Consider now the important spacetime after Minkowski space, the Schwarzschild space-ti th metric tensor ds2=(1-2M/r)dt2-(1-2M/r)-1ar2-r2(dr2+sin2bdl2).(7) It can be shown( 12, Exercise 3.6.8) that the only Killing vector fields K are linear combinations of a and an angular momentum" Killing field A Ke(r, 0, ae+Ko(r, 0, )ao, where Ke and Ko satisfy some additional condi- tions which are unimportant for our purposes. The fields at and A commute, as do their flows. In other words, the only Killing symmetries of Schwarzschild spacetime are the expected ones arising from rotational and time invariance there are no hidden Killing symmetries In this situation, the only natural mathematical candidate for an"energy the conserved quantity corresponding to the Killing field d: for one thing, it the only rotationally invariant choice. It is also physically reasonable in our context of analyzing the motion and fields of charged particles. If we surround a stationary charged particle by a stationary sphere which generates a three- dimensional "tube"as it progresses through time, the integral of the normal component of TOi over the tube between times ti and t2 physically represents the outflow of the conserved quantity corresponding to at between these times When the calculation is carried out, it is seen to be the same as integrating the normal component of the Poynting vector E X B/4T over the sphere and multiplying by a factor proportional to t2-t1. It is usually assumed that the field produced by a stationary charged particle may be taken to be a pure electric field, and Appendix 3 proves this under certain auxiliary hypotheses In other words, B=0, so the integral vanishes, and there is no"radiation"of our conserved quantity. We expect no energy radiation; otherwise we would be ble to garner an unlimited amount of"free"energy, since it takes no energy to hold a particle stationary in a gravitational field Thus it seems eminently reasonable in this situation to identify the conserved quantity associated with at with the energy. We expect a conserved"energy this is the only natural mathematical candidate, and its physical properties turn out to be reasonable However, these arguments lose force when hidden symmetries exist. Consider a metric c(r)2dt2-dz2-dy2 Here c(a) represents the a-dependent speed of light as observed from the co- ordinate frame. Such a metric corresponds to a pseudo-gravitational field in the -direction. By a"pseudo"gravitational field we mean that a stationary particle has a worldline which is accelerated in the r-direction, but the Riemann ensor may happen to vanish for some functions c( ), in which case there is no 2Bya“ stationary" particle we mean one whose worldline is a°→(x°,c1,c2,c3) relative to the static coordinate frame with respect to which the metric is(7), where the c are consta
6 Consider now the most important spacetime after Minkowski space, the Schwarzschild space-time with metric tensor ds2 = (1 − 2M/r)dt2 − (1 − 2M/r) −1 dr2 − r 2 (dθ2 + sin2 θdφ2 ) . (7) It can be shown ([12], Exercise 3.6.8) that the only Killing vector fields K are linear combinations of ∂t and an “angular momentum” Killing field A = Kθ(r, θ, φ)∂θ + Kφ(r, θ, φ)∂φ, where Kθ and Kφ satisfy some additional conditions which are unimportant for our purposes. The fields ∂t and A commute, as do their flows. In other words, the only Killing symmetries of Schwarzschild spacetime are the expected ones arising from rotational and time invariance: there are no hidden Killing symmetries. In this situation, the only natural mathematical candidate for an “energy” is the conserved quantity corresponding to the Killing field ∂t; for one thing, it is the only rotationally invariant choice. It is also physically reasonable in our context of analyzing the motion and fields of charged particles. If we surround a stationary charged particle2 by a stationary sphere which generates a threedimensional “tube” as it progresses through time, the integral of the normal component of T 0i over the tube between times t1 and t2 physically represents the outflow of the conserved quantity corresponding to ∂t between these times. When the calculation is carried out, it is seen to be the same as integrating the normal component of the Poynting vector E × B/4π over the sphere and multiplying by a factor proportional to t2 − t1. It is usually assumed that the field produced by a stationary charged particle may be taken to be a pure electric field, and Appendix 3 proves this under certain auxiliary hypotheses. In other words, B = 0, so the integral vanishes, and there is no “radiation” of our conserved quantity. We expect no energy radiation; otherwise we would be able to garner an unlimited amount of “free” energy, since it takes no energy to hold a particle stationary in a gravitational field. Thus it seems eminently reasonable in this situation to identify the conserved quantity associated with ∂t with the energy. We expect a conserved “energy”, this is the only natural mathematical