1 Introduction and summary

1.1 Invitation

The motion of a point electric charge in flat spacetime was the subject of active investigation since the early work of Lorentz, Abrahams, Poincaré, and Dirac [56Jump To The Next Citation Point], until Gralla, Harte, and Wald produced a definitive derivation of the equations motion [82Jump To The Next Citation Point] with all the rigour that one should demand, without recourse to postulates and renormalization procedures. (The field’s early history is well related in Ref. [154Jump To The Next Citation Point].) In 1960 DeWitt and Brehme [54Jump To The Next Citation Point] generalized Dirac’s result to curved spacetimes, and their calculation was corrected by Hobbs [95Jump To The Next Citation Point] several years later. In 1997 the motion of a point mass in a curved background spacetime was investigated by Mino, Sasaki, and Tanaka [130Jump To The Next Citation Point], who derived an expression for the particle’s acceleration (which is not zero unless the particle is a test mass); the same equations of motion were later obtained by Quinn and Wald [150Jump To The Next Citation Point] using an axiomatic approach. The case of a point scalar charge was finally considered by Quinn in 2000 [149Jump To The Next Citation Point], and this led to the realization that the mass of a scalar particle is not necessarily a constant of the motion.

This article reviews the achievements described in the preceding paragraph; it is concerned with the motion of a point scalar charge q, a point electric charge e, and a point mass m in a specified background spacetime with metric gαβ. These particles carry with them fields that behave as outgoing radiation in the wave zone. The radiation removes energy and angular momentum from the particle, which then undergoes a radiation reaction – its world line cannot be simply a geodesic of the background spacetime. The particle’s motion is affected by the near-zone field which acts directly on the particle and produces a self-force. In curved spacetime the self-force contains a radiation-reaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angular-momentum transport. The self-force is proportional to q2 in the case of a scalar charge, proportional to e2 in the case of an electric charge, and proportional to m2 in the case of a point mass.

In this review we derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely self-contained, and all relevant materials are developed ab initio. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. One must first assimilate the basic theory of bitensors (Part I), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line (Part II). One must next formulate a theory of Green’s functions in curved spacetimes (Part III), and finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how they should act on the particle (Part IV). A dedicated reader, correctly skeptical that sense can be made of a point mass in general relativity, will also want to work through the last portion of the review (Part V), which provides a derivation of the equations of motion for a small, but physically extended, body; this reader will be reassured to find that the extended body follows the same motion as the point mass. The review is very long, but the satisfaction derived, we hope, will be commensurate.

In this introductory section we set the stage and present an impressionistic survey of what the review contains. This should help the reader get oriented and acquainted with some of the ideas and some of the notation. Enjoy!

1.2 Radiation reaction in flat spacetime

Let us first consider the relatively simple and well-understood case of a point electric charge e moving in flat spacetime [154, 101Jump To The Next Citation Point, 171]. The charge produces an electromagnetic vector potential A α that satisfies the wave equation

□A α = − 4 πjα (1.1 )
together with the Lorenz gauge condition ∂ αAα = 0. (On page 294, Jackson [101Jump To The Next Citation Point] explains why the term “Lorenz gauge” is preferable to “Lorentz gauge”.) The vector jα is the charge’s current density, which is formally written in terms of a four-dimensional Dirac functional supported on the charge’s world line: the density is zero everywhere, except at the particle’s position where it is infinite. For concreteness we will imagine that the particle moves around a centre (perhaps another charge, which is taken to be fixed) and that it emits outgoing radiation. We expect that the charge will undergo a radiation reaction and that it will spiral down toward the centre. This effect must be accounted for by the equations of motion, and these must therefore include the action of the charge’s own field, which is the only available agent that could be responsible for the radiation reaction. We seek to determine this self-force acting on the particle.

An immediate difficulty presents itself: the vector potential, and also the electromagnetic field tensor, diverge on the particle’s world line, because the field of a point charge is necessarily infinite at the charge’s position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.

Difficult but not impossible. To find a way around this problem we note first that the situation considered here, in which the radiation is propagating outward and the charge is spiraling inward, breaks the time-reversal invariance of Maxwell’s theory. A specific time direction was adopted when, among all possible solutions to the wave equation, we chose A αret, the retarded solution, as the physically relevant solution. Choosing instead the advanced solution Aα adv would produce a time-reversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition

α 1( α α ) A S = --A ret + Aadv (1.2 ) 2
would restore time-reversal invariance: outgoing and incoming radiation would be present in equal amounts, there would be no net loss nor gain of energy by the system, and the charge would undergo no radiation reaction. In Eq. (1.2View Equation) the subscript ‘S’ stands for ‘symmetric’, as the vector potential depends symmetrically upon future and past.

Our second key observation is that while the potential of Eq. (1.2View Equation) does not exert a force on the charged particle, it is just as singular as the retarded potential in the vicinity of the world line. This follows from the fact that A αret, Aαadv, and A αS all satisfy Eq. (1.1View Equation), whose source term is infinite on the world line. So while the wave-zone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line – all three electromagnetic fields are dominated by the particle’s Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript ‘S’: it stands for ‘singular’ as well as ‘symmetric’.

Because α A S is just as singular as α Aret, removing it from the retarded solution gives rise to a potential that is well behaved in a neighbourhood of the world line. And because A αS is known not to affect the motion of the charged particle, this new potential must be entirely responsible for the radiation reaction. We therefore introduce the new potential

1( ) AαR = Aαret − A αS =-- Aαret − A αadv (1.3 ) 2
and postulate that it, and it alone, exerts a force on the particle. The subscript ‘R’ stands for ‘regular’, because A α R is nonsingular on the world line. This property can be directly inferred from the fact that the regular potential satisfies the homogeneous version of Eq. (1.1View Equation), □A α = 0 R; there is no singular source to produce a singular behaviour on the world line. Since α AR satisfies the homogeneous wave equation, it can be thought of as a free radiation field, and the subscript ‘R’ could also stand for ‘radiative’.

The self-action of the charge’s own field is now clarified: a singular potential Aα S can be removed from the retarded potential and shown not to affect the motion of the particle. What remains is a well-behaved potential α AR that must be solely responsible for the radiation reaction. From the regular potential we form an electromagnetic field tensor F Rαβ = ∂αARβ − ∂βARα and we take the particle’s equations of motion to be

ma = fext + eF Ru ν, (1.4 ) μ μ μν
where uμ = dzμ∕d τ is the charge’s four-velocity [zμ(τ) gives the description of the world line and τ is proper time], aμ = duμ∕d τ its acceleration, m its (renormalized) mass, and f μ ext an external force also acting on the particle. Calculation of the regular field yields the more concrete expression
μ μ 2e2 ( μ μ )dfνext ma = fext +---- δν + u u ν ----, (1.5 ) 3m dτ
in which the second term is the self-force that is responsible for the radiation reaction. We observe that the self-force is proportional to e2, it is orthogonal to the four-velocity, and it depends on the rate of change of the external force. This is the result that was first derived by Dirac [56Jump To The Next Citation Point]. (Dirac’s original expression actually involved the rate of change of the acceleration vector on the right-hand side. The resulting equation gives rise to the well-known problem of runaway solutions. To avoid such unphysical behaviour we have submitted Dirac’s equation to a reduction-of-order procedure whereby daν∕dτ is replaced with m −1dfνext∕dτ. This procedure is explained and justified, for example, in Refs. [112Jump To The Next Citation Point, 70Jump To The Next Citation Point], and further discussed in Section 24 below.)

