Travelling faster than the speed of light

Question

Let's say I fire a bus through space at the speed of light. If I'm inside the bus (sitting on the back seat) and I run up the aisle of the bus will I in fact be traveling faster than the speed of light? Relative to earth that I just took off from.

Answer 1

Your question has to do with addition of velocities in special relativity. For objects moving at low speeds, your intuition is correct: say the bus move at speed $v$ relative to earth, and you run at speed $u$ on the bus, then the combined speed is simply $u+v$.

But, when objects start to move fast, this is not quite the way things work. The reason is that time measurements start depending on the observer as well, so the way you measure time is just a bit different from the way it is measured on the bus, or on earth. Taking this into account, your speed compared to the earth will be $\frac{u+v}{1+ uv/c^2}$. where $c$ is the speed of light. This formula is derived from special relativity.

Some comments on this formula provide direct answer to your question:

1. If both speeds are small compared with the speed of light, they approximately add up as your intuition tells you.

2. If one of the speeds is the speed of light $c$, you can see that adding any other speed to it does not in fact change it: the speed of light is the same in all reference frames.

3. If you add up any two speeds below $c$, you end up still below the speed of light. So, any material object which has a mass (unlike light, which doesn't), moves at a speed less than $c$. Adding to it according to the correct rule makes it closer to the speed of light, but you can never exceed it, or in fact not even reach it.

I'd recommend Wheeler and Taylor's "Spacetime Physics" to read about this. Unlike the reputation of the subject it is actually pretty intuitive (I learned that formula in high school).

Answer 2

No. Relative to Earth your bus will have zero length, so moving from back to the front of the bus will contribute nothing to your speed relative to Earth.

Answer 3

I will have to answer this quickly, for I suspect this question will be closed. However, this thought experiment is similar to what Einstein thought about 10 years before he published his paper on special relativity. The problem is this. If you were on a reference frame moving at the speed of light you would observe that light, or any electromagnetic wave, as a wave of oscillating electric and magnetic fields. However, this would be stationary, which contradicts the Maxwell equations for the propagation of electromagnetic radiation. Einstein worked to fix this contradiction, which lead to special relativity. The conclusion is that you can’t place yourself on a frame where light is observed to have any velocity other than the speed of light $c~\simeq~300,000km/sec$.

Answer 4

Hmm.

Assume that your bus is approaching the speed of light, because if it had reached it, its mass would be infinite and the question becomes metaphysical as far as the contents and passengers.

Generally, momentum conservation insures that the bus would drop back from the speed it has to compensate for your momentum,as long as you are airborn but when you hit the front glass, it will gain it back.