distorts on its journey, (11)always produces the same numerical results for the diated energy-momentum. This leads to considerable conceptual and mathematical simplification in the important special case in which the particle is unaccelerated in the distant past, put into accelerated motion for a while, and re-enters an unaccelerated tate in the distant future. We can take the ends of the tube as any convenient geometrical shape in the rest frame of the unaccelerated particle in the distant ast ( or future), say a sphere of given radius. The field energy-momentum inside the sphere is the infinite energy of the Coulomb field, which is discarded in a mass renormalization. It is unfortunate that the energy is infinite, but at least it is well understood and can be unambiguously calculated in this special case; this is the reason for insisting that the particle be unaccelerated in distant past nd future Picture the two-dimensional surrounding surface as the walls of an elevator with the charged particle at its center. Suppose the elevator is initially at rest nd then both elevator and particle are gently nudged into uniformly accelerated motion,with both particle and elevator at rest in the Rindler frame(10).3The state of rigidly accelerated motion is then maintained for an arbitrarily long period, after which the acceleration is gently removed and the elevator enters a state of uniform motion thereafter. The result of the integral(11)for a spherical elevator of initial radius e is well-known. Denoting the particle's four velocity at proper time r as u(T), the proper acceleration as a(T): =du/dr, and a°aa≤0,itis(⑤],p.160) rds,=-2/a(()+smt(n)-u(n,(3) where q is the particle's charge. The last term on the right is traditionally discarded in a mass renormalization. The energy component of the first term is always positive. We conclude that there is energy radiation through the walls of the elevato This energy radiation can be detected in several ways in an arbitrarily small elevator. First of all. if we believe in conservation of the usual minkowski en- ergy, the pilot of the rocket driving the charge will observe an additional fuel consumption when his payload is a charged particle, relative to the correspond- ing identical motion of an uncharged particle, the additional fuel consumption eing exactly the amount necessary to"pay for the radiated energy.(However the details of how this"borrowed"energy must be repaid may be controversial as discussed in Appendices 1 and 2) A more fundamental way to meaure it, at least in principle, is to divide th elevator walls into a large number of small coordinate patches with an observer It is not essential that the elevator be at rest in the rindler frame. but this case is particularly easy to visualize and calculate. Our argument requires only that the elevator b itially and finally in uniform motion, and that it always surround the particle
11 distorts on its journey, (11) always produces the same numerical results for the radiated energy-momentum. This leads to considerable conceptual and mathematical simplification in the important special case in which the particle is unaccelerated in the distant past, put into accelerated motion for a while, and re-enters an unaccelerated state in the distant future. We can take the ends of the tube as any convenient geometrical shape in the rest frame of the unaccelerated particle in the distant past (or future), say a sphere of given radius. The field energy-momentum inside the sphere is the infinite energy of the Coulomb field, which is discarded in a mass renormalization. It is unfortunate that the energy is infinite, but at least it is well understood and can be unambiguously calculated in this special case; this is the reason for insisting that the particle be unaccelerated in distant past and future. Picture the two-dimensional surrounding surface as the walls of an elevator with the charged particle at its center. Suppose the elevator is initially at rest, and then both elevator and particle are gently nudged into uniformly accelerated motion, with both particle and elevator at rest in the Rindler frame (10).3 The state of rigidly accelerated motion is then maintained for an arbitrarily long period, after which the acceleration is gently removed and the elevator enters a state of uniform motion thereafter. The result of the integral (11) for a spherical elevator of initial radius is well-known. Denoting the particle’s fourvelocity at proper time τ as u(τ ), the proper acceleration as a(τ ) := du/dτ , and a 2 := a αaα ≤ 0, it is ([5], p. 160): Z S(τ1,τ2) T iα dSα = − 2 3 q 2 Z τ2 τ1 a 2 (τ )u i (τ ) dτ + q 2 2 [u i (τ2) − u i (τ1)] , (13) where q is the particle’s charge. The last term on the right is traditionally discarded in a mass renormalization. The energy component of the first term is always positive. We conclude that there is energy radiation through the walls of the elevator. This energy radiation can be detected in several ways in an arbitrarily small elevator. First of all, if we believe in conservation of the usual Minkowski energy, the pilot of the rocket driving the charge will observe an additional fuel consumption when his payload is a charged particle, relative to the corresponding identical motion of an uncharged particle, the additional fuel consumption being exactly the amount necessary to “pay” for the radiated energy. (However, the details of how this “borrowed” energy must be repaid may be controversial, as discussed in Appendices 1 and 2.) A more fundamental way to meaure it, at least in principle, is to divide the elevator walls into a large number of small coordinate patches with an observer 3It is not essential that the elevator be at rest in the Rindler frame, but this case is particularly easy to visualize and calculate. Our argument requires only that the elevator be initially and finally in uniform motion, and that it always surround the particle
