arXiv:gr-qc/9303025 v8 5 Oct 2001 RadiationfromaUniformlyAcceleratedCharge andtheEquivalencePrinciple StephenParrott DepartmentofMathematics UniversityofMassachusettsatBoston 100MorrisseyBlvd. Boston,MA02125 USA March1,2002 Abstract Wearguethatpurelylocalexperimentscandistinguishastationary chargedparticleinastaticgravitational eldfromanacceleratedpar- ticlein(gravity-free)Minkowskispace.Somecommonargumentstothe contraryareanalyzedandfoundtorestonamisidenti cationof\energy". 1Introduction ItisgenerallyacceptedthatanyacceleratedchargeinMinkowskispaceradiates energy.Itisalsoacceptedthatastationarychargeinastaticgravitational eld(suchasaSchwarzschild eld)doesnotradiateenergy.Itwouldseemthat these two facts imply that some forms of Einstein’s Equivalence Principle do not apply to charged particles. To put the matter in an easily visualized physical framework, imagine that the acceleration of a charged particle in Minkowski space is produced by a tiny rocket engine attached to the particle. Since the particle is radiating energy which can be detected and used, conservation of energy suggests that the ra- diated energy must be furnished by the rocket | we must burn more fuel to produce a given accelerating worldline than we would to produce the same world- line for a neutral particle of the same mass. Now consider a stationary charge in Schwarzschild space-time, and suppose a rocket holds it stationary relative to the coordinate frame (accelerating with respect to local inertial frames). In this case, since no radiation is produced, the rocket should use the same amount of fuel as would be required to hold stationary a similar neutral particle. This gives an experimental test by which we can determine locally whether we are 1 2 accelerating in Minkowski space or stationary in a gravitational eld | sim- ply observe the rocket’s fuel consumption. (Further discussion and replies to anticipated objections are given in Appendix 1.) Some authors (cf. [3]) explain this by viewing a charged particle as inextri- cably associated with its electromagnetic eld. They maintain that since the eld extends throughout all spacetime, no measurements on the particle can be considered truly local. To the present author, such assertions seem to di er only in language from the more straightforward: \The Equivalence Principle does not apply to charged particles". Other authors maintain that the Equivalence Principle does apply to charged particles. Perhaps the most in uential paper advocating a similar view is one of Boulware [2], an early version of which formed the basis for the treatment of the problem in Peierls’ book [11]. This paper claims to resolve \the equiv- alence principle paradox" by establishing that \all the radiation [measured by a freely falling observer] goes into the region of space time inaccessible to the co-accelerating observer." A recent paper of Singal [8] claims that there is no radiation at all. Singal’s argument, which we believe awed, is analyzed in [7]. The present work analyzes the problem within Boulware’s framework but reaches di erent conclusions. He shows that the Poynting vector vanishes in the rest frames of certain co-accelerating observers and concludes from this that \in the accelerated frame, there is no energy ux, ... , and no radi- ation". Singal [8] rederives a special case of this result (his equation (7) on page 962), and concludes that \there are no radiation elds for a charge supported in a a gravitational eld, in conformity with the strong principle of equivalence. We obtain a similar result by other means in Appendix 3, but interpret it di erently. We believe that the above quote of [2] incorrectly identi es the \radiated energy in the accelerated frame", and therefore does not resolve what he characterizes as a \paradox". Also, we do not think there is any \paradox" remaining, unless one regards the inapplicability of the Equivalence Principle to charged particles as a \para- dox". Even if the Equivalence Principle does not apply to charged particles, no known mathematical result or physical observation is contradicted. 2 What is \energy"? The identi cation of \energy" in Minkowski or Schwarzschild spacetime may seem obvious, but there is a subtlety hidden in Boulware’s formulation. This section examines this issue with the goal of clearly exposing the subtlety. To deserve the name \energy", a quantity should be \conserved". The follow- ing is a well-known way to construct a conserved quantity from a zero-divergence 3 time space space 3-dim volume at a xed time ? 2-dim boundary of the volume at a xed time 2-dim boundary evolving through time Figure 1: One space dimension is suppressed. The \top" and \bottom" of the box represent three-dimensional spacelike volumes; the \sides" represent two- dimensional surfaces moving through time; the interior is four-dimensional. symmetric tensor T = Tij and a Killing vector eld K = Ki on spacetime. Form the vector vi := Ti K (repeated indices are summed and usually emphasized by Greek and \:=" means \equals by de nition"), and note that its covariant divergence v j vanishes ([12], p. 96). By Gauss’s theorem, the integral of the normal component of v over the three-dimensional boundary of any four-dimensional region vanishes.1 Such a region is pictured in Figure 1, in which one space dimension is suppressed. The particular region pictured is a rectangular \box" with spacelike \ends" lying in the constant-time hyperplanes t = t1 and t = t2 and time-like \sides". (We use t as a time coordinate and assume that it is, in fact, timelike.) The \end" corresponding to time ti, i = 1;2, represents a three-dimensional region of space at that time. The integral of the normal component of v over the end corresponding to t = t2 is interpreted as the amount of a \substance" (such as energy) in this region of space at time t2. The integral of the normal component over the sides is interpreted as the amount of the substance which leaves the region of space between times t1 and t2. Thus the vanishing of the integral over the boundary expresses a law of conservation of the substance. Similar interpretations hold even if the boundary of the region is \curved" and does not necessarily lie in constant coordinate surfaces. We shall take as Tij the energy-momentum tensor of the retarded electro- magnetic eld produced by a charged particle whose worldline is given. That is, if F = Fij is the electromagnetic eld tensor, then Tij := Fi F j (1=4)F F gij ; (1) where gij is the spacetime metric tensor. Given T, to every Killing vector eld 1When there are points at which the boundary has a lightlike tangent vector, this must be interpreted sympathetically; see [5], Section 2.8 for the necessary de nitions. However, we shall only need to integrate over timelike and spacelike surfaces, on which the concept of \normal component" is unambiguous. 4 x t t = x :5 1 1:5 X = :5 X = 1 X = 1:5 D DD BB AA TT J JJ D D DD B B A AA T T % % % DD L AA S S SS e e e Figure 2: The orbits for the ow of the one-parameter family of boosts (3). K corresponds a conserved scalar quantity as described above. We have to decide which such quantity deserves the name \energy". In Minkowski space, the metric is ds2 = dt2 dx2 dy2 dz2 ; (2) and there seems no question that the energy is correctly identi ed as the con- served quantity corresponding to the Killing vector @t generating time trans- lations. (We use the di erential-geometric convention of identifying tangent vectors with directional derivatives.) If this were not true, we would have to rethink the physical interpretation of most of the mathematics of contemporary relativistic physics. Translations in spacelike directions similarly give Killing vectors whose corresponding conserved quantities are interpreted as momenta in the given directions. There are other Killing vector elds which are not as immediately obvious. For example, consider the Killing eld corresponding to the ow of the one- parameter family 7! ( ; ; ; ) of Lorentz boosts (t;x;y;z) := (tcosh +xsinh ;tsinh + xcosh ;y;z) : (3) The relevant timelike orbits of this ow (curves obtained by xing t;x > 0;y;z and letting vary) are pictured in Figure 2. For xed y;z, they are 5 hyperbolas with timelike tangent vectors. Any such hyperbola is the worldline of a uniformly accelerated particle. On any orbit, the positive quantity X satisfying X2 = (tsinh + xcosh )2 (tcosh +xsinh )2 = x2 t2 is constant, and its value is the orbit’s x-coordinate at time t = 0. Thus an orbit is the worldline of a uniformly accelerated particle which had position x = X at time t = 0. Such an orbit can conveniently be described in terms of X as the locus of all points (X sinh ;X cosh ;y;z), as varies over all real numbers. The tangent vector of such an orbit is @ := (X cosh ;X sinh ;0;0) : This is the Killing vector eld, expressed in terms of X and . Its length is X, so that a particle with this orbit has its proper time given by = X ; (4) its four-velocity @ is @ = 1X@ ; (5) and its proper acceleration is 1=X. The conserved quantity corresponding to the Killing vector @ has no rec- ognized name, but it does have a simple physical interpretation which will be given below. We then argue that it is this quantity which [2] (p. 185) identi es (mistakenly, in our view) as the relevant \energy ux" in the accelerated frame. 