Calculating Proper Time of "A" in Schwarzschild Geometry

[Nilupa]

"A" starts a journey from a massive body in Schwarzschild geometry in a radial path and returns back to the starting point while "B" stays at rest. Please explain how to find the proper time of "A".

 

[Mentz114]

See this thread which discusses the same problem

https://www.physicsforums.com/showthread.php?t=615472

 

[elfmotat]

Using a (+---) signature, the proper time of a particle is given by:

$$\Delta \tau =\int \sqrt{g_{\mu \nu}\dot{x}^\mu \dot{x}^\nu}d\lambda$$

where $\dot{x}^\mu =dx^\mu /d\lambda$ and $\lambda$ is some affine parameter. You could use, for example, $\lambda=t$ (t is Schwarzschild coordinate time) which would simplify the integral (in Schwarzschild coordinates) to:

$$\Delta \tau =\int \sqrt{g_{tt}+g_{rr}\dot{r}^2}dt$$

because, in this problem, dΩ=0.

 

[Nilupa]

Using a (+---) signature, the proper time of a particle is given by:

$$\Delta \tau =\int \sqrt{g_{\mu \nu}\dot{x}^\mu \dot{x}^\nu}d\lambda$$

where $\dot{x}^\mu =dx^\mu /d\lambda$ and $\lambda$ is some affine parameter. You could use, for example, $\lambda=t$ (t is Schwarzschild coordinate time) which would simplify the integral (in Schwarzschild coordinates) to:

$$\Delta \tau =\int \sqrt{g_{tt}+g_{rr}\dot{r}^2}dt$$

because, in this problem, dΩ=0.

Thank you..
What is dot{r}?
Is it
1/sqrt(1-(2GM/r(c^2)))

 

[Nilupa]

See this thread which discusses the same problem

https://www.physicsforums.com/showthread.php?t=615472

Thank you..

 

[Nilupa]

For the local velocity(dr/dt) , can we derive an expression using time like geodesics in schwarzschild geomerty?

 

[tom.stoer]

\dot{r} is dr/dt where r and t are the space- and time-coordinates in Schwarzschild geometry; and I think the expression posted by elfmotat should answer your last question

 

[stevendaryl]

Using a (+---) signature, the proper time of a particle is given by:

$$\Delta \tau =\int \sqrt{g_{\mu \nu}\dot{x}^\mu \dot{x}^\nu}d\lambda$$

where $\dot{x}^\mu =dx^\mu /d\lambda$ and $\lambda$ is some affine parameter.

Actually, I think that form for the proper time works for any parametrization, not just affine parametrizations. If the parameter is not affine, then the geodesic equation (resulting from maximizing the proper time) is much more complicated.

 

[Nilupa]

I'm searching for a expression for the local velocity (dr/dt), for a twin who travels vertically from the surface of a massive body in the schwarzschild geometry using time like geodesics.

 

[tom.stoer]

dr/dt as a function of t *is* what you are looking for; now you can play around with arbitrary motion, not just geodesics.

One twin stays at r°=r(t=0)=const., the other one moves with r(t). Do you relly want to use geodesics, i.e. solutions of the e.o.m.? I don't think that this is realistic b/c in order to follow a geodesic the second twin must start at r° with non-zero velocity therefore the initial conditions for the two twins do not coincide.

But if you really want to do that you may find the geodesics in Schwarzsschild coordinates in many GR textbook.