Einstein's Addition of Vectors

[KingNothing]

Hey all, I guess Einstein has some counter-intuitive, at least naturally, vector addition formula whose result could be that two beams of light traveling towards each other is basically the same as one beam going towards a stationary object, right? Well..I hope you knwo waht I mean. That the two beams don't just 'add together' to go twice the speed of light if you make one beam stationary.

Anyhooo...why? I just don't get why. Where is the logic? Why is it like that?

 

[HallsofIvy]

It doesn't have anything directly to do with vectors. If object A is moving toward Observer C at speed v (relative to C) and Object B is moving toward observer C from exactly the opposite direction with speed u (relative to C) then A's speed relative to B (and B's speed relative to A) is (u+v)/(1+uv/c²). That can be derived from the Lorenz contraction formulas.

At the extremes- if u and v are very small compared to c, then we get very close to (u+ v)/(1+0)= u+v, the classical value. If u and c are both equal to c then we get
(c+c)/(1+1)= c. It is impossible to get a value above c.

 

[KingNothing]

Yes, I get the formulas, but I guess I'm just trying to make sense of it all in my head when I actually think about two objects moving towards each other.

 

[Hurkyl]

(In a particular reference frame) If two objects are moving towards each other with speeds u and v, then, indeed, the distance between them is decreasing at a rate of u + v.

That's seems to be the easy part to get.


The next question is "what is the speed of the second object in the frame where the first is stationary?"

The thing that you must remember is that different reference frames measure lengths and durations differently. In particular, there's no reason to think that the rate of change of the distance between the two objects should be the same in two different reference frames.


In SR, the relative velocity between two objects is defined to be this change of distance that gets computed when we change into the reference frame of one of the two objects. Since there's usually a change of reference frames involved in computing this, we should expect that the relative velocity between two objects will usually be different than the sum of the velocities we computed in a different frame.

 

[arildno]

Yes, I get the formulas, but I guess I'm just trying to make sense of it all in my head when I actually think about two objects moving towards each other.

You are not alone in this "predicament"!
But a very important issue here is:
Should the laws of physics be inferred from actual experience, or should only such laws of physics be accepted which intuitively "make sense" in our minds?

A really shocking fact of nature is that the speed of light does not follow the "natural", Galilean transformation of velocities; the speed of light has been measured to be the same irrespective of two observers' relative velocity to each other!

This means, that the addition law of velocities that do "make sense" intuitively (i.e, the Galilean) cannot be a correct law of nature.

 

[baffledMatt]

There is a way to understand this geometrically. The Lorentz transformation is simply a rotation in space-time (try drawing some world lines and you will see what I mean) What you are describing is adding the velocity of something in your frame to the velocity of something else in someone elses frame. When you want to add vectors from a different frame to ones in your own you must apply this 'rotation' first, which is exactly equivalent to all this mucking about with length contrations, time dilations etc.

This is not a new or fancy idea. Say i want to add two vectors together which are defined in two different coordinate systems (say they differ by a rotation), I have to transform one of the vectors (perform the rotation to it) to bring it into the coordinate system of the other before I can add their values together.

The fantastic thing about relativity is that it can all be thought of purely geometrically, with the strange effects you observe simply being the consequnces of transformations between coordinate frames. In fact, I think the only real 'physics' in any of it is in Einstein's field equations.