3 Energy in static space-times Consider a static spacetime whose metric tensor is ds2 = g00(x1;x2;x3)(dx0)2 + 3X I;J=1 gIJ(x1;x2;x3)dxIxJ : (6) The important feature is that the metric coe cients gij do not depend on the timelike coordinate x0, so that @x0 is a Killing eld. Another way to say this is that the spacetime is symmetric under time trans- lation. In general, the ow of a Killing eld can be regarded as a space-time symmetry. The symmetry of time translation was obvious from looking at the metric, but for some metrics there may exist less obvious, \hidden" symme- tries. An example is the Minkowski metric (2), which possesses symmetries corresponding to one-parameter families of boosts which might not be obvious at rst inspection. 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) 1dr2 r2(d 2 + sin2 d 2) : (7) It can be shown ([12], Exercise 3.6.8) that the only Killing vector elds K are linear combinations of @t and an \angular momentum" Killing eld A = K (r; ; )@ + K (r; ; )@ , where K and K satisfy some additional condi- tions which are unimportant for our purposes. The elds @t and A commute, as do their ows. 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 eld @t; for one thing, it is the only rotationally invariant choice. It is also physically reasonable in our context of analyzing the motion and elds of charged particles. If we surround a stationary charged particle2 by a stationary sphere which generates a three- dimensional \tube" as it progresses through time, the integral of the normal component of T0i over the tube between times t1 and t2 physically represents the out ow 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=4 over the sphere and multiplying by a factor proportional to t2 t1. It is usually assumed that the eld produced by a stationary charged particle may be taken to be a pure electric eld, 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 eld. 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)2dt2 dx2 dy2 dz2 : (8) Here c(x) represents the x-dependent speed of light as observed from the co- ordinate frame. Such a metric corresponds to a pseudo-gravitational eld in the x-direction. By a \pseudo" gravitational eld 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 x0 7!(x0;c1;c2;c3) relative to the static coordinate frame with respect to which the metric is (7), where the ci are constants independent of x0. 7 curvature of space-time and no true gravitational eld. It is well known that when the Riemann tensor vanishes, spacetime may be metrically identi ed with a piece of Minkowski space. Routine calculation shows that the only nonvanishing connection coe cients are, in an obvious notation, ttx = txt = c 0 c ; x tt = c 0c : The four-velocity u of a stationary particle is u = c 1@t , so a stationary particle has acceleration (Duu)k = u @ uk + k u u 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. How- ever, the reasonableness of such an identi cation must ultimately be justi ed by its mathematical and physical consequences. We shall argue that such an identi cation is sometimes inappropriate. The \obvious" Killing symmetries of (8) are those associated with time trans- lation, translations in spatial directions perpendicular to the x-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 x by X and t by : ds2 = X2d 2 dX2 dy2 dz2 : (10) The Riemann tensor vanishes for this spacetime, and it can be identi ed with a piece of Minkowski space. If t;x;y;z denote the usual Minkowski coordinates with metric given by (2), then this identi cation is: t = X sinh x = X cosh : Moreover, the present Killing eld @ is the same as the Minkowski space Killing eld @ discussed in Section 2. The part of Minkowski space covered by the map ;X;y;z7!t;x;y;z con- sists of the region jxj > jtj, but we will only be concerned with the smaller region x>jtj, which is called the \Rindler wedge". The coordinates ;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 8 space coordinates as seen by occupants of a rigidly accelerated elevator. Boul- ware [2] uses in place of for the timelike coordinate. We prefer because it seems more natural to reserve = X 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;x = X under the ow (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 col- lection. The spatial distance between two points with the same Rindler \time" coordinates say ;X1;y1;z1 and ;X2;y2;z2, is just the ordinary Euclidean dis- tance [(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 di erent for di erent 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 \ oor" of the elevator at x = 0 is approached; observers nearer the oor 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 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 = 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)q2 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 @ as the \energy". We want to emphasize that these two approaches are essentially di erent and yield di erent 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 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 di erent physical prediction than the corresponding prediction based on Minkowski physics, and the di erence between the two predictions is in principle experimentally testable. It is precisely at this point that we di er 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 di erent coordinate systems. We think it is worth empha- sizing that they are not the same energy measured in di erent systems; instead, they are di erent \energies" derived from di erent Killing elds. The observa- tion that the pseudo-energy radiation is zero does not validate the equivalence principle. 4 Discussion of calculation of radiation We want to brie y discuss what we think is the physically correct way to calcu- 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 given does not apply to non at 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 >jtj in Minkowski space. Suppose we are given the worldline of a (not necessarily uniformly) acceler- ated particle and a proper time . 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 S surrounds the particle. As the particle progresses on its worldline, let S move with it in such a way that the particle is always surrounded. As proper time progresses from an initial value 1 to a later value 2, the surface S generates a three-dimensional manifold S( 1; 2) 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 = Tij over this three- 10 -space 6 time worldline = 2 - tube S( 1; 2) - S( 1; 2) tube S( 1; 2) - S( 1; 2) spacelike 3-volume inside sphere = 1 * : D D D D DD B B BB A AA D D D D DD B B BB % %% DD L AA S S SS e ee D D D D DD B B BB AA E E E E E E EE (((((( ( hhhhhh h Figure 3: Two three-dimensional tubes which coincide at their ends. dimensional manifold will be denotedZ S( 1; 2) Ti dS : (11) The precise mathematical de nition of (11) is discussed in detail in [5]. Since the de nition entails summing vectors in di erent tangent spaces, it does not make sense in general spacetimes, in which there is no natural identi cation of tangent spaces at di erent points. The intuitive meaning is that for xed i, we integrate the normal component of the vector Ti 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 S for 1 2. The energy radiated is (11) with i = 0. Suppose we have two tubes, say S and S , which coincide at the initial and nal proper times 1 and 2: S 1 = S 1 and S 2 = S 2 . 