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Curved Spacetime and Relative Velocity 
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#1
Aug2010, 05:19 AM

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Is it meaningful to talk of relative velocity between two moving points at a distance in curved spacetime? This interesting issue came up in the course of discussion of the thread "Curved Spacetime and the Speed of Light".I remember Dalespam giving some formidable logic with a very good example[Thread :#25]The content of the thread #19 was also very interesting (and similar in some sense). I would like the to repeat the basic idea that could prevent relative velocity from becoming a valid concept in the framework of general relativity :
To calculate relative velocity we need to subtract one velocity vector from another at a distance.For this we have to bring the vectors to a common point by parallel transport. We could keep one vector fixed[let us call this the first vector]and parallel transport the other [the second vector]to the position of the first vector. Now parallel transport may be performed along several routes. If different routes lead to different directions of the second vector in the final position, relative velocity does not have a unique meaning and becomes mathematically unacceptable.Again I refer to the illustration in thread #25. My queries: 1)If two points(or observers) are moving relative to each other in curved spacetime how should the motion of one point appear to the other physically? Is such observation meaningless from the physical point of view if we are unable to interpret it in the existing framework of mathematics? 2)On Parallel transport: We start with a familiar example on parallel transport: A vector e on a globe at point A on the equator is directed to the north along a line of longitude.We parallel transport the vector first along the line of longitude until we reach the north pole N and then (keeping it parallel to itself) drag it along another meridian to the equator.Then (keeping the direction there) subsequently transport it along the equator(moving the vector paralley) until we return to point A. Then we notice that the paralleltransported vector along a closed circuit does not return as the same vector; instead, it has another orientation. " It is interesting to observe that the route followed in parallel transport [in the above example and the example cited by Dalespam in thread #25]involves sharp bends or joints where derivatives cannot be defined. Incidentally from the mathematical point of view the definition of parallel transport[Wald:page 34] involves derivatives[covariant derivatives to state accurately] and we know very well that derivatives do not exist at sharp junction.Can we entertain the above examples to refute the concept of relative velocity in curved space time, considering the fact that such examples use paths containing sharp junctions? Interestingly one may perform the above examples by " rounding off" the north pole edge and the ambiguity will be removed.One may draw "smooth(closed) curves" on "curved surfaces" and carry out examples of parallel transport. The vectors will coincide in their initial and final positions![If one is to perform an experiment by drawing smooth curves on a basket ball he should take care to move the vector parallely without bothering about what angle it is making with the curve after the motion has been started.Angles should be noted only at the initial and the final points/stages.] If the arguments in point (2) are correct I may conclude that 1)The concept of relative velocity is mathematically consistent in relation to the notion of curved space time. 2)The ideas portrayed in the thread "Curved Spacetime and the Speed of Light" are correct. 


#2
Aug2010, 09:42 AM

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I'm not sure I follow you, so let's consider a simple example. Consider two static observers in Schwarzschild spacetime who both have the [itex]\theta[/itex] and [itex]\phi[/itex] values, but who hover at different values of [itex]r[/itex]. Using the method of parallel transport, what is their relative velocity? I think that this is fairly straightforward to compute, at least for geodesic radial paths.



#3
Aug2010, 10:31 AM

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The thread referred to in #1 is http://www.physicsforums.com/showthread.php?t=408994 Post #25 by DaleSpam is here http://www.physicsforums.com/showpos...0&postcount=25



#4
Aug2010, 10:51 AM

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Curved Spacetime and Relative Velocity



#5
Aug2010, 10:58 AM

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#6
Aug2010, 12:20 PM

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I have been ,for quite some time, trying to explore the possibility of breaking the speed barrier within the "confines of relativity"? Locally we cannot do it. The laws are very strong in this context.The only option would be to explore the matter in a "nonlocal" consideration. If I am standing at some point in curved spacetime and a ray of light is coming from a distant point (close to some dense object)it would be my normal interest to know the speed of light at each and every point as it comes towards me[Of course I continue to stand at the same point]. With this idea in mind I wrote "Curved Spacetime and the Speed of Light".
I am repeating the basic aspects of my considerations in the following calculations: Let us consider two points A and B separated by a large distance with different values of the metric coefficients.Observers at A and B consider a light ray flashing past B. Speed of light at B as observed from A= {Spatial separation at B}/{sqrt{g(00)} at A.dt} Speed of light at B as observed from B ie, c ={Spatial separation at B}/{sqrt{g(00)} at B.dt} [Noting that the speed of light is locally "c"] Speed of light at B as observed from A= c sqrt{(g(00) at B)/(g(00) at A)} The left side of the above relation exceeds the speed of light if g(0,0) at B>g(0,0) at A [spatial separation is the same for both the observers while the temporal separations[physical] are differentthe clocks have different rates at the two points] It is important to note that general relativity seems to avoid "nonlocal considerations" [Please do correct me if I am mistaken] and we can always take advantage of this fact to investigate the speed of light at one point as it is observed from another in case, we can find some interesting result. This exercise does not contradict any law in Special or General Relativity. 


