Editing Proper time (section)

===In general relativity===

Proper time is defined in [[general relativity]] as follows: Given a [[pseudo-Riemannian manifold]] with a local coordinates {{math|''x''<sup>''μ''</sup>}} and equipped with a [[metric tensor (general relativity)|metric tensor]] {{math|''g''<sub>''μν''</sub>}}, the proper time interval {{math|Δ''τ''}} between two events along a timelike path {{mvar|P}} is given by the [[line integral]]<ref>{{harvnb|Foster|Nightingale|1978|p=57}}</ref>

{{NumBlk|:|<math>\Delta\tau = \int_P \, d\tau = \int_P \frac{1}{c}\sqrt{g_{\mu\nu} \; dx^\mu \; dx^\nu}.</math>|{{EquationRef|(4)}}|RawN=.}}

This expression is, as it should be, invariant under coordinate changes. It reduces (in appropriate coordinates) to the expression of special relativity in [[flat spacetime]].

In the same way that coordinates can be chosen such that {{math|1=''x''<sup>1</sup>, ''x''<sup>2</sup>, ''x''<sup>3</sup> = const}} in special relativity, this can be done in general relativity too. Then, in these coordinates,<ref>{{harvnb|Landau|Lifshitz|1975|p=251}}</ref>
<math display="block">\Delta\tau = \int_P d\tau = \int_P \frac{1}{c}\sqrt{g_{00}} dx^0.</math>

This expression generalizes definition {{EquationNote|(2)}} and can be taken as the definition. Then using invariance of the interval, equation {{EquationNote|(4)}} follows from it in the same way {{EquationNote|(3)}} follows from {{EquationNote|(2)}}, except that here arbitrary coordinate changes are allowed.