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 o the worldline, Z S( 1; 2) Ti dS = Z S( 1; 2) Ti dS : (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 11 distorts on its journey, (11) always produces the same numerical results for the radiated energy-momentum. This leads to considerable conceptual and mathematical simpli cation 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 eld energy-momentum inside the sphere is the in nite energy of the Coulomb eld, which is discarded in a mass renormalization. It is unfortunate that the energy is in nite, 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 four- velocity at proper time as u( ), the proper acceleration as a( ) := du=d , and a2 := a a 0, it is ([5], p. 160): Z S( 1; 2) Ti dS = 23q2 Z 2 1 a2( )ui( )d + q 2 2 [u i( 2) ui( 1)] ; (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 rst 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 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 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 nally in uniform motion, and that it always surround the particle. 12 stationed on each patch. Instruct the observers to measure the elds, calcu- late 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 out ow n (E B=4 ) S (where n is the outward unit normal vector to the wall in the observer’s rest frame, S the area of his patch, the increment in his proper time, and E and B his electric and magnetic elds, respectively), and nally adding up the total energy out ow 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 nal 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 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 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 @ ; 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; ; (which bear the same relation to rectangular Rindler coordinates X;y;z that ordi- nary spherical coordinates r; ; bear to Euclidean coordinates x;y;z). In the integral, u = u( ;R; ; ) denotes the four-velocity of the point of the elevator located at Rindler spherical coordinates R; ; at its proper time (i.e. Rindler time coordinate = =X ), and n = n(R; ; ) 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 2 1 d R 0 d R2 0 d R 2 sin u T ( n ) : (14) (The minus sign is because the spatial inner product is negative de nite.) Recall from (5) that u = @ = @ =X, so that in Rindler coordinates in which 13 @ =: @0 is associated with the zero’th tensor index, u0 = 1=X, so u0 = X. Hence u T n = XT0 n = XP3J=1T0JnJ. Recalling also from (4) that = X and that K0 = X2, we may rewrite (14) in Rindler coordinates as: pseudo-energy radiation =R 2 1 d R 0 d R2 0 d R 2 sin P3 J=1 K T JnJ ; (15) with i := i=X, i = 1;2. Equation (15) demonstrates that (14) is actually computing the radiation of the conserved quantity corresponding to the Killing vector K = @ . A su cient condition for (15) to vanish is for T0J = 0 for all spatial indices J in Rindler coordinates. Equation (IV.3), p. 185 of [2] establishes that T0J = 0 and from this draws the conclusion that: \in the accelerated frame there is no energy ux, ... , and no radia- tion". That T0J = 0 is essentially the well-known \fact"4 that a stationary charged particle in a static spacetime does not radiate energy, where \energy" is de- ned as the conserved quantity corresponding to translation by the formal time coordinate (in this case, ) in this spacetime. We agree with [2] that there is no radiation of the conserved quantity cor- responding to the Killing vector @ , but we believe that this fact is irrelevant to questions concerning physically observed radiation and to questions about the applicability of the Equivalence Principle. Whether it is Minkowski energy radiation or pseudo-energy radiation which corresponds to energy that must be furnished by the driving forces is an experimental question. In principle, it could be settled by uniformly accelerating a large charge in a rocket and observing if more fuel were required than for a neutral payload of the same mass. We would bet that more fuel would be required, which would mean that Minkowski energy is the physically relevant \energy". On the other hand, in the Schwarzschild spacetime (7), the energy corre- sponding to the analog @t of @ is universally accepted as the physically relevant \energy". The spacetime (10) provides an interface between a Schwarzschild- type spacetime and Minkowski space within which questions about the Equiva- lence Principle can be conveniently addressed. If our hypothesis that Minkowski energy is the physically relevant \energy" in (10) is correct, then the vanish- ing of (15) not only does not validate the Equivalence Principle, but strongly suggests that it does not apply to charged particles. If we treat questions of radiation in the spacetime (10) in exactly the same way that such questions are treated in Schwarzschild space (7), then we are led to the probably incorrect 4We put \fact" in quotes because although this assertion is often made, we know of no proof in the literature, and in fact, it seems unlikely that it has been proved. Appendix 3 discusses this problem and furnishes a proof under certain auxiliary hypotheses. 14 conclusion that the rocket accelerating the charge does not require any extra fuel, since there is no radiation. The assertion that T0J = 0 for spatial indices J implies that \in the acceler- ated frame there is ... no [energy] radiation" merits further discussion because similar arguments are used by other authors (cf. [18]), and we believe that lan- guage such as \energy radiation in the accelerated frame" encourages a subtle error. The 3-vector T0J is the Poynting vector: (T01;T02;T03) = (E B)=4 , so that T0J = 0 says that every elevator observer sees a zero Poynting vector. If we identify seeing a zero Poynting vector with seeing no energy radiation, then this says that no elevator observer sees any energy radiation, which seems to lead to the conclusion that there is no energy radiation \in the elevator frame". Of course, one could obtain this conclusion by taking the vanishing of the Poynting vector in the elevator frame to be the de nition of \no energy radiation in the elevator frame", but we argue that such a de nition would be physically inappropriate. This is the main point of this section: Although each observer in a rigidly accelerating elevator surrounding the particle measures a vanishing Poynting vector in his own private rest frame, nevertheless, taken as a whole there is radiation through the elevator walls. Adding the (zero) energy uxes measured by each observer on the wall in his private rest frame to (incorrectly) con- clude zero total energy radiation is an illegitimate operation because these energy uxes refer to di erent rest frames. 5 Remarks on detecting energy radiation near a particle Reference [2] (unlike [8]) does recognize that Minkowski energy radiation is nonzero but concludes that it cannot be detected within the elevator (and thence that there is no violation of the Equivalence Principle). It discusses and dis- cards several possible methods to observe Minkowski radiation within the eleva- tor. For example, \if one identi es the radiation by the 1=r dependence of the eld along the light cone, one cannot ... remain within [the region covered by the elevator coordinates] and let r become large enough for the radiation eld to dominate." This overlooks the fact that the eld components are analytic functions o the worldline, and an analytic function is uniquely determined by its values on any open set, however small. To pick o the radiation terms that go to zero like 1=r as r!1, we need only evaluate the eld at a few points, which can be as close to the worldline as we want, and perform a few algebraic calculations to nd the coe cients of the 1=r terms. For example, if we write the elds in terms of the retarded distance rret, the eld components in a given direction from the retarded (emission) point are simple quadratic polynomials in 1=rret, whose coe cients can be easily determined. 