The Special Theory of Relativity

The Lorentz Transformation

After Einstein's paper on the special theory of relativity was published in 1905, it was taken for granted that a relativistic sphere would appear to be compressed along an axis in the direction of motion, but in 1959 James Terrell showed that a sphere would actally appear to be rotated. He needed no new physics to show this: he simply took account of the fact that light rays arriving simultaneously at an observer leave points on the sphere at different times. It is a straightforward matter to write an applet that demonstrates the apparent rotation once we know the Lorentz Transformation, so that is where we begin.

The Lorentz Transformation existed before 1905, but Einstein showed that it is a consequence of the postulate that the laws of physics and the velocity of light (c, a universal constant) are the same in all inertial frames. For the velocity of light to be constant, time must enter as a fourth dimension in the transformation of coordinates between inertial frames. The two coordinates transverse to the motion are unaffected by the transformation, so we need only write:

Position	Time	Velocity

The origins of the two frames coincide at t=t'=0, and the primed frame moves along the positive x-axis in the unprimed frame with velocity v. The direct and inverse transformations can be obtained from each other by exchanging the primes and reversing the sign of v, and the velocity transformation follows directly from the coordinate transformations. The following example illustrates the use of the transformation: "A spaceship leaves a spacestation and travels at speed .6c for a time T before firing a missle back toward the station at speed .8c. How long do the inhabitants of the station have to flee between when the flash from the missle firing and the missile itself arrive? How long after the missle is fired does the flash from the exploding station reach the ship?" The Lorentz Transformation enables us to construct the following table of events:

x=1.25(x'+.6ct')	t=1.25(t'+.6x'/c)	x'=1.25(x-.6ct)	t'=1.25(t-.6x/c)	Event
0	0	0	0	1 Ship leaves station
.75 cT	1.25 T	0	T	2 Ship fires missle
0	2 T	-1.5 cT	2.5 T	3 Flash reaches station
0	3.2 T	-2.4 cT	4 T	4 Missle reaches station
4.8 cT	8 T	0	6.4 T	5 Flash reaches ship

The coordinates of the missile-firing event in the station frame have the following meaning: if an observer in the station frame is at that x and carries a watch synchronized with the station clock, he will see the missle fired in front of him at that t. The ship and the observer disagree about the time and position of the firing (the ship thinks it is .6cT from the station), but agree that the ship has been traveling at 6c. The station doesn't "see" the firing - it doesn't know about it until the next event, the arrival of the flash. As expected, the speed with which the flash travels is c in both frames. The ship sees the missle closing on the station at .2c, so its time of flight will be 3T, and the transformation gives the time of arrival in the station frame. Alternately, we can use the velocity transformation to find the speed of the missle in the station frame, and work from that. The flash from the exploding station again travels at speed c in both frames.

The terms "length contraction" and "time dilation" are commonly used to describe the effects of the Lorentz Transformation. These terms are best defined by the following equation:

where the subscript on the bar specifies the variable that is held constant (the notation used in thermodynamics). The table has three events with x=0, and three with x'=0, so time dilation (and its symmetry) can easily be checked by inspection.

Length contraction, like time dilation, is symmetric between the primed and unprimed systems, but one can imagine schemes for capturing a moving object in a box smaller than the object would be at rest. Observer A puts the following proposition to observer B: "Get yourself a long rigid pole of length D. Sit on the back end of it and get up to speed .6c. As you pass me we'll set our watches to zero, and I'll close a gate behind you. There is a second gate .92D ahead of you, and I claim you'll end up caught between gates and that will prove you're the one who is contracted." B doesn't like the sound of this, but reluctantly agrees (he would be even unhappier if he knew the far "gate" was actually a solid wall. Here is the table of events:

x=1.25(x'+.6ct')	t=1.25(t'+.6x'/c)	x'=1.25(x-.6ct)	t'=1.25(t-.6x/c)	Event
0	0	0	0	1 Back gate closed
.8 D	0	D	-.6 D/c	2 Front of rod at t=0
.92 D	0	1.15 D	-.69 D/c	3 Solid wall at t=0
1.25 D	.75 D/c	D	0	4 Front of rod at t'=0
.92 D	.546 D/c	.736 D	0	5 Solid wall at t'=0

Note that Event 2, the front of the rod at t=0, has t and x' specified and the other two coordinates calculated - any two entries in a row enable the other two to be calculated. Looking at Events 1 through 3 it looks like a fair setup from A's perspective. When B, looking somewhat the worse for wear, comes back to see A he is irate. He says that the front of his pole was splattered all over the wall before A closed the gate behind him, so no wonder he was caught. A replies that B was told to get a rigid pole, but B points out that there really is no such thing - a compression wave has to travel along the rod before the back end of the rod knows the front end has hit something, and the wave has to travel slower than the speed of light.

This table has three events with t=0 and three with t'=0, so length contraction (and its symmetry) can easily be checked by inspection. The important thing we learn from this second example is that Simultaneity is a Relative Concept.

Finally we get to our relativistic sphere. The transparent sphere in the applet has parallels of latitude and meridians of longitude inscribed on it at 10 degree intervals. Its polar axis is aligned with the direction of motion. The grid lines are shown in blue on the side near the observer and in red on the far side. With the sphere at rest only the blue lines are visible. The sphere is assumed to be far enough away for rays reaching the observer to be considered parallel. (This is similar to the condition for Fraunhofer diffraction in optics). Hence time differences depend only on differences in distance along the line of sight (perpendicular to the direction of motion). Enough of this - try running the applet.

BBC Basic Source Java Source

If the sphere were solid you would see the red grid lines wrapped around the left end but not the blue ones wrapped around the right end. Please keep in mind that the rotation is apparent, not real. For an object with a depth equal to its length, you can show that the apparent rotation angle should be the angle whose sine is v/c. For v=.6c or .8c the angles are the familiar ones of the 3-4-5 triangle. The easiest number to get from the figure is the number of parallels of latitude between the poles (treating both as being on the near surface). From this you can calculate the angle of the apparent rotation.