To establish that the singular field exerts no force on the particle requires a careful analysis that is presented in the bulk of the paper. What really happens is that, because the particle is a monopole source for the electromagnetic field, the singular field is locally isotropic around the particle; it therefore exerts no force, but contributes to the particle’s inertia and renormalizes its mass. In fact, one could do without a decomposition of the field into singular and regular solutions, and instead construct the force by using the retarded field and averaging it over a small sphere around the particle, as was done by Quinn and Wald [150Jump To The Next Citation Point]. In the body of this review we will use both methods and emphasize the equivalence of the results. We will, however, give some emphasis to the decomposition because it provides a compelling physical interpretation of the self-force as an interaction with a free electromagnetic field.

1.3 Green’s functions in flat spacetime

To see how Eq. (1.5View Equation) can eventually be generalized to curved spacetimes, we introduce a new layer of mathematical formalism and show that the decomposition of the retarded potential into singular and regular pieces can be performed at the level of the Green’s functions associated with Eq. (1.1View Equation). The retarded solution to the wave equation can be expressed as

∫ α α ′ β′ ′ ′ A ret(x) = G+ β′(x,x )j (x)dV , (1.6 )
in terms of the retarded Green’s function G +αβ′(x,x′) = δαβ′δ(t − t′ − |x − x′|)∕ |x − x ′|. Here x = (t,x ) is an arbitrary field point, x ′ = (t′,x ′) is a source point, and dV ′ := d4x′; tensors at x are identified with unprimed indices, while primed indices refer to tensors at x′. Similarly, the advanced solution can be expressed as
α ∫ α ′ β′ ′ ′ A adv(x) = G −β′(x, x )j (x )dV , (1.7 )
in terms of the advanced Green’s function G −αβ′(x,x′) = δαβ′δ (t − t′ + |x − x ′|)∕|x − x′|. The retarded Green’s function is zero whenever x lies outside of the future light cone of x′, and G α (x,x′) +β′ is infinite at these points. On the other hand, the advanced Green’s function is zero whenever x lies outside of the past light cone of ′ x, and α ′ G− β′(x,x ) is infinite at these points. The retarded and advanced Green’s functions satisfy the reciprocity relation
G −β′α(x′,x) = G+αβ′(x, x′); (1.8 )
this states that the retarded Green’s function becomes the advanced Green’s function (and vice versa) when x and x′ are interchanged.

From the retarded and advanced Green’s functions we can define a singular Green’s function by

α ′ 1[ α ′ α ′] G Sβ′(x,x ) = 2- G+ β′(x,x ) + G− β′(x,x ) (1.9 )
and a regular two-point function by
α ′ α ′ α ′ 1[ α ′ α ′] G Rβ′(x,x ) = G+ β′(x,x ) − G Sβ′(x,x ) = -- G+ β′(x,x ) − G −β′(x, x) . (1.10 ) 2
By virtue of Eq. (1.8View Equation) the singular Green’s function is symmetric in its indices and arguments: GSβ′α(x′,x ) = GSαβ′(x,x′). The regular two-point function, on the other hand, is antisymmetric. The potential
∫ A α(x) = G α ′(x,x ′)jβ′(x ′)dV ′ (1.11 ) S Sβ
satisfies the wave equation of Eq. (1.1View Equation) and is singular on the world line, while
∫ α α ′ β′ ′ ′ A R(x) = G Rβ′(x, x)j (x )dV (1.12 )
satisfies the homogeneous equation □A α = 0 and is well behaved on the world line.

Equation (1.6View Equation) implies that the retarded potential at x is generated by a single event in spacetime: the intersection of the world line and x’s past light cone (see Figure 1View Image). We shall call this the retarded point associated with x and denote it z(u); u is the retarded time, the value of the proper-time parameter at the retarded point. Similarly we find that the advanced potential of Eq. (1.7View Equation) is generated by the intersection of the world line and the future light cone of the field point x. We shall call this the advanced point associated with x and denote it z(v); v is the advanced time, the value of the proper-time parameter at the advanced point.

View Image

Figure 1: In flat spacetime, the retarded potential at x depends on the particle’s state of motion at the retarded point z(u) on the world line; the advanced potential depends on the state of motion at the advanced point z (v ).

1.4 Green’s functions in curved spacetime

In a curved spacetime with metric gαβ the wave equation for the vector potential becomes

□A α − R αβA β = − 4 πjα, (1.13 )
where αβ □ = g ∇ α∇ β is the covariant wave operator and R αβ is the spacetime’s Ricci tensor; the Lorenz gauge conditions becomes ∇ αA α = 0, and ∇ α denotes covariant differentiation. Retarded and advanced Green’s functions can be defined for this equation, and solutions to Eq. (1.13View Equation) take the same form as in Eqs. (1.6View Equation) and (1.7View Equation), except that dV ′ now stands for ∘ -----′-4 ′ − g(x )d x.

The causal structure of the Green’s functions is richer in curved spacetime: While in flat spacetime the retarded Green’s function has support only on the future light cone of x′, in curved spacetime its support extends inside the light cone as well; G α+β′(x, x′) is therefore nonzero when x ∈ I+ (x ′), which denotes the chronological future of x′. This property reflects the fact that in curved spacetime, electromagnetic waves propagate not just at the speed of light, but at all speeds smaller than or equal to the speed of light; the delay is caused by an interaction between the radiation and the spacetime curvature. A direct implication of this property is that the retarded potential at x is now generated by the point charge during its entire history prior to the retarded time u associated with x: the potential depends on the particle’s state of motion for all times τ ≤ u (see Figure 2View Image).

View Image

Figure 2: In curved spacetime, the retarded potential at x depends on the particle’s history before the retarded time u; the advanced potential depends on the particle’s history after the advanced time v.

Similar statements can be made about the advanced Green’s function and the advanced solution to the wave equation. While in flat spacetime the advanced Green’s function has support only on the past light cone of x′, in curved spacetime its support extends inside the light cone, and G −αβ′(x,x′) is nonzero when x ∈ I− (x′), which denotes the chronological past of x′. This implies that the advanced potential at x is generated by the point charge during its entire future history following the advanced time v associated with x: the potential depends on the particle’s state of motion for all times τ ≥ v.