15 6 Conclusions Does Einstein’s Equivalence Principle hold for charged particles? We can- not de nitively answer this because a mathematically precise statement of the \equivalence principle" seems elusive | most statements in the literature are not su ciently de nite to be susceptible of proof or disproof. However, we do conclude that most usual formulations seem not to hold in any direct and obvious way for charged particles. We believe that [2], which is widely cited in contexts suggesting that its analysis supports the validity of the Equivalence Principle for charged particles, does not in fact validate any form of the Equivalence Principle. We argue that its conclusion that \in the accelerated frame, there is no energy ux, ... and no radiation", is correct only if \energy" is misidenti ed (in our view) as the conserved quantity associated with a one-parameter family of Lorentz boosts in Minkowski space, instead of with the one-parameter family of time translations. Appendix 1: The relation of the Lorentz-Dirac equation to this problem We anticipate that some readers may be uneasy about our assertion that a uniformly accelerated charge in gravity-free (i.e., Minkowski) space may be lo- cally distinguished from a stationary charge in (say) Schwarzschild space-time by observing how much energy an external force, such as our fanciful rocket, must supply to maintain the worldline. Some may observe that in Minkowski space, the radiation reaction term in the Lorentz-Dirac equation vanishes, so one might think that no more energy would be required in either case than would be needed for a neutral particle of the same mass. This appendix discusses this point, which is important but peripheral to the main text. The Lorentz-Dirac equation[4] for a particle of mass m and charge q in an external eld F = Fij is: mdud = qF(u) + 23q2 da d +a 2u ; (16) where is proper time, u = ui the particle’s four-velocity, a := du=d its proper acceleration, a2 := a a and F(u)i := Fi u . The left side is the rate of change of mechanical energy-momentum, the term qF(u) is the Lorentz force, and the remaining term (2=3)q2[da=d +a2u] is the \radiation reaction" term which describes the e ect of the particle’s radiation on its motion. For a uniformly accelerated particle (i.e., a2 is constant) moving in one space dimension, the radiation reaction term (2=3)q2[da=d + a2u] vanishes identi- 16 cally.5 It is tempting to interpet this as implying that there is no physical radiation reaction for a uniformly accelerated charged particle, by which we mean that a rocket-driven uniformly accelerated charge requires no more en- ergy from the rocket than an otherwise identical neutral charge. However, we believe such an interpretation is unlikely to be correct. An obvious aw in the argument just given is that it is inconsistent with usual ideas of conservation of energy. If we grant that the uniformly accelerated charge does radiate energy into Minkowski space which can be collected and used, as nearly all modern authors ([8] and [21] excepted) seem to agree, then this radiated energy must be furnished by some decrease in energy of other parts of the system. Fulton and Rohrlich [15] suggest that it may somehow come from the eld energy but give no proof. (Since the eld energy in a spacelike hyperplane is in nite for a point electron, it’s not clear what would constitute a proof.) We look at the matter di erently. All derivations of the Lorentz-Dirac equa- tion are motivated by conservation of energy-momentum: the change of energy- momentum of the particle over a given proper-time interval should equal the energy-momentum furnished by the external forces driving the particle minus the radiated energy-momentum, assuming that it is legitimate to absorb in nite terms of a certain structure into a mass renormalization. Although this principle motivates the derivation, the nal equation unfortunately does not guarantee such conservation of energy-momentum in general, but only in certain special cases. One such special case is when the particle is asymptotically free, mean- ing that its proper acceleration vanishes asymptotically in the in nite past and future. Thus it’s not clear that the equation should apply to a particle which is not asymptotically free, such as a particle which is uniformly accelerated for all time.6 Since the equation doesn’t guarantee conservation of energy-momentum for uniform acceleration for all time, the fact that the radiation reaction term vanishes implies nothing about the additional force which the rocket must fur- nish for perpetually uniformly accelerated motion. But we should at least try to understand the case of a particle which is unaccelerated in the distant past, nudged into uniform acceleration, uniformly accelerated for a long time, and nally nudged back into an unaccelerated state. For this case, the Lorentz-Dirac equation does imply conservation of energy- momentum. However, since the radiation reaction term vanishes for the period 5 To see this, write u = ( ;v ;0;0) with v the velocity and := (1 v2) 1=2, and let w := (v ; ;0;0) be an orthogonal unit vector associated with the same spatial direction. By general principles, the proper accelerationa is orthogonal tou, so thata = Aw for some scalar function A, and A2 = a2 is constant. Since w is a unit vector, dw=d is orthogonal to w, and hence da=d = Adw=d is a multiple of u. That the multiple is a2 can be determined by taking the inner product u da =d = d(u a )=d a du =d = a2. 6 Actually, the equation is controversial even for asymptotically free particles, but that brings up issues outside the scope of this article. The reader can nd more information in [16], [5], [6], [17]. 17 of uniform acceleration, the equation implies that all the radiation energy must be furnished at the beginning and ending of the trip, while the particle is nudged into or out of its uniformly accelerated state. In other words, if we believe in the Lorentz-Dirac equation, we need to add a bit of energy to start the uniform acceleration, and thereafter the radiation, which can persist for an arbitrarily long time and add up to an arbitrarily large amount, is \free" until the end of the trip. In e ect, we can \borrow" an arbitrarily large amount of radiated energy (which in principle can meanwhile be collected and used by other observers in Minkowski space), so long as we pay it back at the end of the trip. Although there is no logical contradiction here, this is hard to accept physically, and seems one of many good reasons to question the Lorentz-Dirac equation. Most of the above issues are only peripherally relevant to the present work, and we present them only to dispel potential confusion. The point is that the vanishing of the radiation reaction term does not imply that the rocket accelerating the charged particle in Minkowski space does not have to furnish the radiation energy. The rocket almost certainly does have to supply this energy, and this gives a local experiment which distinguishes certain accelerated motion in Minkowski space from similar motion in Schwarzschild space. Appendix 2: The equation of motion of a charged rocket A noted expert in the eld raised the following interesting objection to the dis- cussion of Appendix 1 in an earlier version of this paper. Consider a charged rocket which undergoes a modest uniform acceleration g (one gravity, say) from just after an inital time i to just before a nal time f. More precisely, the rocket is at rest in some Lorentz frame (the initial frame) up to some initial proper time i, nudged into uniform acceleration over a small proper time inter- val [ i; i+ ], uniformly accelerated up to proper time = f , nudged back into an unaccelerated state over the interval [ f ; f], to remain unaccelerated for > f. He presented a