In this blog I derived, from first principles, the Lorentz transformations which are used in Einstein’s special theory of relativity to relate one frame of reference to another frame of reference which are moving relative to each other with a speed .

So, these relate the length and time in two different reference frames which are moving relative to each other with a velocity . One of the most intriguing and surprising consequences of Einstein’s special theory of relativity is that time is *relative* and not absolute. What this means in simple terms is that two observers in two reference frames and moving relative to each other with a velocity will measure time to be passing at *different rates*.

## Time dilation

This phenomenon is known as time dilation. Let us consider our two reference frames and . We will have a clock in frame , which in that reference frame is stationary (e.g. a clock on a rocket ship, although the rocket ship is moving, the clock is stationary relative to the rocket ship).

Two successive events on the clock in are separated by a time interval which we are going to call the *proper time* . The time interval in the other reference frame, , is . How does this compare to ?.

In the reference frame the clock is stationary, so we can say that the location of the clock in the x-dimension, , does not change. That is, .

Using our equation which relates from above, we can write

and so we can write

This means the time interval in frame will appear to be *dilated* by a factor of compared to the proper time interval .

## Time dilation in Nature

We observe the effects of time dilation every day in Nature. Cosmic rays, high energy particles from space, strike molecules in our atmosphere and create particles from the high energy interactions (this is the same as happens in the Large Hadron Collider). One of the particles created in these reactions are *muons*, which decay very rapidly in about 2 microseconds second (2 millionths of a second). Given the distance between where they are created in the upper atmosphere and the Earth’s surface, they should not survive long enough to make it to the surface of the Earth. But they do. How? Because of time dilation, the muons are moving so quickly that is appreciable more than 1, meaning that 2 microseconds in the muon’s frame of reference is much longer in our frame of reference. So, in the muon’s frame of reference it is indeed decaying in let us say 2 microseconds, but in our frame or reference it could survive for maybe a millisecond (thousandth of a second) or more, long enough to reach the surface of the Earth.

## The symmetry of relativity

One aspect of relativity which confuses a lot of people is that it is symmetrical. Although an observer in frame will think that the clock in frame is ticking more slowly, if an observer in were to look at a clock which was at rest in frame , that observer would think that the clock in frame is moving more slowly. Each would think that their clock is behaving normally, and it is the clock in the other’s reference frame which is showing the effects of time dilation.

## The twin paradox

If a twin sets off on a space trip where the rocket will travel close to the speed of light, then time dilation effects will come into play. This means that e.g. a 20-year old twin can set off on a space trip which for the twin who stays on Earth appears to last for 40 years, but because of time dilation effects maybe only 5 years will appear to pass for the twin on the rocket. Thus, the 60-year old twin who stayed on Earth will be greeted after 40 years by a 25-year old twin!!

In the example I have shown, 40 years for the twin who stays on Earth appears to pass as 5 years for the twin on the rocket. This means the time dilation factor is , and as the time dilation factor is just the Lorentz factor , this means the rocket will need to travel at a speed of of the speed of light.

*HANG ON!!!* you say, what about the symmetry of relativity? Surely the twin in the rocket will think that the twin on Earth is aging more slowly, so why doesn’t he return to find the twin on Earth is only 25 and he is 60? Or maybe, because of the symmetry, they will both be 60 when the travelling twin returns?

No, what one has to realise is that there is no symmetry in this trip. In order for the travelling twin to leave the Earth and travel at close to the speed of light he has to speed up considerably. Also, in order to come back he has to slow down and reverse his direction, speeding up again once he’s turned his rocket around to come back to Earth. And, as he approaches Earth, he will have to slow down again. These large accelerations (changes in speed) which the travelling twin experiences break the symmetry, and so it really is the case that the travelling twin will return younger than the twin who has stayed on Earth. How much younger depends on how close to the speed of light the travelling twin travels.

## Back to the future

Although it is possible therefore to “travel to the future”, as our twin in the example above does, what is not possible is to travel to the past. In order to do this one would need to travel *faster* than the speed of light, which Einstein’s theory does not allow. The results of neutrinos travelling faster than the speed of light, announced back in the Autumn of 2011, proved to be incorrect. One of the reasons that story caused so much interest is that travelling back in time has all kinds of problems associated with it, the movie *“Back to the future”* illustrated some of them. I will discuss time travel more in another blog.

