As you will remember, a general way of accommodating influences on the behavior of a stable particle is to modify the rate at which it “ticks.” We write the modified amplitude associated with an infinitesimal path segment dC in the form Z(dC) = [1:dS/ℏ], where the infinitesimal action increment dS is a function of x, y, z, t, dx, dy, dz, and dt, which satisfies two conditions: it is (i) invariant under Lorentz transformations and (ii) homogeneous.
There are exactly two ways of modifying the amplitude associated with dC without violating these conditions (and without turning the amplitude into a quantity that has multiple components like a vector). The first consists in adding to dS = −mc2ds (the action increment associated with a freely propagating stable particle) a sum of four terms each containing one of the coordinate intervals:
(2.17.1) dS = −mc2ds + (q/c)(Ax dx + Ay dy + Az dz − V dt).
The second consists in the substitution
(2.17.2) c ds → √(Σjk gjk dxj dxk).
The so-called (scalar) potential V and the three components of the so-called vector potential A are “fields,” in the sense (and only in the sense) that they are functions of x,y,z,t. The overall factor q — the particle’s charge — allows the effects on the behavior of a particle encapsulated by V and A to differ between one particle species and another. The indices j and k run from 0 to 3, so the root contains a sum of sixteen terms. The intervals dx0, dx1, dx2, and dx3 stand for c dt, dx, dy, and dz, respectively. (The superscripts are indices, not powers.) The sixteen components gjk of the so-called metric tensor g are also fields in the sense just spelled out. With g00 = 1, gjj = −1 for j=1,2,3, and gjk = 0 for all other index pairs, we recover the original expression c ds from the right-hand side of the substitution (2.17.2).
As previously noted, dS defines a differential geometry, and classical particles follow the geodesics of the differential geometry defined by the action increment dS.
The geodesics defined by (2.17.1) are characterized by the so-called Lorentz force law, a computational tool that encapsulates all electromagnetic effects on the motion of a classical particle. It takes as its input the particle’s charge q, the so-called electric field E, the so-called magnetic field B, and the coordinate intervals dt,dx,dy,dz pertaining to an infinitesimal segment dG of a geodesic, and it yields as its output the change dpk of the particle’s kinetic momentum pk that takes place as the particle travels dG. E and B are defined in terms of the potentials V and A in (2.17.1).
Energy and momentum
What do we mean by the kinetic momentum of a classical particle? We begin by observing that spacetime coordinates are human inventions. They belong to the language we use to describe a physical situation rather than to the physical situation itself. To make the laws of physics as simple as possible, we require of our coordinates that equal coordinate intervals be physically equivalent. This implies, 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. In consequence of the physical equivalence of equal time intervals there is a conserved physical quantity — the particle’s energy E. And in consequence of the physical equivalence of equal intervals of the space coordinates there is another conserved physical quantity — the particle’s momentum p. Because space has three dimensions, p has three components (px, py, and pz). (If a physical system is closed, meaning not subject to any external influences, then the values of its conserved quantities do not change as time passes.)
The definitions of E and p depend on the form of dS. If dS is given by (2.17.1), then
(2.17.3) E = mc²/√1 − v²/c² + qV and p = mv/√1 − v²/c² + (q/c)A.
The terms containing the root are the particle’s kinetic energy and momentum, respectively, while qV and (q/c)A are its potential energy and momentum, respectively. v is the particle’s velocity, and v is the corresponding magnitude, the particle’s speed.
Let us note in passing that if v is so small compared to c that all powers of v2/c2 but the first can be ignored, we obtain the following expressions for the non-relativistic (Newtonian) theory:
E = mc2 + (m/2)v2 + qV, pk = mv.
Since the rest energy mc2 is now equivalent to a constant potential energy, and since only the differences between potential energies at different places and/or times are physically relevant, the rest energy term is usually dropped, so that the kinetic energy is given by (m/2)v2 = pk2/2m.
Influences or effects on the motion of classical particles are thus mathematically represented by a differential geometry. Particles affected or acted on follow the geodesics of that geometry. Figure 2.17.1 illustrates the effects represented by E and B.
Consider first the rectangle at the right, which lies in a spatial plane. A mathematical expression called the flux of the magnetic field through this rectangle determines the difference between the actions of the paths A→B→C and A→D→C. If B vanishes, the actions of the two paths are equal, and the geodesic from A to C — the path with the least action — is the straight line from A to C. If on the other hand B does not vanish, then the path via B (say) is shorter than the path via D — shorter according to the spacetime geometry defined by (2.17.1). As a result, the geodesic from A to C is curved as indicated. According to a story popular among teachers of classical physics (and hence, unfortunately, among their students), the curvature is due to a magnetic “force” that pulls (or pushes) in a direction perpendicular to the particle’s motion.
Consider now the rectangle at the left, which lies in a spacetime plane containing (or parallel to) the time axis. In this case the difference between the actions of the paths A→B→C and A→D→C. is determined by the flux of the electric field through the rectangle. If E vanishes, the actions of the two paths are again equal, and the geodesic from A to C is again the straight line from A to C. If B does not vanish, then the path via B (say) is again shorter than the path via D. As a result, the geodesic from A to C is curved as indicated. But curvature in a spacetime plane containing the time axis means that the particle is accelerating. According to the story just mentioned, the acceleration is due to an electric “force” that pulls (or pushes) in the direction in which the particle is moving.
While the geodesics defined by the action increment (2.17.1) are particle-specific — they depend on the masses and the charges of the particles acted on — the geometry defined by the substitution (2.17.2) is independent of particle-specific parameters. For this reason it has become customary to attribute this geometry to “spacetime itself.” The attribution of a geometry to “spacetime itself,” however, is as uncalled for as the notion of forces bending the paths of moving objects. Both kinds of geometry — the one that represents influences on the motion of a particle in a particle-specific manner (technically known as a Finsler geometry) and the one that represents influences on the motion of a particle in a nonspecific manner (technically known as a pseudo-Riemannian geometry) — are mathematical tools used for representing influences on the motion of particles. The fact that one of these tools is nonspecific as far as the affected particles are concerned, by no means warrants thinking of it as a property of “spacetime itself.”
It is, however, worth an illustration that the influences which Newton attributed to the force of gravity are indeed encapsulated by a universal spacetime geometry. If a ball is thrown so that it rises to a height of about 5 m and covers a distance of 10 m, it hits the ground after 2 seconds. If a bullet is fired so that it rises to a height of about 0.5 mm and covers the same distance, it hits the ground after 20 milliseconds. The curvatures of their trajectories in space are obviously very different, but not the curvatures of their paths in spacetime, as one gathers from Fig. 2.17.2.