Exact Solutions for General Relativity

[Synctime]

I would like to know if there are exact solutions using General Relativity to determine: 1) time dilation from velocity, and 2) velocity from redshift (arising from kinematic movement). I understand that Special Relativity can handle both of these questions, but the implementation of SR is limited to certain situations. Schwartzschild presented an exact solution for GR to calculate time dilation in response to gravity, and I'm wondering if there is a similar exact solution to these questions.

Confirmation that there are no exact solutions for GR with these questions would also be very helpful.

 

[bcrowell]

Welcome to PF!

Schwartzschild presented an exact solution for GR to calculate time dilation in response to gravity, and I'm wondering if there is a similar exact solution to these questions.

This isn't really right. The Schwartzschild metric doesn't lead to any general conclusions about gravitational time dilation, because it's only one example of a spacetime that has gravity, and you can't prove anything from one example. The exact result for gravitational time dilation follows from the equivalence principle, and the result holds for all static spacetimes, regardless of whether those spacetimes can be represented in the form of, say, a vacuum solution that can be expressed in terms of elementary functions.

I would like to know if there are exact solutions using General Relativity to determine: 1) time dilation from velocity, and 2) velocity from redshift (arising from kinematic movement)

Kinematic time dilation isn't a feature of a spacetime, it's a statement about how the same spacetime looks different when described in two different sets of coordinates. The simplest example is flat spacetime (Minkowski space). In a non-flat spacetime, kinematic time dilation is a purely local thing, but otherwise the story is the same -- it describes changes of coordinates, not the spacetime itself. The same holds true for the relationship between velocity and Doppler shifts of light.

 

[Synctime]

Welcome to PF!


This isn't really right. The Schwartzschild metric doesn't lead to any general conclusions about gravitational time dilation, because it's only one example of a spacetime that has gravity, and you can't prove anything from one example. The exact result for gravitational time dilation follows from the equivalence principle, and the result holds for all static spacetimes, regardless of whether those spacetimes can be represented in the form of, say, a vacuum solution that can be expressed in terms of elementary functions.

Kinematic time dilation isn't a feature of a spacetime, it's a statement about how the same spacetime looks different when described in two different sets of coordinates. The simplest example is flat spacetime (Minkowski space). In a non-flat spacetime, kinematic time dilation is a purely local thing, but otherwise the story is the same -- it describes changes of coordinates, not the spacetime itself. The same holds true for the relationship between velocity and Doppler shifts of light.

Thanks for your answer!

For the first part, I suppose I meant a solution that did not involve tensors, which I understand are difficult to work with to provide an exact solution (i.e., in a given situation, you get a discrete single number solution). Schwartzschild was able to simplify the situation to a non-rotating sphere to provide an exact solution that did not involve tensors. I suppose that all exact solutions will involve a limitation of some sort.

For your second point, do you mean that in local environments, SR (but not GR) will be used to determine relative time dilation in response to velocity and relative velocity from redshift and that GR does not cover such questions because the SR results are a (somewhat illusory) result of viewing events from different local coordinate frames? In this way, would you consider that GR uses an overriding coordinate frame that cannot be viewed from different local frames?

 

[PAllen]

I would like to know if there are exact solutions using General Relativity to determine: 1) time dilation from velocity, and 2) velocity from redshift (arising from kinematic movement). I understand that Special Relativity can handle both of these questions, but the implementation of SR is limited to certain situations. Schwartzschild presented an exact solution for GR to calculate time dilation in response to gravity, and I'm wondering if there is a similar exact solution to these questions.

Confirmation that there are no exact solutions for GR with these questions would also be very helpful.

Welcome to Physics Forums!

These questions have nothing to do with the issue of exact solutions. The differences on these between SR and GR have to do with GR including gravitational and cosmological effects, which imply dynamic and curved spacetime.

Even in SR, I sense some confusion in your questions. Time dilation is a function of velocity in some chosen inertial reference frame. It is thus a frame dependent, derived, quantity that is not directly observable. What is observable and frame independent are things like doppler and differential time passage for specific 'clocks' that compare time a specific way (e.g. separate and get back together; or exchange signals in some specified way). These observables are frame independent. The interpretation into who is dilated how much varies by frame, but always leads to the same observable quantities.

Velocity, even in SR, is determined by doppler only given more information e.g. the specification that motion is either directly towards or away from the receiver (that is, source velocity is colinear with light path). Otherwise, you obviously need additional information to relate doppler to relative speed.

In GR, a fundamental difference is that relative velocity at a distance has no unique definition at all. This has nothing to do with exact vs. inexact solution, it is a direct consequence of curvature. If you want to compare two vectors (e.g. velocity vectors), you have to move one to the other. In flat spacetime, you can do this in a unique, direction preserving manner, the result being independent of the path you move them on. In curved spacetime, the answer is potentially different for every possible path you use to bring the vectors together. There is no 'solution' to this problem - this feature is the definition of curvature. Thus, in GR, relative velocity is strictly a coordinate dependent, purely conventional quantity (unlike SR, where relative velocity between to world lines at two specified events is frame independent).

Be that as it may, direct observables like doppler and differential time flow (using a specified physical procedure) are perfectly computable in GR, for any solution. Further, if you pick a particular coordinate system (e.g. cosmological comoving coordinates), coordinate velocity can be related Doppler in the same manner as for SR, except that you may need more information. For example, to factor an observed doppler for an object moving colinear with its light path as you observer it, into cosmological and peculiar velocity, you would need to somehow know its distance, or know of equidistant, nearby objects which you know are comoving with the CMB. But this is just a deficiency of knowledge, not a deficiency in what is computable in GR.

