Editing Proper time (section)

===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.