#7
Aug2010, 12:30 PM

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George's reply seems to favor me if I am not incorrect.And I have liked his signature very much.
A relevant point: If I am standing still at one point V=0. What do I get if I transport a null vector? The nature of the curve will not be a big factor. 


#8
Aug2010, 01:13 PM

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#9
Aug2010, 02:16 PM

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http://www.lightandmatter.com/html_b...tml#Section6.2 , subsection 6.2.5. 


#11
Aug2110, 04:52 AM

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"The definition of a geodesic is that it paralleltransports its own tangent vector, so the velocity vector has to stay constant." If a particle moves solely under the influence of gravity it should follow a geodesic and the above definition(which is common to all texts) settles the issue. Now we may think of forced motions where forces other than gravity are operating and the bodies are constrained to move along lines that are not geodesics. [We may think of an aircraft following a line of latitude instead of a great circle and another one which moves along a path which is neither a line of latitude or longitude or a great circle]In such cases one may follow the logic suggested in thread#1 ,Point 2, to make relative velocity unique. 


#12
Aug2110, 05:44 AM

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It is important to note that the spacetime around the earth is approximately flat and the nature of this spacetime should not be confused with the example of the two aircraft I have given or with the spherical shape of the earth. The aircraft example have been given to emphasize that we may have motion along a geodesic (when gravity is the only agent) and we may have a nongeodesic motion if some other force,I mean some inertial force, is operating. If the effect of the inertial force is taken to be similar/equivalent to gravity we may think of adjusting the original metric to take care of the inertial force. At this point I may refer to Thread #8 of the posting "On the Speed of Light Again!" where the equivalence/similarity of gravity and acceleration has been highlighted with reference to papers in the archives of the "Scientific American" and the "Physical Review" as cited by Robert Resnick.
[If the original metric is altered/adjusted the lines of geodesic should change ] 


#13
Aug2110, 06:10 AM

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Many issues in curved geometry can be worked around or defined away, but the nonuniqueness of parallel transport is something we just have to live with. Distant vectors are in different tangent spaces and cannot be compared. PS when refering to posts in other threads a link would be helpful. 


#14
Aug2110, 08:54 AM

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Let us consider the definition of parallel transport as we know in general relativity:
A vector is parallely propagated along a curve if its covariant derivative along the curve vanishes at each point. So if the velocity vector is parallely propagated along a curve "IT CANNOT CHANGE". Simple as that. Conclusions: 1) The notion of Relative velocity is consistent with the mathematics of curved spacetime. 2)My assertions in the posting "Curved Spacetime and the Speed of Light" are correct. 


#15
Aug2110, 09:12 AM

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Can a spacetime surface be exactly spherical?
Let us see. A particle at the south pole sees several geodesics connecting it to the north pole. Which direction is to follow? It will be in a state of indecision.[We are assuming the presence of gravity only] We may have several geodesics emerging from the same point.But they should not terminate on the same and the identical point on the other side. 


#16
Aug2110, 10:18 AM

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#17
Aug2110, 11:27 AM

P: 621

Well, the spacetime structure referred to in thread #15 was never created by the particles themselves,I mean the particles between whom the relative velocity is to be calculated. The mechanism of creation of the spacetime surface has not been described[and possibly cannot be described] . I have simply assumed its existence to prove that it cannot exist. The particles have been used as "test particles" whose fields are of negligible strength.They should not disrupt the existing field or interact between themselves but they should respond to the existing gravitational field created by some "other means".
Now let us consider a pair of gravitating particles separated by a large distance. The lines of force between them are basically parallel lines and the spacetime structure is "not a sphere". The particles if released from a large distance will move along a straight line. The geodesic is simply a straight line and the space you have described is flat spacetime .[the shortest distance being a straight line]. If you kept the two initial particles fixed and released smaller particles midway between them ,they should be moving along straight lines.Of course, the "smaller particles" must be "small enough" not to disrupt the existing field.This in fact would be a better interpretation of the situation. Now in the first paragraph I have used the term "test particles". It is important that you understand them in relation to the study of gravitational fields/spacetime structure. I would refer to the book "Gravity" by James B. Hartley , Chapter 8,"Geodesics" in understanding the concept of "Test Particles" 


#18
Aug2110, 12:10 PM

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