Self-force on a charge outside a five-dimensional black hole

# Self-force on a charge outside a five-dimensional black hole

Matthew J. S. Beach, Eric Poisson, and Bernhard G. Nickel Department of Physics, University of Guelph, Guelph, Ontario, N1G 2W1, Canada
May 14, 2014
###### Abstract

We compute the electromagnetic self-force acting on a charged particle held in place at a fixed position outside a five-dimensional black hole described by the Schwarzschild-Tangherlini metric. Using a spherical-harmonic decomposition of the electrostatic potential and a regularization prescription based on the Hadamard Green’s function, we express the self-force as a convergent mode sum. The self-force is first evaluated numerically, and next presented as an analytical expansion in powers of , with denoting the event-horizon radius. The power series is then summed to yield a closed-form expression. Unlike its four-dimensional version, the self-force features a dependence on a regularization parameter that can be interpreted as the particle’s radius. The self-force is repulsive at large distances, and its behavior is related to a model according to which the force results from a gravitational interaction between the black hole and the distribution of electrostatic field energy attached to the particle. The model, however, is shown to become inadequate as becomes comparable to , where the self-force changes sign and becomes attractive. We also calculate the self-force acting on a particle with a scalar charge, which we find to be everywhere attractive. This is to be contrasted with its four-dimensional counterpart, which vanishes at any .

###### pacs:
04.50.Gh, 04.70.Bw, 04.25.Nx, 04.40.Nr

## I Introduction and summary

A particle held in place at a fixed position outside a nonrotating black hole of mass requires an external agent to supply an external force that compensates for the black hole’s gravity. When the particle carries an electric charge , the external force is smaller than when the particle is neutral. The difference is accounted for by the particle’s electromagnetic self-force, which originates in a subtle interaction between the particle, the electric field it generates, and the spacetime curvature around the black hole. The electromagnetic self-force acting on a charged particle at rest outside a Schwarzschild black hole was computed by Smith and Will smith-will:80 (). The only nonvanishing component of the force vector is the radial component , and the force invariant , with the sign adjusted so that the sign of agrees with the sign of , is given by

 F=q2R2r3, (1)

where is the event-horizon radius (we use geometrized units, in which ). The positive sign on the right-hand side indicates that the self-force is repulsive, which leads to a smaller when the particle carries a charge.

The repulsive nature of the electromagnetic self-force is a surprising feature that is difficult to explain. An attempt to provide some intuition relies on the fact that the event horizon is necessarily an equipotential surface, which suggests that the black hole should behave as a perfect conductor. This observation leads to the expectation that the self-force could be derived (up to numerical factors) on the basis of an elementary model involving a spherical conductor of radius in flat spacetime. The model features a charge at position , a first image charge at position inside the conductor, and a second image charge at the center. The first image charge produces a grounded conductor with a net charge distributed on its surface, and the second image charge eliminates this net charge, without violating the equipotential condition at the surface. (The black hole does not support a net charge.) The model predicts a self-force resulting from an interaction between the charge and the image dipole inside the conductor, and a simple calculation neglecting corrections of order reveals that the self-force scales as , with a negative sign indicating an attractive force. The model is a complete failure: it fails to produce to the correct scaling with and , and it even fails to produce the correct sign.

Another attempt to provide intuition, proposed in Sec. IV of Ref. burko-etal:00 (), produces a more intelligible picture. This model focuses its attention on the force acting on the black hole instead of the force acting on the charged particle. This force is necessarily gravitational in nature, and according to Newton’s third law, it must be equal in magnitude to the force acting on the particle. (The model features a mixture of Newtonian and relativistic ideas.) The force on the black hole is produced in part by the particle’s mass, but there is also a contribution from the distribution of electrostatic field energy that surrounds the charge. In this view, the charged particle behaves as an infinitely extended body, and the black hole is comparatively much smaller. The force on the black hole is then , with the negative sign indicating that the force is attractive, and denoting the mass within radius associated with the particle and the distribution of field energy. The particle’s mass is identified with , and , where is the density of field energy. With we have that , and the force becomes . The first term is the attractive gravitational force exerted by the particle, and the second term is a repulsive contribution from the field energy. Writing gives us an alternative interpretation: the first term is the gravitational force exerted by the black hole, and the second term is the self-force. This model is a success: it produces the correct scaling with and , and it produces the correct sign. It reproduces the Smith-Will force of Eq. (1) up to a factor of 2.