The physically relevant solution to Eq. (1.13View Equation) is obviously the retarded potential A αret(x), and as in flat spacetime, this diverges on the world line. The cause of this singular behaviour is still the pointlike nature of the source, and the presence of spacetime curvature does not change the fact that the potential diverges at the position of the particle. Once more this behaviour makes it difficult to figure out how the retarded field is supposed to act on the particle and determine its motion. As in flat spacetime we shall attempt to decompose the retarded solution into a singular part that exerts no force, and a regular part that produces the entire self-force.

To decompose the retarded Green’s function into singular and regular parts is not a straightforward task in curved spacetime. The flat-spacetime definition for the singular Green’s function, Eq. (1.9View Equation), cannot be adopted without modification: While the combination half-retarded plus half-advanced Green’s functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Eq. (1.13View Equation), this candidate for the singular Green’s function would produce a self-force with an unacceptable dependence on the particle’s future history. For suppose that we made this choice. Then the regular two-point function would be given by the combination half-retarded minus half-advanced Green’s functions, just as in flat spacetime. The resulting potential would satisfy the homogeneous wave equation, and it would be regular on the world line, but it would also depend on the particle’s entire history, both past (through the retarded Green’s function) and future (through the advanced Green’s function). More precisely stated, we would find that the regular potential at x depends on the particle’s state of motion at all times τ outside the interval u < τ < v; in the limit where x approaches the world line, this interval shrinks to nothing, and we would find that the regular potential is generated by the complete history of the particle. A self-force constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and regular Green’s functions.

The proper definitions were identified by Detweiler and Whiting [53Jump To The Next Citation Point], who proposed the following generalization to Eq. (1.9View Equation):

α ′ 1 [ α ′ α ′ α ′] G Sβ′(x,x ) = -- G +β′(x, x) + G −β′(x,x ) − H β′(x, x) . (1.14 ) 2
The two-point function H αβ′(x,x′) is introduced specifically to cure the pathology described in the preceding paragraph. It is symmetric in its indices and arguments, so that GSαβ′(x,x′) will be also (since the retarded and advanced Green’s functions are still linked by a reciprocity relation); and it is a solution to the homogeneous wave equation, α ′ α γ ′ □H β′(x,x ) − R γ(x)H β′(x,x ) = 0, so that the singular, retarded, and advanced Green’s functions will all satisfy the same wave equation. Furthermore, and this is its key property, the two-point function is defined to agree with the advanced Green’s function when x is in the chronological past of x′: H αβ′(x,x′) = G −αβ′(x,x ′) when x ∈ I− (x′). This ensures that G Sαβ′(x, x′) vanishes when x is in the chronological past of x ′. In fact, reciprocity implies that H α ′(x,x ′) β will also agree with the retarded Green’s function when x is in the chronological future of ′ x, and it follows that the symmetric Green’s function vanishes also when x is in the chronological future of x ′.

The potential AαS(x ) constructed from the singular Green’s function can now be seen to depend on the particle’s state of motion at times τ restricted to the interval u ≤ τ ≤ v (see Figure 3View Image). Because this potential satisfies Eq. (1.13View Equation), it is just as singular as the retarded potential in the vicinity of the world line. And because the singular Green’s function is symmetric in its arguments, the singular potential can be shown to exert no force on the charged particle. (This requires a lengthy analysis that will be presented in the bulk of the paper.)

View Image

Figure 3: In curved spacetime, the singular potential at x depends on the particle’s history during the interval u ≤ τ ≤ v; for the regular potential the relevant interval is − ∞ < τ ≤ v.

The Detweiler–Whiting [53Jump To The Next Citation Point] definition for the regular two-point function is then

[ ] G αRβ′(x,x ′) = G +αβ′(x,x′) − G Sαβ′(x,x′) = 1-G +αβ′(x,x′) − G −αβ′(x,x′) + H αβ′(x,x′) . (1.15 ) 2
The potential A αR(x) constructed from this depends on the particle’s state of motion at all times τ prior to the advanced time v: τ ≤ v. Because this potential satisfies the homogeneous wave equation, it is well behaved on the world line and its action on the point charge is well defined. And because the singular potential α AS (x ) can be shown to exert no force on the particle, we conclude that α A R(x) alone is responsible for the self-force.

From the regular potential we form an electromagnetic field tensor F R = ∇ αAR − ∇ βAR αβ β α and the curved-spacetime generalization to Eq. (1.4View Equation) is

ext R ν ma μ = fμ + eFμνu , (1.16 )
where uμ = dzμ∕d τ is again the charge’s four-velocity, but aμ = Du μ∕dτ is now its covariant acceleration.

1.5 World line and retarded coordinates

To flesh out the ideas contained in the preceding subsection we add yet another layer of mathematical formalism and construct a convenient coordinate system to chart a neighbourhood of the particle’s world line. In the next subsection we will display explicit expressions for the retarded, singular, and regular fields of a point electric charge.

Let γ be the world line of a point particle in a curved spacetime. It is described by parametric relations μ z (τ ) in which τ is proper time. Its tangent vector is μ μ u = dz ∕dτ and its acceleration is aμ = Du μ∕dτ; we shall also encounter ˙aμ := Da μ∕dτ.

On γ we erect an orthonormal basis that consists of the four-velocity uμ and three spatial vectors eμ a labelled by a frame index a = (1,2,3). These vectors satisfy the relations gμνuμu ν = − 1, μ ν gμνu ea = 0, and μ ν gμνeaeb = δab. We take the spatial vectors to be Fermi–Walker transported on the world line: μ μ De a∕d τ = aau, where

aa(τ) = aμeμa (1.17 )
are frame components of the acceleration vector; it is easy to show that Fermi–Walker transport preserves the orthonormality of the basis vectors. We shall use the tetrad to decompose various tensors evaluated on the world line. An example was already given in Eq. (1.17View Equation) but we shall also encounter frame components of the Riemann tensor,
R (τ) = R eμu λeνuρ, R (τ) = R eμuλeνeρ, R (τ ) = R eμeλeνeρ, (1.18 ) a0b0 μλνρ a b a0bc μλνρ a b c abcd μλνρ a b c d
as well as frame components of the Ricci tensor,
μ ν μ ν μ ν R00 (τ ) = R μνu u , Ra0 (τ) = Rμνeau , Rab (τ ) = Rμνeaeb. (1.19 )
We shall use δab = diag(1,1,1) and its inverse δab = diag (1,1,1) to lower and raise frame indices, respectively.

Consider a point x in a neighbourhood of the world line γ. We assume that x is sufficiently close to the world line that a unique geodesic links x to any neighbouring point z on γ. The two-point function σ(x,z ), known as Synge’s world function [169Jump To The Next Citation Point], is numerically equal to half the squared geodesic distance between z and x; it is positive if x and z are spacelike related, negative if they are timelike related, and σ(x, z) is zero if x and z are linked by a null geodesic. We denote its gradient μ ∂ σ∕∂z by σμ(x,z), and μ − σ gives a meaningful notion of a separation vector (pointing from z to x).

