Science Forums 2

[Erasmus00]

Relativity is often discussed in this and the strange claims forum. However, it seems to me many of the people who proffer opinions on the topic don't seem to understand many of the basics. Therefore, I've decided to take it upon myself to present a brief overview of some key concepts, starting with special relativity. I will try to keep the math level to algebra, although at times I have a feeling the only way to get from point A to point B will be calculus.

This first post will develop the idea of Lorentz transformations between reference frames. I'll try to add to it when I have time. My introduction to relativistic physics came largely from Feynman's lectures on physics, Gravity by Hartle, and An Introduction to Mechanics by Kleppner and Kalenkov, so I'm sure this will draw heavily on their presentation/pedagogy.

As I'm not writing history, I will take as a starting point that light moves at a speed c regardless of the reference frame. This was supported in Einstein's time by both theory and experiment (Maxwell's EM theory, and the Michelson interferometer experiments). Up untill Einstein, the equations to move between two observers reference frames were galilean. That is, to move from observer A (who uses x, y,z,t for his reference frame) to observer B (who uses x',y',z',t' and is traveling at velocity V relative to observer A) the following equations are used. (Note, for simplicity, I'm assuming at time 0 the two frames coincide, and I've used my freedom to orient the axis such that all motion is in the x direction).

x' = x-Vt (1)
y' = y (2)
z' = z (3)
t' = t (4)

Taking the derivative of equation 1 with respect to time, however, we get

v'= v- V (5)
This is the addition of velocities that we are used to, however, it is fairly obvious this equation can yield results above c, which we do not want. So what do we do? The only way to keep the velocity of light a constant in all reference frames is to allow time to change as we move from observer to observer.

So, lets postulate the most general transformation between two inertial frames.

x' = Ax+Bt (6)
y' = y (7)
z'= z (8)
t' = Cx+Dt (9)

The transformation is assumed linear, otherwise we couldn't have a simple 1 to 1 correspondance between events in each system. A nonlinear transformation could have acceleration in one system, even if we don't have acceleration in another. Clearly unacceptable.

Now, we do four thought experiments in the hopes of arriving at A, B, C, and D.

Experiment 1: Observer in the (x,y) system sees the origin of (x',y') move along x axis with velocity v.

Coordinates in (x,y) are x = vt (10)
Coordinates in (x',y') are x' = 0. (11)
Using this equation 6 becomes 0 =Avt+Bt, so B= -Av, and equation 6 now reads
x' = A(x-vt).

Experiment 2: Observer in the (x',y') sees the origin of (x,y) move along x' axis with velocity -v.
Coordiantes in (x,y) are x = 0. (12)
Coordinate in (x',y') are x'=-vt' (13)
Working with 6 and 9, we get
A(0-vt) = -v(0+Dt), or A=D.
Now we have x'=A(x-vt) and t'=Cx+At.

Now, experiment 3, A light pulse sent out from the origin along the x axis at time t= 0 . Its location is given by:
x= ct (14)
x'=ct' (15)
Using these in our transformation equations, we get C = -Av/c^2

Now, we just need a value for A. To get this value, we do one last experiment.
Experiment 4: A light pulse is emitted straight up the y axis at t=0.
In the (x,y) system, its position is given by
x = 0 (16)
y = ct (17)
In the (x',y') we get
x'^2 + y'^2 = (ct')^2 (to the moving observer, the light travels at an angle)

Using these and our transform equations, we arrive at A = 1/sqrt(1-v^2/c^2). We use the positive root so that when v=0, x' = x.

So, our general transformation is:

x' = 1/sqrt(1-v^2/c^2) [x-vt] (18)
y'=y
z'=z
t'= 1/sqrt(1-v^2/c^2)[t-vx/c^2] (19)

Notice, that when v/c <<1, then 1-v^2/c^2 is approximately 1, so 1/sqrt(1-v^2/c^2) is approxiamtely 1, and our transformations become
x' = x-vt
t'=t.

This is good. At slow speeds, we should expect that our intuition should be correct, and we have confirmed that it is.

So, to recap, we have mathematically developed the lorentz transformations from the first principle that the speed of light is constant in any reference frame. I'll continue when I have time. I next plan to discuss time dilation, non-simultaneity and time dilation. After that, I hope to discuss the idea of 4-vectors, on my way to developing E=mc^2 and other famous equations. I hope that if I develop 4 vectors using metric terminology, I can introduce in a purely qualitative way the ideas of general relativity.

Questions? Please ask.
-Will