The failure of the electrostatic model at providing a reliable expression for the self-force has been a source of fascination in the literature, and it has motivated a line of inquiry that probes into the mysteries of the self-force in various circumstances. Thus, authors have replaced the black hole with various material bodies unruh:76 (); burko-etal:00 (); shankar-whiting:07 (); drivas-gralla:11 (); isoyama-poisson:12 (); they observed that the Smith-Will behavior of Eq. (1) is universal at large distances, but modified when becomes comparable to the body’s radius . Other authors have replaced the asymptotically-flat boundary conditions of the Schwarzschild spacetime by asymptotic cosmological conditions (specifically, de Sitter or anti de Sitter conditions kuchar-poisson-vega:13 ()); they observed that the Smith-Will behavior continues to hold approximately when the black-hole and cosmological scales are well separated, but is substantially modified when the scales are comparable.

In this paper we continue this line of inquiry, and ask whether the interpretation of the self-force as a gravitational interaction between the black hole and the electrostatic field energy attached to the particle continues to apply in higher dimensions. Extending the model to an -dimensional spacetime, with denoting the number of angular directions, we have that the force on the black hole is now given by . The mass within radius becomes , where is the density of field energy, and is the area of a unit -sphere. With we have that , and we obtain . The second term is identified with the electromagnetic self-force, and relating the black-hole mass to its event-horizon radius via , we arrive at an expected scaling of for the self-force. We wish to know whether this expectation is borne out by an actual computation. Self-forces in higher-dimensional spacetimes were also considered by Frolov and Zelnikov frolov-zelnikov:12a (); frolov-zelnikov:12b (); frolov-zelnikov:12c (); frolov-zelnikov:12d (), who provided concrete results for the specific case of Majumdar-Papapetrou spacetimes.

For reasons that will be explained below, our calculation of the self-force is restricted to the five-dimensional case. We obtain

 F=q2R22r5Ξf3/2, (2)

where is the event-horizon radius, , and

 (3)

with . The self-force depends on an unknown parameter, the dimensionless quantity , which originates in the regularization prescription to be described below. An interpretation for the length scale is that it represents the radius of the particle, which must of course be much smaller than the black hole, so that . The self-force, therefore, is not independent of the particle’s size, and presumably this is an indication that in five dimensions, the self-force cannot be expected to be independent of the details of internal structure. A graph of for selected values of is displayed in Fig. 1.

When is large compared with , the function behaves as , and the self-force becomes

 F∼q2R22r5,(r≫R). (4)

This repulsive behavior matches the expectation from the gravitational model, up to a factor of two that was also seen in the four-dimensional case. When decreases toward , however, the self-force force changes sign and becomes attractive. As approaches the diverging factor begins to dominate, but the divergence is limited by the fact that the particle cannot be closer to the horizon than a distance of order . Taking , we find that the self-force is bounded by

 F>−3128√2q2R31~s3/2ln128~s. (5)

In spite of this bound, the behavior of the self-force very close to the horizon should be viewed with suspicion, because a large implies a large electric field that can no longer be treated as a test field in a fixed background spacetime. The detailed description of the self-force reveals that the interpretation in terms of a gravitational interaction does not hold up to five-dimensional scrutiny. While the large- behavior of the self-force is repulsive and compatible with the model, the agreement does not persist when becomes comparable to .

Unlike Smith and Will, our computation of the five-dimensional self-force does not proceed on the basis of an exact solution to Maxwell’s equations for a point charge in the Schwarzschild-Tangherlini spacetime. Indeed, such a five-dimensional analogue of the Copson solution copson:28 (); linet:76 () is not known. Our method of calculation is therefore more convoluted. We begin in Sec. II with the formulation of Maxwell’s equations in higher-dimensional spacetimes, their specialization to the specific case of a point charge in the Schwarzschild-Tangherlini spacetime, and a presentation of the solution in terms of a decomposition in higher-dimensional spherical harmonics. This leads to a self-force expressed as an infinite and diverging sum over spherical-harmonic modes.

