Special relativity and time dilation

Now that we know what a theory is, we can tackle one of the biggest theories in physics: general relativity, Einstein's masterpiece. We should, however, start with special relativity to prepare the playing field a bit. This post (and the next one) will be a review of the things we discussed on Periscope recently. There will be some maths, just in case you want to check the numbers I give, but you can happily skip all the maths and still understand what I'm saying.

In order to understand Einstein's theories, we need to understand the context in which he developed them; which means we need to start with Newton.

1. Newtonian gravity

Newton invented calculus, a law of
gravity, and ridiculous hair styles.

Newton, as well as inventing calculus, came up with an idea in 1687 that was revolutionary for his time: things fall down. But they don't just fall down, they fall towards each other. Specifically, he realised that any two objects in the universe - no matter how big or small or how far away they were - would attract each other via a gravitational force, as long as they had mass.

This gravitational force is attractive, universal, proportional to the mass of the objects, and inversely proportional to their distance. Which means the closer together and the bigger the objects are, the more gravitational pull they feel.

That is why we are stuck on the surface of our planet: Earth is huge and very close to us, so it attracts us. The Sun is huge and relatively close to Earth, so it keeps Earth in a stable orbit. Gravity is also very weak, which is why something as small as a magnet can pick up a paper clip that is being held down by the whole planet's mass.
Newton's law of gravity was, and still is, very useful for describing gravity in everyday life, but it had it's problems. The first problem Newton encountered was that he couldn't explain why gravity exists, nor how it could affect things at a distance. It also had serious problems explaining the movement of things that travelled really fast, such as light beams. It was incomplete, but it was the best he could come up with in his time.

2. Special relativity
In the second half of the 19th century, scientists were trying to find an aether; a mysterious medium through which light (and gravity) propagated. In this point of view, the laws of physics would depend on how people moved relative to this aether, despite the fact that this contradicted experiments. Then along came Einstein to shake things up a bit, he decided to accept the experimental results as true, and modify the theory, and in doing so he removed the need for an aether.

The first concept we need to understand is that there is no such thing as absolute motion; everything moves relative to one another. As an example, imagine you are on a train travelling at 100 km/h. You could quite happily argue that you are at rest, and everything else is moving. If you were to throw a ball out of the window at 10 km/h in the direction the train is moving, you would see the ball as moving at 10 km/h. An observer outside the train, however, would see the ball moving at 110 km/h, as they would have to add on the speed of the train. Motion depends on the observer; it depends on the frame of reference from which you measure.

Next we can introduce the concept of inertial frames of reference: these are just systems that move at constant velocity, with no acceleration. If you are in a car travelling at 50 km/h constantly, you are in an inertial frame. However, when you slow down at a traffic light, you are changing your speed, so you are no longer in an inertial frame.

Armed with these pieces of knowledge, we can now formulate the postulates of special relativity, as Einstein did in 1905. Note that a postulate is something we use to build a theory; they are facts that have been proven to be true, but we don't necessarily need to know why they are true.

1. The laws of physics must be the same in all inertial frames of reference (thus eliminating the idea of an aether).
2. The speed of light in a vacuum is the same for all observers, regardless of the velocity of the source/observer.

These postulates matched all the observed phenomena, and changed the way people thought about motion. From these two postulates, we reach an obvious conclusion: the idea of simultaneity is relative. If you measure two events as happening at the same time, it doesn't necessarily mean other observers will also see these events happening at the same time. Einstein had a useful thought experiment to illustrate this. Imagine you are on a moving train (at a constant speed!), standing directly at the centre, and you fire two laser beams in opposite directions.

You will see the laser beams reach the ends of the carriage at exactly the same time, as seen in the image on the top. An observer outside the train, however, will see the train as being in motion, so for them the light beams will not reach the ends of the carriage at the same time. This is because from their point of view, the light beam moving in the direction of travel has a greater distance to cover, because it has to 'catch up' with the movement of the train, as seen in the bottom image.

The next big question we need to ask ourselves is how can the speed of light be constant? Let's imagine we are on separate rockets, and for now we'll work from my frame of reference. My rocket is at rest, and your rocket is moving past me at 0.5c (half the speed of light). In your rocket, you are bouncing a light beam off two mirrors; one on the ceiling on one on the floor of your rocket. You will measure the light beam as travelling at the speed of light. From my frame of reference, I will also measure the light beam as travelling at the speed of light, even though the light is travelling a greater distance (because the mirrors are 'running away' from the light beam at the speed the rocket is moving.

Image source: How to: Special Relativity

In the above image we can clearly see that from my point of view, the light beam has a greater distance to travel. Velocity is always defined as distance

travelled over time it takes, so if you and I measure the same velocity, but a different distance, something else also has to change. Specifically, the time it takes to make the journey also has to change.

