- #1
npnacho
- 16
- 0
hi everyone. I'm having trouble understanding the concept of proper time in general relativity.
suppose we have some metric given by a fixed mass distribution, say schwarzschild or something (it's not important) and a test particle go over some path between two events A and B.
if we want to compute the proper time measured by this particle in its travel, we have to consider the metric due its acceleration besides the metric given by the external gravitational field, right? is this doable? how would do you calculate the resulting metric in a problem like this?
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i think i also have problems with the idea of coordinate time... I'm looking at a derivation of the gravitational redshift and it's something like this: suppose you have an uniform gravitational field in the z direction and two observers A and B with positions zA and zB respectively. if a electromagnetic wave travels from A to B, the periods of the wave measured by them are related by:
[itex]\frac{T_{A}}{T_{B}]}=\frac{\Delta\tau_{A}}{\Delta\tau_{B}}=\frac{\sqrt{g_{00}(z_{A})}}{ \sqrt{g_{00}(z_{B})}}\cdot\frac{\Delta t_{A}}{\Delta t_{B}}=\frac{\sqrt{g_{00}(z_{A})}}{\sqrt{g_{00}(z_{B})}}[/itex]
(T goes for period, tau goes for proper time and g00 is the obvious component of the metric tensor)
i can't get the last step, they say that the periods in t (the coordinate time) are the same for both observers so [itex]\frac{\Delta t_{A}}{\Delta t_{B}}=1[/itex]. i could understand it in the framework of special relativity where both observers are at rest with respect to each other and the coordinate time becomes the proper time for both (and, obviously, as the metric remains unchanged, you don't get any redshift), but here? what is the frame that measures that coordinate time and why is the same for both observers?
i'm sorry if my english sounds a little funny, I'm not used to write it.
thanks!
suppose we have some metric given by a fixed mass distribution, say schwarzschild or something (it's not important) and a test particle go over some path between two events A and B.
if we want to compute the proper time measured by this particle in its travel, we have to consider the metric due its acceleration besides the metric given by the external gravitational field, right? is this doable? how would do you calculate the resulting metric in a problem like this?
---------
i think i also have problems with the idea of coordinate time... I'm looking at a derivation of the gravitational redshift and it's something like this: suppose you have an uniform gravitational field in the z direction and two observers A and B with positions zA and zB respectively. if a electromagnetic wave travels from A to B, the periods of the wave measured by them are related by:
[itex]\frac{T_{A}}{T_{B}]}=\frac{\Delta\tau_{A}}{\Delta\tau_{B}}=\frac{\sqrt{g_{00}(z_{A})}}{ \sqrt{g_{00}(z_{B})}}\cdot\frac{\Delta t_{A}}{\Delta t_{B}}=\frac{\sqrt{g_{00}(z_{A})}}{\sqrt{g_{00}(z_{B})}}[/itex]
(T goes for period, tau goes for proper time and g00 is the obvious component of the metric tensor)
i can't get the last step, they say that the periods in t (the coordinate time) are the same for both observers so [itex]\frac{\Delta t_{A}}{\Delta t_{B}}=1[/itex]. i could understand it in the framework of special relativity where both observers are at rest with respect to each other and the coordinate time becomes the proper time for both (and, obviously, as the metric remains unchanged, you don't get any redshift), but here? what is the frame that measures that coordinate time and why is the same for both observers?
i'm sorry if my english sounds a little funny, I'm not used to write it.
thanks!