# Derivation: Time Dilation By Using a Light Clock The time dilation from special relativity describes how time ticks in different reference frames. The time dilation can be derived simply with the help of a light clock. A light clock basically consists of two mirrors at a fixed distance $$L$$ from each other. A photon (light particle) is reflected back and forth between the two mirrors. This photon always moves with constant speed of light $$c$$.

To get the time dilation, you have to look at the light clock from two different inertial frames (unaccelerated reference frames). The one inertial frame represents the rest observer $$\text{B}$$, which is motionless relative to the light clock. The other inertial frame is the $$\text{B}'$$, which moves relative to the light clock with the $$-v$$ to the left (in the negative x-direction). From the point of view of $$\text{B}'$$, the clock moves to the right (in the positive x-direction) with velocity $$v$$. Let us now describe the motion of the photon by putting ourselves first in the observer $$\text{B}$$ and then in the observer $$\text{B}'$$.

From the view of $$\text{B}$$ you see the light clock at rest. There the photon oscillates vertically upwards and then downwards (let's say: along the y-axis). From one to the other mirror the photon needs the following time from the point of view of the rest observer $$\text{B}$$:

Distance covered in reference frame A
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For a period $$T$$ (i.e. once back and forth) it needs twice the time: $$T ~=~ 2\Delta t$$.

Now you switch to a moving inertial system $$\text{B}'$$, which moves to the left with the velocity $$-v$$. Now you observe something completely different! While the photon flies from one mirror to the other, the light clock moves in the positive x-direction (to the right) with the velocity $$v$$.

From the point of view of $$\text{B}'$$, the photon does not fly straight up (as in the case of resting light clock), but makes a zigzag motion (see illustration 1). The distance $$L'$$, which the photon travels from one mirror to the other, is from this point of view, LONGER.

The velocity of the photon is also equal to $$c$$ from the point of view of $$\text{B}'$$. This is one of the postulates on which the special theory of relativity is based: The speed of light is the same in all inertial frames. However, the photon in the reference frame $$\text{B}'$$ must travel a longer distance $$L'$$ at the speed of light:

Distance covered in reference frame B
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Since $$L'$$ is greater than $$L$$, $$\Delta t'$$ must be greater than $$\Delta t$$, because $$c$$ is constant in both reference frames:

Duration in reference frame B is greater than in frame A
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The photon arrives at the opposite mirror from the point of view of $$\text{B}'$$ after the time $$\Delta t'$$, while from the point of view of $$\text{B}$$ it arrives after the shorter time $$\Delta t$$.

The question now is: How are $$\Delta t$$ and $$\Delta t'$$ related? If you combine the observation where the light clock is resting ($$c\, \Delta t$$) and the observation where the light clock is moving $$c \, \Delta t'$$ and $$v \, \Delta t'$$, you get a right triangle to which you can apply Pythagoras' theorem $$c^2 = a^2 + b^2$$:

Pythagorean theorem applied to distances of the light clock
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The equation 4 can then be rearranged for example for $$\Delta t'$$, i.e. for the time the photon needs to reach the other mirror in the reference frame $$B'$$. For this, divide the equation by $$c^2$$:

Time in reference frame B squared
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Bring the second summand containing $$\Delta t'$$ to the left side:

Time in reference frame A squared
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Now you can factor out $$\Delta t'^2$$:

Time in reference frame A squared is equal to time in B scaled
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Bring the factor before the $$\Delta t'^2$$ to the right side and take the square root on both sides:

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The times $$\Delta t$$ and $$\Delta t'$$ in two different reference frames are linked by the so-called Lorentz factor (lorentz term) $$\gamma$$:

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The Lorentz factor $$\gamma$$ is always greater than 1 and becomes greater the faster the light clock moves relative to the observer (i.e. with increasing $$v$$). As a result, the time $$\Delta t'$$ becomes even greater. The time dilation has a larger effect.

With the help of time dilation the second effect of special relativity can be derived, namely the length contraction. ## The Most Useful Physics Formula Collection on the Internet

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