Adrian C Ottewill
email: adrian.ottewill@ucd.ie

Peter Taylor
email: peter.taylor@ucd.ie

Renormalized Vacuum Polarization and Stress Tensor on the Horizon of a Schwarzschild Black Hole Threaded by a Cosmic String

Keywords: numbers: 04.62.+v, 04.70.Dy, 11.27.+d

We calculate the renormalized vacuum polarization and stress tensor for a massless, arbitrarily coupled scalar field in the Hartle-Hawking vacuum state on the horizon of a Schwarzschild black hole threaded by an infinte straight cosmic string. This calculation relies on a generalized Heine identity for non-integer Legendre functions which we derive without using specific properties of the Legendre functions themselves.

Introduction

There is a long and successful history of calculating the renormalized vacuum polarization and stress-energy tensor on spherically-symmetric black holes (see [1,2,3,4,5,6,7,8]), however, in the astrophysically significant case of the Kerr-Newman black hole, such calculations have proved intractable. This is due to the fact that all the calculations in the spherically symmetric case have relied heavily on specific properties of the Legendre functions (mainly the Legendre Addition Theorem and the Heine Identity) and also the applicability of the Watson-Sommerfeld Formula to obtain a mode-sum expression that is numerically tractable. In the axially symmetric case, however, addition theorems for the angular functions do not exist nor do we have an analog of the Heine identity for the angular functions that arise. We will show, in the framework of the Schwarzschild black hole threaded by a cosmic string, that we can proceed in the axially symmetric case by obtaining useful summation formulae based on the Hadamard structure of the Green function without using specific properties of the mode-functions themselves. In this paper we will adopt this approach in order to derive a generalization of the standard Heine identity, adapted to the non-integer Legendre functions which arise in the separation of the wave equation on the background metric of a Schwarzschild black hole threaded by a cosmic string. Furthermore, we apply this formula to the calculation of the vacuum polarization and renormalized stress tensor for a massless scalar field on the horizon of the black hole.

The Green's Function

The cosmic string is modeled by introducing an azimuthal deficit parameter, α, into the standard Schwarzschild metric. We may describe this in coordinates (t, r, θ, φ), where φ is periodic with period 2πα, so we may take φ ∈ [0, 2πα), in which the line element is given by ds 2 = -(1 -2M/r)dt 2 + (1 -2M/r) -1 dr 2 + r 2 dθ 2 + r 2 sin 2 θd φ2 .

For GUT scale cosmic strings α = 1 -4µ where µ is the mass per unit length of the string so we shall assume 0 < α ≤ 1. We can alternatively define a new azimuthal coordinate by

φ = αφ (2)
so that φ is periodic with period 2π and we may take φ ∈ [0, 2π). In coordinates (t, r, θ, φ), the line element is given by ds 2 = -(1 -2M/r)dt 2 + (1 -2M/r) -1 dr 2 + r 2 dθ 2 + α 2 r 2 sin 2 θdφ 2 .

(

) 3
We shall consider a massless, minimally coupled scalar field, ϕ, in the Hartle-Hawking vacuum state. Since this is a thermal state, it is convenient to work with the Euclidean Green's function, performing a Wick rotation of the temporal coordinate t → -iτ and eliminating the conical singularity at r = 2M by making τ periodic with period 2π/κ where κ = 1/(4M) is the surface gravity of the black hole. The massless, minimally coupled scalar field, satisfies the homogenous wave-equation ϕ(τ, r, θ, φ) = 0, (4) which can be solved by a separation of variables by writing ϕ(τ, r, θ, φ) ∼ e inκτ +imφ P (θ)R(r) ( 5 ) where P (θ) is regular and satisfies

1 sin θ d dθ sin θ d dθ - m 2 α 2 sin 2 θ
+ λ(λ + 1) P (θ) = 0 (6) while R(r) satisfies

d dr (r 2 -2Mr) d dr -λ(λ + 1) - n 2 κ 2 r 4 r 2 -2Mr R(r) = 0. ( 7
)
The λ(λ + 1) term arises as the separation constant. The choice of λ is arbitrary for ϕ to satisfy the wave equation but requires a specific choice in order for the modefunction P (θ) to satisfy the boundary conditions of regularity on the poles. In the Schwarzschild case (no cosmic string, α = 1), regularity on the poles means that λ = l, i.e the separation constant is l(l + 1). In the cosmic string case, the appropriate choice of λ that guarantees regularity of the angular functions on the poles is

λ = l -|m| + |m|/α. (8)
(We note here the dependence of λ on l and m. However, to avoid being typographically cumbersome, we will leave it implicit in the remainder of this paper.) With this choice of λ, the angular function is the Legendre function of both non-integer order and noninteger degree,viz.,

P (θ) = P -|m|/α λ (cos θ). (9)
For completeness, we have shown in Appendix A that these angular functions satisfy the following normalization condition (an alternative proof of this normalization can be found in MacRobert's treatise [START_REF] Macrobert | Spherical Harmonics: An Elementary Treatise on Harmonic Functions with Applications[END_REF]) ,

The periodicity of the Green's function with respect to (τ -τ ) and (φ -φ ) with periodicity 2π/κ and 2π, respectively, combined with Eq.( 10) allow us to write the mode-sum expression for the Green's function as

G(x, x ) = κ 8π 2 ∞ n=-∞ e inκ(τ -τ ) ∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )χ nλ (r, r ), (11)
where χ nλ (r, r ) satisfies the inhomogeneous equation,

d dr (r 2 -2Mr) d dr -λ(λ + 1) - n 2 κ 2 r 4 r 2 -2Mr χ nλ (r, r ) = - 1 α δ(r -r ). ( 12
)
It is convenient to write the radial equation in terms of a new radial variable η = r/M -1, the radial equation then reads

d dη (η 2 -1) d dη -λ(λ + 1) - n 2 (1 + η) 4 16(η 2 -1) χ nλ (η, η ) = - 1 αM δ(η -η ), (13)
where we have used the fact that κ = 1/4M. For n = 0, the two solutions of the homogeneous equation are the Legendre functions of the first and second kind. For n = 0, the homogeneous equation cannot be solved in terms of known functions and must be solved numerically. We denote the two solutions that are regular on the horizon and infinity (or some outer boundary) by p |n| λ (η) and q |n| λ (η), respectively. A near-horizon Frobenius analysis for n = 0 shows that the indicial exponent is ±|n|/2, and so we have the following asymptotic forms:

p |n| λ (η) ∼ (η -1) |n|/2 η → 1, q |n| λ (η) ∼ (η -1) -|n|/2 η → 1. ( 14
)
Defining the normalizations by these asymptotic forms and using the Wronskian conditions one can obtain the appropriate normalization of the Green's function:

χ nλ (η, η ) =      1 αM P λ (η < )Q λ (η > ) n = 0, 1 2|n|αM p |n| λ (η < )q |n| λ (η > ) n = 0. ( 15
)

Renormalization

From the asymptotic forms [START_REF] Smith | Formation and Evolution of Cosmic Strings[END_REF], it is clear that taking one point on the horizon (with the other outside) means that all the n = 0 modes vanish. The unrenormalized Green's function then reduces to

G(x, x ) = 1 32π 2 M 2 α ∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )Q λ (η). ( 16
)
In the absence of the cosmic string, i.e. α = 1, this sum can be done using a combination of the Legendre Addition Theorem and the Heine Identity, yielding a closed form expression [1], that, of course, diverges in the coincidence limit.

To renormalize we follow the spirit of the Christensen-DeWitt [START_REF] Christensen | [END_REF] point-separation method subtracting the geometrical singular part of the Green's function for small separations in the coordinates. This method relies on the universal Hadamard singularity structure of the Green's function which ensures that once we have subtracted the geometric singular part, 'U/σ + V ln σ', the coincidence limit of the remainder 'W ' is finite. To calculate φ2 ren in the massless Ricci-flat case, the only subtraction term needed is

G sing (x, x ) = 1 4π 2 s 2 (17)
where s is the geodesic distance between x and x . Having already taken one point (x say) on the horizon we choose to separate radially, placing x at 2M + ; the geodesic distance is then given by

s(2M, 2M + ) = 2M + 2M dr (1 -2M/r ) 1/2 = (2M ) 1/2 2 + 6M - 2 80M 2 + 3 448M 3 - 5 4 9216M 4 + 7 5 45056M 5 + O( 6 ) . ( 18
)
Note that the leading term behaves like (∆x) 1/2 rather that the usual ∆x since the metric is singular at r = 2M so g rr = O( ), but this is simply a coordinate singularity and in no way affects the validity of the geometrical subtraction. The subtraction terms up to O( ) are

G sing = 1 32π 2 M - 1 192π 2 M 2 + O( ) = 1 32π 2 M 2 (η -1) - 1 192π 2 M 2 + O(η -1). (19)
Taking the partial coincidence limit φ → φ , θ → θ in (16), subtracting the renormalization terms [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF] and taking the limit as η → 1 ( → 0) gives us the renormalized expression for the vacuum polarization. However, we now face our fundamental challenge as the limit cannot be taken in a meaningful way in the forms given. In order to take the limit, we must either invent a way to write the divergent terms as an appropriate mode-sum and do a mode-by-mode subtraction or alternatively, we must attempt to perform the sum in ( 16) so that we can write the Green's function in closed form. In [12] we follow the former route to obtain numerical values of φ2 ren outside our black hole. However, when possible, the latter is clearly preferred since it can give us a simple closed-form solution for the renormalized vacuum polarization. In the next section, we derive a formula that will allow us to perform the sum (16).

Generalized Heine Identity

We recall that for real arguments, the Heine Identity for the Legendre Functions is [START_REF] Oberhettinger | Higher Transcendental Functions[END_REF]

] ∞ l=0 (2l + 1)P l (Ψ)Q l (ζ) = 1 (ζ -Ψ) , (20)
valid for |Ψ| < |ζ|. Writing Ψ = cos γ = cos θ cos θ + sin θ sin θ cos(φ -φ ), and using the Legendre Addition Theorem,

P l (cos γ) = l m=-l e im(φ-φ ) (l -m)! (l + m)! P m l (cos θ)P m l (cos θ ), (21)
we can re-write the Heine Identity as

∞ l=0 l m=-l e im(φ-φ ) (2l + 1) (l -m)! (l + m)! P m l (cos θ)P m l (cos θ )Q l (ζ) = 1 (ζ -cos γ) . ( 22
)
Below we shall generalize this form of the Heine Identity to the non-integer Legendre functions that arise in the mode-sum of a cosmic string space-time. It is this form rather than a generalization of (20) that is useful to us since the fundamentaly axiallysymmetric nature of the Green's function for α = 1 means that its radial part will always depend on m (through λ), and therefore the m-sum over the angular functions can never be done independently of the radial part.

In Appendix B, we have derived other identities involving Legendre Functions of non-integer order and/or degree. We have done so since, although they are not directly useful here, they turn out to be useful in other contexts, for example, determining the Hadamard singularity structure of a 3D Green's function on a cosmic string black hole space-time [12].

Returning to the problem at hand, we want to construct a generalization of (22). In addition, we would prefer to do this in such a way that the derivation does not rely on specific properties of the Legendre functions themselves, since ultimately we wish to apply the same techniques in Kerr space-time. There is a very natural way to do this by equating equivalent expressions for the same Green's function on a conveniently chosen space-time. In our case, the most appropriate case to consider is that of an infinite cosmic string in otherwise flat space-time. Of course, normally in this case one would exploit translational invariance along the string and work in cylindrical polar coordinates but for our purposes we choose to work in spherical polar coordinates, so

ds 2 = dτ 2 + dr 2 + r 2 dθ 2 + α 2 r 2 sin 2 θdφ 2 , ( 23
)
where φ is periodic with period 2π. The Green's function for a massless, minimally coupled scalar field on this space-time has a closed-form solution, which can be obtained, for example, by separating in cylindrical polar coordinates [START_REF] Smith | Formation and Evolution of Cosmic Strings[END_REF] G

