OK. Now we are about to see just how Einstein confined an age-old and apparently obvious and valid mathematical principle to the scrap heap.

But first, here’s a little reminder about how we have got to this point:

It all starts with your marble that is rolling along the corridor of one of the glass-roofed carriages of the majestic Glacier Express as it makes it way through the Alps. Remember we talked about your Grandad on top of one of the magnificent peaks. We calculated that he sees your marble roll at a total speed of (speed of marble) + (speed of train). Click here to read that again.

Einstein called this the “Theorem of the Addition of Velocities”. I recommend that you bandy this phrase about next time you’re in the pub for a bit of kudos. All we’re doing when we apply the Theorem of the Addition of Velocities is taking two figures and adding them together. This is the only way to take into account the momentum that the movement of the carriage gives to the speed of the marble after you’ve given it a nudge with your finger. Seems straightforward enough.

The principle is actually perfectly valid usually

For the (relatively) tiny speeds that we are on Earth are able to achieve, the Theorem of the Addition of Velocities works perfectly well. It’s when the calculations get bigger that the principle starts to become inaccurate, for example when you move from maths involving 100km/h to maths involving 200,000km/h.

How did Einstein think through the Theorem of the Addition of Velocities?

Einstein imagined two forks of lightning striking the embankment next to a railway line at Points A and B. I don’t know about you, but I find place-markers named after letters a bit clinical and uninteresting, so instead I’m going to imagine that the bolts of lightning strike a junction box and an abandoned bathtub. Let’s imagine that your girlfriend is sitting on the grass of the embankment next to the railway line. The weather is taking a cloudy turn and she decides to shelter for the moment under a shady oak tree. The oak tree is located exactly at the mid-point between the junction box and the bathtub.

Got the picture? You know what’s coming, of course. Our two bolts of lightning are going to strike the junction box and the bathtub at the same time, and very obviously we appreciate that the light from each of those strikes is going to reach your girlfriend’s eyes at the same time. In other words, your girlfriend will see both lightning strikes at the same time.

Now here comes the Glacier Express:

Now let us compare your girlfriend’s experience of the lighting strikes with how you experience the same strikes from where you are on the moving train. You look up from your marble-rolling and glance through a carriage window at exactly the point when you pass by your girlfriend under the oak tree. Let’s give the Glacier Express a speed of 100 kilometres per hour which doesn’t sound unreasonable. (And we’ll not ask ourselves why someone has abandoned a bathtub in the middle of the beautiful Swiss Alps.)

The question for us will be: for you on the moving train, do the two lightning strikes happen at the same time as each other? Which, of course, is the same thing as asking whether the rays of light from both lightning strikes reach your eyes at the same time. You do realise, of course, that because I am asking that very question, the answer is going to be No

At the moment of the lightning strikes, as the junction box and the bathtub burst into flames, you are moving. When you are level with your girlfriend under the tree, you are travelling towards the beam of light coming from the bathtub and away from the beam of light coming from the junction box. You are travelling at a speed of 100 km/h because that is the speed of the train. Pause for a moment here, and get your head round the fact (it took me a while) that however quickly you freeze-frame that image of yourself whizzing along by the train window, you cannot escape the fact that you are moving whilst your girlfriend is not.

Here we arrive at an important realisation –

  1. We know what the speed of light is, and therefore we can calculate how long it takes the light from the strikes to travel any distance we like;
  2. We know that you are moving away from the junction box and towards the bathtub;
  3. We therefore know that the distance between the (lightning strike on the bathtub and your eyes) is less than the distance between the (lightning strike on the junction box and your eyes);
  4. We know the trusty and reliable formula from school that (distance) = (speed) x (time), which we can reformulate to read (time) = (distance) / (speed);
  5. We therefore know that the time taken for the light to travel to your eyes from the bathtub is less that the time that the light takes to travel to your eyes from the junction box.


What if we trade up from a trusty old train to an intergalactic spacecraft?

Granted: the time difference between the two strikes on the bathtub and the junction box is going to be tiny. This is a feature of the slow speed of the train. The difference (and I’m totally guessing here in a flamboyant way to make the point) will be something like 0.00000000000000001%. So we don’t really notice it …

… until we we multiply up the rather-pathetic-by-intergalactic-standards speed of the Glacier Express. Let’s put you instead in a spaceship travelling at 200,000 kilometres per hour. Imagine that the junction box and the bathtub are really nearby stars. That tiny and barely noticeable time difference suddenly becomes pretty big. Your increased velocity means you are further past that midpoint at the moment when the light sources are emitted. That’s why (unlike with the Glacier Express story) the time difference between you seeing the light from the star ahead of you, and then the light from the star behind you, is much greater and very much more noticeable.

But what about your girlfriend? 

Ah yes. Let’s not forget about your girlfriend. We need to come full circle in our thinking and finish where we began. We sat your girlfriend on the embankment under a tree exactly at the midpoint between the junction box and the bathtub, remember? And we realised the very obvious fact that she saw both lightning strikes at exactly the same time.

Precisely the same principle applies if we were to sit her at the window of an intergalactic spacestation exactly at the midpoint between our two stars. Let’s imagine that she sees you zoom by in your spaceship in much the same way that in the shady tree example she sees you zip by on the Glacier Express. If events occurred on those two stars at the same time and caused an emission of light (I’m not at all sure that you can get lightning strikes in space, so this is kind of an equivalent), your girlfriend would therefore say that the light was emitted from both sources at the same time.

Are you thinking, so what? Well, think of it like this.

  1. On board your moving spacecraft you say that the light coming from the star ahead of you was emitted before the light from the star behind you;
  2. On board the stationary space station, your girlfriend says the light from both stars was emitted at the same time.

And this difference is explained, here is Einstein’s theory, simply by the fact that one observer is moving relative to the other. This is a very important point in our thinking. What is Time? When does an event occur? Did you think that Time is a forward-marching force of Physics that is forever reliably steady and constant? Is the progress of Time something that everyone can agree on, whatever their position and viewpoint?

Be prepared to have your faith shaken! The example of You on the Spacecraft and Your Girlfriend on the Space Station shows that the same event does not always happen at the same time for all observers. Time is flexible and is different for everyone. It’s just that here on Earth with our non-intergalactic and rather humdrum Earthly speeds, the difference is too tiny for us to notice. You can only understand the Time that events occur if you know where you are judging that event from. Is it from the spaceship or the space station, for example?