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paultsui
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In relativity, proper time along a world-line is be defined by [itex]d\tau^{2} = ds^{2} / c^{2}[/itex]
However, proper time can also be understood as the time lapsed by an observer who carries a clock along the world-line.
In special relativity, this can easily be proven:
The line element in special relativity is given by [itex]ds^{2} = (cdt)^{2} - dx^{2} - dy^{2} - dz^{2}[/itex], therefore in a frame that moves along the world line, we have [itex]dx^{2} = dy^{2} = dz^{2} = 0[/itex], giving us [itex]ds^{2} = (cd\tau)^{2}[/itex]
Tn general relativity, things seem to be a little tricker because of the metric element [itex]g_{tt}[/itex]. Repeating the derivation ends up with [itex]ds^{2} = g_{tt}(cd\tau)^{2}[/itex] instead.
I found a proof here: http://arxiv.org/pdf/gr-qc/0005039v3.pdf
However, on p.2, the author states that [itex]g_{t't'}[/itex] can always be chosen as 1, hence completing the proof. This baffles me as I always think that [itex]g_{t't'}[/itex] is defined by the geometry of space-time, which cannot be chosen arbitrarily.
Can anyone give me a hint on where my logic go wrong?
Thank you!
However, proper time can also be understood as the time lapsed by an observer who carries a clock along the world-line.
In special relativity, this can easily be proven:
The line element in special relativity is given by [itex]ds^{2} = (cdt)^{2} - dx^{2} - dy^{2} - dz^{2}[/itex], therefore in a frame that moves along the world line, we have [itex]dx^{2} = dy^{2} = dz^{2} = 0[/itex], giving us [itex]ds^{2} = (cd\tau)^{2}[/itex]
Tn general relativity, things seem to be a little tricker because of the metric element [itex]g_{tt}[/itex]. Repeating the derivation ends up with [itex]ds^{2} = g_{tt}(cd\tau)^{2}[/itex] instead.
I found a proof here: http://arxiv.org/pdf/gr-qc/0005039v3.pdf
However, on p.2, the author states that [itex]g_{t't'}[/itex] can always be chosen as 1, hence completing the proof. This baffles me as I always think that [itex]g_{t't'}[/itex] is defined by the geometry of space-time, which cannot be chosen arbitrarily.
Can anyone give me a hint on where my logic go wrong?
Thank you!
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