(x, x ) = 1 8π 2 α sinh(χ/α) ρρ sinh χ(cosh(χ/α) -cos(φ -φ )) , (24)
where

cosh χ = (τ -τ ) 2 + r 2 + r 2 -2rr cos θ cos θ 2ρρ (25)
and ρ = r sin θ is the cylindrical polar coordinate. One can also obtain the spherical polar mode-sum expression for this Green's function:

G(x, x ) = 1 4π 2 α ∞ 0 cos ω(τ -τ )dω ∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )R ωλ (r, r ), (26)
where R ωλ (r, r ) is the 1D Green's function satisfying d dr

r 2 d dr -ω 2 r 2 -λ(λ + 1) R ωλ = -δ(r -r ). ( 27
)
The solution of this equation can be given in terms of Modified Bessel Functions,

R ωλ (r, r ) = I λ+1/2 (ωr < )K λ+1/2 (ωr > ) (rr ) 1/2 , ( 28
)
where I and K are the Modified Bessel Functions of the first and second kinds, respectively. The integral over ω can now be performed using the identity [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF] ∞

0 cos ω(τ -τ )I λ+1/2 (ωr < )K λ+1/2 (ωr > )dω = 1 2(rr ) 1/2 Q λ (ζ) (29)
valid for λ > -1, where Q λ is the Legendre function of the second kind and

ζ = (τ -τ ) 2 + r 2 + r 2 2rr . (30)
Thus, the spherical polar mode-sum expression for the Green's Function may be written as

G(x, x ) = 1 8π 2 αrr ∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )Q λ (ζ). ( 31
)
We now equate this Green's Function with the equivalent closed form expression (24) to obtain the following generalized Heine Identity:

∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )Q λ (ζ) = sinh(χ/α) sin θ sin θ sinh χ(cosh(χ/α) -cos(φ -φ )) (32)
where

cosh χ = ζ -cos θ cos θ sin θ sin θ ⇔ ζ = cos γ + (cosh χ -cos(φ -φ )) sin θ sin θ . ( 33
)
This identity, valid for ζ > | cos γ|, is completely analagous to (22) and reduces to it in the α → 1 limit.

Calculation of φ2

ren Returning now to our calculation of the vacuum polarization on the horizon of the Schwarzschild black hole threaded by a cosmic string, we see that our generalized Heine Identity (32) has precisely the right form to enable us to perform the mode-sum in the Green's function expression (11). The Green's function, where one point has been taken to lie on the horizon, may now be written as

G(x, x ) = 1 32π 2 M 2 α sinh(χ/α) sin θ sin θ sinh χ(cosh(χ/α) -cos(φ -φ )) (34)
For purely radial separation, we take φ → φ, θ → θ, to yield

G(r, θ; 2M, θ) = 1 32π 2 M 2 α sinh(χ/α) sin 2 θ sinh χ(cosh(χ/α) -1) (35)
where we now have

cosh χ = 1 + (η -1) sin 2 θ . ( 36
)
From Eqs.( 19) and ( 35), the renormalized vacuum polarization is given by

φ2 horizon ren = lim η→1 G(r, θ; 2M, θ) -G sing (r; 2M) = lim η→1 1 32π 2 M 2 α sin 2 θ sinh(χ/α) sinh χ(cosh(χ/α) -1) - 1 32π 2 M 2 (η -1) - 1 192π 2 M 2 . ( 37
)
It is now convenient first expand in terms of χ

sinh(χ/α) sinh χ(cosh(χ/α) -1) = 2α χ 2 + 1 6α - α 3 + O(χ 2 ). ( 38
) then cosh χ = 1 + (η -1) sin 2 θ ⇒ 1 χ 2 = sin 2 θ 2(η -1) + 1 12 + O(η -1) 2 (39) so we obtain sinh(χ/α) sinh χ(cosh(χ/α) -1) = α sin 2 θ (η -1) + 1 6α (1 -α 2 ) + O(η -1) 2 . ( 40
)
Substituting ( 40) into (37) and taking the limit, we arrive at the renormalized vacuum polarization on the horizon,

φ2 horizon ren = 1 192π 2 M 2 1 + 1 -α 2 α 2 sin 2 θ . (41)
The first term here is simply the Candelas result [1], which, of course we recover when α → 1, and the second term represents the contribution due to the presence of the cosmic string. The result (41) has in fact been calculated by Davies and Sahni [11] but for a restricted set of α values such that (1/α) is an integer. For this restricted class of values, the cosmic string solution is obtained as a sum of (1/α) images of the solution in the absence of a string. The method of images cannot be applied for general azimuthal deficits, however, so our method presents a generalization and independent derivation of the result of Davies and Sahni.

We have plotted the vacuum polarization in Figure 1 for a range of values of α. We see that the presence of the string increases φ2 horizon ren everywhere. In the equatorial plane it is increased by a factor of 1/α 2 from its value in the absence of a string. As we approach the axis φ2 horizon ren diverges in a non-integrable manner as

φ2 horizon ren ∼ 1 48π 2 (1 -α 2 ) α 2 1 (2M sin θ) 2 sin θ → 0, (42)
as one would expect a distance (2M sin θ) from a flat-space cosmic string. The region over which the string dominates may be characterised by the range of cos θ (near θ = 0) at which φ2 horizon ren exceeds twice its value in the equatorial plane, this is given by (in units M = 1) as a function of cos θ for a range of values of α. For α = 1, φ2 horizon ren diverges at the poles since there is a curvature singularity there due to the presence of the string. φ2 horizon ren increases everywhere as α decreases.

1 -(cos θ) 2 = 1 - 1 (2 -α 2 ) 1/2 = (1 -α) + O((1 -α) 2 )). ( 43

The Stress Tensor for a Massless Scalar Field with Arbitrary Coupling

In this section, we shall employ our generalized Heine identity [START_REF] Weinstein | [END_REF] in order to calculate the renormalized expectation value of the stress energy tensor for a massless scalar field with arbitrarily coupling to the Ricci scalar in the Hartle-Hawking vacuum state. This quantity is of particular physical interest in obtaining the back-reaction on the spacetime geometry in the semi-classical approximation to quantum gravity.