candidate, and its physical properties turn out to be reasonable. However, these arguments lose force when hidden symmetries exist. Consider a metric ds2 = c(x) 2 dt2 − dx2 − dy2 − dz2 . (8) Here c(x) represents the x-dependent speed of light as observed from the coordinate frame. Such a metric corresponds to a pseudo-gravitational field in the x-direction. By a “pseudo” gravitational field we mean that a stationary particle has a worldline which is accelerated in the x-direction, but the Riemann tensor may happen to vanish for some functions c(·), in which case there is no 2By a “stationary” particle we mean one whose worldline is x 0 7→ (x 0, c 1, c 2, c 3) relative to the static coordinate frame with respect to which the metric is (7), where the c i are constants independent of x 0
curvature of space-time and no true gravitational field. It is well known that when the Riemann tensor vanishes, spacetime may be metrically identified with a piece of Minkowski space Routine calculation shows that the only nonvanishing connection coefficients e in an obvious notation The four-velocity u of a stationary particle is u dt, so a stationary particle has acceleration(D2u)=u°ax+rbau°u3 given b That is, the acceleration is in the x-direction with a magnitude given by the relative rate of change of c in the x-direction. This acceleration Du u is what w mean by acceleration with respect to local inertial frames It might seem reasonable, even natural, to identify the conserved quantity associated with a, with energy, in analogy with Schwarzschild spacetime. How- ever, the reasonableness of such an identification must ultimately be justified by its mathematical and physical consequences. We shall argue that such an identification is sometimes inappropriate. The"obvious"Killing symmetries of (8)are those associated with time trans- lation, translations in spatial directions perpendicular to the z-axis, and rota- tions about the x-axis. Only for very special choices of c( will there exist other, "hidden"symmetries. One such choice yields the following metric, in which for later purposes we replace the coordinate symbol a by X and t by a (10) The Riemann tensor vanishes for this spacetime, and it can be identified with a piece of Minkowski space. If t, 1, y, a denote the usual Minkowski coordinates with metric given by(2), then this identification is t Sinh入 X cosh a Moreover, the present Killing field a is the same as the Minkowski space Killing field d discussed in Section 2 The part of Minkowski space covered by the map A, X,,a region r>t, which is called the"Rindler wedge, oncerned with the smaller sists of the region z>t, but we will only be The coordinates A, X, y, z for this portion of Minkowski space are known as Rindler coordinates( 13, Section 8.6). They are also sometimes known as levator coordinates because we shall see below that X, 3, z may be regarded as
7 curvature of space-time and no true gravitational field. It is well known that when the Riemann tensor vanishes, spacetime may be metrically identified with a piece of Minkowski space. Routine calculation shows that the only nonvanishing connection coefficients are, in an obvious notation, Γ t tx = Γ t xt = c 0 c , Γ x tt = c 0 c . The four-velocity u of a stationary particle is u = c −1∂t , so a stationary particle has acceleration (Duu) k = u α∂αu k + Γ k αβu αu β given by Duu = c 0 c ∂x , (9) That is, the acceleration is in the x-direction with a magnitude given by the relative rate of change of c in the x-direction. This acceleration Duu is what we mean by “acceleration with respect to local inertial frames”. It might seem reasonable, even natural, to identify the conserved quantity associated with ∂t with energy, in analogy with Schwarzschild spacetime. However, the reasonableness of such an identification must ultimately be justified by its mathematical and physical consequences. We shall argue that such an identification is sometimes inappropriate. The “obvious” Killing symmetries of (8) are those associated with time translation, translations in spatial directions perpendicular to the x-axis, and rotations about the x-axis. Only for very special choices of c(·) will there exist other, “hidden” symmetries. One such choice yields the following metric, in which for later purposes we replace the coordinate symbol x by X and t by λ: ds2 = X 2 dλ2 − dX2 − dy2 − dz2 . (10) The Riemann tensor vanishes for this spacetime, and it can be identified with a piece of Minkowski space. If t, x, y, z denote the usual Minkowski coordinates with metric given by (2), then this identification is: t = X sinh λ x = X cosh λ . Moreover, the present Killing field ∂λ is the same as the Minkowski space Killing field ∂λ discussed in Section 2. The part of Minkowski space covered by the map λ, X, y, z 7→ t, x, y, z consists of the region |x| > |t|, but we will only be concerned with the smaller region x > |t|, which is called the “Rindler wedge”. The coordinates λ, X, y, z for this portion of Minkowski space are known as Rindler coordinates ([13], Section 8.6). They are also sometimes known as elevator coordinates because we shall see below that X, y, z may be regarded as
