will be the same as the Schwarzschild t coordinate? I'm pretty sure the answer to these questions are no and no. In which case, each would have a different concept of ruler distance.

 

[PAllen]

It seems to me that the ruler-distance formula given by PassionFlower
<br /> s = \int \sqrt{g_{rr}}dr<br />
can be interpreted in two ways

1. an observer using an infinitesimal ruler placed end-to-end ( following what path ?)
2. a number of peaks in a monchromatic light ray traveling between the two shell observers.
The ruler's proper length or the wavelength measured in the starting frame are used in calibration.

Are these the same thing ?

(PassionFlower, please don't exercised about this, I also use this formula, but it seems good time to analyse what it means operationally).

And for whom is it meaningful? See my post #41. It seems to me any observers with different perceptions of simultaneity will want to use different ruler defintions. What is a ruler but a spacelike path perceived as simultaneous by *some observer*.

 

[Passionflower]

Let me ask you this:

Consider 3 observers momentarily adjacent (but with different relative velocities) at 2R. One is free falling, the other is orbiting (unstably, since we 2R is well inside of the closest stable orbit), the last is head static by a rocket. Do you think they will agree on simultaneity? Do you think there is any reason any of their perceptions of simultaneity will be the same as the Schwarzschild t coordinate? I'm pretty sure the answer to these questions are no and no. In which case, each would have a different concept of ruler distance.

First of all the problem statement you gave concerned stationary observers now you want to talk about observer who are not stationary. Would you agree that it is a lot better to get the first problem resolved first without complicating the matter by introducing non stationary observers? As discussed a few weeks ago on this forum there are several notions of distance for non stationary observers in a Schwarzschild solution.

 

[Mentz114]

And for whom is it meaningful? See my post #41. It seems to me any observers with different perceptions of simultaneity will want to use different ruler defintions. What is a ruler but a spacelike path perceived as simultaneous by *some observer*.

Seems obvious 'for whom it is meaningful'. Both definitions are counts of events along a worldline and therefore agreed by all observers.

An observer at any given time only has one 'perception of simultaneity'.

"What is a ruler but a spacelike path perceived as simultaneous by *some observer*".

That's what you'd like it to be. So give a rigorous definition before you wear out your wrists with all the handwaving.

 

[bcrowell]

Below is a plot of light speeds between pairs of static observers (o1, o2) separated a fixed ruler distance of 1 with the radar distance as measured by a clock at observer o1. In the plot you can see the ruler distance (which is 1 for each pair) divided by the radar distance, this ratio is larger for pairs closer to the EH. This is a coordinate free plot as only the ruler distance and proper time is used.

The proper time and the thing you're calling ruler distance are both coordinates.

 

[PAllen]

First of all the problem statement you gave concerned stationary observers now you want to talk about observer who are not stationary. Would you agree that it is a lot better to get the first problem resolved first without complicating the matter by introducing non stationary observers? As discussed a few weeks ago on this forum there are several notions of distance for non stationary observers in a Schwarzschild solution.

The only reason I brought up other observers was to accentuate the issue of observer dependence of simultaneity. I am only intersted in the static observer. However, do you know of any reason this non-inertial observer will perceive of Schwarzschild t coordinate as their simultaneity? It seems implausible to me, but I could be convinced by a demonstration or argument.

 

[PAllen]

Seems obvious 'for whom it is meaningful'. Both definitions are counts of events along a worldline and therefore agreed by all observers.

An observer at any given time only has one 'perception of simultaneity'.

"What is a ruler but a spacelike path perceived as simultaneous by *some observer*".

That's what you'd like it to be. So give a rigorous definition before you wear out your wrists with all the handwaving.

Everyone may agree this is the invariant length of this world line (actually, it isn't a world line, as it is a spacelike path along r at fixed t). However, observers with different states of motion will disagree on whether it is functionally a ruler.

Consider SR. For one observer, the path (t,x) : (0,0),(0,1) is a ruler. For a different observer, the invariant length of this path won't change, but they will not remotely view it as a ruler because its ends are not simultaneous. So I am wondering why an arbitrary (non-inertial) static observer sees a path defined by constant Schwarzschild t as meaningful ruler. I don't know that it is not, but I haven't seen the question addressed.

 

[Mentz114]

Everyone may agree this is the invariant length of this world line. However, observers with different states of motion will disagree on whether it is functionally a ruler.

I agree, but a ruler is not required to have global significance, it is defined by some observer.

path defined by constant Schwarzschild t as meaningful ruler. I don't know that it is not, but I haven't seen the question addressed.

Am I right that we can find a observer where t and \tau coincide ? For such an observer this objection would disappear.

Maybe this ruler is a chimera. There seems to be no satisfactory operational definition on a large scale.
It's perhaps not surprising there's disagreement over the finer points.