We first draw some general conclusions about the quantity we wish to calculate. We know that, by definition, the Hartle-Hawking vacuum respects the symmetries of the spacetime and that the metric given by Eq.( 3) is symmetric under the transformations

τ → -τ φ→ -φ θ→ π -θ, (44)
and under the translational symmetries

τ → τ + τ 0 φ → φ + φ 0 . (45)
We note, however, that unlike the spherically symmetric case, we do not have a translational symmetry in the θ-direction, but only the discrete reflective symmetry in the equatorial plane. As such, we have the possibility of non-zero off-diagonal components T r θ and T θ r . All other off-diagonal components are identically zero for the Hartle-Hawking vacuum. For a massless scalar field propagating in a Ricci-flat background, we have the following expression for the classical stress tensor

T a b = (1 -2ξ)g ac ϕ ;c ϕ ;b + (2ξ -1 2 )δ a b g cd ϕ ;c ϕ ;d -2ξg ac ϕϕ ;cb + 2ξδ a b ϕ ϕ (46)
It will prove convenient to write this tensor as

T a b = [τ a b (ϕ(x)ϕ(x ))] ≡ lim x →x τ a b (ϕ(x)ϕ(x )) (47)
where τ a b = τ a b (x, x ) is a differential operator which may be defined in any way provided it gives (46) in the coincidence limit. We shall adopt the following definition,

τ a b = (1 -2ξ)g ac ∇ b ∇ c + (2ξ -1 2 )δ a b g cd ∇ c ∇ d -2ξg ac ∇ c ∇ b + 2ξδ a b ∇ c ∇ c (48)
where g ab are the bivectors of parallel transport. Using point separation, the regularized, unrenormalized quantum stress tensor in the Hartle-Hawking vacuum state arises when this differential operator acts on the thermal Euclidean Green's function (11),

T a b unren = (1 -2ξ)g ac G ;c b + (2ξ -1 2 )δ a b g cd G ;cd -2ξg ac G ;cb + 2ξδ a b G. ( 49
)
This is clearly divergent in the coincident limit and must be renormalized prior to relaxing the regularization. To achieve this we note that this Green's function is of Hadamard form and may be written as

G(x, x ) = 1 8π 2 ∆ 1/2 σ + V ln(σ) + W (50)
where σ(x, x ) is half the square of the geodesic distance between x and x and

∆ = -g -1/2 (x) det(σ ;ab )g -1/2 (x ) ( 5 1 )
is the Van Vleck-Morette determinant. V (x, x ) and W (x, x ) are regular symmetric biscalar functions with V (x, x) being geometrical (local) while the determination of W (x, x ) requires (global) boundary conditions. Thus V (x, x) captures the (local) ultraviolet divergences of the quantum theory while W (x, x ) captures the state dependence.

The divergent part of the stress tensor is then the result of the operator (48) acting on the singular part of the Green's function, i.e.

T a b div = 1 8π 2 τ a b ∆ 1/2 σ + V ln(σ) . ( 52
)
The renormalized stress tensor then essentially has to be given by the coincident limit of the differential operator (48) acting on the regular part W ,

1 8π 2 [τ a b W (x, x )] . (53)
Rather than calculating W (x, x ) itself, we calculate (53) by subtracting Eq.( 52) from (49) and then take thing coincidence. There is one caveat, however, since the application of the differential operator τ a b to W (x, x ) results in a stress tensor which is not conserved since W (x, x ) is not a solution to the homogeneous wave equation but rather an inhomogeneous one where the source is geometrical. It is easy to redeem this situation by adding a regular geometrical term to the definition of the stress tensor (see [16,17] for example),

T a b ren = T a b unren -T a b div + 1 4π 2 v 1 δ a b ( 54
)
where for a massless scalar field in a Ricci-flat background, the geometrical scalar v 1 has the very simple form

v 1 = 1 720 C pqrs C pqrs . ( 55
)
On the horizon of our spacetime this reduces to v 1 (2M) = κ 2 /(60M 2 ).

In this paper, we are concerned with the calculation of the stress tensor on the horizon only. In this case, it is most convenient to split in the radial direction, taking the inner point (say x ), to lie on the horizon. Where derivatives of the Green's function are concerned, we must first perform the derivatives before we take partial coincidence limits. The bivectors of parallel transport have been calculated for radial separation in reference [1] for the Schwarzschild spacetime and are trivially extended to the cosmic string case by multiplying the azimuthal component by an α 2 factor. We list them here for completeness:

g τ τ = (1 -2M/r) 1/2 (1 -2M/r ) 1/2 g rr = (1 -2M/r) -1/2 (1 -2M/r ) -1/2
g θθ = rr g φφ = rr α 2 sin 2 θ.

The Derivatives of the Green's Function

Examination of (49) reveals that there is essentially two types of terms we need to evaluate: those of the form g ac G ;c b and those of the form g ac G ;cb . Considering the former case first, such terms involve a covariant derivative at each spacetime point but since G(x, x ) is a scalar at both x and x , we are in fact only dealing with partial derivatives. For the angular terms g θθ G ;θθ and g φφ G ;φφ , it is clear from the asymptotic forms ( 14) that taking x to lie on the horizon means that only the n = 0 terms will contribute in the limit η → 1. Therefore, we can differentiate directly Eq.(34) which is straight forward. Taking partial coincidence limits and expanding about the horizon, we obtain

[g θθ G ;θθ ] r = κ 2 8π 2 M 2 (η -1) 2 - κ 2 16π 2 M 2 (η -1) + κ 2 32π 2 M 2 - κ 2 (α 2 -1) 1440π 2 M 2 (11α 2 + 1) + 30α 2 cos 2 θ α 4 sin 4 θ + O(η -1) (56)
and

