## Time dilation

Required math: algebra, geometry

Required physics: basics of relativity

We saw in an earlier post that the events with a particular interval ${\Delta s^{2}}$ between them lie on an invariant hyperbola, and that this hyperbola can be used to calibrate the time or space intervals on the coordinate axes of two observers. We noted in passing that one effect of the postulates of relativity is time dilation, in which one observer believes the other observer’s clocks run slow relative to himself.

We’ll review the effect here and then resolve a common so-called paradox that many people believes invalidates relativity.

Consider the situation shown in the figure.

\par

Observer ${O_{2}}$ moves with a speed ${v}$ in the ${x}$ direction relative to observer ${O_{1}}$ as usual. The coordinate systems of the two observers are shown on the diagram, and the dark blue hyperbola (with equation ${x_{1}^{2}-t_{1}^{2}=-4}$) passing through the points ${A}$ and ${B}$ defines all events with an interval ${\Delta s^{2}=-4}$. This means that the point ${A}$, where the hyperbola intersects the ${t_{1}}$ axis, is the point where ${O_{1}}$‘s clock with world line ${0A}$ reads ${t_{1}=2}$. Observer ${O_{2}}$‘s clock whose world line is ${0B}$ reads ${t_{2}=2}$ at point ${B}$.

Since ${O_{1}}$ measures all events on a horizontal line passing through ${A}$ as being simultaneous, and since ${B}$ is not on this line, he will say that ${A}$ and ${B}$ occur at different times. As we saw in the earlier post, we can work out the coordinates of event ${B}$ (by solving for the intersection of the hyperbola and the line ${0B}$), and they turn out to be

 $\displaystyle x_{1}$ $\displaystyle =$ $\displaystyle \frac{2v}{\sqrt{1-v^{2}}}$ $\displaystyle t_{1}$ $\displaystyle =$ $\displaystyle \frac{2}{\sqrt{1-v^{2}}}$

That is, ${O_{1}}$ measures the time of event ${B}$ as ${t_{1}=t_{2}/\sqrt{1-v^{2}}}$ while ${O_{2}}$ measures it as ${t_{2}=2}$. Since ${t_{1}>t_{2}}$, ${O_{1}}$ believes that more time has elapsed than ${O_{2}}$ does, so he believes that ${O_{2}}$‘s clock runs slow. This is the time dilation effect.

The paradox (or so it is sometimes believed) is this: if ${O_{1}}$ believes ${O_{2}}$‘s clock runs slow, then surely ${O_{2}}$ must believe that ${O_{1}}$‘s clock runs fast. However, the principle of relativity (all inertial frames are equivalent) says that each observer should believe the same thing about the other, so that ${O_{2}}$ should measure ${O_{1}}$‘s clock as slow, not fast, relative to himself.

The reason why this apparent paradox occurs is that the measurement process which gives rise to the time dilation effect is not symmetric between the two observers. To see this, we must consider carefully what it is we are actually measuring.

The disparity occurs because we are comparing what the two observers are measuring as the time of event ${B}$. ${O_{2}}$ does this with one clock: the clock with world line ${0B}$. ${O_{1}}$, however, uses two clocks to do the measurement. The first clock is one whose world line is ${0A}$. This clock coincides with ${O_{2}}$‘s clock at the origin, where both observers agree that the time of this event is ${t_{1}=t_{2}=0}$. However, after this event, the two clocks diverge and follow different world lines, so when the observers want to measure the time of event ${B}$, ${O_{1}}$ can’t use the same clock that was used to measure the time at the origin, since that clock’s world line doesn’t go through event ${B}$. Instead he has to use the clock with world line ${FB}$. This looks fine to ${O_{1}}$ since he measures events 0 and ${F}$ as simultaneous (they both lie on his ${x_{1}}$ axis so they both occur at ${t_{1}=0}$). However, ${O_{2}}$ disagrees that events 0 and ${F}$ are simultaneous. According to ${O_{2}}$, the events that are simultaneous with event 0 are those that lie on the ${x_{2}}$ axis (the red line in the diagram), and it is clear that event ${F}$ occurs before any event on this axis. Thus to ${O_{2}}$, ${O_{1}}$‘s clock ticks off more time in travelling from ${F}$ to ${B}$ than ${O_{2}}$‘s clock does in going from 0 to ${B}$. Thus both ${O_{1}}$ and ${O_{2}}$ will agree, after doing this analysis, that ${O_{1}}$‘s clock should read a later time than ${O_{2}}$‘s clock at event ${B}$.

