# Addition of velocities: What actually happens?

1. Aug 12, 2007

### Holocene

Say a spacecraft is traveling just a few miles per hour under the speed of light. Now say a person on board this spacecraft starts running towards the front of the ship. If the craft is already moving just under light speed, and the person inside is running at a speed of say 8MPH, isn't the person inside the ship actually traveling faster than light, relative to space?

2. Aug 12, 2007

### arildno

No, because Galilean velocity addition is invalid when you are moving close to the speed of light.

3. Aug 12, 2007

### JesseM

See How Do You Add Velocities in Special Relativity? from John Baez's site. There is no such thing as absolute velocity so it doesn't make sense to talk about a ship moving at "just under light speed" in any absolute sense, but if the ship was moving at 0.9c in the frame of the Earth, and the runner was moving at 0.2c in the frame of the ship, then in the Earth's frame the runner would not be traveling at 1.1c, but only at (0.9c + 0.2c)/(1 + 0.9*0.2) = 1.1c/1.18 = 0.9322c, using the formula for relativistic velocity addition given in the link. This has to do with the fact that both the Earth and the ship use rulers and clocks at rest relative to themselves to measure velocities in their own frame, but each one observes the other guy's rulers to be shrunk and the other guy's clocks to be slowed-down and out-of-sync.

4. Aug 12, 2007

### jostpuur

Sounds dangerous terminology. Try to get rid of the idea of some kind of background aether.

5. Aug 12, 2007

### robphy

Of course, technically and special-relativistically speaking, Galilean velocity addition is always invalid.
As an approximation, it is okay for speeds much less than that of light.

6. Aug 13, 2007

### country boy

As an illustration of the problem, ask yourself whether its 8 mph faster for him or for the observer that sees him going nearly at the speed of light. The first can be done, but not the second.

7. Aug 13, 2007

### jcsd

It's 100% accurate if the velocities you are adding are zero-valued :tongue2:

8. Aug 13, 2007

### yogi

its all a consequence of how time and distance are measured in Special Relativity. Clocks moving at high speeds appear to slow down, and distances appear to shrink in the direction of motion when observed by an observer that is "at rest" The "at rest observer, however, has no privileged location - he simply considers himself at rests and makes observations about what is going on with clocks and rod lengths that are moving relative to him.

Last edited: Aug 13, 2007