[g φφ G ;φφ ] r = κ 2 8π 2 M 2 (η -1) 2 - κ 2 16π 2 M 2 (η -1) + κ 2 32π 2 M 2 + κ 2 480π 2 M 2 (α 2 -1)(α 2 + 1) α 4 sin 4 θ + O(η -1), (57)
where we have adopted square bracket notation [..] r to indicate that we have taken the partial coincidence limit (τ → τ , η → 1, θ → θ , φ → φ ). For g τ τ G ;τ τ and g rr G ;rr , we must differentiate the full Green's function given by Eqs. (11) and (15) before we can take x to lie on the horizon. Considering g τ τ G ;τ τ first, we have

g τ τ G ;τ τ = 1 (1 -2M/r) 1/2 (1 -2M/r ) 1/2 κ 3 8π 2 ∞ n=-∞ n 2 e inκ(τ -τ ) ∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )χ nλ (η, η ) (58)
Trivially, the n = 0 term will vanish. Moreover, using the asymptotic forms Eq.( 14), it is clear that taking x to the horizon only the terms corresponding n = ±1 survive. We now have, in the partial coincidence limit,

[g τ τ G ;τ τ ] r = κ 2 16π 2 M 2 α (η + 1) 1/2 2 1/2 (η -1) 1/2 ∞ m=-∞ ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) (59)
We will denote the double sum by

F (η, cos θ) ≡ ∞ m=-∞ ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) [P -|m|/α λ (cos θ)] 2 q 1λ (η), (60)
and return to its evaluation later; this will involve both numerical and analytic parts.

We follow a similar argument to obtain an expression for g rr G ;rr . We have

g rr G ;rr = (1 -2M/r) 1/2 (1 -2M/r ) 1/2 κ 8π 2 ∞ n=-∞ e inκ(τ -τ ) ∞ m=-∞ e im(φ-φ ) × ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ ) ∂ 2 ∂η∂η χ nλ (η, η ). ( 61
)
Splitting up the n = 0 and n = 0 terms and writing in terms of radial variable η, we have

g rr G ;rr = (η -1) 1/2 (η -1) 1/2 (η + 1) 1/2 (η + 1) 1/2 κ 8π 2 αM 3 ∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) ×P -|m|/α λ (cos θ)P -|m|/α λ (cos θ ) dP λ (η ) dη dQ λ (η) dη + ∞ n=1
e inκ(τ -τ ) dp nλ (η ) dη dq nλ (η) dη .

(62)

Now using the fact that

(η 2 -1) 1/2 dP λ (η ) dη = P 1 λ (η ) → 0 as η → 1 ( 6 3 )
implies that the n = 0 term vanishes. Also, using the asymptotic forms Eq.( 14), we have for n = 0,

(η -1) 1/2 1 |n| dp nλ (η ) dη = 1 2 (η -1) |n|/2-1/2 (64)
which will vanish for all but the n = ±1 terms in the η → 1 limit. Therefore, the only contribution to g rr G rr comes from the n = ±1 terms, and in the partial coincidence limit we have

[g rr G ;rr ] r = κ 2 4π 2 M 2 α (η -1) 1/2 2 1/2 (η + 1) 1/2 × ∞ m=-∞ ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) [P -|m|/α λ (cos θ)] 2 dq 1λ (η) dη = κ 2 4π 2 M 2 α (η -1) 1/2 2 1/2 (η + 1) 1/2 dF (η, cos θ) dη . (65)
Finally, we require g rr G ;r θ . Similar calculations to those above show that only the n = ±1 term survives when we differentiate with respect to r and then take this point to lie on the horizon, yielding

g rr G ;r θ = κ 2 4π 2 Mα (η -1) 1/2 2 1/2 (η + 1) 1/2 × ∂ ∂θ ∞ m=-∞ ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )q 1λ (η). ( 66
)
In order to take the θ → θ limit, we make use of the fact that

lim θ→θ ∞ m=-∞ ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) ∂ ∂θ P -|m|/α λ (cos θ) P -|m|/α λ (cos θ ) = 1 2 ∂ ∂θ ∞ m=-∞ ∞ l=|m| (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ) 2 . ( 67
)
Therefore, in the partial coincidence limit, we have

[g rr G ;r θ ] r = κ 2 8π 2 Mα (η -1) 1/2 2 1/2 (η + 1) 1/2 ∂ ∂θ F (η, cos θ). (68)
We now consider derivatives of the form g ac G ;cb . For such terms, we have two derivatives at the same spacetime point, which will involve the Christoffel symbols,

G ;ab = G ,ab -Γ c ab G ,c , (69)
where the Christoffel symbols are the same as those of Schwarzschild spacetime except

Γ r φφ = -α 2 (r -2M) sin 2 θ Γ θ φφ = -α 2 sin θ cos θ . (70)
In spite of this minor added complication, things are significantly easier in this case since only the n = 0 term will contribute for each derivative. This is because we can always choose the differentiation to act on the outer point, this will not affect the asymptotic structure at the other spacetime point and so taking η → 1 and using the asymptotic forms [START_REF] Smith | Formation and Evolution of Cosmic Strings[END_REF], it is clear that only the n = 0 term will contribute in the limit. Now, the derivatives in (69) can be obtained by direct differentiation of (34). Performing the derivatives and using the appropriate Christoffel symbols, followed by taking the partial coincidence limits and expanding about the horizon, we obtain

[g τ τ G ;τ τ ] r = - κ 2 8π 2 M 2 (η -1) 2 + κ 2 8π 2 M 2 (η -1) - 3κ 2 32π 2 M 2 + κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ + O(η -1), ( 71
) [g rr G ;rr ] r = 3κ 2 8π 2 M 2 (η -1) 2 - κ 2 8π 2 M 2 (η -1) + κ 2 32π 2 M 2 + κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ + O(η -1), ( 72
) [g θθ G ;θθ ] r = - κ 2 8π 2 M 2 (η -1) 2 + κ 2 32π 2 M 2 + κ 2 48π 2 M 2 (α 2 -1) α 2 sin 2 θ - κ 2 1440π 2 M 2 (α 2 -1)(49α 2 -1) α 4 sin 4 θ + O(η -1), ( 73
) [g φφ G ;φφ ] r = - κ 2 8π 2 M 2 (η -1) 2 + κ 2 32π 2 M 2 - κ 2 480π 2 M 2 (α 2 -1)(α 2 + 1) α 4 sin 4 θ + κ 2 48π 2 M 2 (α 2 -1) α 2 cos 2 θ sin 4 θ + O(η -1), ( 74
) [g rr G ;rθ ] r = - κ 2 720π 2 M 3 (α 2 -1)(11α 2 + 1) cos θ α 4 sin 5 θ - κ 2 96π 2 M 3 (α 2 -1) cos θ α 2 sin 3 θ + O(η -1).
(75)

