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According to Einstein, do observers in relative motion agree on the time order of all events?

I don't think they would agree on the timing of events, but I am having trouble figuring out why they wouldn't agree. Any thoughts?

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possible duplicate of why is time order invariant in timelike interval? –  Qmechanic Jan 23 '12 at 21:27
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up vote 3 down vote accepted

Observers in relative motion would not agree on the time order of all events. They may disagree on the order of events connected by a space-like interval, i.e. for events such that

\begin{equation} -c^2\cdot \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 > 0 \end{equation}

Note that since the time interval between such events is too short for any communication between the events to occur there may be no causal relationship between them.

All observers do agree however on the time order of events connected by a time-like interval, i.e. for events such that

\begin{equation} -c^2\cdot \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 < 0 \end{equation}

These events may have causal connection.

To see how this happens consider three events which in certain inertial frame of reference O have the following (t, x, y, z) coordinates:

\begin{equation} A = [0, 0, 0, 0] \end{equation} \begin{equation} B = [c, 0, 0, 0] \end{equation} \begin{equation} C = [0, 1, 0, 0] \end{equation}

Events A and B are connected by a time-like interval (in fact they correspond to the same place in O at different points in time) while A and C are connected by a space-like interval (in O they correspond to different places at the same moment in time).

Now, to obtain spacetime coordinates of A, B and C in a frame of reference O' moving relative to O at velocity v>0 along the x axis, we can use the following Lorentz transformation

\begin{equation} \Lambda_x(v) = \left[ \begin{array}{cccc} \gamma & -\beta \gamma & 0 & 0 \newline -\beta \gamma & \gamma & 0 & 0 \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{array} \right] \end{equation}

where

\begin{equation} \beta = \frac{v}{c}>0 \end{equation} \begin{equation} \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}>0 \end{equation}

In the frame of reference O' moving with velocity v along the x axis the coordinates of the three events are

\begin{equation} A = [0, 0, 0, 0] \end{equation} \begin{equation} B = [\gamma c, -\beta \gamma c, 0, 0] \end{equation} \begin{equation} C = [-\beta \gamma, \gamma, 0, 0] \end{equation}

In the frame of reference O'' moving with velocity -v along the x axis (i.e. the same speed as O', but the opposite direction) the coordinates of the three events are

\begin{equation} A = [0, 0, 0, 0] \end{equation} \begin{equation} B = [\gamma c, \beta \gamma c, 0, 0] \end{equation} \begin{equation} C = [\beta \gamma, \gamma, 0, 0] \end{equation}

since only β changes sign.

We see that from the perspective of an observer stationary in the O reference frame events A and C are simultaneous, from the perspective of an observer stationary in the O' reference frame event C occurs before event A while from the perspective of an observer stationary in the O'' reference frame event A occurs before event C. In all frames A occurs before B.

See also relativity of simultaneity.

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