In relativity, proper time along a world-line is be defined by [itex]d\tau^{2} = ds^{2} / c^{2}[/itex](adsbygoogle = window.adsbygoogle || []).push({});

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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# Derivation of Proper Time in General Relativity

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