Candelas Method for Obtaining a Series Solution for F (η, cos θ)

In order to proceed with the evaluation of the stress tensor on the horizon, we require a series solution for the function we have called F (η, cos θ) (60) about the horizon η = 1. Fortunately, Candelas [1] has obtained the series solution in the Schwarzschild case and the method described here follows an almost identical path, so we refer the reader to the Candelas paper for more details. Following Candelas, we may capture the asymptotic behaviour of q 1λ (η) as η → 1 by writing

q 1λ (η) = - √ 2 Q 1 λ (η) + β λ p 1λ (η) -f (η)Q λ (η) + q1λ (η) ( 7 6 )
where the β λ are constants depending on l and m which can be evaluated numerically and the function f (η) and the β λ are chosen such that q1λ does not contribute as η → 1.

The β λ coefficients are global in nature and depend on the boundary conditions imposed, as we shall see in Appendix C. Further following Candelas, we may take

f (η) = √ 2 32 η 1 ((ξ + 1) 4 -16) (ξ 2 -1) 3/2 dξ, (77)
so, in particular, have

f (η) = (η -1) 1/2 + 1 32 (η -1) 5/2 + O(η -1) 7/2 . ( 78
)
A Frobenius analysis on this differential equation about η = 1 now shows that q1λ (η) satisfies the asymptotic condition

q1λ (η) ∼ 3 1792 (η -1) 7/2 ln(η -1) as η → 1, (79)
which is uniformly valid in l and m. As a consequence, Candelas [1] has shown that for this choice of f (η) and the coefficients β λ , the remainder term q1λ (η) does not contribute in the limit η → 1; although Candelas's analysis was for λ integer, the argument applies immediately to non-integer λ. The contribution coming from q1λ (η) + β l p 1λ (η) can now be calculated to leading order in (η -1), using the asymptotic forms [START_REF] Smith | Formation and Evolution of Cosmic Strings[END_REF] for p 1λ and the fact that q1λ does not contribute in the limit, as

∞ l=0 l m=-l (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) [P -|m|/α λ (cos θ)] 2 [q 1λ (η) + β λ p 1λ (η)] = (η -1) 1/2 A(cos θ) + O(η -1) 3/2 , (80)
where

A(cos θ) = ∞ l=0 l m=-l (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) [P -|m|/α λ (cos θ)] 2 β λ . ( 81
)
This sum is highly convergent and easily evaluated numerically. As we show in more detail in Appendix C, the β λ coefficients are determined by equating the asymptotic expansions for two equivalent representations of the function q 1λ (η), one being an integral representation obtained from the Wronskian condition and the other being the form (76). Returning now to the evaluation of F (η, cos θ),

F (η, cos θ) = ∞ l=0 l m=-l (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) [P -|m|/α λ (cos θ)] 2 q 1λ (η) = ∞ l=0 l m=-l (2λ + 1) Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) [P -|m|/α λ (cos θ)] 2 β λ p 1λ (η) -f (η)Q λ (η) - √ 2(η 2 -1) 1/2 dQ λ (η) dη + q1λ (η) . (82)
For the second and third terms in the expression above, we can use the generalized Heine identity (32) and its derivative respectively. As a series expansion in (η -1), we obtain

F (η, cos θ) = 2α (η -1) 3/2 - α 2(η -1) 1/2 + (η -1) 1/2 A(cos θ) - α 16 + (α 2 -1) 6α sin 2 θ + (1 + 10α 2 -11α 4 ) 90α 3 sin 4 θ + O(η -1) 3/2 . ( 83
)
We can now calculate the asymptotic series expansions for g τ τ G ;τ τ , g rr G ;rr and g rr G ;r θ . Employing Eq.(83) in expressions (59), ( 65) and (68) yields

[g τ τ G ;τ τ ] r = κ 2 8π 2 M 2 (η -1) 2 - κ 2 64π 2 M 2 + κ 2 96π 2 M 2 (α 2 -1) α 2 sin 2 θ - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ + κ 2 16π 2 M 2 α A(cos θ) + O(η -1), ( 84
) [g rr G ;rr ] r = - 3κ 2 8π 2 M 2 (η -1) 2 + κ 2 8π 2 M 2 (η -1) - 3κ 2 64π 2 M 2 + κ 2 96π 2 M 2 (α 2 -1) α 2 sin 2 θ - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ + κ 2 16π 2 M 2 α A(cos θ) + O(η -1), (85)
[g rr G ;r θ ] r = O(η -1).

(86)

Renormalized Stress Tensor

We have now calculated all the derivatives of the Green's function we require in order to calculate the unrenormalized components of the stress tensor. The term g ab G ;ab appearing in the stress tensor (49) is found by summing the expressions (84), ( 85), ( 56) and (57), yielding

[g ab G ;ab ] r = κ 2 8π 2 M 2 α A(cos θ) - κ 2 24π 2 M 2 (α 2 -1) α 2 cos 2 θ sin 4 θ + O(η -1). (87)
Moreover, the term G appearing in (49) vanishes since, by definition, G(x, x ) for a massless, Ricci-flat scalar field is a solution to the inhomogeneous wave-equation G = -δ. There is, however, a contribution to the divergent subtraction terms (52) coming from G div since this is non-zero. Combining these results, we arrive at the following expression for the unrenormalized stress tensor components near the horizon:

T τ unren = κ 2 8π 2 M 2 1 (η -1) 2 -2ξ κ 2 8π 2 M 2 1 (η -1) + (14ξ -1) κ 2 64π 2 M 2 - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ -(2ξ -1 2 ) κ 2 24π 2 M 2 (α 2 -1) α 2 sin 4 θ + (ξ -1 6 ) κ 2 16π 2 M 2 (α 2 -1) α 2 sin 2 θ , (88)
T r r unren = -

