Editing Proper time (section)

== Mathematical formalism ==

The formal definition of proper time involves describing the path through [[spacetime]] that represents a clock, observer, or test particle, and the [[metric tensor (general relativity)|metric structure]] of that spacetime. Proper time is the [[pseudo-Riemannian manifold|pseudo-Riemannian]] arc length of [[world line]]s in four-dimensional spacetime. From the mathematical point of view, coordinate time is assumed to be predefined and an expression for proper time as a function of coordinate time is required. On the other hand, proper time is measured experimentally and coordinate time is calculated from the proper time of inertial clocks.

Proper time can only be defined for timelike paths through spacetime which allow for the construction of an accompanying set of physical rulers and clocks. The same formalism for spacelike paths leads to a measurement of [[proper distance]] rather than proper time. For lightlike paths, there exists no concept of proper time and it is undefined as the spacetime interval is zero. Instead, an arbitrary and physically irrelevant [[geodesics|affine parameter]] unrelated to time must be introduced.<ref>{{harvnb|Lovelock|Rund|1989|pp=256}}</ref><ref>{{harvnb|Weinberg|1972|pp=76}}</ref><ref>{{harvnb|Poisson|2004|pp=7}}</ref><ref>{{harvnb|Landau|Lifshitz|1975|p=245}}</ref><ref>Some authors include lightlike intervals in the definition of proper time, and also include the spacelike proper distances as imaginary proper times e.g {{harvnb|Lawden|2012|pp=17, 116}}</ref><ref>{{harvnb|Kopeikin|Efroimsky|Kaplan|2011|p=275}}</ref>

===In special relativity===
With the [[timelike]] convention for the [[metric signature]], the [[Minkowski metric]] is defined by
<math display="block">\eta_{\mu\nu} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix} ,</math>
and the coordinates by
<math display="block">(x^0, x^1, x^2, x^3) = (ct, x, y, z)</math>
for arbitrary Lorentz frames.

In any such frame an infinitesimal interval, here assumed timelike, between two events is expressed as
{{NumBlk||<math display="block">ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = \eta_{\mu\nu} dx^\mu dx^\nu,</math> |{{EquationRef|(1)}}|RawN=.}}
and separates points on a trajectory of a particle (think clock{?}). The same interval can be expressed in coordinates such that at each moment, the particle is ''at rest''. Such a frame is called an instantaneous rest frame, denoted here by the coordinates <math>(c\tau,x_\tau,y_\tau,z_\tau)</math> for each instant. Due to the invariance of the interval (instantaneous rest frames taken at different times are related by Lorentz transformations) one may write
<math display="block">ds^2 = c^2 d\tau^2 - dx_\tau^2 - dy_\tau^2 - dz_\tau^2 = c^2 d\tau^2,</math>
since in the instantaneous rest frame, the particle or the frame itself is at rest, i.e., <math>dx_\tau = dy_\tau = dz_\tau = 0</math>. Since the interval is assumed timelike (ie. <math>ds^2 > 0</math>), taking the square root of the above yields<ref>{{harvnb|Zwiebach|2004|p=25}}</ref>
<math display="block">ds = cd\tau,</math>
or
<math display="block">d\tau = \frac{ds}{c}.</math>
Given this differential expression for {{mvar|τ}}, the proper time interval is defined as
{{Equation box 1
|equation = <math>\Delta\tau = \int_P d\tau = \int_P \frac{ds}{c}.</math>{{spaces|10}}{{EquationRef|(2)}}
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
Here {{mvar|P}} is the worldline from some initial event to some final event with the ordering of the events fixed by the requirement that the final event occurs later according to the clock than the initial event.

Using {{EquationNote|(1)}} and again the invariance of the interval, one may write<ref>{{harvnb|Foster|Nightingale|1978|p=56}}</ref>
{{Equation box 1
|equation = <math>\begin{align}
\Delta\tau
&= \int_P \frac{1}{c} \sqrt{\eta_{\mu\nu}dx^\mu dx^\nu}
\\
&= \int_P \sqrt {dt^2 - {dx^2 \over c^2} - {dy^2 \over c^2} - {dz^2 \over c^2}}
\\
&= \int_a^b \sqrt {1 - \frac{1}{c^2} \left [ \left (\frac{dx}{dt}\right)^2 + \left (\frac{dy}{dt}\right)^2 + \left ( \frac{dz}{dt}\right)^2 \right] }dt
\\
&= \int_a^b \sqrt {1 - \frac{v(t)^2}{c^2}} dt
\\
&= \int_a^b \frac{dt}{\gamma(t)},\end{align}</math>{{spaces|10}}{{EquationRef|(3)}}
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where
<math display="block"> (x^0, x^1, x^2, x^3 ) : [ a , b ] \rightarrow P </math>
is an arbitrary bijective parametrization of the worldline {{mvar|P}}
such that
<math display="block">(x^0(a), x^1(a), x^2(a), x^3(a))\quad\text{and}\quad (x^0(b), x^1(b), x^2(b), x^3(b))</math> 
give the endpoints of {{mvar|P}} and a < b; {{math|''v''(''t'')}} is the coordinate speed at coordinate time {{mvar|t}}; and {{math|''x''(''t'')}}, {{math|''y''(''t'')}}, and {{math|''z''(''t'')}} are space coordinates. The first expression is ''manifestly'' Lorentz invariant. They are all Lorentz invariant, since proper time and proper time intervals are coordinate-independent by definition.

If {{math|''t'', ''x'', ''y'', ''z''}}, are parameterised by a [[parameter]] {{mvar|λ}}, this can be written as
<math display="block"> \Delta\tau
= \int \sqrt {\left (\frac{dt}{d\lambda}\right)^2 - \frac{1}{c^2} \left [ \left (\frac{dx}{d\lambda}\right)^2 + \left (\frac{dy}{d\lambda}\right)^2 + \left ( \frac{dz}{d\lambda}\right)^2 \right] } \,d\lambda.</math>

If the motion of the particle is constant, the expression simplifies to
<math display="block"> \Delta \tau = \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} - \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}},</math>
where Δ means the change in coordinates between the initial and final events. The definition in special relativity generalizes straightforwardly to general relativity as follows below.

===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.