The explanation of how we get to the equation γ = 1 / √1-v²/c² is through a diagram where light bounces back and forth between two mirrors. While the light is bouncing back and forth, a clock is ticking.
The distance can be worked out through distance = speed x time. The speed is the speed of light (c) and the time taken is t’ (tau) or τ which is a symbol for the wristwatch time. One tick of the wristwatch clock = 2τ.
Now, let’s create the same scenario However, this time, you are stationary and a clock with light bouncing against mirrors are moving past you with a speed of v.
You can see from the diagram that the distance between the two mirrors has stayed the same at cτ (tau) with τ being wristwatch (observer’s) clock time. However, due to the clock pasting you at speed of v with the light and mirrors, the distance the light has to travel has become longer. However, the speed has not increased, it is still c. Therefore, t (normal t) is longer to accommodate the longer distance the light has to travel. From this, the moving clock travelling past you at speed v ticks more slowly than the observer’s wristwatch clock.
- The motion of the light clock (travelling at the same speed as the light and mirrors) does not affect the speed of light.The speed of light is a constant.
- The horizontal distance moved due to speed v is x = vt.
We can use Pythagoras’s theorem to come up with the equation:
c²t² = c²τ² = v²t²
We can arrange that to center the equation around τ.
τ² = t²(c² – v²) / c²
Where τ is the wristwatch time and t is the observed time. We can use the above equation to show that t = γτ where γ = 1 / √1-v²/c²:
- τ² = t²(c² – v²) / c²
- τ² = (t²c² – t²v²) / c²
- τ² = t²- t²v² / c²
- τ² = t²(1- v²/c²)
- τ = t √ (1- v²/c²)
- If t = γτ, the γ = 1 / √1-v²/c².
- As v = 0, γ =1. As v = c, γ = infinity
Therefore, if τ = 1 nano second and v = 0.75c, what will t be? Well, after using the above equation in bold, the answer would be 1.5 nano seconds. Therefore, at wristwatch time, it will be at 1 nano second. However, the observed time, due to speed of v, will tick 1.5 nano seconds.
Summary
- The speed of light is the constant in the universe. Time slows down as you approach the speed of light.
- We can use the equation τ = t √ (1- v²/c²) to work out how time dilates. The wristwatch time measure the time of the moving object relative to wristwatch. The observed time is the time observed by a clock of the same speed as the moving object.
- As v = 0, γ =1. As v = c, γ = infinity