3κ 2 8π 2 M 2 1 (η -1) 2 + κ 2 8π 2 M 2 1 (η -1) + (2ξ -3) κ 2 64π 2 M 2 +ξ κ 2 8π 2 M 2 α A(cos θ) - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ -(2ξ -1 2 ) κ 2 24π 2 M 2 (α 2 -1) α 2 sin 4 θ + (ξ -1 6 ) κ 2 16π 2 M 2 (α 2 -1) α 2 sin 2 θ , (89)
T θ θ unren = κ 2 8π 2 M 2 1 (η -1) 2 -(1 -2ξ) κ 2 16π 2 M 2 1 (η -1) + (1 -4ξ) κ 2 32π 2 M 2 +(2ξ -1 2 ) κ 2 8π 2 M 2 α A(cos θ) - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ +ξ κ 2 24π 2 M 2 (α 2 -1) α 2 sin 4 θ , (90)
T φ φ unren = κ 2 8π 2 M 2 1 (η -1) 2 -(1 -2ξ) κ 2 16π 2 M 2 1 (η -1) + (1 -4ξ) κ 2 32π 2 M 2 +(2ξ -1 2 ) κ 2 8π 2 M 2 α A(cos θ) + κ 2 480π 2 M 2 (α 2 -1)(α 2 + 1) α 4 sin 4 θ -(ξ -1 6 ) κ 2 8π 2 M 2 (α 2 -1) α 2 cos 2 θ sin 4 θ , (91)
T r θ unren = ξ κ 2 360π 2 M (α 2 -1)(11α 2 + 1) cos θ α 4 sin 5 θ + ξ κ 2 48π 2 M (α 2 -1) α 2 cos θ sin 3 θ . ( 92
)
Note that the off-diagonal component is regular and vanishing for minimal coupling.

The geometrical subtraction terms are found by obtaining a series expansion for the singular part of the Green's function, then applying the differential operator (48) and taking partial coincidence limits; this is completely equivalent to, for example, Christensen's method [START_REF] Christensen | [END_REF]. The results are

T τ τ div = κ 2 8π 2 M 2 1 (η -1) 2 -2ξ κ 2 8π 2 M 2 1 (η -1) + (ξ -1 16 ) κ 2 4π 2 M 2 , ( 93
) T r r div = - 3κ 2 8π 2 M 2 1 (η -1) 2 + κ 2 8π 2 M 2 1 (η -1) + (ξ -3 4 ) κ 2 16π 2 M 2 , ( 94
)
T θ θ div = κ 2 8π 2 M 2 1 (η -1) 2 -(1 -2ξ) κ 2 16π 2 M 2 1 (η -1) -(ξ -7 20 ) κ 2 16π 2 M 2 , ( 95
)
T φ φ div = κ 2 8π 2 M 2 1 (η -1) 2 -(1 -2ξ) κ 2 16π 2 M 2 1 (η -1) -(ξ -7 20 ) κ 2 16π 2 M 2 , ( 96
) T r θ div = 0. ( 97
)
Finally, from Eqs.( 54), ( 88)-( 92) and ( 93)-( 97), we arrive at the renormalized stress tensor for a massless scalar field in the Hartle-Hawking vacuum on the horizon of the Schwarzschild black hole threaded by an infinite cosmic string:

T τ τ ren = ( 2 15 -ξ) κ 2 32π 2 M 2 + ξ κ 2 8π 2 M 2 α A(cos θ) - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ -(2ξ -1 2 ) κ 2 24π 2 M 2 (α 2 -1) α 2 sin 4 θ + (ξ -1 6 ) κ 2 16π 2 M 2 (α 2 -1) α 2 sin 2 θ , (98)
T r r ren = ( 2 15 -ξ)

κ 2 32π 2 M 2 + ξ κ 2 8π 2 M 2 α A(cos θ) - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ -(2ξ -1 2 ) κ 2 24π 2 M 2 (α 2 -1) α 2 sin 4 θ + (ξ -1 6 ) κ 2 16π 2 M 2 (α 2 -1) α 2 sin 2 θ , (99)
T θ θ ren = ( 13 60 -ξ) κ 2 16π 2 M 2 + (2ξ -1 2 ) κ 2 8π 2 M 2 α A(cos θ) - κ 2 1440π 2 M 2 (α 2 -1)(11α 2 + 1) α 4 sin 4 θ +ξ κ 2 24π 2 M 2 (α 2 -1) α 2 sin 4 θ , (100)
T φ φ ren = ( 13 60 -ξ) κ 2 16π 2 M 2 + (2ξ -1 2 ) κ 2 8π 2 M 2 α A(cos θ) + κ 2 480π 2 M 2 (α 2 -1)(α 2 + 1) α 4 sin 4 θ -(ξ -1 6 ) κ 2 8π 2 M 2 (α 2 -1) α 2 cos 2 θ sin 4 θ , ( 101
) T r θ ren = ξ κ 2 360π 2 M (α 2 -1)(11α 2 + 1) cos θ α 4 sin 5 θ + ξ κ 2 48π 2 M (α 2 -1) α 2 cos θ sin 3 θ . ( 102
)
We note from these expressions that T τ τ ren = T r r ren , which we would have expected from the symmetries of the black hole horizon. We also see that T θ θ ren = T φ φ ren but that we retrieve equality between these components in the α → 1 limit.

1 0.5 0.5 1 cosΘ 0.8 0.6 0.4 0.2 0.2 0.4 0.6 8Π 2 M 2 Κ 2 T a b ren 8Π 2 M 2 Κ 2 T Τ Τ ren 8Π 2 M 2 Κ 2 1 r T r Θ ren 8Π 2 M 2 Κ 2 T Θ Θ ren 8Π 2 M 2 Κ 2 T Φ Φ ren Cos Θ Figure 2. (8π 2 M 2 /κ 2 ) T a b horizon ren
for a conformally coupled scalar field, ξ = 1/6, as a function of cos θ for α = 0.95.