simple estimate showing that the energy required to accom- plish the nal deceleration, as measured in the nal rest frame at = f, is modest and independent of the period of uniform acceleration. This can be an- ticipated without calculation, since from the viewpoint of the nal rest frame, going backwards in time from = f to = f only requires nudging the rocket back up to a modest uniform acceleration, and this cannot not require an unbounded energy change. Thus it would seem that from the point of view of the rocket’s pilot, only a modest amount of fuel must be burned to start the acceleration at the beginning of the trip and stop it at the end, with no excess fuel (relative to an uncharged rocket) required during the period of uniform 18 x t ti = 0 tf - - - - - - - - - - - - - - - - - - - - - - -2tf Figure 4: The worldline of a particle at rest up to time ti and uniformly accel- erated from time ti + to tf , where is the length of a small time interval during which the particle is nudged into or out of uniform acceleration. At time tf the acceleration is reversed in a time-symmetric way so as to bring the particle back to rest at time 2tf. acceleration (which can be arbitrarily long).7 The modest amount of energy used in the nal frame (along with its asso- ciated momentum) can Lorentz-transform into a large amount of energy in the initial rest frame at = i, so there is no apparent violation of conservation of energy from the standpoint of the initial frame. However, we can obtain what might appear to be a violation if we imagine reversing the proper acceleration a at the nal time f in a time-symmetric way (i.e., a( f + ) = a( f )) to eventually bring the rocket back to rest in the initial frame at = 2 f, as depicted in Figure 4. The expert’s estimate shows that the excess fuel used over the entire trip from = i to = 2 f is modest and independent of the duration of the uniform acceleration. At the beginning and end of the trip the rocket is at rest in the initial frame, so the energy of the radiation plus the exhaust should 7This was produced in evidence for the widely held belief (which we think incorrect) that there is no radiation reaction for a uniformly accelerated charge in Minkowski space. This line of reasoning suggests that we could allow the uniform acceleration to continue inde nitely without using any more fuel. (By extension, perpetual uniform acceleration would presumably require no fuel at all.) That would violate conservation of energy, assuming that the radiation energy is physically accessible, but proponents of this view sometimes use arguments similar to those criticized in Section 5 to assert that radiation cannot be observed within a Rindler elevator. 19 equal the rest-mass loss of the rocket (fuel used). If the loss of rest mass is nite and independent of the duration of the uniform acceleration (hence independent of the arbitrarily large energy radiation), we have a violation of conservation of energy in the initial frame. The situation was clari ed by actually solving the equation of motion for the radiating rocket, using the Lorentz-Dirac radiation reaction expression. It turns out that with a xed amount of initial fuel, one cannot obtain an arbi- trarily large period of uniformly accelerated motion (i.e., arbitrarily large f) unless one allows the rocket mass to go negative. Put another way, a charged rocket will run out of fuel if it uniformly accelerates long enough, so our time- symmetric motion is impossible with xed initial fuel and arbitrarily large f. This is in contradistinction to a uniformly accelerated uncharged rocket which can accelerate forever, assuming that all of its mass can be used as fuel. In retrospect, this conclusion seems natural and the analysis leading to it elementary, but I found the expert’s objection su ciently troubling to feel it necessary to actually work it out. Having done so, perhaps including it here may save readers with similar questions some work. The rocket will always move in the positive x-direction, and the other two constant space coordinates will be suppressed. If its initial-frame velocity is v, its initial-frame \rapidity" is de ned by := tanh 1(v). Then its four-velocity u is given in initial-frame coordinates by u = (cosh ;sinh ) : The scalar proper acceleration A is de ned by du=d = Aw where w is the unit vector w := (sinh ;cosh ) orthogonal to u. The (four-vector) proper acceleration is a := Aw. The scalar proper acceleration is related to the rapidity by A = d =d . In particular for constant scalar proper acceleration A( ) g, we have ( ) = g + (0). The Lorentz-Dirac expression for the proper-time rate of energy-momentum radiation of a charge q is: rate of energy-momentum radiation = (2q2=3)(da=d + a2u) = (2q2=3)(dA=d )w : (17) The second line follows from the rst in a fashion similar to that of the rst footnote in Appendix 1. We eliminate the constant factor by choosing units so that 2q2=3 = 1. Let m( ) denote the rocket’s rest mass, so that dm=d is the rate of ejection of rest mass into the exhaust. (This is not the same as the rate at which the exhaust acquires rest mass, as will be apparent from the expressions to follow. Rest mass is not conserved in general.) 20 There are two parameters which can be used to control the rocket’s worldline: the exhaust velocity and the rate dm=d of ejection of rest mass into the exhaust. The analysis to follow assumes that the exhaust velocity as seen from the rocket is always constant and that dm=d is varied so as to produce the desired worldline. The rocket is always moving to the right in the initial frame, so the exhaust is always moving left. We allow dm=d to have either sign. A positive dm=d means that the rocket is taking on mass. This could physically be accomplished by shooting bullets into it from the right, with the bullets constituting the \exhaust". Let denote the exhaust rapidity in the rocket’s instantaneous rest frame. That is, is positive, and an exhaust particle has velocity tanh( ) in the this rest frame. Then the exhaust’s four-velocity is: four-velocity of exhaust = ucosh wsinh : (18) Let = ( ) denote the proper-time rate at which the exhaust rest mass is increasing. Let Rw = R( )w( ( )) denote the proper-time rate at which electromagnetic energy-momentum is being emitted. It is assumed that this rate is a multiple of w because the above Lorentz-Dirac expression is of this form. It is convenient to allow arbitrary R because this enables us to simultaneously treat the case of an uncharged rocket by setting R 0. The uncharged case is also worked out in [19], in a similar fashion with identical results. The equation of energy-momentum balance is: 0 = d(mu)d + (ucosh wsinh ) + Rw = (dmd + cosh )u+ (mA sinh + R)w : (19) The rst term in the rst line is the proper-time rate of change of energy- momentum of the rocket, the second term the proper-time rate at which the exhaust is acquiring energy-momentum, and the third the proper-time rate of energy-momentum radiation. Since u and w are orthogonal, the last line separates into two independent equations: = 1cosh dmd ; (20) and mA sinh + R = 0 : (21) Equation (20) may be regarded as de ning , and then (21) becomes, setting := 1=tanh : dm d + mA + R = 0 : (22) 21 Recalling that A = d =d , we can immediately write down the solution with zero initial rapidity in terms of : m( ) = e ( )m( i) e ( ) Z i e (s)R(s)ds : (23) For R 0, corresponding to an uncharged rocket, we see that m( ) decreases exponentially with ( ). It also decreases exponentially with during the rst period of uniformly accelerated motion, since in that period, ( ) = ( i )g + ( i + ). In particular, m can never vanish for an uncharged rocket. An uncharged rocket can uniformly accelerate forever, assuming that all of its rest mass can be used as fuel. Now consider a charged rocket with R given by the Lorentz-Dirac expression R( ) := dA=d . Then (23) becomes: m( ) = e ( )m( i) e ( ) Z i e (s) dAds ds : (24) To dispel the notion that the charged rocket can uniformly accelerate for an arbitrarily long period without using any more fuel than would an uncharged rocket, we want to show that if A decreases monotonically from a constant value g down to 0 over a nal proper-time interval [ f ; f] of xed length , then m( f) must become negative for large f. That is, for such an A and for a xed initial mass m( i), we cannot nd positive-mass solutions de ned for arbitrarily large proper times f. This can be seen from the following simple estimates, in which it is helpful to remember that both and dA=ds are positive.8 First observe that from the Mean Value Theorem, for f s f, ( f) (s) ( f) (s) f s = d d (^ ) for some ^ with s ^ f = A(^ ) g : Using this, we have: e ( f) Z f f e (s) dAds ds = Z f f e ( ( f) (s)) dAds ds e g Z f f dAds ds ; 8That dA=ds is positive follows from the previous assumption, made for simplicity, that A decreases monotonically from g to 0. If we agree to eject mass at a positive rate (i.e., dm=d < 0) until A = 0, and if we de ne f to be the rst time after deceleration that A = 0, then this assumption follows from (22) with R := dA=d . 