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The speed of light in a gravitational field 
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#1
Nov1010, 02:26 PM

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Does a photon gain speed if it travels in "freefall" toward the earth? A particle of mass gains speed as it freefalls to the earth, and yet, mass is just another form of energy. Light is energy. So, does light gain speed?



#2
Nov1010, 04:14 PM

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. 


#3
Nov1010, 04:31 PM

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.
[tex] 3^{1/2} [/tex] 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 [tex] 2^{1/2} [/tex] 


#4
Nov1010, 09:08 PM

P: 161

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 earthobserver's frame, the observer would measure the photon takes less time to reach the earth for the premeasured distance traveled by the photon because the clock at the earth's surface would tick slower nonlocally 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 earthobserver'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? 


#5
Nov1010, 09:33 PM

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#6
Nov1010, 10:02 PM

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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. 


#7
Nov1110, 08:51 AM

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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. 


#8
Nov1110, 09:48 AM

P: 1,555

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. 


#9
Nov1110, 10:17 AM

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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: [tex] \frac{d\mathbf{p}}{dt} = \frac{E}{c^2} \, \mathbf{g} \, \left ( 1 + \frac{v^2}{c^2}} \right ) [/tex] 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. 


#10
Nov1110, 10:29 AM

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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. 


#11
Nov1110, 05:31 PM

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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* noninertial 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. 


#12
Nov1110, 06:40 PM

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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. 


#13
Nov1110, 07:03 PM

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Then, you can use a colocated and instantaneously comoving inertial frame to measure the speed of light, because you've limited yourself to a region of spacetime 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 oneway 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. 


#14
Nov1110, 07:09 PM

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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. 


#15
Nov1110, 07:11 PM

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I also explicitly said it wasn't exactly Rindler, just similar until tidal effects come into play. 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). 


#16
Nov1110, 07:32 PM

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#17
Nov1110, 07:57 PM

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Then the ruler distance between them is: [tex] \rho = R \left( \sqrt {3}\sqrt {2}\sqrt {2}+\ln \left( \sqrt {3}+\sqrt {3} \sqrt {2}\sqrt {2}+2 \right) \right) [/tex] Now the radar distance T in coordinate time between them is: [tex] T = R+R\ln \left( 2 \right) [/tex] The radar distance in proper time for R2 is: [tex] \tau_{R2} = 1/2\, \left( R+R\ln \left( 2 \right) \right) \sqrt {2} [/tex] And for R3: [tex] \tau_{R3} = 1/3\, \left( R+R\ln \left( 2 \right) \right) \sqrt {6} [/tex] 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. 


#18
Nov1110, 08:14 PM

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[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 3metric on some spatial hypersurface for the noninertial observer at 2R. 


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