Please continue asking questions for further clarification.

 

[Synctime]

Be that as it may, direct observables like doppler and differential time flow (using a specified physical procedure) are perfectly computable in GR, for any solution. Further, if you pick a particular coordinate system (e.g. cosmological comoving coordinates), coordinate velocity can be related Doppler in the same manner as for SR, except that you may need more information. For example, to factor an observed doppler for an object moving colinear with its light path as you observer it, into cosmological and peculiar velocity, you would need to somehow know its distance, or know of equidistant, nearby objects which you know are comoving with the CMB. But this is just a deficiency of knowledge, not a deficiency in what is computable in GR.

Thanks for the clarification on relative velocity.

Am I correct that the use of tensors is so complex that only approximations can be achieved normally - hence the distinction that Schwartzschild was the first to provide an exact solution to GR (albeit for a limited situation)? If it is possible to calculate exact solutions for Doppler shift and time passage that do not involve tensors, are there equations for these (which also may be limited to certain situations)?

 

[bcrowell]

Am I correct that the use of tensors is so complex that only approximations can be achieved normally - hence the distinction that Schwartzschild was the first to provide an exact solution to GR (albeit for a limited situation)?

No, this is not true. The reason exact solutions in GR are hard to find is not that the field equations involve tensors, it's that the field equations are nonlinear.

If it is possible to calculate exact solutions for Doppler shift and time passage that do not involve tensors, are there equations for these (which also may be limited to certain situations)?

When you say "exact solutions" to a relativist, it has the connotation of a spacetime that is an solution to the Einstein field equations, expressible in closed form in terms of elementary functions such as sin, cos, and exponentials.

None of this has anything to do with the results for kinematic Doppler shifts and kinematic time dilation, which are simple and exact, and have nothing to do with GR, only SR. You can find the equations on Wikipedia.

 

[pervect]

I would like to know if there are exact solutions using General Relativity to determine: 1) time dilation from velocity, and 2) velocity from redshift (arising from kinematic movement). I understand that Special Relativity can handle both of these questions, but the implementation of SR is limited to certain situations. Schwartzschild presented an exact solution for GR to calculate time dilation in response to gravity, and I'm wondering if there is a similar exact solution to these questions.

Confirmation that there are no exact solutions for GR with these questions would also be very helpful.

There is a fairly simple example in SR that shows that time dilation can be viewed as "gravitational" by one observer, and as "due to velocity" by another.

Thus the idea of time dilation is observer dependent. In other words, there is no absolute time in special relativity. I'm a bit concerned that you may be one of the millions of people asking detailed questions about relativity who have somehow managed to miss this pint on the non-existence of "absolute time".

The basic example is fairly simple - it's just Einstein's elevator. There's a rather short summar at http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

To summarize, you've got an elevator (or sometimes a rocketship), it accelerates, you've got one observer at the head, another at the tail.

The highlights of what's observable:

The round trip path delay of a light signal from tail to nose and back stays constant

Light going from tail to nose is redshifted - if you have a cesium clock at the tail, when light from it reaches the nose it won't be resonant with the cesium atom anymore. This "redshift" is what's referred to as time dilation - if this isn't obvious, perhaps more questions are in order.

Light going from nose to tail is blueshiftd.

Light traveling from tail to nose and back is not frequency shifted at all.

 

[tom.stoer]

What about a few equations?

The proper times of two observers A and B moving along two arbitrary timelike curves C_A,B in a certain metric g are calculated as the invariant length

S[C_{A,B}] = \int_{C_{A,B}} ds = \int_{C_{A,B}} \sqrt{g_{\mu\nu}\,dx^\mu\,dx^\nu}

If we chose the two curves such that they have common start and end points in spacetime, then the difference of the proper times between start and end can be calculated as

\Delta S_{AB} = \int_{C_{A}} ds - \int_{C_{B}} ds

This formula contains effects due to velocity (and acceleration) and all gravitational effects.

 

[Synctime]

What about a few equations?

The proper times of two observers A and B moving along two arbitrary timelike curves C_A,B in a certain metric g are calculated as the invariant length

S[C_{A,B}] = \int_{C_{A,B}} ds = \int_{C_{A,B}} \sqrt{g_{\mu\nu}\,dx^\mu\,dx^\nu}

If we chose the two curves such that they have common start and end points in spacetime, then the difference of the proper times between start and end can be calculated as

\Delta S_{AB} = \int_{C_{A}} ds - \int_{C_{B}} ds

This formula contains effects due to velocity (and acceleration) and all gravitational effects.

Thanks for the response. I don't know GR well enough to understand the equations (and what is embedded in the terms). However, I do have a question about whether these equations are solvable for an exact problem. For example, if you have a defined start and end point and know the velocities for each observer, can you determine the relative time dilations using these equations for both Minkowski and non-Minkowski conditions? And if so, are the values the same as would be calculated using SR under Minkowski conditions?

 

[tom.stoer]

Regarding your last question: the proper times depend on the metric g; so the Minkowski metric from SR is a very special metric; for different metrics you will get different proper times.

Regarding exact solutions: you can solve these equations exactly for not too complicated curves in flat Minkowski spacetime; you can solve them for some special curves in a black hole spacetime i.e. Schwarzschild metric (this is the case for gravitational redshift); you can solve them for not too complicated curves in a cosmological models i.e. expanding spacetime (this is the case for cosmological redshift).

Please note that in all cases you can chose curves which 'mix' effects due to velocity and gravitation.