Some weeks ago, I was involved in a dispute with someone over some physics ideas. This person had said that Einstein’s energy-mass equivalence equation (that wonderful E=mc² thing) is functionally equivalent to the Lorentz transformation.
They are not.
“Why is this?”, I hear you ask. I’m glad you asked. Here’s why:
The Lorentz transformation is a complex set of equations aimed at describing – in mathematical terms – how people in two different inertial frames of reference would encounter the same event when one of these frames is moving. The transformation involves two frames of reference in standard configuration, meaning that the x-axis in a three-dimensional orthogonal set of axes in the stationary frame is aligned with the x-axis of the moving one, whilst the moving frame traverses space along the corresponding axis in the stationary frame.
To demonstrate this sort of thing to yourself, you could stretch out the thumb and the index and middle fingers of your right hand – all at right-angles to one another (which is what orthogonal means) – and (with the remaining fingers curled in, of course,) the arrangement of the fingers shows the arrangement of the axes in an orthogonal system. The index finger represents the x-axis, the middle finger represents the y-axis and the thumb points vertically upward, demonstrating the direction (both the orientation and the sense) of the z-axis. If you imagine an identical copy of that arrangement of fingers suddenly moving off in the direction of increasing x, that is pretty much what a pair of inertial reference frames in standard configuration sort-of looks like. Increasing x is the direction in which the index finger points. So that’s the physical set-up of the axes.
The mathematical bit comes up now. The stationary frame contains the axes x, y and z. The convention for labelling the axes in the moving frame is to use a dash, to differentiate the moving frame’s axes from those in the stationary one. The moving frame moves off at velocity v, and the distance between the respective z-axes (z and z’) increases with time, and the equation in non-Einsteinian relativistic mechanics is:
x’ = x – vt,
in which we are signifying the x-coordinate of the event being seen in these reference frames. As I said, this is not the version used in the Einsteinian version of relativity; this is the Galileian version. The Einsteinian relativistic version includes the Lorentz factor. This is a factor, formulated by Hendrik Lorentz, labelled as γ – or gamma – whose value is determined as being:
γ = 1 / √(1 – v²/c²),
which is one divided by the square-root of the difference between 1 and the quotient of the square of the velocity of the moving frame of reference and the square of the velocity of light. The equation for x’ will come shortly. In Einsteinian relativistic mechanics, with frames in standard configuration, the motion of the moving frame is entirely within the x-axis. This means that y’ = y, and z’ = z. Time in the moving frame is given by the equation t’ = t√(1 – v²/c²). The equation for x’ is this:
x’ = (x – vt)/√(1 – v²/c²).
Together, then, the Lorentz transformation – rather simplified – is as follows:
x’ = (x – vt)/√(1 – v²/c²)
y’ = y
z’ = z
t’ = t√(1 – v²/c²)
As I said, this transformation is aimed at the determination of spatial and temporal coordinates of an event happening in space-time in a frame of reference moving with respect to another frame, which is at rest. The aim of the energy-mass equivalence equation – E=mc² – is to demonstrate the way in which differences in mass can also be seen in terms of differences in rest-mass energy. The sets equations are entirely different, and their aims are correspondingly different. Ergo, the energy-mass equation and the Lorentz transformation are not, never have been and never will be functional equivalents of each other.