To construct a coordinate system in this neighbourhood we locate the unique point x′ := z(u) on γ which is linked to x by a future-directed null geodesic (this geodesic is directed from x′ to x); we shall refer to ′ x as the retarded point associated with x, and u will be called the retarded time. To tensors at ′ x we assign indices ′ α, ′ β, …; this will distinguish them from tensors at a generic point z(τ) on the world line, to which we have assigned indices μ, ν, …. We have σ(x,x ′) = 0 and − σ α′(x, x′) is a null vector that can be interpreted as the separation between x′ and x.

View Image

Figure 4: Retarded coordinates of a point x relative to a world line γ. The retarded time u selects a particular null cone, the unit vector Ωa := ˆxa∕r selects a particular generator of this null cone, and the retarded distance r selects a particular point on this generator.

The retarded coordinates of the point x are a (u, ˆx ), where a a α′ ˆx = − eα′σ are the frame components of the separation vector. They come with a straightforward interpretation (see Figure 4View Image). The invariant quantity

∘ ---a--b α′ r := δabˆx xˆ = uα′σ (1.20 )
is an affine parameter on the null geodesic that links x to x′; it can be loosely interpreted as the time delay between x and x ′ as measured by an observer moving with the particle. This therefore gives a meaningful notion of distance between x and the retarded point, and we shall call r the retarded distance between x and the world line. The unit vector
Ωa = ˆxa∕r (1.21 )
is constant on the null geodesic that links x to ′ x. Because a Ω is a different constant on each null geodesic that emanates from ′ x, keeping u fixed and varying a Ω produces a congruence of null geodesics that generate the future light cone of the point x ′ (the congruence is hypersurface orthogonal). Each light cone can thus be labelled by its retarded time u, each generator on a given light cone can be labelled by its direction vector Ωa, and each point on a given generator can be labelled by its retarded distance r. We therefore have a good coordinate system in a neighbourhood of γ.

To tensors at x we assign indices α, β, …. These tensors will be decomposed in a tetrad (eα,eα) 0 a that is constructed as follows: Given x we locate its associated retarded point x′ on the world line, as well as the null geodesic that links these two points; we then take the tetrad (uα′,eα′) a at x′ and parallel transport it to x along the null geodesic to obtain (eα,eα) 0 a.

1.6 Retarded, singular, and regular electromagnetic fields of a point electric charge

The retarded solution to Eq. (1.13View Equation) is

∫ A α(x) = e G +αμ(x,z)u μdτ, (1.22 ) γ
where the integration is over the world line of the point electric charge. Because the retarded solution is the physically relevant solution to the wave equation, it will not be necessary to put a label ‘ret’ on the vector potential.

From the vector potential we form the electromagnetic field tensor F αβ, which we decompose in the tetrad (eα,eα) 0 a introduced at the end of Section 1.5. We then express the frame components of the field tensor in retarded coordinates, in the form of an expansion in powers of r. This gives

F (u, r,Ωa) := F (x)eα(x )eβ (x ) a0 αβ a 0 = e-Ω − e(a − a ΩbΩ ) + 1eR Ωb ΩcΩ − 1e(5R Ωb + R Ωb Ωc) r2 a r a b a 3 b0c0 a 6 a0b0 ab0c 1--( b c ) 1- 1- b tail + 12e 5R00 + RbcΩ Ω + R Ωa + 3eRa0 − 6 eRabΩ + F a0 + O (r), (1.23 ) a α β Fab (u, r,Ω ) := Fαβ(x)ea(x )eb (x ) e( ) 1- ( ) c = r aaΩb − Ωaab + 2 e Ra0bc − Rb0ac + Ra0c0Ωb − ΩaRb0c0 Ω 1 ( ) − -e Ra0 Ωb − ΩaRb0 + F taabil+ O (r), (1.24 ) 2
where
F tail= Fta′il′(x′)eα′uβ′, F tail= F tai′l′(x′)eα′eβ′ (1.25 ) a0 α β a ab αβ a b
are the frame components of the “tail part” of the field, which is given by
tail ′ ∫ u− ′ μ F α′β′(x ) = 2e ∇ [α′G+ β′]μ(x ,z)u dτ. (1.26 ) −∞
In these expressions, all tensors (or their frame components) are evaluated at the retarded point x ′ := z(u) associated with x; for example, aa := aa(u) := aα′eα′ a. The tail part of the electromagnetic field tensor is written as an integral over the portion of the world line that corresponds to the interval − ∞ < τ ≤ u− := u − 0+; this represents the past history of the particle. The integral is cut short at u − to avoid the singular behaviour of the retarded Green’s function when z(τ) coincides with ′ x; the portion of the Green’s function involved in the tail integral is smooth, and the singularity at coincidence is completely accounted for by the other terms in Eqs. (1.23View Equation) and (1.24View Equation).

The expansion of F αβ(x) near the world line does indeed reveal many singular terms. We first recognize terms that diverge when r → 0; for example the Coulomb field Fa0 diverges as − 2 r when we approach the world line. But there are also terms that, though they stay bounded in the limit, possess a directional ambiguity at r = 0; for example Fab contains a term proportional to Ra0bcΩc whose limit depends on the direction of approach.

This singularity structure is perfectly reproduced by the singular field S F αβ obtained from the potential

∫ A α(x) = e G α(x,z)uμd τ, (1.27 ) S γ Sμ
where α GSμ(x, z) is the singular Green’s function of Eq. (1.14View Equation). Near the world line the singular field is given by
FS (u,r,Ωa) := F S(x )eα(x)eβ(x) a0 αβ a 0 -e e( b ) 2- 1- b c 1- ( b b c) = r2Ωa − r aa − abΩ Ωa − 3 e˙aa + 3eRb0c0Ω Ω Ωa − 6 e 5Ra0b0Ω + Rab0cΩ Ω 1 ( b c ) 1 b + --e 5R00 + RbcΩ Ω + R Ωa − -eRab Ω + O (r), (1.28 ) S a S12 α β 6 Fab(u,r,Ω ) := Fαβ(x )ea(x)eb(x) e( ) 1 ( ) c = --aa Ωb − Ωaab + --e Ra0bc − Rb0ac + Ra0c0Ωb − ΩaRb0c0 Ω r ( 2 ) − 1e Ra0 Ωb − ΩaRb0 + O(r). (1.29 ) 2
Comparison of these expressions with Eqs. (1.23View Equation) and (1.24View Equation) does indeed reveal that all singular terms are shared by both fields.

