time dilation

Time dilation is the difference in time intervals between inertial frames of reference moving with respect to each other. This is usually studied in Physics, but here is the mathematics behind it.Consider a train moving with speed $u$ and an observer standing still. Let the station (where observer is standing) be inertial frame of reference $S$, and train be inertial frame of reference $S^{\prime }$. A light source and a mirror are placed vertically above each other in the train as shown.

Figure 1: Path of light according to a person in the train

Let $\mathrm{\Delta }{t}_0$ denote the time taken for light to travel from the source back to the source according to a person on the train. In other words, in frame of reference $S^{\prime }$ the time taken for light to travel is given by:

$$\mathrm{\Delta }{t}_0=\frac{2d}{c}$$

where $c$ is the speed of light. The round-trip time measured by the observer in frame $S$ is a different interval $\mathrm{\Delta }t$. This is what he would observe:

Figure 2: Path of light according to observer

To the observer, the events of light leaving the source and coming back occur at different points in space. The distance travelled by light in this case is $2l$, and by the Pythagorean theorem:

$$l=\sqrt{d^2+{\left(\frac{u\mathrm{\Delta }t}{2}\right)}^2}$$

Now the time for light to travel, according to observer, is:

$$\mathrm{\Delta }t=\frac{2l}{c}=\frac{2}{c}\sqrt{d^2+{\left(\frac{u\mathrm{\Delta }t}{2}\right)}^2}$$

But we know from the equation above that $\mathrm{\Delta }{t}_0=2d/c$, and by rearranging, $d=c\mathrm{\Delta }{t}_0/2$. Now we substitute to get:

$$\mathrm{\Delta }t=\frac{2}{c}\sqrt{{\left(\frac{c\mathrm{\Delta }{t}_0}{2}\right)}^2+{\left(\frac{u\mathrm{\Delta }t}{2}\right)}^2}$$

Now let us rearrange and solve for $\mathrm{\Delta }t$:

$$c^2\mathrm{\Delta }{t}^2=c^2\mathrm{\Delta }{t}_0^2+u^2\mathrm{\Delta }{t}^2$$

$$(c^2-u^2)\mathrm{\Delta }{t}^2=c^2\mathrm{\Delta }{t}_0^2$$

$$\mathrm{\Delta }{t}^2=\frac{c^2\mathrm{\Delta }{t}_0^2}{c^2-u^2}$$

Dividing the numerator and denominator by $c^2$ and taking the square root yields:

$$\mathrm{\Delta }t=\frac{\mathrm{\Delta }{t}_0}{\sqrt{1-u^2/c^2}}$$

Since the expression $1/\sqrt{1-u^2/c^2}$ occurs quite frequently in relativity, sometimes it is preferable to use the letter $\gamma$ to represent it. Therefore:

$$\gamma =\frac{1}{\sqrt{1-u^2/c^2}}$$

The equation for time dilation can then be written this way:

$$\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0$$

History and Uses

This is a very famous result in relativity, and in fact it was the basis for the evolvement of relativistic mechanics, in which Albert Einstein defiantly challenged Newton’s equations for objects with very high speeds. From this, mechanics was classified into two branches: Newtonian mechanics and Relativistic mechanics. It was also the basis of Einstein’s theory of relativity $E=m{c}^2$, and many other equations in relativity. The time dilation equation can be used to calculate the difference in time intervals between two inertial frames moving with respect to each other.

Title	time dilation
Canonical name	TimeDilation
Date of creation	2013-03-22 18:08:17
Last modified on	2013-03-22 18:08:17
Owner	curious (18562)
Last modified by	curious (18562)
Numerical id	6
Author	curious (18562)
Entry type	Topic
Classification	msc 83A05