Time Dilation And Its Meaning | Month of Equations

Talking about equations in physics and not remembering Prof. Einstein would be a sin. I don't want him to be offended so here we are with one of the flagship equations of special relativity: Time Dilation. Science fiction and pop science loves this concept. We have seen time dilation is the blockbuster interstellar too. According to relativity, there are two types of this phenomenon: Velocity time dilation and gravitational time dilation. In our article, we will discuss the former.

Meaning of equation

Time Dilation

The faster you travel through space, the slower you travel through time

Understanding the time dilation

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.

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Since light traveled equal distance for both the observers, they'll always say that speed of light is constant but argue on the fact of simultaneity. Thus, time is not absolute. It is different for people in different reference frames.

Einstein comes with an idea

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.

Why does time dilate?

Einstein's special theory of relativity is based on two postulates:
1. The laws of physics are same in all the inertial frames of reference.
2. The speed of light is constant for everyone

If you are moving with the speed say c/2 with a torch in your hand, then the speed of light for an observer in front of you won't be c + c/2. It will be c only. 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.

Applications of time dilation

One may wonder if there is any application of this strange phenomenon. Yes, there are many and in this article I will mention one from particle physics. There are particles known as muons. They are much like the electrons except they have much greater mass and a very short lifetime of just 2.2 microseconds. These muons are found in the upper atmosphere. When cosmic rays, which are primarily high energy protons, bombard the atmosphere, one of the end product is a muon travelling at relativistic speeds.

Suppose a muon is created at 10 Km above the land. If it travels at the speed of 0.98 c (c = speed of light), then according to Newtonian mechanics, the time taken to reach the ground is 34 microseconds. This is much longer than the lifetime of the muon and thus most of them must decay long before they reach the Earth. However, lot of muons reach the ground and they are detected in cloud chambers and other particle detectors. How is this possible? Do the muons travel through time to reach the ground?

Muons and time dilation
Time dilation explains the Muon paradox

No we don't need science fiction to explain this anomaly. This is where Einstein's special relativity comes into picture. The lifetime of the muon depends on the observer's frame. To us, on the ground, time in muon's reference frame dilates and thus we see them living longer than they actually do. But, in the frame of muon, this number isn't 34 microseconds. Using the above equation, this number comes out to be 11.1 microseconds. With a mean lifetime of 11.1 microseconds, a lot more muons have time to travel to Earth. This perfectly explains the muon flux on the ground.

Previous in series: The Poisson's Equation (Electrodynamics)

Author's message

Special relativity is the most wonderful theory that has so many applications in today's world. Without this theory, the large particle detectors such as the LHC would not have been possible. Special relativity is the backbone of particle physics and high energy physics. The muon decay problem is one of the experimental proofs of time dilation and thus the special relativity. I hope you enjoyed learning this equation. If you did, do comment below and ask your doubts, if you have any. See you tomorrow with a new equation.

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