The difference between the retarded and singular fields defines the regular field R Fαβ(x). Its frame components are

F R = 2e˙aa + 1eRa0 + Ftail+ O(r), (1.30 ) a0 3 3 a0 F Rab = F taabil+ O (r), (1.31 )
and at ′ x the regular field becomes
( ) R ( ) 2 γ′ 1 γ′ δ′ tail F α′β′ = 2eu[α′g β′]γ′ + uβ′]uγ′ 3 ˙a + 3R δ′u + F α′β′, (1.32 )
where ′ ′ ˙aγ = Da γ ∕dτ is the rate of change of the acceleration vector, and where the tail term was given by Eq. (1.26View Equation). We see that F R (x) αβ is a regular tensor field, even on the world line.

1.7 Motion of an electric charge in curved spacetime

We have argued in Section 1.4 that the self-force acting on a point electric charge is produced by the regular field, and that the charge’s equations of motion should take the form of ma = fext+ eF Ru ν μ μ μν, where ext fμ is an external force also acting on the particle. Substituting Eq. (1.32View Equation) gives

( ) ∫ τ− μ μ 2( μ μ ) -2--Df-νext- 1- ν λ 2 [μ ν] ( ′) λ′ ′ ma = fext + e δ ν + u uν 3m dτ + 3 R λu + 2e uν −∞ ∇ G +λ′ z(τ),z(τ) u dτ ,(1.33 )
in which all tensors are evaluated at z (τ ), the current position of the particle on the world line. The primed indices in the tail integral refer to a point z(τ′) which represents a prior position; the integration is cut short at τ′ = τ− := τ − 0+ to avoid the singular behaviour of the retarded Green’s function at coincidence. To get Eq. (1.33View Equation) we have reduced the order of the differential equation by replacing ν ˙a with −1 ˙ν m fext on the right-hand side; this procedure was explained at the end of Section 1.2.

Equation (1.33View Equation) is the result that was first derived by DeWitt and Brehme [54Jump To The Next Citation Point] and later corrected by Hobbs [95Jump To The Next Citation Point]. (The original version of the equation did not include the Ricci-tensor term.) In flat spacetime the Ricci tensor is zero, the tail integral disappears (because the Green’s function vanishes everywhere within the domain of integration), and Eq. (1.33View Equation) reduces to Dirac’s result of Eq. (1.5View Equation). In curved spacetime the self-force does not vanish even when the electric charge is moving freely, in the absence of an external force: it is then given by the tail integral, which represents radiation emitted earlier and coming back to the particle after interacting with the spacetime curvature. This delayed action implies that in general, the self-force is nonlocal in time: it depends not only on the current state of motion of the particle, but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that the physical process that leads to Eq. (1.33View Equation) is simply an interaction between the charge and a free electromagnetic field R Fαβ; it is this field that carries the information about the charge’s past.

1.8 Motion of a scalar charge in curved spacetime

The dynamics of a point scalar charge can be formulated in a way that stays fairly close to the electromagnetic theory. The particle’s charge q produces a scalar field Φ (x) which satisfies a wave equation

( ) □ − ξR Φ = − 4π μ (1.34 )
that is very similar to Eq. (1.13View Equation). Here, R is the spacetime’s Ricci scalar, and ξ is an arbitrary coupling constant; the scalar charge density μ(x) is given by a four-dimensional Dirac functional supported on the particle’s world line γ. The retarded solution to the wave equation is
∫ Φ (x) = q G+ (x,z )d τ, (1.35 ) γ
where G+ (x,z) is the retarded Green’s function associated with Eq. (1.34View Equation). The field exerts a force on the particle, whose equations of motion are
ma μ = q(gμν + uμu ν)∇ νΦ, (1.36 )
where m is the particle’s mass; this equation is very similar to the Lorentz-force law. But the dynamics of a scalar charge comes with a twist: If Eqs. (1.34View Equation) and (1.36View Equation) are to follow from a variational principle, the particle’s mass should not be expected to be a constant of the motion. It is found instead to satisfy the differential equation
dm--= − quμ∇ μΦ, (1.37 ) dτ
and in general m will vary with proper time. This phenomenon is linked to the fact that a scalar field has zero spin: the particle can radiate monopole waves and the radiated energy can come at the expense of the rest mass.

The scalar field of Eq. (1.35View Equation) diverges on the world line and its singular part ΦS (x) must be removed before Eqs. (1.36View Equation) and (1.37View Equation) can be evaluated. This procedure produces the regular field ΦR (x), and it is this field (which satisfies the homogeneous wave equation) that gives rise to a self-force. The gradient of the regular field takes the form of

( ) 1 ( ) 1 ν 1 ν λ tail ∇ μΦR = − 12(1 − 6ξ)qRu μ + q gμν + uμuν 3-˙a + 6R λu + Φ μ (1.38 )
when it is evaluated on the world line. The last term is the tail integral
∫ − tail τ ( ′ ) ′ Φ μ = q ∇ μG+ z(τ),z(τ ) dτ , (1.39 ) −∞
and this brings the dependence on the particle’s past.

Substitution of Eq. (1.38View Equation) into Eqs. (1.36View Equation) and (1.37View Equation) gives the equations of motion of a point scalar charge. (At this stage we introduce an external force fμ ext and reduce the order of the differential equation.) The acceleration is given by

[ ν ∫ τ− ] ma μ = f μ + q2(δμ + uμu ) -1--Df-ext-+ 1-Rν uλ + ∇ νG (z(τ),z(τ′))dτ′ (1.40 ) ext ν ν 3m dτ 6 λ − ∞ +
and the mass changes according to
∫ τ− ( ) dm--= − -1-(1 − 6ξ)q2R − q2uμ ∇ μG+ z(τ),z(τ′) dτ′. (1.41 ) d τ 12 −∞
These equations were first derived by Quinn [149Jump To The Next Citation Point]. (His analysis was restricted to a minimally coupled scalar field, so that ξ = 0 in his expressions. We extended Quinn’s results to an arbitrary coupling counstant for this review.)

In flat spacetime the Ricci-tensor term and the tail integral disappear and Eq. (1.40View Equation) takes the form of Eq. (1.5View Equation) with q2∕(3m ) replacing the factor of 2e2∕(3m ). In this simple case Eq. (1.41View Equation) reduces to dm ∕dτ = 0 and the mass is in fact a constant. This property remains true in a conformally flat spacetime when the wave equation is conformally invariant (ξ = 1 ∕6): in this case the Green’s function possesses only a light-cone part and the right-hand side of Eq. (1.41View Equation) vanishes. In generic situations the mass of a point scalar charge will vary with proper time.

1.9 Motion of a point mass, or a small body, in a background spacetime

The case of a point mass moving in a specified background spacetime presents itself with a serious conceptual challenge, as the fundamental equations of the theory are nonlinear and the very notion of a “point mass” is somewhat misguided. Nevertheless, to the extent that the perturbation hαβ(x ) created by the point mass can be considered to be “small”, the problem can be formulated in close analogy with what was presented before.