The mode-sum evaluation of the self-force requires regularization, and we carry out the necessary steps in Sec. III. We adopt a regularization prescription based on Hadamard’s Green’s function hadamard:23 (), a local expansion of the electrostatic potential that identifies the singular part that must be subtracted before the mode sum is evaluated. While Hadamard regularization can be formulated in any spacetime dimension, its practical implementation becomes increasingly difficult as the number of dimensions increases, for the simple reason that the electric field of a point charge becomes increasingly singular at the position of the particle. With the techniques at our disposal we were able to handle the five-dimensional case with relative ease, and this motivated our restriction to five dimensions. An extension to higher dimensions is possible, but would require a substantial amount of additional work.

Unlike the situation in four dimensions, the five-dimensional Hadamard Green’s function features a logarithmic dependence on the separation between field and source points. This is the source of the term in the self-force, with the length parameter interpreted as the particle’s radius. Unlike its four-dimensional version, the five-dimensional self-force depends on the details of the particle’s internal structure. This dependence is likely to be even more dramatic in higher dimensions, because the field of a point charge becomes increasingly singular and requires additional regularization. It would be interesting to pursue these matters by performing a self-force calculation in six dimensions.

The calculation of the self-force proceeds in Sec. IV with a numerical evaluation of the regularized mode sum, and an analytical evaluation presented as an expansion in powers of . We carry out this expansion to a very high order, and manage to sum the series to the closed-form expression displayed in Eq. (2).

In Sec. V we exploit the same methods to calculate the self-force acting on a scalar charge at a fixed position in the five-dimensional Schwarzschild-Tangherlini spacetime. Our final result is displayed in Eq. (141) below. When is much larger than we find that the scalar self-force behaves as

 Fscalar∼−38q2R4r7ln2r√~sR. (6)

The self-force is attractive everywhere, and its scaling with can be contrasted with the scaling of the electromagnetic self-force. This result can also be contrasted with Wiseman’s four-dimensional expression wiseman:00 (): . Like its electromagnetic counterpart, the scalar self-force is bounded by Eq. (5) when the particle is close to the horizon.

What would happen to the five-dimensional self-force if the topology of the event horizon were changed from the topology examined here to the topology of a black string? The regularization techniques developed in this paper could be adapted to this new situation, and a fresh calculation of the self-force could be attempted. Would the self-force continue to diverge as the event horizon is approached? We hope to return to this question in future work. In the remainder of the paper we present the detailed calculations that lead to the results summarized in this introductory section.

## Ii Electrostatics in a higher-dimensional black-hole spacetime

In this section we formulate Maxwell’s equations in a curved spacetime of arbitrary dimensionality, and specialize them to the description of a static electric charge in the spacetime of a nonrotating black hole. This spacetime is static and spherically symmetric, and we denote the number of angular directions by ; the total number of spatial dimensions is then , and is the number of spacetime dimensions.

### ii.1 Maxwell’s equations and Lorentz force

Maxwell’s equations in a curved, -dimensional spacetime are expressed in covariant form as

 ∇βFαβ=Ωnjα,∇[αFβγ]=0, (7)

where is the electromagnetic field tensor, is the current density, is the covariant derivative operator, and is the area of an -dimensional unit sphere — an explicit expression is given in Eq. (144); indices enclosed within square brackets are antisymmetrized. The sourcefree Maxwell equations can be solved by expressing the electromagnetic field in terms of a vector potential,

 Fαβ=∇αΦβ−∇βΦα. (8)

In a given Lorentz frame in flat spacetime, the components of a static electric field are given by with , and Maxwell’s equations reduce to Gauss’s law , where is the charge density. The field produced by a point charge at the spatial origin of the coordinate system is given by , where is a unit vector in the direction of the field point , and the associated potential is given by .

The current density of a point charge moving on a world line described by the parametric relations (with denoting proper time) is given by

 jα(x)=q∫uαδ(x,z(τ))dτ, (9)

where is a scalarized Dirac distribution defined by when lies within the domain of integration; is the metric determinant evaluated at , and is an arbitrary test function.

Formally, the electromagnetic self-force acting on this point charge is given by the Lorentz force

 Fα=qFα βuβ, (10)

where is the electromagnetic field produced by the charge. Since this field diverges at the position of the particle, the equation has only formal validity, and the field must be regularized before the self-force is computed.

