A Derivation of the Lorentz Transformation of Space and Time Coordinates
G.R.Dixon, August 3, 2008
Given two inertial frames of reference (say K and K’) with the usual relative motion, three of the basic tenets of Special Relativity Theory are (1) an observer, using the rectangular coordinate grid and clocks of K, will measure the grid of K’ to be contracted in the x direction; (2) the same observer will measure the clocks at rest in K’ to run more slowly than those of K; and (3) the clocks in K’, with different x’ coordinates, will not be synchronized.
The intriguing thing is that the same things will be found for the grid and clocks of K, when the grid and clocks of K’ are used to make the measurements. A "symmetry of disagreement" exists, and it is not possible to definitively demonstrate that one party is correct and the other is wrong.
The contraction of another frame’s grid, and the slower rate of its clocks are both specified by the factor
The amount by which the other frame’s clocks are out of synch is less obvious, but derivable. Such a derivation is provided in this article.
We begin by imagining that we are an observer in K, watching an observer in K’ attempt to synchronize the K’ clocks. (K’ moves in the positive x direction of K at speed v.) We shall consider three K’ clocks: (1) the clock at x’=D; (2) the K’ origin clock; and (3) the clock at x’=-D. The K’ observer sends out a pulse of light in the positive and negative x’ directions. Since (according to him) the speed of light is c in all directions, relative to K’, he posits that the clocks at D and –D are synchronized with the origin clock if they are set to D/c upon reception of the pulse.
Let us call (1) the sending out of the pulse(s) "Event 1"; (2) the reception of the pulse at x’=D "Event 2"; (3) the reception of the pulse at x’=-D "Event 3." The K’ space-time coordinates of Event 1 are
And let us begin by considering the arrival of the pulse at x’=+D.
The K’ observer says that the K’ space-time coordinates of Event 2 are
From the perspective of frame K, the same events have the coordinates
At time t=0, the distance to the K’ clock at x’=D is only g-1D (owing to length contraction). And since the pulse speed is finite, the clock moves to the right while the pulse is en route. Specifically, if the time for the pulse to propagate from the K’ origin clock to the targeted K’ clock is Dt then when the pulse arrives that K’ clock will be at
We can solve for Dt by also noting that
And of course t2 equals Dt:
Substitution into Eq. 5 produces
Now while the pulse of light was en route, the K’ origin clock has been running. According to K, when the pulse arrives at the targeted x’ clock, the K’ origin clock will have advanced from to’=0 to
According to K, this is what the targeted K’ clock should be set to if it is to be truly synchronized with the K’ origin clock. The actual setting of (D/c) is less than this value. K therefore perceives that the K’ clock at x’=D runs behind the K’ origin clock by the amount
And this lack of synchronization will (in the opinion of K) persist as time passes.
Let us now consider a random event, with space-time coordinates x and t. Our only requirement for now will be that x is greater than vt (the location of the K’ origin clock when the event occurs). This being the case, the distance between the event and the K’ origin clock will, in the opinion of K, be (x-vt). But using the grid of K’, this maps to
At time t the K’ origin clock will read g-1t. And the K’ clock at x’ will read x’v/c2 behind that. Therefore
Or, in terms of K parameters,
Let us now consider the synchronization of the K’ clock at x’=-D. We shall again say that Event 1 is the sending out of a light pulse from the K’ origin clock. Thus once again
Event 3, the reception of the light pulse by the K’ clock at x’=-D, has coordinates
Again we have
And the distance between the K’ origin clock and the targeted K’ clock is (according to K) only g-1D. If Dt again denotes the pulse’s time of flight, then in this case
so that Dt solves to
Once again, t3 equals Dt:
Substitution for Dt in Eq. 18 produces
When the pulse reaches the targeted K’ clock, the K’ origin clock will have advanced from zero to
According to K, this is what the targeted K’ clock should be set to in order to be synchronized with the K’ origin clock. But in this case the actual setting, D/c, is greater. Thus K concludes that the K’ clock at x’=-D runs ahead of the K’ origin clock by the amount
Let us again consider a random event with coordinates x and t. This time we address the case where x is less than vt (the location of the K’ origin clock when the event occurs). The distance between the event and the K’ origin clock is (vt-x). In K’ terms this maps to a distance of g(vt-x). Or, since x’ is negative,
At K time t the K’ origin clock will read g-1t. But the x’ clock will read –xv/c2 more than that. Thus
which is identical to Eq. 13. Therefore
In conclusion, given an event with K space-time coordinates x and t, the event’s K’ space-time coordinates will be
These two equations are the celebrated Lorentz transformations for space and time coordinates.
Of course from the perspective of K’, it is the K clocks that are not synchronized. K’ contends that the K clocks on the negative x axis run behind the K origin clock, whereas the K clocks on the positive x axis run ahead of the K origin clock. In this case it is readily demonstrated that, given an event with K’ space-time coordinates x’ and t’, the K coordinates will be