I'm doing some sort of notes on special relativity and i got this question, because proper time and time dilation have a relation. In fact we have that proper time is mathematically the arc length of a timelike curve. I will use geometrical units and the Minkowski Metric with the -+++ signature.(adsbygoogle = window.adsbygoogle || []).push({});

For example, we have two events [tex]A,B[/tex] separated timelike by a straight worldline which can be parametrized by (in the particular case of [tex]A[/tex] being at our origin of reference and at time [tex]x^{0}=0[/tex] and [tex]B[/tex] having only [tex]x^{1}[/tex] nonzero, the general case is obtained via Poincaré transformation).

[tex]\sigma^{\alpha}\left(x^{0}\right)=\left(x^{0},\frac{x_{B}^{1}}{x_{B}^{0}}x^{0},0,0\right)[/tex]

Then

[tex]\frac{d\,\sigma^{\alpha}}{d\,x^{0}}=\left(1,\frac{x_{B}^{1}}{x_{B}^{0}},0,0\right)[/tex]

and therefore proper time function is

[tex]\tau=\int\sqrt{\left\vert\eta_{\alpha\gamma}\frac{d\,\sigma^{\alpha}}{d\,x^{0}}\frac{d\,\sigma^{\gamma}}{d\,x^{0}}\right\vert}dx^{0}=\sqrt{1-\left(\frac{x_{B}^{1}}{x_{B}^{0}}\right)^{2}}x^{0}[/tex]

but we have that

[tex]\frac{x_{B}^{1}}{x_{B}^{0}}=v[/tex]

that is the speed with sign, therefore

[tex]\tau=\sqrt{1-v^{2}}x^{0}=\frac{1}{\gamma}x^{0}[/tex]

or in differential form

[tex]dx^{0}=\gamma d\tau[/tex]

For a general timelike curve parametrized by

[tex]\sigma^{\alpha}\left(x^{0}\right)=\left(x^{0},\sigma^{a}\left(x^{0}\right)\right)[/tex]

we can write

[tex]\tau=\int\sqrt{\left\vert\eta_{\alpha\gamma}\frac{d\,\sigma^{\alpha}}{d\,x^{0}}\frac{d\,\sigma^{\gamma}}{d\,x^{0}}\right\vert}dx^{0}=\int\sqrt{\left\vert\delta_{ag}\frac{d\,\sigma^{a}}{d\,x^{0}}\frac{d\,\sigma^{g}}{d\,x^{0}}-1\right\vert}dx^{0}=\int\sqrt{1-\delta_{ag}\frac{d\,\sigma^{a}}{d\,x^{0}}\frac{d\,\sigma^{g}}{d\,x^{0}}}dx^{0}[/tex]

but [tex]\delta_{ag}\frac{d\,\sigma^{a}}{d\,x^{0}}\frac{d\,\sigma^{g}}{d\,x^{0}}[/tex] is the squared norm of the instantaneous 3-velocity on the curve. Therefore we have in differential form

[tex]dx^{0}=\gamma(x^{0}) d\tau[/tex]

and through this we have derived time dilation in general. In fact, for inertial observers (those that their frames of reference in spacetime are related through Poincaré transformations) the time that every observer actually measures is their proper time and the worldlines of the observers viewed by others are straight lines, then we can use

[tex]dx^{0}=\gamma d\tau[/tex]

for derive

[tex]dx^{0}=\gamma dx^{\hat{0}}[/tex]

or

[tex]dx^{\hat{0}}=\gamma dx^{0}[/tex]

for two arbitrary inertial observers.

Therefore I formulate my original question: do you know similar derivation for the length contraction through proper length function (equivalent of the proper time for spacelike curves)?

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# Is there any derivation of length contraction through proper lenght?

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