Relativity of Event Order

Theorem: Relativity of Event Order

Suppose $S$ and $S^{'}$ are two inertial frames of reference and that $S^{'}$ is moving with constant velocity $u$ relative to $S$ .

Two events will be perceived in the same order in $S$ as they are perceived in $S^{'}$ if and only if the locations $(x_{1}^{'}, y_{1}^{'}, z_{1}^{'})$ and $(x_{2}^{'}, y_{2}^{'}, z_{2}^{'})$ and the times $t_{1}^{'}$ and $t_{2}^{'}$ at which they happen in $S^{'}$ obey
$sign {c^{2} (t_{2}^{'} - t_{1}^{'}) + u_{x} (x_{1}^{'} - x_{2}^{'}) + u_{y} (y_{1}^{'} - y_{2}^{'}) + u_{z} (z_{1}^{'} - z_{2}^{'})} = sign {t_{2}^{'} - t_{1}^{'}}$

NOTE

This means that it is possible for an event which happens after another event in $S^{'}$ to be perceived as happening before it in $S$ and vice versa.

PROOF

If $t_{1}$ and $t_{2}$ are the times at which the events happen in $S$ , then the Lorentz transformation gives us
$t_{1} = \frac{γ}{c} (c t_{1}^{'} - \frac{u _{x}}{c} x_{1}^{'} - \frac{u _{y}}{c} y_{1}^{'} - \frac{u _{z}}{c} z_{1}^{'})$ $t_{2} = \frac{γ}{c} (c t_{2}^{'} - \frac{u _{x}}{c} x_{2}^{'} - \frac{u _{y}}{c} y_{2}^{'} - \frac{u _{z}}{c} z_{2}^{'})$
If $t_{2}^{'} > t_{1}^{'}$ , then we want $t_{2} > t_{1}$ and so
$\frac{γ}{c} (c t_{2}^{'} - \frac{u _{x}}{c} x_{2}^{'} - \frac{u _{y}}{c} y_{2}^{'} - \frac{u _{z}}{c} z_{2}^{'}) - \frac{γ}{c} (c t_{1}^{'} - \frac{u _{x}}{c} x_{1}^{'} - \frac{u _{y}}{c} y_{1}^{'} - \frac{u _{z}}{c} z_{1}^{'}) > 0$ $c^{2} (t_{2} - t_{1}) + u_{x} (x_{1}^{'} - x_{2}^{'}) + u_{y} (y_{1}^{'} - y_{2}^{'}) + u_{z} (z_{1}^{'} - z_{2}^{'}) > 0$
Alternatively, if $t_{2}^{'} < t_{1}^{'}$ , then we want $t_{2} < t_{1}$ and so
$c^{2} (t_{2} - t_{1}) + u_{x} (x_{1}^{'} - x_{2}^{'}) + u_{y} (y_{1}^{'} - y_{2}^{'}) + u_{z} (z_{1}^{'} - z_{2}^{'}) < 0$
This can be summarised as
$sign {c^{2} (t_{2}^{'} - t_{1}^{'}) + u_{x} (x_{1}^{'} - x_{2}^{'}) + u_{y} (y_{1}^{'} - y_{2}^{'}) + u_{z} (z_{1}^{'} - z_{2}^{'})} = sign {t_{2}^{'} - t_{1}^{'}}$

Theorem: Relativity of Simultaneity

Suppose $S$ and $S^{'}$ are two inertial frames of reference and that $S^{'}$ is moving with constant velocity $u$ relative to $S$ .

Two events which happen at locations $(x_{1}^{'}, y_{1}^{'}, z_{1}^{'})$ and $(x_{2}^{'}, y_{2}^{'}, z_{2}^{'})$ and at times $t_{1}^{'}$ and $t_{2}^{'}$ in $S^{'}$ , will be perceived as simultaneous in $S$ if and only if
$u_{x} (x_{2}^{'} - x_{1}^{'}) + u_{y} (y_{2}^{'} - y_{1}^{'}) + u_{z} (z_{2}^{'} - z_{1}^{'}) = c^{2} (t_{2}^{'} - t_{1}^{'}),$
where $c$ is the speed of light in a vacuum.

NOTE

This means that two events which are perceived as simultaneous in one inertial frame of reference need not be perceived as simultaneous in another and two events which are not simultaneous in one inertial frame of reference may be simultaneous in another:

Two events which happen at the same time $t^{'}$ in $S^{'}$ will also be perceived as simultaneous in $S$ if and only if their locations $(x_{1}^{'}, y_{1}^{'}, z_{1}^{'})$ and $(x_{2}^{'}, y_{2}^{'}, z_{2}^{'})$ in $S^{'}$ obey

$u_{x} (x_{2}^{'} - x_{1}^{'}) + u_{y} (y_{2}^{'} - y_{1}^{'}) + u_{z} (z_{2}^{'} - z_{1}^{'}) = 0$

Even if two events are not simultaneous in $S^{'}$ , they will be perceived as simultaneous in $S$ if they obey the first equation but not the second.

PROOF

If $t_{1}$ and $t_{2}$ are the times at which the events happen in $S$ , then the Lorentz transformation gives us
$t_{1} = \frac{γ}{c} (c t_{1}^{'} - \frac{u _{x}}{c} x_{1}^{'} - \frac{u _{y}}{c} y_{1}^{'} - \frac{u _{z}}{c} z_{1}^{'})$ $t_{2} = \frac{γ}{c} (c t_{2}^{'} - \frac{u _{x}}{c} x_{2}^{'} - \frac{u _{y}}{c} y_{2}^{'} - \frac{u _{z}}{c} z_{2}^{'})$
For the events to be perceived as simultaneous in $S$ , we want $t_{1}$ to be equal to $t_{2}$ and so
$\frac{γ}{c} (c t_{1}^{'} - \frac{u _{x}}{c} x_{1}^{'} - \frac{u _{y}}{c} y_{1}^{'} - \frac{u _{z}}{c} z_{1}^{'}) = \frac{γ}{c} (c t_{2}^{'} - \frac{u _{x}}{c} x_{2}^{'} - \frac{u _{y}}{c} y_{2}^{'} - \frac{u _{z}}{c} z_{2}^{'})$ $c t_{1}^{'} - \frac{u _{x}}{c} x_{1}^{'} - \frac{u _{y}}{c} y_{1}^{'} - \frac{u _{z}}{c} z_{1}^{'} = c t_{2}^{'} - \frac{u _{x}}{c} x_{2}^{'} - \frac{u _{y}}{c} y_{2}^{'} - \frac{u _{z}}{c} z_{2}^{'}$ $\frac{u _{x}}{c} x_{2}^{'} + \frac{u _{y}}{c} y_{2}^{'} + \frac{u _{z}}{c} z_{2}^{'} - \frac{u _{x}}{c} x_{1}^{'} - \frac{u _{y}}{c} y_{1}^{'} - \frac{u _{z}}{c} z_{1}^{'} = c (t_{2} - t_{1})$ $\frac{u _{x}}{c} (x_{2}^{'} - x_{1}^{'}) + \frac{u _{y}}{c} (y_{2}^{'} - y_{1}^{'}) + \frac{u _{z}}{c} (z_{2}^{'} - z_{1}^{'}) = c (t_{2} - t_{1})$ $u_{x} (x_{2}^{'} - x_{1}^{'}) + u_{y} (y_{2}^{'} - y_{1}^{'}) + u_{z} (z_{2}^{'} - z_{1}^{'}) = c^{2} (t_{2} - t_{1})$

Razon

Explorer

Relativity of Event Order

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