Beginner needs answers to basic questions about the speed of light

[ksinelli]

So I've been reading Brian Greene's The Elegant Universe, and I'm not quite sure I understand general relativity. This is the way it was roughly presented in the book.

Two friends are throwing a baseball back and forth at 20 m/s. However, when one of the friends begins to run away from the ball at 12 m/s when the ball is approaching him, the ball really only approaches him at 8 m/s.

Seems simple enough so far.

But if a beam of light approaches you at 670 million mph and you hop in your spaceship and fly away from that beam of light at 100 million mph, the beam of light will still be approaching you at 670 million mph.

This seems to imply that the beam of light "speeds up" by 100 million mph to maintain it's approach of 670 million mph.

But shortly after this description, the book says that the speed of light never speeds up or slows down, it is always constant.

Well how can that be? If the beam of light didn't "speed up" in order to maintain an approaching speed of 670 million mph towards the space ship, then the spaceship would still be speeding away at 100 million mph and the beam of light would only be approaching at 570 million mph.

Can somebody clarify this please? Thanks much.

 

[Cyosis]

It's the second postulate of special relativity. No matter in what reference frame you are the speed of light always travels at a constant c. Seeing as objects with mass act very differently it's weird to accept this for light, but accept it we must. This is the reason for time dilation and Lorentz contraction.

 

[JesseM]

Different observers measure the speed of anything using rulers and clocks which are at rest relative to themselves, and synchronized in their own frame using the Einstein clock synchronization convention. But the rulers of different observers don't measure distance the same way (each observer measures rulers that are moving relative to themselves to be shrunk by a factor of sqrt(1 - v^2/c^2) relative to their own ruler/clock system, the effect known as length contraction), the clocks of different observers don't measure time the same way (each observer measures clocks that are moving relative to themselves to have the time between ticks expanded by a factor of 1/sqrt(1 - v^2/c^2), the effect known as time dilation), and clocks that are synchronized in their own frame will be measured to be out-of-sync in other frames (clocks that are synchronized and have a distance x between them in their own rest frame will be out-of-sync by vx/c^2 in another frame where they are moving at speed v, due to what's called the relativity of simultaneity). So, it's because of all these differences that two observers can each think light is moving at c relative to themselves (also, in your baseball example you actually wouldn't measure the ball to move at exactly 8 m/s in relativity, the relativistic velocity addition formula says it'd be slightly different, although very close at velocities that are such a tiny fraction of light speed).

For a numerical example of how all these effects come together to ensure both observers measure a single light ray to move at c, see my post #6 on this thread.

 

[diazona]

You've stumbled on to exactly one of the things that confuses so many people about relativity

According to our experience, it makes sense that if one thing is moving at velocity u and another thing is moving at velocity v (in the same direction), then one sees the other moving at velocity u - v. This is called a Galilean transformation. So you'd think that a spaceship is flying at 100 million mph (relative to, say, a planet) and a light beam is traveling in the same direction at 670 million mph (relative to the same planet), the ship sees the light beam traveling at 570 million mph - and that if the ship actually sees the light moving at 670 million mph, it means the light must have sped up.

Einstein's insight was to say that that's wrong. He started from the assumption that everybody sees light traveling at 670 million mph, but threw everything else out, including the Galilean transformation. If you assume, as Einstein did, that the light beam is seen as traveling at 670 million mph by both the ship and the planet, you find that space and time have to be distorted in order to keep the speed of light the same. If you're familiar with length contraction and time dilation, that's what I'm talking about. It turns out that there's a new velocity addition rule that replaces the Galilean transformation:
$$u' = \frac{u - v}{1 + uv/c^2}$$
where c is the speed of light. In this case, u is the speed of light relative to the planet, v is the speed of the ship, and u' is the speed of light relative to the ship. If you work through the math, you find that both u and u' are equal to c.

 

[neopolitan]

$$u' = \frac{u - v}{1 + uv/c^2}$$
where c is the speed of light. In this case, u is the speed of light relative to the planet, v is the speed of the ship, and u' is the speed of light relative to the ship. If you work through the math, you find that both u and u' are equal to c.

Little correction - you gave something like the velocity addition equation, but that should be:

$$u' = \frac{u + v}{1 + uv/c^2}$$

I think you have to define what the speed of the ship is relative to as well (planet makes sense in your description).

cheers,

neopolitan