## Time for a photon

I will finish this blog with a question about photons (particles of light). Remember that Einstein’s theory of special relativity is based on the premise that light *always* travels at the same speed in a vacuum. The nearest star system beyond our Solar System is the Proxima Centauri system, which is 4.2 light years away. That means it takes light 4.2 years to travel from this system to us, which in terms of kilometres is 40 trillion kilometres ( kilometres!). Now you know why we use light years for such large distances.

So if light takes 4.2 years to travel the 40 trillion kilometres from Proxima Centauri to Earth, my question to you is

*how long would it seem to take if you were a photon moving at the speed of light?*

Answers on a postcard, or in the comment section below.

on 30/05/2013 at 07:01 |john gribbinJust one point. It is the special theory of relativity, not the theory of special relativity. The theory is special, not the relativity.

on 30/05/2013 at 07:29 |RhEvansGood point, I shouldn’t try and write blogs at 5am. I will correct it forthwith.

on 30/05/2013 at 10:45 |Michael MerrifieldPhotons make lousy comedians: no sense of timing at light speed.

on 30/05/2013 at 10:47 |RhEvansAnd professors of astrophysics? What’s their excuse?

on 31/05/2013 at 01:12 |Rati MehrotraI’m looking forward to your time travel blog. I wrote a story where my protagonist travels back in time using a wormhole, resulting in a split timeline. But I kind of got lost in it…

on 31/05/2013 at 01:52 |RhEvansSplit timelines are, as far as I know, the only way to avoid all the paradoxes which would ensue.

on 01/06/2013 at 12:46 |Phillip Helbig“These large accelerations (changes in speed) which the travelling twin experiences break the symmetry, and so it really is the case that the travelling twin will return younger than the twin who has stayed on Earth.”Yes, but this implies that the accelerations are somehow responsible for the effect (which seems plausible since acceleration is like a gravitational field and a gravitational field also produces time dilation). However, make the journey much longer, but otherwise the same, in particular the accelerations are the same. However, the time dilation is greater. Does this show that the accelerations are not really responsible?

on 01/06/2013 at 19:32 |RhEvansAre you asking because you don’t know, or because you are going to offer a clearer explanation? I’d welcome one if you have one đź™‚

You are right, acceleration produces time dilation because acceleration is equivalent to gravity (Einstein’s

“principle of equivalence”). But as you also say, the time dilation I’ve mentioned, travelling at 99.2% of the speed of light, would lead to a“greater”age difference if the twin in the rocket were to go on a longer trip (as long as he maintained the same speed).There is, I seem to remember, a very good explanation of the twin paradox in the first chapter of Bernard Schutz’s book

“General Relativity”. I’ll have to dig out my copy.on 10/06/2013 at 11:03 |RhEvansI found the discussion I was looking about the “twin paradox” for in Bernard Schutz’s

“A first course in general relativity”. It is discussed in a fair amount of detail in an appendix to the first chapter, on pages 28-30.on 07/06/2013 at 04:59 |T.G.I.F! | Premium Thoughts[…] Einstein and time travel (thecuriousastronomer.wordpress.com) […]

on 15/01/2015 at 23:26 |Alex Watt“how long would it seem to take if you were a photon moving at the speed of light?”

is the answer to your question “no time at all”?

I had always thought that photons were timeless entities in a sense, as fresh as the day they were born.

on 16/01/2015 at 07:26 |RhEvansYes, you’re correct, for a photon travelling at the speed of light time stands still and so any journey seems instantaneous to it đź™‚

on 16/01/2015 at 08:48 |RhEvansOf course, an alternative way to think about the trip from the photon’s perspective is to consider the distance. Because of length contraction the distance in the photon’s frame of reference is zero, and so a photon is effectively everywhere simultaneously!

on 11/02/2015 at 11:00 |Time dilation in General Relativity | thecuriousastronomer[…] at the heart of SR, and show why time dilation occurs when one travels near the speed of light. In this blog here, I worked through some examples of time dilation in SR. But, what about time dilation in […]

on 28/01/2016 at 07:30 |Is Tim Peake getting younger or older? | thecuriousastronomer[…] I showed in this blog, the time dilation due to SR can be calculated using the […]