Finally, there are the events on the light cone; those for which
−t2 + (x2 + y2 + z2) = 0.
Those are the points that a light signal, starting at the origin, would reach. A person who is at a lightlike event relative to the origin would see the flash of light.
1.6 Historical Perspective
1.6.1 Einstein
People often wonder whether Einstein’s declaration that “c is a law of physics” was based on theoretical insight or prior experimental results—in particular the Michelson-Morley experiment. Of course, we can’t be certain of
the answer. No one really knows what’s in another person’s mind. Einstein himself claimed that he was not aware of Michelson’s and Morley’s result when he wrote his 1905 paper. I think there’s every reason to believe him.
Einstein took Maxwell’s equations to be a law of physics. He knew that they give rise to wavelike solutions. At age sixteen, he puzzled over what would happen if you moved along with a light ray. The “obvious” answer is that you’d see a static electric and magnetic field with a wavelike structure that doesn’t move. Somehow, he knew that was wrong—that it was not a solution to Maxwell’s equations. Maxwell’s equations say that light moves at the speed of light. I’m inclined to believe that, consistent with Einstein’s own account, he didn’t know of the Michelson-Morley experiment when he wrote his paper.
In modern language we would explain Einstein’s reasoning a little differently. We would say that Maxwell’s equations have a symmetry of some kind—some set of coordinate transformations under which the equations have the same form in every reference frame. If you take Maxwell’s equations, which contain x’s and t’s, and plug in the old Galilean rules,
x′ = x − vt t′ = t,
you would find that these equations take a different form in the primed coordinates. They don’t have the same form as in the unprimed coordinates.
However, if you plug the Lorentz transformation into Maxwell’s equations, the transformed Maxwell equations have exactly the same form in the primed coordinates as in the unprimed coordinates. In modern language, Einstein’s great accomplishment was to recognize that the symmetry structure of Maxwell’s equations is not the Galileo transformation but the Lorentz transformation. He encapsulated all of this in a single principle. In a sense, he didn’t need to actually know Maxwell’s equations (though he did know them, of course). All he needed to know is that Maxwell’s equations are a law of physics, and that the law of physics requires light to move with a certain velocity. From there he could just work with the motion of light rays.
1.6.2 Lorentz
Lorentz did know about the Michelson-Morley experiment. He came up with the same transformation equations but interpreted them differently. He envisioned them as effects on moving objects caused by their motion through the ether. Because of various kinds of ether pressures, objects would be squeezed and therefore shortened.
Was he wrong? I suppose you could say that in some way he wasn’t wrong.
But he certainly didn’t have Einstein’s vision of a symmetry structure—the symmetry required of space and time in order that it agree with the principle of relativity and the motion of the speed of light. Nobody would have said that Lorentz did what Einstein did.15 Furthermore, Lorentz didn’t think it was exact.
He regarded the transformation equations as a first approximation. An object moving through a fluid of some kind would be shortened, and the first approximation would be the Lorentz contraction. Lorentz fully expected that the Michelson-Morley experiment was not exact. He thought there would be corrections to higher powers of v/c, and that experimental techniques would eventually become precise enough to detect differences in the velocity of light.
It was Einstein who said this is really a law of physics, a principle.
1 Sometimes we’ll use the abbreviation IRF for inertial reference frame.
2 We could also describe this as “my trajectory in your frame.”
3 This statement may sound glib. The fact is that many of the world’s most talented physicists tried to make things work out without giving up on the equation t′ = t. They all failed.
4 This is actually a slight variation of Einstein’s approach.
5 You can think of the t axis being calibrated in light-seconds instead of seconds if you prefer, but it amounts to the same thing.
6 Once again, this is actually a slight variation of Einstein’s approach.
7 For v/c = 10−5 one finds
8 We’re talking about a fixed difference in orientation, not a situation where either frame is rotating with a nonzero angular velocity.
9 If you don’t believe me, ask the guy who sold it to me on Canal Street for twenty-five bucks.
10 In what follows we will pretend that Art and Lenny were both born at the same spacetime event (labeled O) in Fig. 1.8.
11 We’re using the term spacetime distance in a generic sense. Later on, we’ll switch to the more precise terms proper time and spacetime interval.
12 We use the label t′ in two slightly different ways in this discussion. Its primary meaning is “Lenny’s t′
coordinate.” But we also use it to label the t′ axis.
13 Sign conventions in relativity are not as consistent as we would like; some authors define s2 to have the same sign as τ2.
14 Once again, we’re using the shorthand term “spacelike event” to mean “event that is spacelike separated from the origin.”
15 Including Lorentz himself, I believe.