22 = e g ( A( f) + A( f )) ; = e g g : (25) Substituting (25) in (24), we see that to obtain a positive mass solution for arbitrarily large f (and arbitrarily large radiated energy), we need arbitrarily great rocket mass (i.e., fuel) m( i) to start with. In other words, a charged rocket in Minkowski space which starts with a nite amount of fuel cannot uniformly accelerate for an arbitrarily long time, after which the acceleration is removed. Unlike a corresponding uncharged rocket, it must eventually run out of fuel. What is peculiar is that if it has su cient fuel to get into the uniformly accelerated state, it will not run out of fuel until after the uniform acceleration is removed! It can uniformly accelerate for an arbitrarily long period, radiating all the while, but the physical contradiction of running out of fuel followed by the mass going negative will not be revealed until after the uniform acceleration is removed. We emphasize that this is a rigorous mathematical conclusion from the given assumptions|there are no approximations in the analysis which led to it. Phys- ically, it is very hard to believe. The most questionable assumption seems to be the Lorentz-Dirac expression (17) for the radiated energy. It is enlightening to follow the solution further to the nal resting state at = 2 f, but before doing this let’s think about what we would expect for an uncharged rocket. Since our formulation assumes that the exhaust velocity cannot be varied, the deceleration after = f is accomplished by taking in mass (and momentum), so we will have dm=d > 0 for f < < 2 f . In e ect, deceleration is accomplished by returning some of the previous exhaust energy-momentum to the rocket. For an uncharged rocket, the symmetry of the situation suggests that this energy-momentum return will be accomplished in time-symmetric fashion, and we can anticipate without calculation that all the exhaust energy-momentum will have been returned to the rocket at the nal resting time = 2 f. In particular, the nal rest mass should be the same as the initial rest mass. Indeed, this is what equation (24) does give if the radiation term dA=d is omitted. However, the result is quite di erent for the charged rocket. In this case, m(2 f) di ers from m( i) by the amount of the second term containing the integral. We have (2 f) = 0, so the exponential factor in front of the integral doesn’t contribute. The mass de cit at the end is m( i) m(2 f) = Z i+ i e (s)dAd ds Z f+ f e (s)dAd ds Z 2 f 2 f e (s)dAd ds: The rst and third integrals are of moderate size, while the second integral over the interval [ f ; f + ] is large for large f because e is large on this 23 interval. In e ect, the large initial-frame energy furnished over [ f ; f + ] (corresponding to a small loss of rest mass at f with high initial- frame velocity) has been transferred to the same large energy loss caused by a correspondingly large initial-frame rest mass loss. To put it more physically, by observing his fuel gauge, the charged rocket pilot sees only a modest excess fuel loss over [ f ; f + ] (relative to an uncharged rocket), but he does observe this loss, and he can gure out that because he is going very fast in the initial frame, it corresponds to a large initial-frame energy loss. Moreover, as he decelerates back to rest at = 2 f, this modest rest mass loss grows exponentially to an initial-frame excess rest mass loss large enough to pay for the radiated energy. This last observation may sound strange, but properly viewed it is to be expected. The rest mass of an uncharged rocket, will increase exponentially during the period [ f + ;2 f ] of uniform deceleration, and the same is true of the charged rocket. Over this period, the charged rocket behaves identically to an uncharged rocket with the same rest mass at = f + . However, the uncharged rocket which started with initial rest mass mi at = 0 does not have exactly the same rest mass at f + as the charged rocket with the same worldline and initial mass. There is a di erence due to the radiation in the time interval [0; f + ]. This di erence is modest even when the radiation is large. If f is large enough to give large radiation, this modest di erence in rest masses at = f + is ampli ed by the exponential growth to a correspondingly large di erence in rest masses at = 2 f . This analysis provides additional insight into the discussion of Appendix 1. It demonstrates by explicit calculation that contrary to widely held beliefs, there is indeed physical radiation reaction for a particle which is uniformly accelerated for a nite time even though the Lorentz-Dirac radiation reaction expression vanishes identically during the period of uniform acceleration.9 However, if we believe in the Lorentz-Dirac equation (and many experts don’t), we must accept the very strange conclusion that all of this radiation reaction occurs at the beginning (t ti) and end (t tf) of the trip while the particle is being nudged into or out of its uniformly accelerated state. 9Whether there is radiation reaction for a perpetually uniformly accelerated particle de- pends on one’s de nition of \radiationreaction". The \radiation reaction" term in the Lorentz- Dirac equation does vanish identically, but there is no good physical reason to identify this term with physically observed radiation reaction. Instead, it seems more reasonable to obtain the answer for uniform acceleration for all time as a limit of whatever answer is eventually generally accepted for uniform acceleration for nite times. There is probably no reasonable way to do this without rejecting the Lorentz-Dirac equation, since the above answer for uni- form acceleration for nite times (which is a consequence of the Lorentz-Dirac equation) is so strange. 24 Appendix 3: The eld of a stationary particle in a static spacetime It is often stated in the literature (e.g., [2]) that a charged particle which is stationary with respect to the coordinate frame in a static spacetime generates a pure electric eld in that frame; since the Poynting vector vanishes, there is no radiation. However, we know of no proof in the literature, and the matter seems to us not as simple as it apparently does to the authors who make this assertion. Implicit in such statements is that the eld generated by the particle is the \retarded eld" for its worldline. The problem is that there is no gener- ally accepted, mathematically rigorous de nition of \retarded eld" in general spacetimes. In Minkowski space one can de ne the retarded eld via the usual explicit formula, but no similar closed-form expressions are known for general spacetimes. A retarded- eld construction should be a rule which assigns to each charged particle worldline 7! z( ) (de ned as a curve in spacetime with unit-norm tangent u( ) := dz=d ) a 2-form F = F(x) satisfying Maxwell’s equations with source the distribution current associated with the worldline. Symbolically, these equations are dF = 0 ( d F)(x) = Z (x z( ))qu[( )d ; where d is the di erential operator on alternating forms, the Hodge duality op- eration, the four-dimensional Dirac delta distribution, q the particle’s charge, and u[ the 1-form corresponding