space coordinates as seen by occupants of a rigidly accelerated elevator. Boul- ware 2 uses T in place of A for the timelike coordinate. We prefer A because it seems more natural to reserve T= AX for the proper time on the worldlines of points of the elevator For constant y and z, a curve X= constant is the orbit oft=0, I= X under the Hlow (3). This curve is also the worldline of a uniformly accelerated particle with proper acceleration 1/X. The set of all such curves for all X, y, z may be regarded as the worldlines of a collection of uniformly accelerated observers of whom are at rest in the minkowski frame at time t=0 The Rindler coordinates X, y, z specify the particular worldline in the col- lection. The spatial distance between two points with the same Rindler"time coordinates say A, X1, 91, 21 and A, X2, 32, 22, is just the ordinary Euclidean dis- tance (X2-X1)2+(v-y)2+( 211/2. Moreover Catial displacement vector is orthogonal to the worldlines of constant X, y his says that an observer following such a worldline sees at any given moment other such worldlines at a constant distance in his rest frame at that moment Thus we may take a collection of such worldlines and imagine connecting them with rigid rods(the rods can be rigid because the proper distances are constant) obtaining a rigid accelerating structure which we might call an"elevator However, it would be misleading to call it a uniformly accelerating elevator Though every point on it is uniformly accelerating, the magnitude 1/X of the uniform acceleration is different for different points. Because of this, the every day notion of a uniformly accelerating elevator gives a potentially misleading physical picture. A more nearly accurate picture is obtained by thinking of each point of the elevator as separately driven on its orbit through Minkowski space by a tiny rocket engine. Observers moving with the elevator experience a pseudo-gravitational force which increases without limit as the"floor"of the elevator at a=0 is approached; observers nearer the floor need more powerful ockets than those farther up We have two ways to view the physics of such an elevator On the one hand since the elevator is a subset of Minkowski space, we can transform the well understood physics of Minkowski space into elevator coordinates to derive what residents of the elevator should observe. In particular, if a particle of charge q is situated at X= l, say, its motion being driven by a tiny rocket attached to it, then the energy required by the rocket per unit proper time would be the required for an uncharged particle of the same mass plus the radiated he proper-time rate of radiated energy being(2/3)q2 as required by the Law for proper acceleration 1/X=l A second approach would be to emphasize the analogy of the metric(10) with the Schwarzschild metric(7), interpreting the conserved quantity correspondin to a as the "energy". We want to emphasize that these two approaches are essentially different and yield different physical predictions. We'll see below that the second approach(which seems similar to that of I yields a conserved quantity whose integral over the "walls"of (say) a spherical
8 space coordinates as seen by occupants of a rigidly accelerated elevator. Boulware [2] uses τ in place of λ for the timelike coordinate. We prefer λ because it seems more natural to reserve τ = λX for the proper time on the worldlines of points of the elevator. For constant y and z, a curve X = constant is the orbit of t = 0, x = X under the flow (3). This curve is also the worldline of a uniformly accelerated particle with proper acceleration 1/X. The set of all such curves for all X, y, z may be regarded as the worldlines of a collection of uniformly accelerated observers all of whom are at rest in the Minkowski frame at time t = 0. The Rindler coordinates X, y, z specify the particular worldline in the collection. The spatial distance between two points with the same Rindler “time” coordinates say λ, X1, y1, z1 and λ, X2, y2, z2, is just the ordinary Euclidean distance [(X2 − X1) 2 + (y2 − y1) 2 + (z2 − z1) 2 ] 1/2 . Moreover, the corresponding spatial displacement vector is orthogonal to the worldlines of constant X, y, z. This says that an observer following such a worldline sees at any given moment other such worldlines at a constant distance in his rest frame at that moment. Thus we may take a collection of such worldlines and imagine connecting them with rigid rods (the rods can be rigid because the proper distances are constant), obtaining a rigid accelerating structure which we might call an “elevator”. However, it would be misleading to call it a uniformly accelerating elevator. Though every point on it is uniformly accelerating, the magnitude 1/X of the uniform acceleration is different for different points. Because of this, the everyday notion of a uniformly