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Nilupa
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"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".
elfmotat said:Using a (+---) signature, the proper time of a particle is given by:
[tex]\Delta \tau =\int \sqrt{g_{\mu \nu}\dot{x}^\mu \dot{x}^\nu}d\lambda [/tex]
where [itex]\dot{x}^\mu =dx^\mu /d\lambda[/itex] and [itex]\lambda[/itex] is some affine parameter. You could use, for example, [itex]\lambda=t[/itex] (t is Schwarzschild coordinate time) which would simplify the integral (in Schwarzschild coordinates) to:
[tex]\Delta \tau =\int \sqrt{g_{tt}+g_{rr}\dot{r}^2}dt[/tex]
because, in this problem, dΩ=0.
Mentz114 said:See this thread which discusses the same problem
https://www.physicsforums.com/showthread.php?t=615472
elfmotat said:Using a (+---) signature, the proper time of a particle is given by:
[tex]\Delta \tau =\int \sqrt{g_{\mu \nu}\dot{x}^\mu \dot{x}^\nu}d\lambda [/tex]
where [itex]\dot{x}^\mu =dx^\mu /d\lambda[/itex] and [itex]\lambda[/itex] is some affine parameter.
Proper time is the time measured by a clock that is stationary with respect to a given point in space. It is the time experienced by an observer who is at rest relative to the gravitational field.
Proper time is calculated using the Schwarzschild metric, which is a mathematical formula that describes the curvature of spacetime around a massive object. The formula takes into account the mass and radius of the object, as well as the distance from the object. By plugging in these values, the proper time can be calculated.
Proper time is significant because it is a measure of the actual time experienced by an observer in the presence of a massive object. It is also used to calculate important physical quantities such as the time dilation effect and the gravitational redshift.
Proper time and coordinate time are two different ways of measuring time in Schwarzschild geometry. Proper time is the actual time experienced by an observer, while coordinate time is a mathematical construct used to describe the behavior of objects in a curved spacetime. Proper time is always measured in seconds, while coordinate time can have different units depending on the reference frame.
Yes, an observer outside the event horizon can measure proper time. However, as the observer gets closer to the event horizon, the effects of gravitational time dilation become more significant, causing the proper time to slow down. At the event horizon, proper time effectively stops, and an observer would need infinite time to reach the event horizon.