3. Time dilation in special relativity
If we impose that the speed of light has to be constant, and different observers see light travelling different distances, the clocks must also tick differently. This is the main idea of time dilation: time does not pass the same for people moving at different speeds. Recovering our example of the rockets above, if you are travelling at 0.5c, I will see clocks on your ship as running slower. By the time I've seen your clocks tick one second, mine will have ticked 1.15 seconds. You will still feel time as running normally, but I will see time passing slower for you.

We can see some specific examples of this if we look at the formula on the right: if I see you travelling at 0.5c, one second for you is 1.15 seconds for me. If you are travelling at 0.9c (90% the speed of light), one second for you will be 2.3 seconds for me. If you travel at 99.9% the speed of light, one second for you is 22.3 seconds for me. The faster you travel, the more I see your clocks slowing down.

This effect, however, is not enough to keep the speed of light constant; there is an analogous effect known as 'length contraction', where lengths increase for observers moving fast (1 meter for me is 1.15 meters for you), but I won't go into that effect today. This idea of time dilation for moving frames of reference has some important consequences:

• Space and time are not separate entities, they are united. Einstein replaced the notion of space and time, and instead introduced spacetime; two sides of the same coin.
• Time slows down as we approach the speed of light, so if we reached the speed of light, time would stop completely. You can extrapolate to say that if you move faster than the speed of light; time would move backwards. But nothing can travel faster than the speed of light; as such clocks can't tick backwards. This means that time travel to the past is not possible.

Don't believe me? Check the maths!

Another argument one could naively make is the following. If I'm travelling at 0.6c, and you are travelling at 0.6c in the opposite direction, surely I can argue that you are travelling away from me at 1.2c, therefore you are travelling faster than the speed of light from my frame of reference. This was true from a classical approach, but completely wrong from a relativity approach: you can't just add velocities together like that. When dealing with high speeds, we need to take into consideration time dilation and length contraction. If we take into consideration these effects, you would actually see me moving away at 0.95c, so still within the allowed speed limit.

You might be thinking at this point that we've done everything from my frame of reference. What happens if we change to your frame of reference? Actually, as we've seen that motion is relative, you could argue that you are at rest, and my rocket is moving. In this case, you will see my clocks as running slow. Time dilation caused by motion is also relative: you will see time passing slower for me, and I will see time passing slower for you, which leads us to an interesting paradox.

4. Twin paradox
Imagine we have twin sisters, Elizabeth and Laura. Laura leaves on a round-trip mission to Alpha Centauri, the closest star to the Solar System (except the Sun, of course). This star is 4 light years away (this means it takes light four years to make the journey), and Laura will be travelling at a constant speed of 0.8c. Elizabeth will be staying on Earth, monitoring her sister's journey.
In Earth time, which is the same as Elizabeth's time, the round trip takes 10 years:

$t=\frac{2*d}{v}= \frac{2*4}{0.8}=10$y.

We are interested in how much Laura will have aged from Elizabeth's point of view. We can use the first formula I introduced above to calculate this (or skip the calculation and trust my numbers!):

$t = \frac{t'}{\gamma}= t' \sqrt{1-\frac{v^2}{c^2}} = 10*\sqrt{1-\frac{(0.8c)^2}{c^2}} = 10*0.6 = 6$y

This means that from Elizabeth's point of view, Laura's ship has only felt 6 years, even though Elizabeth felt 10 years. From Elizabeth's view point, when they reunite on Earth, Elizabeth will be 10 years older, while Laura will only be 6 years older.

Now comes the paradox. If we said there are no preferred frames of reference, Laura could argue that she is stationary, and Elizabeth is the one moving. Therefore, from Laura's point of view, she will age 10 years, and Elizabeth will only age 6 years. They can't both be right, so who is correct?

While this result might seem paradoxical, it actually isn't. Laura is naively applying the conditions of special relativity, however we stated above that special relativity works in inertial frames of reference. Laura's ship, however, is not an inertial frame of reference. This is because her ship turns around, and during that time she is changing her velocity. Changing velocity means acceleration, which means it's not an inertial frame.

So how do we study acceleration? Einstein saw that his idea was valid for inertial frames, so he tried to apply it to non-inertial frames. To do that, he needed to think bigger: he needed general relativity. For now, rest assured that the paradox does have a solution: Laura will be younger than Elizabeth after the trip, in the same way that astronauts on the ISS age slower than people on Earth. We'll find out how to deal with acceleration in the next post.


In the second part, we'll discuss general relativity, time dilation from a GR point of view, and we'll revisit the twin paradox, armed with new tools to tackle it better. I'll also discuss some specific examples like the ISS, a possible Mars colony, and of course, everyone's favourite time dilation movie; Interstellar. Be sure to check it out!