Now let’s look at event ${B}$ from another viewpoint. According to ${O_{2}}$, events that are simultaneous with ${B}$ must lie on the line passing through ${B}$ and parallel to the ${x_{2}}$ axis; this line is shown in dark green in the figure. This line intersects the ${t_{1}}$ axis at event ${D}$, so ${O_{2}}$ measures the time of event ${D}$ as ${t_{2}=2}$. What time does ${O_{1}}$ assign to this event? We can work this out by finding the coordinates of ${D}$. The equation of the dark green line which has slope ${v}$ (as we saw in an earlier post), using the point-slope form of a straight line, is (we’ll work it out for a general value of ${t_{2}}$ since we’re trying to show the time dilation effect is symmetric):

$\displaystyle t_{1}-\frac{t_{2}}{\sqrt{1-v^{2}}}=v\left(x_1-\frac{t_{2}v}{\sqrt{1-v^{2}}}\right)$
where we used the coordinates of ${B}$ given above as the point through which this line passes. This line crosses the ${t_{1}}$ axis at ${x_{1}=0}$, so the time of event ${D}$ as measured by ${O_{1}}$ is thus

 $\displaystyle t_{1}$ $\displaystyle =$ $\displaystyle \frac{t_{2}}{\sqrt{1-v^{2}}}-\frac{t_{2}v^{2}}{\sqrt{1-v^{2}}}$ $\displaystyle$ $\displaystyle =$ $\displaystyle t_{2}\frac{1-v^{2}}{\sqrt{1-v^{2}}}$ $\displaystyle$ $\displaystyle =$ $\displaystyle t_{2}\sqrt{1-v^{2}}$ $\displaystyle t_{2}$ $\displaystyle =$ $\displaystyle \frac{t_{1}}{\sqrt{1-v^{2}}}$

That is, ${O_{2}}$ now sees ${O_{1}}$‘s clock as running slow compared to his own. So the time dilation effect really does work both ways, and each observer really does see the other’s clocks as running slow. In this last analysis, note that we are using one of ${O_{1}}$‘s clocks (the one with world line ${0D}$) and two of ${O_{2}}$‘s clocks (one with world line ${0B}$ to measure the time at the origin, and the other with world line ${GD}$ (light green) to measure the time of event ${D}$). In this case, ${O_{2}}$ measures events ${0}$ and ${G}$ as simultaneous, but ${O_{1}}$ thinks ${G}$ occurred before ${0}$, so he thinks ${O_{2}}$ is measuring a longer time than ${O_{1}}$.

The key in understanding time dilation, and resolving the paradox, is to understand that the measurements involved in analyzing the times determined for a given event are not symmetric between the two observers: one observer always has to use one clock, and the other observer always has to use two clocks. This is because the two observers are moving relative to each other, so when one of ${O_{1}}$‘s clocks coincides with one of ${O_{2}}$‘s clocks, those two clocks can never be at the same place again, so one of the observers has to use a different clock to compare times between the two frames. The time dilation effect most definitely does not imply that all clocks in one frame run at a different rate from all clocks in another frame. You must be very careful about which clocks are being compared, and at which events the comparison takes place.

One final note: nowhere in this discussion of time have we made any assumption about the nature of the clocks being used to make the measurements. They could be mechanical clocks or atomic clocks or whatever. These effects arise entirely from the postulates of relativity, and the difference between relativity and Galilean physics is due entirely to the assumption of the constancy of the speed of light. The time dilation effect is real and is a property of time itself, not of the clocks used to measure it.