General Relativity.

Son Goku

Basically:
Objections, questions, opinions and ideas.

Whatever you want to talk about on the subject.

Very vague, but I'd like to keep it open to anybody who wants to talk about anything to do with it or has ideas and questions.

Turdle

Ok here's the deal. Now I know my understanding of relativity is weak at best but I've got one major concern with it that, maybe somebody could help me with.
Relativity states that matter, and for that matter energy cannot travel faster than the speed of light. In fact as the velocity of matter approaches the speed of light, its mass approaches infinity. Okay, I understand that much. Relativity also states that the laws of physics are the same for every observer and that the observations made by two moving observers on each other are dependent on their relative velocities. Fine
Now, suppose we launched two space craft from Earth. Each of the craft were capable of accelerating to just over half the speed of light and we sent them on their way in opposite directions. This is the bit that confuses me When these two craft each reach their top speed of just over half the speed of light (relative to Earth), would they not be traveling faster than the speed of light relative to each other. According to relativity the observer on craft A could consider him/herself to be stationary, look out the window and see the Earth accelerate away and also craft B move off, eventually disappearing as it appeared to go beyond the speed of light. Is this not a contradiction?

Sev

Well special relativity isnt like your normal galilean concept of relativity, you cannot just add velocities, there is a more complicated expression you must use.

Based on the postulates of relativity you can derive the lorentz transformations, I'm sure there are many resources on the internet that give derivations, but I think theres 2 main approaches, ones kind of a geometric way, and the other more algebraic.

The lorentz transformations are... x' = γ(x-ut), y' = y, z'=z, t' = γ(t-xb/c)

It relates the coordinates x,y,z,t in a stationary frame of rerence to coordinates x',y',z',t' in a moving frame of reference where u is the velocity of the moving frame (which is only in the x direction), γ = 1/√(1-β²) and β = u/c

Given these transformations, by performing a bit of differentiation and swapping things around you can derive velocity transformations.

Essentially what you want to know is if an object has a velocity of dx/dt = -0.5c in the stationary frame, and the velocity of the moving frame is u = 0.5c, then what is the observed velocity of the object, dx'/dt' in the moving frame?

You can try work this out yourself by differentiation and shuffling around the lorentz transformations I gave you above.

You want to get an expression v' = dx'/dt' in terms of u and v = dx/dt. You can try it yourself if your differentiation is sound, have a go. I tried but then realised it was a bigger job than I expected, and gave up, but you should be able to do it.

Well actually it can be a little more complicated than that. The result of the above calculations is just the expression for velocity in the x direction, which is all your problem is concerned with. But a full expression of velocity would be √[(dx/dt)² + (dy/dt)² + (dz/dt)²].

But take a look on the internet, just type "velocity transformations" into google, theres hundreds of hits.

Turdle

Ah, so is the observed velocity of the stationary frame being <c due to time dialation?

Spear

Turdle wrote:

Ah, so is the observed velocity of the stationary frame being <c due to time dialation?

No. Time in the frame of each ship, as viewed from the frame of the Earth will appear to be slower. Whereas the ships will see time move slower on the earth, and slower still on the opposite ship. That's the result of time dilation. Any velocity, or observation thereof will always be less than c due the application of transformations from one frame to another.