We take the metric gαβ of the background spacetime to be a solution of the Einstein field equations in vacuum. (We impose this condition globally.) We describe the gravitational perturbation produced by a point particle of mass m in terms of trace-reversed potentials γαβ defined by

γ = h − 1(gγδh )g , (1.42 ) αβ αβ 2 γδ αβ
where h αβ is the difference between g αβ, the actual metric of the perturbed spacetime, and g αβ. The potentials satisfy the wave equation
□γ αβ + 2Rγαδβγγδ = − 16πT αβ + O (m2 ) (1.43 )
together with the Lorenz gauge condition αβ γ ;β = 0. Here and below, covariant differentiation refers to a connection that is compatible with the background metric, □ = gαβ∇ α∇ β is the wave operator for the background spacetime, and Tαβ is the energy-momentum tensor of the point mass; this is given by a Dirac distribution supported on the particle’s world line γ. The retarded solution is
∫ γ αβ(x) = 4m G α+βμν(x, z)uμuνdτ + O (m2 ), (1.44 ) γ
where αβ G+ μν(x, z) is the retarded Green’s function associated with Eq. (1.43View Equation). The perturbation hαβ (x ) can be recovered by inverting Eq. (1.42View Equation).

Equations of motion for the point mass can be obtained by formally demanding that the motion be geodesic in the perturbed spacetime with metric gαβ = gαβ + h αβ. After a mapping to the background spacetime, the equations of motion take the form of

1( )( ) aμ = − --g μν + u μuν 2hνλ;ρ − h λρ;ν uλu ρ + O (m2 ). (1.45 ) 2
The acceleration is thus proportional to m; in the test-mass limit the world line of the particle is a geodesic of the background spacetime.

We now remove hS (x) αβ from the retarded perturbation and postulate that it is the regular field R h αβ(x) that should act on the particle. (Note that S γαβ satisfies the same wave equation as the retarded potentials, but that R γαβ is a free gravitational field that satisfies the homogeneous wave equation.) On the world line we have

( ) hR = − 4m u R + R u uρu ξ + htail, (1.46 ) μν;λ (μ ν)ρλξ μρνξ λ μνλ
where the tail term is given by
∫ τ− ( ) ( ) htail = 4m ∇ λ G+ μνμ′ν′ − 1-gμνG ρ ′ ′ z(τ),z(τ′) uμ′uν′d τ′. (1.47 ) μνλ −∞ 2 + ρμ ν
When Eq. (1.46View Equation) is substituted into Eq. (1.45View Equation) we find that the terms that involve the Riemann tensor cancel out, and we are left with
μ 1-( μν μ ν)( tail tail) λ ρ 2 a = − 2 g + u u 2h νλρ − h λρν u u + O (m ). (1.48 )
Only the tail integral appears in the final form of the equations of motion. It involves the current position z(τ) of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions z(τ′), at which tensors with primed indices are evaluated. As before the integral is cut short at τ′ = τ− := τ − 0+ to avoid the singular behaviour of the retarded Green’s function at coincidence.

The equations of motion of Eq. (1.48View Equation) were first derived by Mino, Sasaki, and Tanaka [130Jump To The Next Citation Point], and then reproduced with a different analysis by Quinn and Wald [150Jump To The Next Citation Point]. They are now known as the MiSaTaQuWa equations of motion. As noted by these authors, the MiSaTaQuWa equation has the appearance of the geodesic equation in a metric g + htail αβ αβ. Detweiler and Whiting [53Jump To The Next Citation Point] have contributed the more compelling interpretation that the motion is actually geodesic in a spacetime with metric R gαβ + hαβ. The distinction is important: Unlike the first version of the metric, the Detweiler-Whiting metric is regular on the world line and satisfies the Einstein field equations in vacuum; and because it is a solution to the field equations, it can be viewed as a physical metric — specifically, the metric of the background spacetime perturbed by a free field produced by the particle at an earlier stage of its history.

While Eq. (1.48View Equation) does indeed give the correct equations of motion for a small mass m moving in a background spacetime with metric gαβ, the derivation outlined here leaves much to be desired – to what extent should we trust an analysis based on the existence of a point mass? As a partial answer to this question, Mino, Sasaki, and Tanaka [130Jump To The Next Citation Point] produced an alternative derivation of their result, which involved a small nonrotating black hole instead of a point mass. In this alternative derivation, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: it must move according to Eq. (1.48View Equation). This alternative derivation (which was given a different implementation in Ref. [142Jump To The Next Citation Point]) is entirely free of singularities (except deep within the black hole), and it suggests that that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body’s internal structure can be ignored). This derivation, however, was limited to the case of a non-rotating black hole, and it relied on a number of unjustified and sometimes unstated assumptions [83Jump To The Next Citation Point, 144Jump To The Next Citation Point, 145Jump To The Next Citation Point]. The conclusion was made firm by the more rigorous analysis of Gralla and Wald [83Jump To The Next Citation Point] (as extended by Pound [144Jump To The Next Citation Point]), who showed that the MiSaTaQuWa equations apply to any sufficiently compact body of arbitrary internal structure.

It is important to understand that unlike Eqs. (1.33View Equation) and (1.40View Equation), which are true tensorial equations, Eq. (1.48View Equation) reflects a specific choice of coordinate system and its form would not be preserved under a coordinate transformation. In other words, the MiSaTaQuWa equations are not gauge invariant, and they depend upon the Lorenz gauge condition αβ 2 γ ;β = O (m ). Barack and Ori [17Jump To The Next Citation Point] have shown that under a coordinate transformation of the form α α α x → x + ξ, where α x are the coordinates of the background spacetime and ξα is a smooth vector field of order m, the particle’s acceleration changes according to aμ → aμ + a[ξ]μ, where

( 2 ν ) μ ( μ μ ) D--ξ- ν ρ ω λ a [ξ] = δ ν + u uν dτ2 + R ρωλu ξ u (1.49 )
is the “gauge acceleration”; 2 ν 2 ν μ ρ D ξ ∕dτ = (ξ ;μu );ρu is the second covariant derivative of ν ξ in the direction of the world line. This implies that the particle’s acceleration can be altered at will by a gauge transformation; ξα could even be chosen so as to produce aμ = 0, making the motion geodesic after all. This observation provides a dramatic illustration of the following point: The MiSaTaQuWa equations of motion are not gauge invariant and they cannot by themselves produce a meaningful answer to a well-posed physical question; to obtain such answers it is necessary to combine the equations of motion with the metric perturbation hαβ so as to form gauge-invariant quantities that will correspond to direct observables. This point is very important and cannot be over-emphasized.

