Special Relativity & Time Dilation

As a belated Hallowe’en present, I’m going to spook you with special relativity! These are mostly from my lecture notes, but completely re-written to make better sense to more people. There is less maths, more conceptual discussion!

The Principles of Special Relativity

Most people have heard of some sort of problem involving a train/vehicle moving at constant velocity: if the train has absolutely nothing surrounding it, can it tell it’s moving or not? Another version of the question involved another train, usually stationary. Does each train know which ones moving and which ones not, or do they both think the other train is moving?

These concepts are a “little taster” to special relativity. With no acceleration, there’s no resultant force on either train, so they don’t feel anything in principle. Hence, neither can actually tell which train’s moving and which one’s not.

Here’s three things you need to remember, regardless of what you learn about in special relativity. These are that:

  1. All velocities are relative, regardless of your reference frame. By extension, accelerations in each frame are the same, so while you can’t measure absolute velocity you can measure absolute acceleration.
  2. The laws of physics are identical in all non-accelerating inertial frames.
  3. The speed of light, c, in a vacuum is a constant for observers in inertial frames.

This is a good start, but there’s something else we need to consider: What if the velocity was approaching light speed? This is where fun effects start to take place!

Time Dilation

Time dilation is a simple concept in its essence, but it can get fairly complex quickly. Basically, when you’re moving at relativistic speeds (i.e close to the speed of light), you appear to be moving through time a lot slower than someone who is “stationary”.

Take the example of a ticking clock and a mirror. Let’s say a photon bounces between the clock and the mirror, and whenever the photon bounces off the clock it ticks.

For a stationary clock, the photon will just move back and forth some distance, let’s call this d.

Now let’s move the clock sideways. The light path is now much longer than d, so it will take longer for the clock to tick. You could say this clock is moving through time slower, but interestingly the clock won’t realise it’s moving slower!

Math Time!

If we remember from primary school, speed = distance / time, then you can do this!

If the clock moves at speed v, we can rearrange the above equation and find the distance the clock moves between each tick is vt. So we can say the time between a stationary clock tick is:

It’s doubled, because the photon bounces back and forth!

That’s nice and all, but we want to know what happens to moving clocks. We’ll call the time the clock experiences t, and the time a stationary observer experiences t’. This is sometimes called proper time and is written with a tau. Don’t get them mixed up!

The time between ticks of a moving clock is delta t’:

We’re basically using Pythagoras to find the distance the light travels, and then divide by speed to get time.

With a bit of rearranging we can make delta t’ in terms of delta t, we can get this beauty!

Now he have a term that can show you how much dilation has occurred! Let’s look at this further.

The Lorentz Factor

That v/c squared term can be turned into a simple identity called the Lorentz factor:

So if we sub that back into the original equation we found, we get this:

Now doesn’t that look good! It has the stationary observer’s time delta t’ equal to the moving clock’s time delta t multiplied by a factor gamma. For those super high velocities, the Lorentz factor is large and so that time dilation will be more drastic.

So time stretches, but what about space? That’s a topic for next time!

3 comments

  1. I’m really glad you’re covering this. I get relativity, for the most part, but it’s such a mind-bending topic. I need regular refresher courses or I’ll start getting mixed up about it again.

    Liked by 1 person

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