Special Relativity & the Lorentz Transformation

In Special Relativity the Lorentz Transformation allows one to convert measurements such as distances and times made in one frame of reference to those of another. It is easiest to derive the transformation when the situation is symmetric, i.e., changing between the two frames of reference doesn't alter any of the parameters involved. The same transformation can then be used for both reference frames. Consider two spacecraft headed directly towards each other with a relative velocity of v. We can use unprimed variables for measurements made by the first spacecraft and primed variables for the second. In both cases the transformation is L.


The transformation is assumed to be linear and can be represented by four components of a matrix which only depend on the relative velocity.





By noting that a point in the second spacecraft doesn't move relative to itself while it appears to be moving with velocity -v to the first spacecraft, we can deduce B. The method can be generalized to find the any velocity, V', as it appears to the second spacecraft if its value for the first spacecraft, V, is known. This is the formula for the addition of velocities.



















With L the same for both reference frames we can use it twice to make a transformation from the first to the second spacecraft and then back again to the first. Since we should get the original values back the result is the identity matrix. Doing the multiplications and equating terms gives two more terms of the transformation leaving only one unknown, A.














This is as far as symmetry will take us. To go further Einstein had to assume that the speed of light was a universal constant. This is not unreasonable if space is homogeneous. We just have to be careful about directions though. A ray of light moving along the common line of the spacecraft will appear to be moving in different directions to the two observers. Let's say that it moves away from the first and towards the second. This allows us to simplify the expression for C and determine an expression for A.














We then have to use the minus sign so that the direction of time will be the same in both frames of motion.









So we have found the transformation in this particular case. We can use other transformations to convert to situations that are less symmetrical.














This derivation indicates that Relativity doesn't impose any constraints on time travel. There are transformations which will convert positive changes in time to negative ones. But at the same time they will also convert positive energies into negative ones. So it seems likely that if one could travel back in time one would find oneself in an antimatter universe which would be extremely hazardous.