### ii.2 Schwarzschild-Tangherlini spacetime

We specialize the general formulation of Maxwell’s equations to the case of a charge held at a fixed position in a higher-dimensional analogue of the Schwarzschild spacetime, often named the Tangherlini spacetime tangherlini:63 (). Its metric is given by

 ds2=−fdt2+f−1dr2+r2dΩ2n, (11)

where

 f:=1−(R/r)n−1 (12)

and is the metric on a unit -sphere — refer to the Appendix for a fuller description of the notation employed here and below. The gravitational radius marks the position of the event horizon, and it is related to the gravitational (ADM) mass by .

The fixed position of the particle is described by and . To condense the notation it is helpful to represent the angular coordinates by a unit vector defined in such a way that the relation between the spherical polar coordinates and quasi-Cartesian coordinates is given by the usual . In this notation the variable position of a point on a hypersurface is represented by , and the fixed position of the charge is designated by .

For this static situation the only nonvanishing component of the vector potential is , and Maxwell’s equations reduce to the single equation

 r2∂rrΦ+nr∂rΦ+1fD2Φ=r2Ωnjt, (13)

in which denotes a partial derivative with respect to , and is the Laplacian operator on the unit -sphere (refer to the Appendix). The charge density can be obtained from the general expression of Eq. (9) by switching integration variables from to . We get

 jt=qδ(r−r0)rn0δ(Ω,Ω0), (14)

where is the angular Dirac distribution introduced in Eq. (152).

A formal expression for the self-force acting on the charged particle can be obtained from Eq. (10). Its only nonvanishing component is

 Fr=q√f0∂rΦ(r0,Ω0), (15)

where . It is useful to remove the dependence on the coordinate system by working instead with the invariant , with the sign selected so that . This gives

 F=q∂rΦ(r0,Ω0), (16)

which represents the magnitude of the force actually measured by a static observer at .

### ii.3 Decomposition in spherical harmonics

To proceed we decompose the potential and charge density in the higher-dimensional spherical harmonics introduced in the Appendix. We write

 Φ(r,Ω)=∑ℓ,jψℓ,j(r)Yℓ,j(Ω) (17)

and

 δ(Ω,Ω0)=∑ℓ,j¯Yℓ,j(Ω0)Yℓ,j(Ω), (18)

where an overbar indicates complex conjugation, are the spherical harmonics, labelled by an integer degree () and a degeneracy index that ranges over a number of distinct values — refer to Eq. (147). Making the substitution returns the sequence of ordinary differential equations

 r2ψ′′ℓ,j+nrψ′ℓ,j−ℓ(ℓ+n−1)fψℓ,j=qΩnrn−20¯Yℓ,j(Ω0)δ(r−r0) (19)

for the expansion coefficients ; a prime indicates differentiation with respect to .

Without loss of generality we may place the particle on the polar axis. According to Eq. (161), this ensures that only the axisymmetric mode contributes to . Defining , we find that the differential equations become

 r2ψ′′ℓ+nrψ′ℓ−ℓ(ℓ+n−1)fψℓ=qN(n,ℓ)rn−20δ(r−r0). (20)

Making use of Eq. (159), we also find that the scalar potential can be expressed as

 Φ(r,χ)=∑ℓψℓ(r)Pℓ(cosχ), (21)

where is the angle from the polar axis, and are the generalized Legendre polynomials introduced in the Appendix.

The substitutions

 ξ:=2(r/R)n−1−1,ψℓ:=√fP(ξ) (22)

bring Eq. (20) to the form of an associated Legendre equation with parameters and . The linearly independent solutions to the homogeneous version of Eq. (20) are therefore

 ψinℓ=√fP1ν(ξ),ψoutℓ=√fQ1ν(ξ). (23)

The inner solution is regular at () but singular at infinity, while the outer solution is singular at but regular at infinity. The solution to Eq. (20) can be obtained by combining these solutions and enforcing the appropriate junction conditions at . With denoting the solution for , and denoting the solution for , we have

 ψ<ℓ=qN(n,ℓ)rn0Wℓψoutℓ(r0)ψinℓ(r),ψ>ℓ=qN(n,ℓ)rn0Wℓψinℓ(r0)ψoutℓ(r), (24)

where the Wronskian is evaluated at . Making use of Eq. (8.18) of Ref. abramowitz-stegun:72 (), we find that

 Wℓ=ℓ(ℓ+n−1)2(n−1)Rn−1rn0. (25)

Our final expression for the solution to Eq. (20) is then

 ψ<ℓ=2(n−1)N(n,ℓ)ℓ(ℓ+n−1)qRn−1ψoutℓ(r0)ψinℓ(r) (26)

and

 ψ>ℓ=2(n−1)N(n,ℓ)ℓ(ℓ+n−1)qRn−1ψinℓ(r0)ψoutℓ(r). (27)

Complete expressions can be obtained by inserting Eq. (23) with and .