accelerating elevator gives a potentially misleading physical picture. A more nearly accurate picture is obtained by thinking of each point of the elevator as separately driven on its orbit through Minkowski space by a tiny rocket engine. Observers moving with the elevator experience a pseudo-gravitational force which increases without limit as the “floor” of the elevator at x = 0 is approached; observers nearer the floor need more powerful rockets than those farther up. We have two ways to view the physics of such an elevator. On the one hand, since the elevator is a subset of Minkowski space, we can transform the wellunderstood physics of Minkowski space into elevator coordinates to derive what residents of the elevator should observe. In particular, if a particle of charge q is situated at X = 1, say, its motion being driven by a tiny rocket attached to it, then the energy required by the rocket per unit proper time would be the energy required for an uncharged particle of the same mass plus the radiated energy, the proper-time rate of radiated energy being (2/3)q 2 as required by the Larmor Law for proper acceleration 1/X = 1. A second approach would be to emphasize the analogy of the metric (10) with the Schwarzschild metric (7), interpreting the conserved quantity corresponding to ∂λ as the “energy”. We want to emphasize that these two approaches are essentially different and yield different physical predictions. We’ll see below that the second approach (which seems similar to that of [2]) yields a conserved quantity whose integral over the “walls” of (say) a spherical
elevator surrounding the particle is zero. That is, there is no radiation of this conserved quantity, which we'll call the "pseudo-energy" to distinguish it from the above Minkowski energy. If we interpreted this pseudo-energy as energy radiation as seen by observers in the elevator(such as the pilot of the rocket accelerating the charge), then by conservation of energy we should conclude that no additional energy is required by the rocket beyond that which would be required to accelerate an uncharged particle of the same mass. This is a different physical prediction than the corresponding prediction ased on Minkowski physics, and the difference between the two predictions is in principle experimentally testable. It is precisely at this point that we differ from [ 2. That reference does distinguish between the Minkowski energy and the pseudo-energy, but it gives the impression that they are somehow the same energy"measured in different coordinate systems. We think it is worth empha- sizing that they are not the same energy measured in different systems; instead they are different "energies"derived from different Killing fields. The observa- tion that the pseudo-energy radiation is zero does not validate the equivalence 4 Discussion of calculation of radiation We want to briefly discuss what we think is the physically correct way to cald late the energy radiated by an accelerated charge in Minkowski space. Almost everything we shall say is well known, but we want to present it in a way which will make manifest its applicability to the present problem. The analysis to be iven does not apply to nonfat spacetimes for reasons which will be mentioned later. It applies to any simply connected subset of Minkowski space. In par- ticular, it applies to the rindler" elevator"described in Rindler coordinates by the metric(10) with X>0, and alternately as the Rindler wedge x>t in Minkowski space. Q Suppose we are given the worldline of a(not necessarily uniformly)acceler- ated particle and a proper time T. Surround the particle by a two-dimensional surface S. It may be useful to think of S as a sphere but we don't assume any metrical properties for S, such as rigidity. All we assume is that Sr surrounds As the particle progresses on its worldline, let Sr move with it in such a way that the particle is always surrounded. As proper time progresses from an initial value Ti to a later value T2, the surface S generates a three-dimensional manifold S(1, T2) in Minkowski space which is customarily called a"tube space in which one space dimension is suppressed. The integral of the energy-momentum tensor T= Ta over this three-
9 elevator surrounding the particle is zero. That is, there is no radiation of this conserved quantity, which we’ll call the “pseudo-energy” to distinguish it from the above Minkowski energy. If we interpreted this pseudo-energy as energy radiation as seen by observers in the elevator (such as the pilot of the rocket accelerating the charge), then by conservation of energy we should conclude that no additional energy is required by the rocket beyond that which would be required to accelerate an uncharged particle of the same mass. This is a different physical prediction than the corresponding prediction based on Minkowski physics, and the difference between the two predictions is in principle experimentally testable. It is precisely at this point that we differ from [2]. That reference does distinguish between the Minkowski energy and the pseudo-energy, but it gives the impression that they are somehow the same “energy” measured in different coordinate systems. We think it is worth emphasizing that they are not the same energy measured in different systems; instead, they are different “energies” derived from different Killing fields. The observation that the pseudo-energy radiation is zero does not validate the equivalence principle. 