16 Invariant speed and local conservation

In Newton’s theory, grav­i­ta­tional effects are simul­ta­neous with their causes: the Sun attracts the Earth towards the Sun’s present posi­tion. This is often seen as the reason why Newton was in no posi­tion to “frame hypotheses” (about the mech­a­nism or nat­ural process by which gravity acts).

Elec­tro­mag­netic effects, on the other hand, are retarded. The ear­liest time at which a solar flare can affect us is about eight min­utes later — the dis­tance between the Sun and the Earth divided by the so-​​called speed of light (c). According to a widely held belief, the retar­da­tion of elec­tro­mag­netic effects made it pos­sible to explain how — by what mech­a­nism or nat­ural process — elec­tric charges act on elec­tric charges.

Although we have pre­vi­ously (here and here) dis­posed of such an “expla­na­tion” as a mere sleight-​​of-​​hand, it is worth taking a look at what changed and what did not change when Ein­stein real­ized that the invariant speed was finite (namely, c) rather than infi­nite, as Newton had held. (Reminder: any­thing that “travels” with this speed in one iner­tial frame, does so in every other iner­tial frame.)

The exis­tence of an invariant speed implies a spe­cial kind of spa­tiotem­poral rela­tion between events: either the rela­tion of being simul­ta­neous, which is absolute (that is, inde­pen­dent of the iner­tial frame used) in Newton’s (non-​​relativistic) theory, or the rela­tion of being sit­u­ated on each other’s light cone, which is absolute in Einstein’s (rel­a­tivistic) theory.

Sup­pose that an event e1 at (x1,t1) is the cause of an event e2 at (x2,t2). The fact that e2 hap­pens at t2, rather than at any other time, has two pos­sible expla­na­tions. If the action of e1 on e2 is medi­ated, t2 is deter­mined by the speed of medi­a­tion. This could be the speed of a mate­rial object trav­eling from (x1,t1) to (x2,t2), the speed of a signal prop­a­gating in an elastic medium, or what have you. But if the action of e1 on e2 is unmedi­ated, t2 is deter­mined by the spe­cial kind of spa­tiotem­poral rela­tion that the exis­tence of an invariant speed implies. In Newton’s theory, in which simul­taneity is absolute, t2 is equal to t1, whereas in the rel­a­tivistic theory, t2 is retarded by |x2–x1|/​c. So if an effect e2 at x2 hap­pens a time span Δt = |x2–x1|/​c after its cause e1 at x1, it means that the effect that e1 has on e2 is unmedi­ated; by no means does it follow that e2 is brought about through the medi­a­tion of some­thing that travels from e1 to e2 with the invariant speed c.

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Noether’s the­orem

Space­time coor­di­nates, we said, are human inven­tions. To make the laws of physics as simple as pos­sible, we intro­duce them in such a way that equal coor­di­nate inter­vals are phys­i­cally equiv­a­lent. This means, among other things, that freely moving clas­sical par­ti­cles travel equal space inter­vals Δx,Δy,Δz in equal time inter­vals Δt; the ratios formed of Δx, Δy, Δz, and Δt are con­stants. The phys­ical equiv­a­lence of equal time inter­vals implies a con­served phys­ical quan­tity — energy; and the phys­ical equiv­a­lence of equal inter­vals of the space coor­di­nates implies another con­served phys­ical quan­tity — momentum.

These results have been gen­er­al­ized by Noether’s the­orem. Sup­pose that we have a theory that is defined by a Lagrangian L, and sup­pose that L is invariant under some con­tin­uous trans­for­ma­tion of the fields on which it depends. Noether’s the­orem then implies a locally con­served quan­tity Q. This means that for any region R of space, the total amount of Q inside R increases (or decreases) by the amount of Q that flows into R (or out of R) through the boundary of R.

If, for instance, L is invariant under trans­la­tions in space (which it can only be if equal space inter­vals are phys­i­cally equiv­a­lent), then the the­orem implies the local con­ser­va­tion of momentum, and if L is invariant under time trans­la­tions (which it can only be if equal time inter­vals are phys­i­cally equiv­a­lent), then the the­orem implies the local con­ser­va­tion of energy. More com­pactly, if L is invariant under space­time trans­la­tions, Noether’s the­orem implies the local con­ser­va­tion of energy-​​momentum.

A gauge trans­for­ma­tion is another con­tin­uous trans­for­ma­tion of the fields on which L depends. If L is invariant under such a trans­for­ma­tion, the locally con­served quan­tity implied by this invari­ance is charge — elec­tric charge in the case of this gauge trans­for­ma­tion, or the weak (or flavor) charge asso­ci­ated with par­ti­cles inter­acting via the weak force, or the strong (or color) charge asso­ci­ated with par­ti­cles inter­acting via the strong force.

So are we now to imagine that energy, momentum, and charge are kinds of stuff that con­tin­u­ously “slosh around” in space or space­time? Of course not. The local con­ser­va­tion laws, like the Lagrangians that imply them, are cal­cu­la­tional tools. They ensure that, for every scat­tering event and for every iner­tial frame, the ener­gies, momenta, and charges of the incoming par­ti­cles equal the ener­gies, momenta, and charges (respec­tively) of the out­going par­ti­cles. (If some energy–momentum escapes unde­tected, then it also war­rants the fol­lowing con­di­tional: if the escaped energy–momentum were detected, it would agree with the local con­ser­va­tion law for energy-​​momentum.)

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