The gravitational self-force possesses a physical significance that is not shared by its scalar and electromagnetic analogues, because the motion of a small body in the strong gravitational field of a much larger body is a problem of direct relevance to gravitational-wave astronomy. Indeed, extreme-mass-ratio inspirals, involving solar-mass compact objects moving around massive black holes of the sort found in galactic cores, have been identified as promising sources of low-frequency gravitational waves for space-based interferometric detectors such as the proposed Laser Interferometer Space Antenna (LISA [115]). These systems involve highly eccentric, nonequatorial, and relativistic orbits around rapidly rotating black holes, and the waves produced by such orbital motions are rich in information concerning the strongest gravitational fields in the Universe. This information will be extractable from the LISA data stream, but the extraction depends on sophisticated data-analysis strategies that require a detailed and accurate modeling of the source. This modeling involves formulating the equations of motion for the small body in the field of the rotating black hole, as well as a consistent incorporation of the motion into a wave-generation formalism. In short, the extraction of this wealth of information relies on a successful evaluation of the gravitational self-force.

The finite-mass corrections to the orbital motion are important. For concreteness, let us assume that the orbiting body is a black hole of mass m = 10 M ⊙ and that the central black hole has a mass 6 M = 10 M ⊙. Let us also assume that the small black hole is in the deep field of the large hole, near the innermost stable circular orbit, so that its orbital period P is of the order of minutes. The gravitational waves produced by the orbital motion have frequencies f of the order of the mHz, which is well within LISA’s frequency band. The radiative losses drive the orbital motion toward a final plunge into the large black hole; this occurs over a radiation-reaction timescale (M ∕m )P of the order of a year, during which the system will go through a number of wave cycles of the order of 5 M ∕m = 10. The role of the gravitational self-force is precisely to describe this orbital evolution toward the final plunge. While at any given time the self-force provides fractional corrections of order m ∕M = 10− 5 to the motion of the small black hole, these build up over a number of orbital cycles of order M ∕m = 105 to produce a large cumulative effect. As will be discussed in some detail in Section 2.6, the gravitational self-force is important, because it drives large secular changes in the orbital motion of an extreme-mass-ratio binary.

1.10 Case study: static electric charge in Schwarzschild spacetime

One of the first self-force calculations ever performed for a curved spacetime was presented by Smith and Will [163Jump To The Next Citation Point]. They considered an electric charge e held in place at position r = r0 outside a Schwarzschild black hole of mass M. Such a static particle must be maintained in position with an external force that compensates for the black hole’s attraction. For a particle without electric charge this force is directed outward, and its radial component in Schwarzschild coordinates is given by r 1 ′ fext = 2mf, where m is the particle’s mass, f := 1 − 2M ∕r0 is the usual metric factor, and a prime indicates differentiation with respect to r0, so that f′ = 2M ∕r20. Smith and Will found that for a particle of charge e, the external force is given instead by fr = 1mf ′ − e2M f 1∕2∕r3 ext 2 0. The second term is contributed by the electromagnetic self-force, and implies that the external force is smaller for a charged particle. This means that the electromagnetic self-force acting on the particle is directed outward and given by

e2M frself = --3-f 1∕2. (1.50 ) r0
This is a repulsive force. It was shown by Zel’nikov and Frolov [186Jump To The Next Citation Point] that the same expression applies to a static charge outside a Reissner–Nordström black hole of mass M and charge Q, provided that f is replaced by the more general expression 2 2 f = 1 − 2M ∕r0 + Q ∕r0.

The repulsive nature of the electromagnetic self-force acting on a static charge outside a black hole is unexpected. In an attempt to gain some intuition about this result, it is useful to recall that a black-hole horizon always acts as perfect conductor, because the electrostatic potential φ := − At is necessarily uniform across its surface. It is then tempting to imagine that the self-force should result from a fictitious distribution of induced charge on the horizon, and that it could be estimated on the basis of an elementary model involving a spherical conductor. Let us, therefore, calculate the electric field produced by a point charge e situated outside a spherical conductor of radius R. The charge is placed at a distance r0 from the centre of the conductor, which is taken at first to be grounded. The electrostatic potential produced by the charge can easily be obtained with the method of images. It is found that an image charge ′ e = − eR∕r0 is situated at a distance r′0 = R2 ∕r0 from the centre of the conductor, and the potential is given by φ = e∕s + e′∕s ′, where s is the distance to the charge, while s′ is the distance to the image charge. The first term can be identified with the singular potential φ S, and the associated electric field exerts no force on the point charge. The second term is the regular potential φR, and the associated field is entirely responsible for the self-force. The regular electric field is ErR = − ∂rφR, and the self-force is fsrelf = eErR. A simple computation returns

r -----e2R------ fself = − r3(1 − R2 ∕r2). (1.51 ) 0 0
This is an attractive self-force, because the total induced charge on the conducting surface is equal to e′, which is opposite in sign to e. With R identified with M up to a numerical factor, we find that our intuition has produced the expected factor of e2M ∕r3 0, but that it gives rise to the wrong sign for the self-force. An attempt to refine this computation by removing the net charge e′ on the conductor (to mimic more closely the black-hole horizon, which cannot support a net charge) produces a wrong dependence on r 0 in addition to the same wrong sign. In this case the conductor is maintained at a constant potential ′ ϕ0 = − e ∕R, and the situation involves a second image charge − e′ situated at r = 0. It is easy to see that in this case,
r -----e2R3----- fself = − r5(1 − R2 ∕r2). (1.52 ) 0 0
This is still an attractive force, which is weaker than the force of Eq. (1.51View Equation) by a factor of (R∕r0)2; the force is now exerted by an image dipole instead of a single image charge.

The computation of the self-force in the black-hole case is almost as straightforward. The exact solution to Maxwell’s equations that describes a point charge e situated r = r0 and 𝜃 = 0 in the Schwarzschild spacetime is given by

φ = φS + φR, (1.53 )
where
S e (r − M )(r0 − M ) − M 2cos 𝜃 φ = ---[---------------------------------------------------------2-]1∕2, (1.54 ) r0r (r − M )2 − 2(r − M )(r0 − M )cos 𝜃 + (r0 − M )2 − M 2sin 𝜃
is the solution first discovered by Copson in 1928 [43Jump To The Next Citation Point], while
φR = eM-∕r0- (1.55 ) r
is the monopole field that was added by Linet [114Jump To The Next Citation Point] to obtain the correct asymptotic behaviour φ ∼ e∕r when r is much larger than r 0. It is easy to see that Copson’s potential behaves as e(1 − M ∕r )∕r 0 at large distances, which reveals that in addition to e, S φ comes with an additional (and unphysical) charge − eM ∕r0 situated at r = 0. This charge must be removed by adding to φS a potential that (i) is a solution to the vacuum Maxwell equations, (ii) is regular everywhere except at r = 0, and (iii) carries the opposite charge +eM ∕r0; this potential must be a pure monopole, because higher multipoles would produce a singularity on the horizon, and it is given uniquely by R φ. The Copson solution was generalized to Reissner–Nordström spacetime by Léauté and Linet [113], who also showed that the regular potential of Eq. (1.55View Equation) requires no modification.