4 Discussion of calculation of radiation We want to briefly discuss what we think is the physically correct way to calculate the energy radiated by an accelerated charge in Minkowski space. Almost everything we shall say is well known, but we want to present it in a way which will make manifest its applicability to the present problem. The analysis to be given does not apply to nonflat spacetimes for reasons which will be mentioned later. It applies to any simply connected subset of Minkowski space. In particular, it applies to the Rindler “elevator” described in Rindler coordinates by the metric (10) with X > 0, and alternately as the Rindler wedge x > |t| in Minkowski space. Suppose we are given the worldline of a (not necessarily uniformly) accelerated particle and a proper time τ . Surround the particle by a two-dimensional surface Sτ . It may be useful to think of Sτ as a sphere, but we don’t assume any metrical properties for S, such as rigidity. All we assume is that Sτ surrounds the particle. As the particle progresses on its worldline, let Sτ move with it in such a way that the particle is always surrounded. As proper time progresses from an initial value τ1 to a later value τ2, the surface Sτ generates a three-dimensional manifold S(τ1, τ2) in Minkowski space which is customarily called a “tube”, because it looks like a tube surrounding the worldline in a picture of Minkowski space in which one space dimension is suppressed. The integral of the energy-momentum tensor T = T ij over this three-
worldline 7 tube S(1, T2) time tube S(T1, T2) S(r1,T2) spacelike 3-volume inside sphere igure 3: Two three-dimensional tubes which coincide at their ends dimensional manifold will be denoted Tia ds (11) The precise mathematical definition of (11)is discussed in detail in 5. Since e definition entails summing vectors in different tangent spaces, it does not make sense in general spacetimes, in which there is no natural identification of tangent spaces at different points The intuitive meaning is that for fixed i, we integrate the normal component of the vector T over the tube, the integration being with respect to the natural volume element on the tube induced from Minkowski space. Physically, (11)is nterpreted as the energy-momentum radiated through S for T1 T< T2. The energy radiated is(11) with i=0 Suppose we have two tubes, say S and Sr, which coincide at the initial and final proper times Ti and T2: Sr= Sr, and Sr,= Sr. Such a situation is pictured in Figure 3, in which two space dimensions are suppressed. Taken together, they form the boundary of a four-dimensional region, and since T has vanishing divergence off the worldline Tc ds Tg ds In other words, the calculated radiation is independent of the tube, so long as the tubes coincide at their ends. Put another way, no matter how the sphere
10 space ✲ ✻ time worldline ✟ τ = τ2 ✙✟ ✲ tube S¯(τ1, τ2) ✲ ✛ S¯(τ1, τ2) tube S(τ1, τ2) ✲ ✛ S(τ1, τ2) spacelike 3-volume inside sphere τ = τ1 ✟✯✟ ✘✘✘✘✘✿ ☎ ☎ ☎ ☎ ☎ ☎ ✂ ✂ ✂ ✂ ✁ ✁ ✁ ✁ ✁ ❉ ❉ ❉ ❉ ❉ ❉ ❇ ❇ ❇❇ ❆ ❆❆ ☎ ☎ ☎ ☎ ☎ ☎ ✂ ✂ ✂ ✂ ❉ ❉ ❉ ❉ ❉ ❉ ❇ ❇ ❇ ❇ ☎ ☞ ✁ ✁ ✓ ✓ ✓ ✓ ✜ ✪ ✪✪ ❉ ▲ ❆❆ ❙ ❙ ❙❙ ❭ ❡ ❡❡ ☎ ☎ ☎ ☎ ☎ ☎ ✂ ✂ ✂ ✂ ✁ ✁ ❉ ❉ ❉ ❉ ❉ ❉ ❇ ❇ ❇❇ ❆ ❆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ✭✭✭✭✭✭✭ ❤❤❤❤❤❤❤ Figure 3: Two three-dimensional tubes which coincide at their ends. dimensional manifold will be denoted Z S(τ1,τ2) T iα dSα . (11) The precise mathematical definition of (11) is discussed in detail in [5]. Since the definition entails summing vectors in different tangent spaces, it does not make sense in general spacetimes, in which there is no natural identification of tangent spaces at different points. The intuitive meaning is that for fixed i, we integrate the normal component of the vector T iα over the tube, the integration being with respect to the natural volume element on the tube induced from Minkowski space. Physically, (11) is interpreted as the energy-momentum radiated through Sτ for τ1 ≤ τ ≤ τ2. The energy radiated is (11) with i = 0. Suppose we have two tubes, say Sτ and S¯ τ , which coincide at the initial and final proper times τ1 and τ2: Sτ1 = S¯ τ1 and Sτ2 = S¯ τ2 . Such a situation is pictured in Figure 3, in which two space dimensions are suppressed. Taken together, they form the boundary of a four-dimensional region, and since T has vanishing divergence off the worldline, Z S(τ1,τ2) T iα dSα = Z S¯(τ1,τ2) T iα dSα . (12) In other words, the calculated radiation is independent of the tube, so long as the tubes coincide at their ends. Put another way, no matter how the sphere