# A Roller Coaster Ride through Relativity/Adding speeds together

If I see your spaceship travelling past at a speed u and you see your bullets travel away from your ship at a speed v it turns out that I will see your bullets travelling at a speed:

${\displaystyle w={u+v \over 1+uv/c^{2}}}$

(For a proof of this formula see Appendix D)

If either u or v is much smaller than the speed of light the term 1 + uv/c2 on the bottom is essentially equal to 1 and the formula reduces to w = u + v which is what we would expect. If, however, you put u = c then you will find that a remarkable thing happens. A factor of c + v cancels out and you are left with just c. (And if you try adding together speeds which are greater than c the sum is actually less that the speeds you started with! But this is, of course, impossible anyway.) The speed of light is a kind of ceiling beyond which you just cannot go.

Suppose you had a multistage rocket each of whose stages could reach 80% of the speed of light. A two stage rocket would reach 80% ++ 80% = 97.56%. A three stage rocket would reach 97.56% ++ 80% = 99.72% and a four stage rocket, 99.97% of the speed of light. In fact you would need an infinite number of stages to reach 100%!

But what happens to the 1g rocket? That goes on accelerating at a constant rate so it must be getting faster and faster. Surely then it must eventually reach and exceed the speed of light, mustn't it?

You're right about it getting faster and faster with respect to the universe outside and it is also true to say that from the point of view of the occupants of the rocket, the acceleration is constant in the sense that they experience a constant artificial gravity of 1g. But the rocket never exceeds the speed of light. It turns out that the speed of the rocket increases according to the formula

${\displaystyle v=c\tanh {a \over c}t}$

(The details are given in Appendix A on the 1g rocket problem.)

This formula describes a curve which increases exponentially to a limit and for a rocket accelerating at 1g, it looks like this:

The speed of a rocket accelerating at 1g.

Just in case you think you are getting the hang of this relativity business, I should like to remind you of another consequence of Special Relativity.

Bizarre consequence number 10
Although it is impossible to travel faster than the speed of light, it is perfectly possible to travel to a distant star in less time than it takes for light to get there.

Oh, Come on! you cry, That doesn't make sense at all! You have just proved to me that you can't travel faster than light so how can you say, in the next sentence, that you can travel faster than light!

But I didn't say that.

Yes you did

No, I said it was perfectly possible to travel to a distant star in less time than it takes light to get there.

Well that's just the same thing!

No it isn't because it depends on whose time we are talking about. I will admit that I did phrase my statement in such a way as to make it sound a bit paradoxical, but the truth of the matter is that if you travel to Alpha Centauri at 80% of the speed of light, you will get there in 3 years - 3 of your years, that is. Light takes 4 years to get there. (For more details, see Appendix E)

Well, aren't you travelling faster than light then?

No. If you set out at the same time as a beam of light, the light will still get there first because it is travelling faster than you. That it only takes 3 of your years to get there can be explained in either of two ways. From my point of view (left behind on Earth) your clocks are ticking slowly because of time dilation. From your point of view the distance from Earth to Alpha Centauri is shrunk because of length contraction. Which explanation you chose depends on your point of view.

So how long does it take light to get there from the light beams point of view?

Now there you do have an interesting question. In fact it was this very question that started Einstein thinking about the whole business when he was still a student. At the speed of light time stops still and the whole universe shrinks to a point. I suppose it is meaningless to contemplate what the world looks like from a photon's point of view. Indeed, I rather doubt if a photon has a point of view; but if we must pursue the idea at all we have to conclude that, to a photon, the whole universe is nothing but a single point which exists for zero time. If God exists, He is a photon.

## The spiral

The roller coaster now enters a series of tight curves which cause it to go round in a horizontal circle faster and faster and faster. Some distance away a siren goes off and you are not surprised to hear the pitch of the siren wail up and down as first you approach and then recede from it.

Well, at least the Doppler effect works as normal you say.

Yes, but we are not travelling very quickly at the moment. See what happens to the colour of that green light on the railway track over there as we increase our speed.

After a while you remark I can't see anything wrong with that either. When we approach the light it goes blue and when we recede it goes red.

True enough. But there is something strange about the amount by which the colour of the light is shifted. In sound there are two quite distinct Doppler effects: the 'moving source' effect and the 'moving observer' effect. But in light there is only one Doppler effect and it isn't quite the same as either of them!

Interesting consequence number 11
There is only one Doppler effect in light

It is easy to see why. In sound, there are three bodies to consider – the source, the observer and, thirdly, the air through which the sound moves. Both the source and the observer can move through the air at different speeds so there are two Doppler effects, one for each. But in light, there is only relative motion between the source of the light and the observer. Therefore there can only be one Doppler effect in light.

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