to u (see below). To make the eld \retarded", it is also required that the value of F(x) at any spacetime point x o the worldline should depend only on the part of the worldline on or within the backward light cone with vertex x. In other words, any two worldlines which are identical inside this cone should yield the same F(x). Other assumptions might also reasonably be imposed. For example, one expects that for x o the worldline, the components of F(x) would be an or- dinary in nitely di erentiable 2-form (a priori it is only a distribution). This assumption is not necessary for our purposes, but it does no harm and simpli es thought. One very plausible assumption which we shall need is that in a static spacetime, the retarded eld for a stationary particle is time-independent. Unfortunately, no mathematically rigorous retarded- eld construction seems to be known for general spacetimes or even for static spacetimes. The discussion of Section 5.6 of [20], p. 220 gives the avor of the mathematical di culties. Despite the lack of rigorous mathematical proof, most physicists seem pre- pared to believe that in any given spacetime, a unique retarded- eld construc- 25 tion with the above properties ought to exist. Under this meta-mathematical assumption, we can show that the retarded eld of a stationary particle in a static spacetime (6) is a pure electric eld and that consequently the particle does not radiate. More precisely, there is no radiation through a stationary closed surface (stationary with respect to the \static" coordinates of (6)) sur- rounding the particle. The idea is very simple. Given a retarded eld, we can project out the electric part of it (relative to the static coordinates), and this projected electric part will still be \retarded". It is not obvious that it will satisfy Maxwell’s equations (with the particle’s worldline as source as above), but we shall show that it does. It follows that the electric part is also a retarded eld. If we believe in the uniqueness of the retarded eld construction for the given spacetime, then this shows that the original retarded eld was already a pure electric eld. If we are not willing to make the uniqueness assumption, then at least we have shown that there exists a retarded eld construction for static spacetimes for which the retarded eld is pure electric and the particle does not radiate. If the retarded eld construction is not unique, then we need additional physics to select the physically relevant retarded eld in order to answer the question of whether a stationary charge radiates. Now we prove the above assertion that the pure electric part of the retarded eld for a stationary particle in a static spacetime is itself a solution of the above Maxwell’s equations. As mentioned above, we assume that the retarded eld is time-independent, and this is the only use of the \retarded eld " assumptions. Thus we are really proving that the pure electric part of a time-independent solution is itself a solution. Consider a particle stationary at the origin in a spacetime with the static metric (6). The four-velocity of the particle will be denoted u(= ui), and the corresponding index-lowered one-form as u[(= ui := gi u ). Explicitly, u = g 1=200 @x0, and u[ = g1=200 dx0. Suppose we have a time-independent distribution 2-form F = Fij satisfying the Maxwell equations dF = 0 d F = 3u[ ; (26) where denotes the Hodge duality operation, d the di erential operator on al- ternating forms, and 3(x;y;z) := (x) (y) (z) is the three-dimensional Dirac delta distribution.10 Time-independence means that the coe cients Fij = Fij(x1;x2;x3) do not depend on the coordinate time x0. We may uniquely write F = E[^u[ + ; (27) 10De nitions of the di erential-geometric quantities such as the Hodge dual can be found in [5], Chapter 2. The proof can be given within the rigorous framework of distribution theory, but we write it in the traditional physics language of Dirac delta \functions". 26 where E = P3I=1EI@xI is a purely spatial vector eld, E[i := gi E the cor- responding index-lowered 1-form, and is a purely spatial 2-form. (We use bold-face for vectors in 4-space which are purely spatial with respect to the coordinate system used in (6), and we generally use capital Roman letters for space indices. All index lowering and raising is with respect to the spacetime metric rather than the Euclidean 3-space metric.) To say that is purely spatial means that = 3X I;J=1 IJdxIdxJ : Physically, is the 3-space Hodge dual of the 1-form corresponding to the magnetic eld vector B. The proof of (27) follows routinely from expanding F as a linear combination of dx ^dx , noting that u is proportional to dx0, and collecting terms involving u. We shall now show that if F satis es (26), then the electric part E[^u[ of F also satis es (26). (a) Consider the rst Maxwell equation 0 = dF = d(E[^u) + d . We want to show that d(E[^u) = 0. By routine calculation (directly, or cf. [5] , Section 5.4), du[ = u[^a[; where a := du=d is the acceleration of a stationary observer. Hence d(E[^u[) = d(u[^E[) = du[^E[ + u[^dE[ = u[^(E[^a[ + dE[) : (28) The point is that d(E[^u[) = u[^(something) and hence is orthogonal to any purely spatial 3-form (with respect to the inner product on two- forms induced by the spacetime metric). On the other hand, d is a purely spatial 3-form because its coe cients are time-independent by assumption. Hence d(E[^u[) and d must separately vanish. (b) Now we consider the other Maxwell equation d F = 3u[ and try to prove that this can happen only if d = 0. The 2-form is purely spatial, so its Hodge dual = u[^S[ for some purely spatial vector S. Apply the argument of part (a) with S in place of E to conclude that d = u[^(something). Now take a Hodge dual to see that d is a purely spatial 1-form; i.e. d is the 1-form corresponding (under index- raising) to a vector orthogonal to u. On the other hand, (E[^u[) is purely spatial with time-independent co- e cients, hence d (E[^u[) is a purely spatial 3-form, hence d (E[^u[) is a multiple of u[. Thus we have 3u[ = d (E[^u[) + d 27 with the rst term on the right a multiple of u[ and the second term orthogonal to u[; this can happen only if d (E[^u[) = 3u[ and d = 0. This completes the proof that E[^u[ satis es the Maxwell equations (26). However, the eld E is not usually a Coulomb eld, contrary to impressions given by [2] and other authors.11 To see that E is not necessarily a Coulomb eld, consider a metric of the special form (8), for which (9) gives the acceleration as a = (c0=c)@x6= 0. A Coulomb eld C := (x@x + y@y +z@z)=(x2 + y2 +z2)3=2 would satisfy dC[ = 0 except at the spatial origin (i.e. r C = 0 in 3-space), but this is inconsistent with the vanishing of (28) because u[^C[^a[ = (c0=c2)dt^dx^C[6= 0 : (29) Finally, we note that with F := E[^u[, the energy-momentum tensor (1) has T0J = 0 for J = 1;2;3, which says that the Poynting vector vanishes and there is no radiation through any stationary closed surface surrounding the particle. This was worked out in [2] for the metric (8), and [8] obtains a special case of the same result in di erent language. Appendix 4: Later references This body of this paper was originally posted in the Internet archive www.arXiv.org/abs/gr-qc/9303025 in March, 1993, in response to a discussion in the Internet newsgroup sci.physics concerning the applicability of the Equiv- alence Principle to charged particles. Appendices 2 and 3 were added later in response to questions from readers. Appendix 2 was posted in July, 1994. In the interim, minor errors in the body of the paper have been corrected, and minor stylistic changes made. A minor revision with the addition of Appendix 3 and references to Singal’s 1995 paper [8] was posted in January, 1996. The present Appendix 4 comments on work published since 1994 and bearing on the substance of this paper. It was noted above that Singal’s 1995 paper [8] in General Relativity and Gravitation (GRG) expresses a view opposite to that of the present work: he believes that a perpetually uniformly accelerated charged particle would not radiate. In response, our 1997 GRG paper [7] pointed out that Singal’s