The Lorentz Transformation This is a derivation of the Lorentz transformation of Special Relativity. The basic idea is to derive a relationship between the spacetime coordinates, y, z, t as seen by observer O and the coordinates, y, z, t seen by observer O moving at a velocity V with respect to O along the positive ais. y y V O O These observers are assumed to be inertial. In other words, they are moving at a constant velocity with respect to each other and in the absence of any eternal forces or accelerations which is somewhat redundant). In particular, there is no rotational motion or gravitational field present. Our derivation is based on two assumptions:. The Principle of Relativity: Physics is the same for all observers in all inertial coordinate systems. 2. The speed of light c in a vacuum is the same for all observers independently of their relative motion or the motion of the light source. We first show that this transformation must be of the form t + b = dt + e z = z a) b) where we assume that the origins coincide at t = t = 0. The above figure shows the coordinate systems displaced simply for ease of visualization. The first thing to note is that the y and z coordinates are the same for both observers. This is only true in this case because the relative motion is along the ais only. If the motion were in an arbitrary direction, then each spatial coordinate of O would depend on all of the spatial coordinates of O. However, this is the case that is used in almost all situations, at least at an elementary level.) To see that this is necessary, suppose there is a yardstick at the origin of each coordinate system aligned along each of the y- and y -aes, and suppose there is a paintbrush at the end of each yardstick pointed towards the other.
If O s yardstick along the y -ais gets shorter as seen by O, then when the origins pass each other O s yardstick will get paint on it. But by the Principle of Relativity, O should also see O s yardstick get shorter and hence O would get paint on his yardstick. Since this clearly can t happen, there can be no change in a direction perpendicular to the direction of motion. The net thing to notice is that the transformation equations are linear. This is a result of space being homogeneous. To put this very loosely, things here are the same as things there. For eample, if there is a yardstick lying along the ais between = and = 2, then the length of this yardstick as seen by O should be the same as another yardstick lying between = 2 and = 3. But if there were a nonlinear dependence, say goes like 2, then the first yardstick would have a length that goes like 2 2 2 = 3 while the second would have a length that goes like 3 2 2 2 = 5. Since this is also not the way the world works, equations ) must be linear as shown. We now want to figure out what the coefficients a, b, d and e must be. First, let O look at the origin of O i.e., = 0). Since O is moving at a speed V along the -ais, the coordinate corresponding to = 0 is = V t. Using this in b) yields = 0 = dt + ev t or d = ev. Similarly, O looks at O i.e., = 0) and it has the coordinate = V t with respect to O since O moves in the negative direction as seen by O ). Then from b) we have V t = dt and from a) we have t, and hence t = dt/v = at av = d = ev and thus also a = e and d = av. Using these results in equations ) now gives us t + ba ) 2a) = a V t) 2b) Now let a photon move along the -ais and hence also along the -ais) and pass both origins when they coincide at t = t = 0. Then the coordinate of the photon as seen by O is = ct, and the coordinate as seen by O is = ct. Note that the value of c is the same for both observers. This is assumption 2). Using these in equations 2) yields t + ba ) ct = at + bc ) a = act V t) = cat V ) c cat V ) = = ct = cat + bc ) c a 2
and therefore V/c = bc/a or So now equations 2) become b a = V. t Vc ) 2 = a V t) 3a) 3b) We still need to determine a. To do this, we will again use the Principle of Relativity. Let O look at a clock situated at O. Then = V t and from 3a), O and O will measure time intervals related by t Vc ) 2 = a V 2 ) t. Now let O look at a clock at O so = 0). Then = V t so 3b) yields V t = = a0 V t) = av t and hence t = a t. By the Principle of Relativity, the relative factors in the time measurements must be the same in both cases. In other words, O sees O s time related to his by the factor a V 2 / ), and O sees O s time related to his by the factor /a. This means that a = a V 2 ) or a = V 2 and therefore the final Lorentz transformation equations are t = t V c 2 V 2 = V t V 2 4a) 4b) 3
It is very common to define the dimensionless variables β = V c and γ = =. V 2 β 2 In terms of these variables, equations 4) become t = γ t βc ) = γ βct) 5a) 5b) Since c is a universal constant, it is essentially a conversion factor between units of time and units of length. Because of this, we may further change to units where c = so time is measured in units of length) and in this case the Lorentz transformation equations become t = γt β) = γ βt) 6a) 6b) Note that 0 β γ <. We also see that γ 2 = β 2 γ 2 γ 2 β 2 =. Then recalling the hyperbolic trigonometric identities and cosh 2 θ sinh 2 θ = tanh 2 θ = sech 2 θ we may define a parameter θ sometimes called the rapidity) by β = tanhθ γ = coshθ 4
and γβ = sinhθ. In terms of θ, equations 6) become t = cosh θ)t sinhθ) = sinhθ)t + cosh θ) z = z which looks very similar to a rotation in the t-plane, ecept that now we have hyperbolic functions instead of the usual trigonometric ones. Note al both sinh terms have the same sign. 5