Let’s talk time dilation. We recently all experienced how March 2020 felt like it lasted something like 20 years, and how August to Christmas went in a blink of an eye. That’s almost relativity for you! What Einstein’s theory explained was how time changes as one begins to move.
We don’t seem to experience it in day-to-day life, but actually, the faster you go, the slower time moves for you. This is known as the velocity time dilation. And it is used in this very handy calculator from Omni.
The Time Dilation Formula
The velocity time dilation formula is not among the scariest we might find in physics, but it still requires a little explanation of the details. What we are calculating is how much a time interval (symbolized by the capital Greek letter delta and the letter t) is slowed down by moving at a certain velocity (symbol v). And the reason why this happens is that the speed of light is finite.
The simplest way to demonstrate that this is the case doesn’t require more than a little bit of school maths: Pythagoras’ theorem.
As shown in this above figure, you start with two mirrors, and light bouncing between them. This is like a clock, and your time interval (like a second or a minute) is just the time it takes the light to bounce from one mirror to the other and then back.
But if this clock (your two mirrors) is moving, the light will be taking a longer path to bounce to the other and back. So your new interval of time (“Delta t prime”) is longer compared to the original. You can see how it forms two nice right-angled triangles. The distance, D, can be expressed via Pythagoras' theorem in terms of L, the clock velocity, and the time interval. So combining the two sides gives you the neat little formula from the calculator.
Reciprocity, Perspective, Gravity, And A Big Paradox
All of this is more or less straightforward, but there is something a bit more difficult to wrap your head around. Let’s keep using our mirror clocks. You have one, and I get the other. You are still, and I am moving. You will notice that my clock goes slower. But the kicker is that from my point of view, your clock is the one going slower because I (being self-obsessed) can measure myself as still and the rest of the universe including you, moving.
This concept known as reciprocity seems a bit off, doesn’t it? Clearly, it is the person we can see moving that has a slower clock! But it is easier to imagine the situation like perspective. If you see a distant person, they will appear small. But to them, they are normal size and you are the distant small one.
This property led to the most famous problem related to time dilation: The twin paradox. It goes that you start with two twins, one getting a very fast spaceship and the other one remaining on Earth. When the first twin returns, it turns out that they have aged less than their sibling.
How’s this possible if time dilation is reciprocal? Well, this is due to another type of time dilation due to its acceleration, which is not reciprocal, and that’s enough to produce a dramatic difference. This is also the case in gravitational systems, with gravity producing a time dilation effect.
Researchers have calculated that due to this effect, the core of our planet is about 2.5 years younger than its surface.