un- usual method implies that a charged particle uniformly accelerated for only a 11 This is probably more a question of language than of substance. For instance, although [2] states on page 172 that for the metric (10), \the accelerated observer ... only detects a Coulomb eld", the expressions derived for the eld are not precisely Coulomb elds in either the accelerated or Minkowski frames. Probably what was meant was something like \Coulomb-type" eld. 28 nite time (as in Appendix 2) does radiate, in quantitative accordance with the Larmor law. Thus the answer given by Singal’s method for uniform acceleration for all time is di erent than the answer which would be obtained from the same method by calculating radiation from uniform acceleration over a long but nite time , and then taking a limit as !1. This suggests that the question of radiation for a perpetually accelerated particle may be too singular for traditional math- ematical analysis. This is because two reasonable methods, whose mathematics are unchallenged, lead to di erent conclusions. For details, see [7]. In the same issue of GRG in which [7] appeared, Singal published a sequel [9] entitled \The Equivalence Principle and an Electric Charge in a Gravita- tional Field II. A Uniformly Accelerated Charge Does Not Radiate". Although [7] and [9] happened to appear simultaneously in GRG, the two authors were corresponding and were familiar with each others’ work during the acceptance process. As its title suggests, Singal was unconvinced by [7], though he has not challenged its mathematics. Singal’s sequel [9] comments on various aspects of the problem, but does not address the analysis of the comment [7] on his original paper [8]. It should be emphasized that our disagreement with Singal’s work, is solely a matter of de nition. Our [7] does not question the mathematics of [8] (which indeed we have checked carefully and believe correct), and [9] does not question [7]. In 1999, Shariati and Khorrami published [21]. This work comes to conclu- sions opposite to those of the present paper (which is cited, but not discussed in detail). They de ne a \supported" observer as one whose worldline is that of a point on a Rindler \elevator" in Minkowski space (i.e., a point stationary in the Rindler frame (10)), while an unaccelerated observer is \freely falling". They conclude: \A supported charge does not radiate according to another sup- ported observer." They identify the truth of this statement with the mathematical fact that any supported observer sees a vanishing Poynting vector12 (except in the singular case when the observer coincides with the charge). This is precisely Boulware’s argument quoted above in Section 4, that \in the accelerated frame there is no energy ux, ... , and no radia- tion". Since regarding this point, Shariati and Khorrami’s analysis is essentially that of Boulware, our concluding remarks for Section 4 apply verbatim to their analysis also: 12Their argument that the Poynting vector vanishes for stationary observers in a static spacetime (of which \supported" observers in a Rindler elevator is an instance) is basically that of Appendix 3, expressed in more traditional notation. 29 \Of course, one could obtain this conclusion [of no energy radiation] by taking the vanishing of the Poynting vector in the elevator frame to be the de nition of ‘no energy radiation in the elevator frame’, but we argue that such a de nition would be physically inappropriate." We feel that the main contribution of the present paper was pointing out that Boulware’s conclusion rested on the hidden assumption of this de nition, which we regard as unlikely. It is not clear whether Shariati and Khorrami recognize this assumption; if they do, evidently they disagree that it is unlikely. Thus mathematically, the issue reduces to a question of de nition. The physically correct de nition could be determined by doing experiments discused above, such as comparing the fuel consumption of a uniformly acceler- ating charged rocket with that of an uncharged rocket. Shariati and Khorrami state that they believe the fuel consumptions would be identical: \In the previous section, it was shown that a uniformly acceler- ated charge in a Minkowski spacetime does not radiate, in the sense that for the Rindler observer the Poynting vector vanishes, and an energy-like quantity for the electromagnetic eld is constant. This means that, according to Rindler observers, no extra force is needed to maintain the uniform acceleration of such a charged particle (of course no extra force beside the force needed for a neutral parti- cle of the same mass to have that acceleration). In other words, the world-line of the charged particle will be the same as that of a neutral particle." No one can prove or disprove this assertion without doing the experiment. Their \energy-like quantity" is what we called the \pseudo-energy". Shariati and Khorrami [21] also maintain that \ ::: in a static spacetime, :::, a freely falling charge do[es] not radiate, in the sense that no extra force is needed to maintain [its] world-line the same as that of a neutral particle." This is an issue not addressed in the present work, but those interested in the analysis of Shariati and Khorrami should be aware that it may be inconsistent with the DeWitt/Brehme/Hobbs (DBH) equation. The DBH equation is a generalization of the Lorentz-Dirac equation to ar- bitrary spacetimes. A similar equation was originally derived by DeWitt and Brehme [3], but an error in the very complicated derivation eliminated impor- tant terms. The error was corrected eight years later by Hobbs [14]. Shariati and Khorrami assume without proof a di erent equation of motion (their equation (20)) for charged particles. It di ers from the DBH equation by the omission of terms involving the Ricci tensor, and also omission of a so-called nonlocal "tail" term. They observe that their equation implies that a freely 30 falling charge (i.e., a charge with zero acceleration) experiences no radiation reaction. This conclusion does not seem to follow from the DBH equation. A particle satisfying the DBH equation can fall freely only if some mathematical miracle causes the terms omitted by the Shariati/Khorrami equation to vanish. Shariati and Khorrami do not discuss this issue; indeed, they do not mention the DBH equation.13 Acknowledgment Because [2] is so widely cited (always favorably, to my knowledge), I chose to focus on it as representative of a point of view with which I have come to disagree. The scienti c criticisms presented are in no way intended to denigrate its important contributions to our understanding of these issues. One such contribution is recognizing the role of the metric (10) as an interface between the physics of Minkowski space and the physics of more general spacetimes within which they can conveniently be compared. The present paper is deeply indebted to this advance. References [1] W. B. Bonnor, A new equation of motion for a radiating charged particle, Proc. Roy. Soc. London A337, 591-598 (1974) [2] D. Boulware, \Radiation from a uniformly accelerated charge", Annals of Physics 124, 169-187 (1980) [3] B. DeWitt, and R. Brehme, \Radiation damping in a gravitational eld", Annals of Physics 9, 220-259 (1960) [4] P. A. M. Dirac, \Classical theories of radiating electrons", Proc. Royal Soc. London A167, 148 . (1938) [5] S. Parrott, Relativistic Electrodynamics and Di erential Geometry, Springer, New York, 1987 13This should not be interpretedas our endorsement of the DBH equation. Like the proposed equation of Shariati and Khorrami, it reduces to the Lorentz-Dirac equation in at spacetime, and hence implies the usual zoo of bizarre predictions of that equation, such as those worked out in Appendix 2. But those who do believe in the Lorentz-Dirac equation typically also believe in the DBH equation. It seems strange that Shariati and Khorrami base their analysis on yet another variant of the Lorentz-Dirac equation, which may not have been proposed before, and certainly is not widely accepted. 31 [6] S. 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