A mathematical kludge and its repair

If you are in a train going x miles an hour and throw a paper airplane forward at x feet per second (or x * 3600/5280 miles per hour, relative to someone outside the train sees the plane move a bit faster than x miles an hour.  Well that’s the whole idea of the Galilean transformation.  Except that they don’t really see velocities adding that way for really fast velocities (close to the speed of light).

Relativity says that there are no privileged sites of observation and that no matter how fast two observer frames are moving relative to each other light will zing past both at the same speed (3 x 10^8 meters/second, 186,000 miles/second).

All of Newton’s mechanics and force laws obeys the Galilean transformation (e.g. velocities add).  Maxwell conceived a series of 4 laws linking electricity and magnetism together, which predicted new phenomena (such as radio waves, and the fact that light was basically a form of wave traveling through space).

Even though incredibly successful, Maxwell’s laws led to an equation (the wave equation) which didn’t obey the Galilean transformation.  This led Lorentz to modify it so the wave equation did obey Galileo.  If you’ve got some mathematical background an excellent exposition of all this is to be found in “The Geometry of Spacetime” by James J. Callahan pp. 22 – 27.

The Lorentz transformation is basically a kludge which makes things work out.  But he had no understanding of why it worked (or what it meant).  The equations produced by the Lorentz transformation are ugly.

Here are the variables involved.

t’ is time in the new frame, t in the old, x’ is position in the new frame x in the old. v is the constant velocity at which the two observation frames are moving relative to each other. c is the common symbol for the velocity of light.

Here are the two equations

t’ =  ( t – vz/c^2 )/ sqrt (1 – v^2/c^2)

x’ = ( z – vt ) /  sqrt (1 – v^2/c^2)

Enter Einstein — he derived them purely by thought.  I recommend Appendix 1 in Einstein’s book “Relativity”.  Amazingly you do not need tensors or even calculus to understand his derivation — just high school algebra (and not much of that — no trigonometry etc. etc.)  You will have the pleasure of watching the power of a great mind at work.

One caveat.  The first few equations won’t make much sense if you hit the appendix without having read the rest of the book (as I did).

Light travels at c miles/hour, so multiplying c by time gives you where it is after t seconds.  In equations x = ct.  This is also true for another reference frame x’ = ct’.

This implies that both x – ct =  0 and x’ – ct’ = 0

Then Einstein claims that these two equations imply that

(x – ct) = lambda * (x’ – ct’) ; lambda is some nonzero number.

Say what?  Is he really saying  0 = lambda * 0.

This is mathematical fantasy.  Lambda could be anything and the statement lacks mathematical content.

Yes, but . . .

It does not lack physical content, which is where the rest of the book comes in.

This is because the two frames (x, t) and (x’ , t’) are said to be in ‘standard configuration which is a complicated state of affairs. We’ll see why y, y’, z, z’ are left out shortly

The assumptions of the standard configuration are as follows:

  • An observer in frame of reference K defines events with coordinates t, x
  • Another frame K’ moves with velocity v relative to K, with an observer in this moving frame K’ defining events using coordinates t’, x’
  • The coordinate axes in each frame of reference are parallel
  • The relative motion is along the coincident xx’ axes (y = y’ and z = z’ for all time, only x’ changes, explaining why they are left out)
  • At time t = t’ =0, the origins of both coordinate systems are the same.

Another assumption is that at time t = t’ = 0 a light pulse is emitted by K at the origin (x = x’ = 0)

The only possible events in K and K’ are observations of the light pulse. Since the velocity of light (c) is independent of the coordinate system, K’ will see the pulse at time t’ and x’ axis location ct’, NOT x’-axis location ct’ – vt’ (which is what Galileo would say). So whenever K sees the pulse at time t and on worldline (ct, t), K’ will see the pulse SOMEWHERE on worldline (ct’, t’).

The way to express this mathematically is by (3) (x – ct) = lambda * (x’ – ct’)

This may seem trivial, but I spent a lot of time puzzling over equation (3)

Now get Einstein’s book and watch him derive the complicated looking Lorentz transformations using simple math and complex reasoning.

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