The special case must be handled separately. Here we find

 ψ<0=−1n−1qrn−10,ψ>0=−1n−1qrn−1. (28)

The electric field associated with this solution vanishes for and is equal to for ; these expressions are compatible with the presence of a charge at .

In terms of the mode decomposition, the (formal) self-force acting on the charged particle is obtained by inserting Eq. (21) within Eq. (16) and setting , . This gives

 F=q∑ℓψ′ℓ(r0) (29)

after making use of the normalization condition for the generalized Legendre polynomials — refer to Eq. (158). Because the electric field is actually infinite at the position of the charge, this mode sum does not converge and the computation of the self-force requires regularization.

### iii.1 Regularization and renormalization

We wish to turn Eq. (29) into a meaninful expression for the self-force. We begin by generalizing the context to a charged particle held at a fixed position in any static, ()-dimensional spacetime with metric

where the lapse and spatial metric depend on the spatial coordinates only. The electromagnetic self-force acting on this particle is expressed formally as

 Fa=qN−1hab∂bΦ, (31)

in which all quantities are evaluated at . We wish to turn this formal statement into something meaningful.

We assert quinn-wald:97 () that the physical self-force acting on the particle is

 Fa=qN−1hab⟨∂bΦ⟩ren, (32)

in which is the average of on a small surface surrounding the particle, from which all contributions that diverge in the limit are removed. The average is defined precisely by working in Riemann normal coordinates around , and denotes proper distance from the particle; the averaging is therefore performed on a surface of constant proper distance. We shall see that the diverging terms are proportional to the particle’s acceleration, so that they can be absorbed into a redefinition of the particle’s mass.

For a practical implementation of this regularization procedure, it is convenient to introduce a singular potential , a solution to Maxwell’s equations for a point charge at , constructed locally with no regards to boundary conditions imposed at infinity or anywhere else. The singular potential is just as singular as at , and the difference is smooth. We write

 ⟨∂bΦ⟩ren=∂bΦ−∂bΦS+⟨∂bΦS⟩ren, (33)

omitting the average sign on the difference because it is smooth in the limit .

For the next step we return to the specific context of spherically-symmetric spacetimes, and express the metric in the general form of

 ds2=−e2ϕdt2+f−1dr2+r2dΩ2n, (34)

in which and are arbitrary functions of the radial coordinate, and is the metric on a unit -sphere. With the particle placed on the polar axis , we decompose and as in Eq. (21), and write the self-force as

 F=Fmode+FS, (35)

where

 Fmode:=qe−ϕ0f1/20∑ℓ[ψ′ℓ(r0)−ψS′ℓ(r0)] (36)

is a convergent sum over -modes, and

 FS:=qe−ϕ0f1/20⟨∂rΦS⟩ren (37)

is the regularized contribution from the singular potential. We introduced the notation and .

Our computation of the self-force is based on Eq. (35). We identify the singular potential with the Hadamard Green’s function associated with the differential equation satisfied by an electrostatic potential in a static, -dimensional spacetime. After introducing the main equations we review Hadamard’s construction in an arbitrary number of dimensions, and then specialize it to the specific case of a five-dimensional spacetime (). We next construct the Hadamard Green’s function as a local expansion around the base point, and calculate . Then we specialize the results to a spherically-symmetric spacetime, decompose in generalized Legendre polynomials, and calculate the modes that appear in Eq. (36); these give rise to the ubiquitous regularization parameters of the self-force literature barack-ori:00 (); barack-etal:02 (); barack-ori:03a (). This long computation will return all the ingredients required in the evaluation of Eq. (35).

### iii.2 Green’s function in a static spacetime

The metric of a static, -dimensional spacetime is expressed as in Eq. (30). We introduce the vector field

 Aa:=∂alnN, (38)

and write Maxwell’s equation for the potential as

 ∇2Φ−Aa∂aΦ=Ωnμ, (39)

where and is the Laplacian operator in the -dimensional space with metric ; is the covariant derivative operator in this space.