The identification of Copson’s potential with the singular potential φS is dictated by the fact that its associated electric field S S F tr = ∂rφ is isotropic around the charge e and therefore exerts no force. The self-force comes entirely from the monopole potential, which describes a (fictitious) charge +eM ∕r0 situated at r = 0. Because this charge is of the same sign as the original charge e, the self-force is repulsive. More precisely stated, we find that the regular piece of the electric field is given by

R eM ∕r0 F tr = − --r2---, (1.56 )
and that it produces the self-force of Eq. (1.50View Equation). The simple picture described here, in which the electromagnetic self-force is produced by a fictitious charge eM ∕r0 situated at the centre of the black hole, is not easily extracted from the derivation presented originally by Smith and Will [163Jump To The Next Citation Point]. To the best of our knowledge, the monopolar origin of the self-force was first noticed by Alan Wiseman [185Jump To The Next Citation Point]. (In his paper, Wiseman computed the scalar self-force acting on a static particle in Schwarzschild spacetime, and found a zero answer. In this case, the analogue of the Copson solution for the scalar potential happens to satisfy the correct asymptotic conditions, and there is no need to add another solution to it. Because the scalar potential is precisely equal to the singular potential, the self-force vanishes.)

We should remark that the identification of S φ and R φ with the Detweiler–Whiting singular and regular fields, respectively, is a matter of conjecture. Although φS and φR satisfy the essential properties of the Detweiler–Whiting decomposition – being, respectively, a regular homogenous solution and a singular solution sourced by the particle – one should accept the possibility that they may not be the actual Detweiler–Whiting fields. It is a topic for future research to investigate the precise relation between the Copson field and the Detweiler–Whiting singular field.

It is instructive to compare the electromagnetic self-force produced by the presence of a grounded conductor to the self-force produced by the presence of a black hole. In the case of a conductor, the total induced charge on the conducting surface is ′ e = − eR ∕r0, and it is this charge that is responsible for the attractive self-force; the induced charge is supplied by the electrodes that keep the conductor grounded. In the case of a black hole, there is no external apparatus that can supply such a charge, and the total induced charge on the horizon necessarily vanishes. The origin of the self-force is therefore very different in this case. As we have seen, the self-force is produced by a fictitious charge eM ∕r0 situated at the centre of black hole; and because this charge is positive, the self-force is repulsive.

1.11 Organization of this review

After a detailed review of the literature in Section 2, the main body of the review begins in Part I (Sections 3 to 7) with a description of the general theory of bitensors, the name designating tensorial functions of two points in spacetime. We introduce Synge’s world function σ (x,x′) and its derivatives in Section 3, the parallel propagator gαα′(x, x′) in Section 5, and the van Vleck determinant Δ (x, x′) in Section 7. An important portion of the theory (covered in Sections 4 and 6) is concerned with the expansion of bitensors when x is very close to ′ x; expansions such as those displayed in Eqs. (1.23View Equation) and (1.24View Equation) are based on these techniques. The presentation in Part I borrows heavily from Synge’s book [169Jump To The Next Citation Point] and the article by DeWitt and Brehme [54Jump To The Next Citation Point]. These two sources use different conventions for the Riemann tensor, and we have adopted Synge’s conventions (which agree with those of Misner, Thorne, and Wheeler [131Jump To The Next Citation Point]). The reader is therefore warned that formulae derived in Part I may look superficially different from those found in DeWitt and Brehme.

In Part II (Sections 8 to 11) we introduce a number of coordinate systems that play an important role in later parts of the review. As a warmup exercise we first construct (in Section 8) Riemann normal coordinates in a neighbourhood of a reference point x′. We then move on (in Section 9) to Fermi normal coordinates [122], which are defined in a neighbourhood of a world line γ. The retarded coordinates, which are also based at a world line and which were briefly introduced in Section 1.5, are covered systematically in Section 10. The relationship between Fermi and retarded coordinates is worked out in Section 11, which also locates the advanced point z(v) associated with a field point x. The presentation in Part II borrows heavily from Synge’s book [169]. In fact, we are much indebted to Synge for initiating the construction of retarded coordinates in a neighbourhood of a world line. We have implemented his program quite differently (Synge was interested in a large neighbourhood of the world line in a weakly curved spacetime, while we are interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally his.

In Part III (Sections 12 to 16) we review the theory of Green’s functions for (scalar, vectorial, and tensorial) wave equations in curved spacetime. We begin in Section 12 with a pedagogical introduction to the retarded and advanced Green’s functions for a massive scalar field in flat spacetime; in this simple context the all-important Hadamard decomposition [88] of the Green’s function into “light-cone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in Section 13 along with its restrictions on the past and future null cones of a reference point x′. The retarded, advanced, singular, and regular Green’s functions for the scalar wave equation are introduced in Section 14. In Sections 15 and 16 we cover the vectorial and tensorial wave equations, respectively. The presentation in Part III is based partly on the paper by DeWitt and Brehme [54Jump To The Next Citation Point], but it is inspired mostly by Friedlander’s book [71]. The reader should be warned that in one important aspect, our notation differs from the notation of DeWitt and Brehme: While they denote the tail part of the Green’s function by − v(x,x′), we have taken the liberty of eliminating the silly minus sign and call it instead +V (x, x′). The reader should also note that all our Green’s functions are normalized in the same way, with a factor of − 4π multiplying a four-dimensional Dirac functional of the right-hand side of the wave equation. (The gravitational Green’s function is sometimes normalized with a − 16 π on the right-hand side.)

In Part IV (Sections 17 to 19) we compute the retarded, singular, and regular fields associated with a point scalar charge (Section 17), a point electric charge (Section 18), and a point mass (Section 19). We provide two different derivations for each of the equations of motion. The first type of derivation was outlined previously: We follow Detweiler and Whiting [53Jump To The Next Citation Point] and postulate that only the regular field exerts a force on the particle. In the second type of derivation we take guidance from Quinn and Wald [150Jump To The Next Citation Point] and postulate that the net force exerted on a point particle is given by an average of the retarded field over a surface of constant proper distance orthogonal to the world line — this rest-frame average is easily carried out in Fermi normal coordinates. The averaged field is still infinite on the world line, but the divergence points in the direction of the acceleration vector and it can thus be removed by mass renormalization. Such calculations show that while the singular field does not affect the motion of the particle, it nonetheless contributes to its inertia.

In Part V (Sections 20 to 23), we show that at linear order in the body’s mass m, an extended body behaves just as a point mass, and except for the effects of the body’s spin, the world line representing its mean motion is governed by the MiSaTaQuWa equation. At this order, therefore, the picture of a point particle interacting with its own field, and the results obtained from this picture, is justified. Our derivation utilizes the method of matched asymptotic expansions, with an inner expansion accurate near the body and an outer expansion accurate everywhere else. The equation of motion of the body’s world line, suitably defined, is calculated by solving the Einstein equation in a buffer region around the body, where both expansions are accurate.

Concluding remarks are presented in Section 24, and technical developments that are required in Part V are relegated to Appendices. Throughout this review we use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [131].


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