 

[Passionflower]

I am worried at attempts to move the goalposts.

Are we still talking about the distance between two stationary observers in a Schwarzschild solution where the EH=R and the observers are located at 2R and 3R in Schwarzschild coordinates?

I gave a description on how to calculate such a distance, it has been met by non-responses and 'This definition doesn't relate to what I am asking'.

That is where we are at right?

 

[PAllen]

I am worried at attempts to move the goalposts.

Are we still talking about the distance between two stationary observers in a Schwarzschild solution where the EH=R and the observers are located at 2R and 3R in Schwarzschild coordinates?

I gave a description on how to calculate such a distance, it has been met by non-responses and 'This definition doesn't relate to what I am asking'.

That is where we are at right?

Yes. The fact that the observer is static doesn't address what their reasonable definition of simultaneity is. If we weren't trying to set up a framework for non-circular measurement of speed of light, I would trivially define simultaneity of different events for the static observer using the standard radar convention. But for the current purpose it seems absurd - if we use light to set up a coordinate system, how do we independently measure the speed of light?

 

[yuiop]

I've seen a number of derivations of the idea that an observer away from an event horizon 'sees light as going slower' closer to the event horizon. I would guess that, properly defined, there is little dispute about this (e.g. MTW and Sean Carroll both have such derivations). I think it is also accepted that any sufficiently local measurement of lightspeed will be c (almost all posters here, plus a couple of GR texts say this). The seemingly open question here is whether there is a physically meaningful, preferred, way to talk about a non-local measurement of lightspeed in the radial direction by a (non-inertial) static observer at some fixed position above the event horizon. If there is well defined answer to this, it would not surprise me that this comes out different from c.

In Schwarzschild coordinates the Schwarzschild observer "at infinity" claims the vertical speed of light is c*(1-2m/r) while a stationary local observer at r claims the speed of light is simply c. Now while we are generally used to observers in relativity having different points of view of the same set of events it is easy to see that the points of view are physically and conceptually in contradiction at the event horizon. The local measurement of the speed of light is normally taken to the physically "real" measurement and this implies the speed of light is c at the event horizon (but we assume that it not possible to have a stationary local observer exactly the event horizon) while the measurement of the speed of light by the Schwarzschild observer at infinity implies that the speed of light at the event horizon is exactly zero. The observation by the observer at infinity implies light cannot pass through the event horizon while the observation by the local observer concludes that the light passes through the event horizon without any difficulty. These are physically different conclusions and one must be "right" and the other must be "wrong", but which? Normally it is concluded that the conclusions of the observer at infinity are the "wrong" conclusions because they are the conclusions of a distant observer who just a "bookeeper" (and no one like accountants, right? ) and coordinate measurements are just an abstraction without physical meaning. The event horizon is said to be a "coordinate horizon" without physical significance and the coordinate speed in Schwarzschild coordinates is just as arbitrary as plotting the velocity of vehicles on the surface of the Earth in terms of degrees latitude or longitude per hour.

Yes. The fact that the observer is static doesn't address what their reasonable definition of simultaneity is. If we weren't trying to set up a framework for non-circular measurement of speed of light, I would trivially define simultaneity of different events for the static observer using the standard radar convention. I doubt that would match coordinate t. But for the current purpose it seems absurd - if we use light to set up a coordinate system, how do we independently measure the speed of light?

To try and shed some light on this situation, I would like to try and present "a reasonable definition of simultaneity" in the gravitational field of a non rotation gravitational mass that seems to give some reality to coordinate measurements and see how these arguments are countered. First we should consider a "gravitational twin experiment". A pair of twins are located at R1. One sibling is dropped and freefalls to R2. At some later time (say 50 years as measured at R1) the second sibling freefalls to R2 and comes to rest with their twin at R2. The ages of the twins when they are once again alongside each other, differ by an amount that exactly agrees with the gravitational time dilation measured by the Schwarzschild observer at infinity. Therefore we can conclude that the coordinates measurements of the observer at infinity do have physical significance and time really does slow down lower down in the gravitational field and the speed of light really does slow down lower down in the gravitational field (with the implication that light stops at the event horizon).

To set up a "reasonable definition of simultaneity" in this situation, we can speed up the stationary clocks lower down in the field by the gravitational gamma factor and stationary local observers using the coordinated clocks would measure the local speed of light to be slower than in flat space. Before this synchronisation procedure, observers lower down say that clocks higher up appear to be running fast and observers higher up say that clocks lower down appear to be running slow. (Note that the red or blue shift is not reciprocal as in SR). After the synchronisation procedure, observers at any radius agree that clocks at any other radius are running at exactly the same rate. We would now seem to have a reasonable definition of simultaneity, and having defined simultaneity this way, we would seem to able to conclude that light (and any other physical process) really does slow down lower down in a physically meaningful way. Any thoughts?