The field equation can be solved by means of a Green’s function that satisfies

 ∇2G(x,x′)−Aa∂aG(x,x′)=−Ωnδ(x,x′), (40)

where is a scalarized Dirac distribution defined by when lies within the domain of integration; is the determinant of the spatial metric evaluated at , and is an arbitrary test function of the spatial coordinates. In terms of the Green’s function the solution to Eq. (39) is

 Φ(x)=−∫G(x,x′)μ(x′)√h′dn+1x′. (41)

Notice that the source term in the equation for comes with a positive sign, while it comes with a negative sign in the equation for ; this difference, which is entirely a matter of convention, explains the appearance of a negative sign on the right-hand side of Eq. (41).

For a static charge at a fixed position , the current density of Eq. (9) yields , and Eq. (41) reduces to

 Φ(x)=−qN(x0)G(x,x0). (42)

The Hadamard Green’s function is a local solution to Eq. (40) that incorporates the singularity structure implied by the Dirac distribution, but does not enforce boundary conditions that we might wish to impose on the potential (for example, a falloff condition at spatial infinity). The theory of such objects was developed by Hadamard (who called them “elementary solutions” hadamard:23 ()), and it is conveniently summarized in a number of references dewitt-brehme:60 (); friedlander:75 (); poisson-pound-vega:11 (). We provide a brief description of the construction here, but include no derivations.

The local theory of Green’s functions relies heavily on Synge’s world function , which is half the squared geodesic distance between the field point and the base point ; it is assumed that is sufficiently close to that the geodesic joining them is unique. The gradient of with respect to , denoted , is tangent to the geodesic, and the same is true of , the gradient with respect to ; the vectors point in opposite directions. The mathematical theory of two-point tensors (or bitensors), of which , , and are examples, is developed systematically in Refs. synge:60 (); dewitt-brehme:60 () and summarized in Ref. poisson-pound-vega:11 (). Our developments below rely heavily on these techniques.

The structure of the Hadamard Green’s function depends critically on the dimensionality of the space. When is even ( odd), the Green’s function can be expressed as

 GH(x,x′)=1n−1U(x,x′)(2σ)12(n−1), (43)

where is a biscalar that is assumed to be smooth in the coincidence limit . When is odd ( even) we have instead

 GH(x,x′)=1n−1U(x,x′)(2σ)12(n−1)+V(x,x′)ln2σλ2+W(x,x′), (44)

where and are other smooth biscalars, and is an arbitrary length parameter that makes the argument of the logarithm dimensionless. In both cases must be normalized by to ensure that Eq. (43) satisfies Eq. (40).

When is even, is constructed as an expansion in powers of ,

 U(x,x′)=∞∑p=0Up(x,x′)(2σ)p. (45)

Substitution in Eq. (43) and then Eq. (40) reveals that each expansion coefficient must satisfy

 (2σa∇a−Aaσa+∇2σ+2p−n−1)Up=−2p−n+1(n−1)2(∇2−Aa∇a)Up−1. (46)

This is a recursion relation for , and the differential operator on the left-hand side indicates that each equation is a transport equation that can be integrated along each geodesic that emanates from the base point . The equation for is integrated with a zero right-hand side, and a unique solution is selected by enforcing the coincidence limit . Hadamard proved hadamard:23 () that the expansion of Eq. (45) converges in a sufficiently small neighborhood around .

When is odd the construction must be modified to account for the fact that the right-hand side of Eq. (46) vanishes when . The expansion for must then be truncated to

 U(x,x′)=12(n−3)∑p=0Up(x,x′)(2σ)p, (47)

and the additional terms in Eq. (44) are inserted to ensure that the Green’s function continues to be a solution to Eq. (40). The biscalars and are also constructed as expansions in powers of ,

 V(x,x′)=∞∑p=0Vp(x,x′)(2σ)p,W(x,x′)=∞∑p=0Wp(x,x′)(2σ)p, (48)

and substitution in Eq. (44) and (40) produces the recursion relations

 (2σa∇a−Aaσa+∇2σ−2)V0=−12(n−1)(∇2−Aa∇a)U12(n−3), (49)
 (2σa∇a−Aaσa+∇2σ+2p−2)Vp=−12p(∇2−Aa∇a)Vp−1, (50)

and

 (2σa∇a−Aaσa+∇2σ+2p−2)Wp=−1p(2σa∇a−Aaσa+∇2σ+4p−2)Vp−12p(∇2−Aa∇a)Wp−1. (51)

The recursion relation (46) continues to apply in the odd case. Equation (49) determines from the last coefficient in the expansion for , and Eq. (50) determines the remaining coefficients . Equation (51) permits the determination of for , but there is no equation that determines , which must remain arbitrary. The expansions of Eq. (48) are also known to converge hadamard:23 () in a sufficiently small neighborhood around .