stationed on each patch. Instruct the observers to measure the fields, calcu- ate the corresponding energy-momentum tensor, and approximate to arbitrary ccuracy the energy component of the integral (11) We want to emphasize that this is not the same as having each observer calculate his local energy outflow n(Ex B/4)ASAT(where n is the outward unit normal vector to the wall in the observer s rest frame. As the area of his patch, AT the increment in his proper time, and E and B his electric and magnetic fields, respectively), and finally add p the total energy outHow of all the observers. For arbitrary motion (i.e. elevator allowed to distort), this last procedure would have no invariant meaning because each observer has his own private rest frame at each instant of his proper time. The"energy"obtained as the final result of this procedure would in general depend on the construction of the elevator. For instance, if on the same trip we had a small elevator surrounded by a larger one, there is no reason to suppose that the observers on the large elevator would obtain the same number for energy"radiation as those on the smaller. Neither number would be expected to be related in any simple way to the additional energy required by the rocket for a charged versus uncharged In the procedure just described, the observers are not measuring"energy they are measuring something else. It may seem tempting to call it something like"energy as measured in the(curvilinear) elevator frame", but it is conceptu- ally and experimentally distinct from the usual Minkowski energy. For arbitrary motion, it is not a conserved quantity and therefore probably does not deserve the name "energy". For the special case of an elevator with constant spatial Rindler coordinates, it does happen to be independent of the elevators shape(in fact, it's zero for all! ), but it is still not" energy"as the term is normally used We'll show below that it is the conserved quantity corresponding to the Killing vector for ax: i.e. the quantity which we previously named the "pseudo-energy The pseudo-energy as physically measured by the procedure just described for a spherical elevator S of radius R in Rindler coordinates is mathematically given by the following integral in spherical Rindler coordinates R, 0, o(which bear the same relation to rectangular Rindler coordinates X, y, z that ordi nary spherical coordinates r, 0, o bear to Euclidean coordinates a, y, 2). In the integral,u=u(T, R, 0, denotes the four-velocity of the point of the elevator located at Rindler spherical coordinates R, 6, o at its proper time T(i.e. Rindler time coordinate A=T/X), and n= n(r, 0, o)is the spatial unit normal vector to the sphere at the indicated point (i.e, n is orthogonal to u and normal to the sphere, so that in Rindler coordinates, n=(0, n)) pseudo-energy radiat f, dr jo dojo doR utap(-nB (14) (The minus sign is because the spatial inner product is negative definite. Recall from(5)that u=a=d/X, so that in Rindler coordinates in which
12 stationed on each patch. Instruct the observers to measure the fields, calculate the corresponding energy-momentum tensor, and approximate to arbitrary accuracy the energy component of the integral (11). We want to emphasize that this is not the same as having each observer calculate his local energy outflow n·(E× B/4π)∆S∆τ (where n is the outward unit normal vector to the wall in the observer’s rest frame, ∆S the area of his patch, ∆τ the increment in his proper time, and E and B his electric and magnetic fields, respectively), and finally adding up the total energy outflow of all the observers. For arbitrary motion (i.e. elevator allowed to distort), this last procedure would have no invariant meaning because each observer has his own private rest frame at each instant of his proper time. The “energy” obtained as the final result of this procedure would in general depend on the construction of the elevator. For instance, if on the same trip we had a small elevator surrounded by a larger one, there is no reason to suppose that the observers on the larger elevator would obtain the same number for “energy” radiation as those on the smaller. Neither number would be expected to be related in any simple way to the additional energy required by the rocket for a charged versus uncharged payload. In the procedure just described, the observers are not measuring “energy”; they are measuring something else. It may seem tempting to call it something like “energy as measured in the (curvilinear) elevator frame”, but it is conceptually and experimentally distinct from the usual Minkowski energy. For arbitrary motion, it is not a conserved quantity and therefore probably does not deserve the name “energy”. For the special case of an elevator with constant spatial Rindler coordinates, it does happen to be independent of the elevator’s shape (in fact, it’s zero for all!), but it is still not “energy” as the term is normally used. We’ll show below that it is the conserved quantity corresponding to the Killing vector for ∂λ; i.e. the quantity which we previously named the “pseudo-energy”. The pseudo-energy as physically measured by the procedure just described for a spherical elevator S of radius R in Rindler coordinates is mathematically given by the following integral in spherical Rindler coordinates R, θ, φ (which bear the same relation to rectangular Rindler coordinates X, y, z that ordinary spherical coordinates r, θ, φ bear to Euclidean coordinates x, y, z). In the integral, u = u(τ, R, θ, φ) denotes the four-velocity of the point of the elevator located at Rindler spherical coordinates R, θ, φ at its proper time τ (i.e. Rindler time coordinate λ = τ/X ), and n = n(R, θ, φ) is the spatial unit normal vector to the sphere at the indicated point (i.e., n is orthogonal to u and normal to the sphere, so that in Rindler coordinates, n = (0, n)): pseudo-energy radiation = R τ2 τ1 dτ R π 0 dθ R 2π 0 dφ R2 sin θ uαT αβ(−nβ) . (14) (The minus sign is because the spatial inner product is negative definite.) Recall from (5) that u = ∂τ = ∂λ/X, so that in Rindler coordinates in which