October 1: Velocity Time Dilation
We begin our Month of Equations with the famous equation of Time Dilation. The mathematical equation of velocity time dilation is given above. Now let us try to understand what it really means.
Meaning of Equation
The faster you travel through space, the slower you travel through time
To understand this, let us go back in the year 1905 and meet a patent clerk at Bern, Switzerland who is trying to find a solution to a question no one was asking. So, Einstein had this question: Suppose I have a master clock in a town that sends signals to synchronize all the other clocks in the town, at the speed of light. The clocks will synchronize. No problem so far. But what happens if we try to send the signal to a clock on a moving train? Light will have to speed up or slow down to catch up with the train. But according to Maxwell's equations of electrodynamics, light always travels at the same speed. So either Newton is right (in saying time is absolute, same for everyone) or Maxwell is. It can't be both. Einstein wanted to know who was right between the two.
Einstein then came up with another brilliant thought experiment. Imagine yourself standing on a platform of a station. Two lighting bolts strike in front of you as shown above. The bolts were 300 m apart. For you, on the platform, they were simultaneous. But if a person views the 2 bolts by standing in the middle of a 300 m long moving train, they would not be simultaneous because he would be moving towards one bolt and away from the other. So then who is right? The person on the platform who says the bolts struck at the same time or the person on the train who says the bolts were not simultaneous? It turns out that- both are correct. Absolute simultaneity is not possible. Thus, time is not absolute. It means Newton is the one who gets it between the eyes.
So time dilation is an outcome of the fact that light travels at the same speed in all reference frames. The flow of time in that frame adjusts itself to keep the speed of light constant. So moving clocks run slower than the stationary clocks and this is exactly what the equation tells us. If v, the speed of moving clock gets larger, Δt' gets larger, which means the time duration between two events gets longer and hence time slows down. So moving clocks run slower than the stationary clocks and this is exactly what the equation tells us.