The Hadamard Green’s function for odd is subjected to two types of ambiguities. The first concerns the choice of length parameter , which is arbitrary, and the second concerns the choice of function , which is also arbitrary. These ambiguities are not independent. In fact, the freedom to choose is merely a special case of the freedom to choose . To see this, suppose that an initial choice for is shifted to , where is an alternate choice of length parameter. The shift is then propagated to each by the recursion relations (51), and we find that , which implies that

 W→W′=W+2Vln(λ/λ′). (52)

This, finally, is equivalent to a shift in the Hadamard form of Eq. (44).

### iii.4 Local expansion for n=3

We now set and use Eqs. (44), (47), (48) and the recursion relations of Eqs. (46), (49), (50), (51) to construct the Hadamard Green’s function as a local expansion about the base point . To address the ambiguities discussed in the preceding paragraph, we specifically set for some arbitrary choice of . This choice is justified on the basis that is a smooth contribution to the Hadamard Green’s function that cancels out when it is incorporated in Eq. (33). The remaining terms in the expansion for play no role in the regularization prescription, because they vanish in the limit .

Equation (47) indicates that when , and Eq. (46) implies that satisfies the transport equation

 (2σa∇a−Aaσa+∇2σ−4)U0=0 (53)

with . To integrate this equation we postulate the existence of a local expansion of the form

 U0=1+aa′σa′+12aa′b′σa′σb′+16aa′b′c′σa′σb′σc′+O(ϵ4), (54)

in which , , and are tensors defined at the base point . We let be a measure of distance between and , so that . Noting also that , we see that a truncation of the expansion at order implies that the contribution to the Green’s function is computed through order ; we shall maintain this degree of accuracy in the remaining calculations.

The base-point tensors are determined by inserting Eq. (54) within Eq. (53) and solving order by order in . These manipulations are aided by the identities and satisfied by the world function, as well as the standard expansions

 σ;ab =h a′ah b′b[ha′b′−13Ra′c′b′d′σc′σd′+14Ra′c′b′d′;e′σc′σd′σe′+O(ϵ4)], (55a) σ;a′b =−hb′ b[ha′b′+16Ra′c′b′d′σc′σd′−112Ra′c′b′d′;e′σc′σd′σe′+O(ϵ4)], (55b) Aa =h a′a[Aa′−Aa′;c′σc′+12Aa′;c′d′σc′σd′+O(ϵ3)], (55c) ha b′;c =12ha a′hc′ cRa′ b′c′d′σd′+O(ϵ2); (55d)

here is the parallel propagator, which takes a vector at and returns the parallel-transported vector at , is the spatial Riemann tensor (defined with respect to the spatial metric ) evaluated at , and a semicolon indicates covariant differentiation. A straightforward computation returns

 aa′ =−12Aa′, (56a) aa′b′ =12Aa′;b′+14Aa′Ab′+16Ra′b′, (56b) aa′b′c′ =−12A(a′;b′c′)−34A(a′Ab′;c′)−18Aa′Ab′Ac′−14A(a′Rb′c′)−14R(a′b′;c′), (56c)

where is the spatial Ricci tensor at , and indices enclosed within round brackets are fully symmetrized. While these calculations were carried out specifically for , it is easy to show that the end result for is actually independent of .

We next compute the contribution to the Green’s function through order (formally treating the logarithm as a quantity of order unity), and this requires expanded through order . We write

 V0=b+ba′σa′+O(ϵ2) (57)

and determine the coefficients and by inserting Eqs. (54) and (57) within Eq. (49). Another straightforward computation produces and , and substitution of Eqs. (56) gives

 b =−116(Aa′ ;a′−12Aa′Aa′+13R′), (58a) ba′ =132(∇′2Aa′+Ab′ ;b′Aa′−Ab′Aa′;b′−12Ab′Ab′Aa′−Ab′Ra′b′+13R′Aa′+13R′;a′), (58b)

where is the Ricci scalar at , and is the Laplacian operator with respect to the variables .