0.06 8Π 2 M 2 Κ 2 T a b ren 8Π 2 M 2 Κ 2 T Τ Τ ren 8Π 2 M 2 Κ 2 1 r T r Θ ren 8Π 2 M 2 Κ 2 T Θ Θ ren 8Π 2 M 2 Κ 2 T Φ Φ ren Cos Θ Figure 3. (8π 2 M 2 /κ 2 ) T a b horizon ren
(zoomed in plot) for a conformally coupled scalar field, ξ = 1/6, as a function of cos θ for α = 0.95. Furthermore, in the conformally coupled case, ξ = 1/6, we reproduce the Candelas [1] result in the α → 1 limit. Eqs.( 98)-( 101) also, of course, yield new results for the renormalized stress tensor on the horizon of a Schwarzschild black hole for general ξ, these are trivially obtained from by taking the limit α → 1.

In Fig. 2 and Fig. 6 we have plotted the components of the renormalized stress tensor for the massless, conformally coupled scalar field for α = 0.95 and α = 0.6, respectively. We have further plotted the minimally coupled case in Fig. 4 and Fig. 7 for α = 0.95 and α = 0.6, respectively. We see that for small azimuthal deficits, the values only move away from the constant Schwarzschild values very close to the poles where the curvature singularity dominates. For this reason, we have included magnified plots Fig. 3 and Fig. 5 for the α = 0.95 case. For larger deficits (α = 0.6 for example), we observe that the effects of the string are apparent even relatively close to the equator.

1.0 8Π 2 M 2 Κ 2 T a b ren 8Π 2 M 2 Κ 2 T Τ Τ ren 8Π 2 M 2 Κ 2 T Θ Θ ren 8Π 2 M 2 Κ 2 T Φ Φ ren Cos Θ Figure 4. (8π 2 M 2 /κ 2 ) T a b horizon ren
for a minimally coupled scalar field, ξ = 0, as a function of cos θ for α = 0.95.

0.15 8Π 2 M 2 Κ 2 T a b ren 8Π 2 M 2 Κ 2 T Τ Τ ren 8Π 2 M 2 Κ 2 T Θ Θ ren 8Π 2 M 2 Κ 2 T Φ Φ ren Cos Θ Figure 5. (8π 2 M 2 /κ 2 ) T a b horizon ren
(zoomed in plot) for a minimally coupled scalar field, ξ = 0, as a function of cos θ for α = 0.95.

8Π 2 M 2 Κ 2 T a b ren 8Π 2 M 2 Κ 2 T Τ Τ ren 8Π 2 M 2 Κ 2 1 r T r Θ ren 8Π 2 M 2 Κ 2 T Θ Θ ren 8Π 2 M 2 Κ 2 T Φ Φ ren Cos Θ Figure 6. (8π 2 M 2 /κ 2 ) T a b horizon ren
for a conformally coupled scalar field, ξ = 1/6, as a function of cos θ for α = 0.6.

1 0.5 0.5 1 cosΘ 2 1 1 8Π 2 M 2 Κ 2 T a b ren 8Π 2 M 2 Κ 2 T Τ Τ ren 8Π 2 M 2 Κ 2 T Θ Θ ren 8Π 2 M 2 Κ 2 T Φ Φ ren Cos Θ Figure 7. (8π 2 M 2 /κ 2 ) T a b horizon ren
for a minimally coupled scalar field, ξ = 0, as a function of cos θ for α = 0.6.

Checking Conservation and the Trace Anomaly

The Conservation Equations: The Minimally Coupled Case

In this section, we give an example of how the stress tensor we have calculated may be shown to satisfy the conservation equations. For simplicity, we consider the minimally coupled case ξ = 0. The conservation equations are

∇ a T a b ren = ∂ a T a b ren + Γ a ac T c b ren -Γ c ab T a c ren = 0. (103)
In particular taking b = θ we have

∇ a T a θ ren = ∂ θ T θ θ ren + ∂ r T r θ ren + 1 r T r θ ren + (r -2M) T θ r ren + cot θ T θ θ ren -cot θ T φ φ ren = 0. (104) Now (r -2M) T θ r ren = 1 r T r θ ren → 0 as r → 2M (105)
for the minimally coupled case. However, the derivative ∂ r T r θ ren is non-zero on the horizon, as we shall see below. For ξ = 0, the classical expression for this term is given by

∂ r T r θ ren = ∂ r (g rr ϕ ,r )ϕ ,θ + g rr ϕ ,r ϕ ,rθ (106)
and in the regularized version terms like ϕ ,r ϕ ,θ are reexpressed with a radial derivative of the Green's function at one spacetime point and a θ derivative at the other spacetime point. We consider the first term on the right-hand side of Eq.( 106), taking the radial derivatives with respect to the outer point and taking the inner point to lie on the horizon, then only the n = 0 term survives and we can differentiate our Heine identity (32) directly,

∂ r (g rr ϕ ,r )ϕ ,θ = κ 2 2π 2 M 2 α ∂ ∂η (η -1) (η + 1) ∂ 2 ∂η∂θ sinh(χ/α) sin θ sin θ sinh χ(cosh(χ/α) -1) (107)
where

cosh χ = η -cos θ cos θ sin θ sin θ , (108)
and we have taken τ → τ and φ → φ . Finally, performing the differentiation and taking θ → θ , we have

[∂ r (g rr ϕ ,r )ϕ ,θ ] r = - κ 2 360π 2 M 2 (α 2 -1)(11α 2 + 1) cos θ α 4 sin 4 θ + O(η -1) 1/2 .( 109
)
In taking the θ → θ limit, we have made use of Eq.(67).

For the second term on the right-hand side of Eq.( 106), we have

g rr ϕ ,r ϕ ,rθ = (1 -2M/r) 1/2 (1 -2M/r ) 1/2 ∂ ∂r ∂ ∂θ ∂ ∂r κ 8π 2 ∞ n=-∞ ∞ m=-∞ ∞ l=|m| (2λ + 1) × Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )χ nλ (r, r ) (110)
where we have taken the partial limits φ → φ , τ → τ . Taking the 'primed' coordinates to correspond the inner spacetime point, we can see from the asymptotic forms ( 14) that taking r to lie on the horizon eliminates all but the n = ±1 modes. The contribution coming from these modes is

κ 2 8 √ 2 π 2 M 2 α (η 2 -1) 1/2 (η + 1) ∂ ∂η ∂ ∂θ F (η, cos θ) ( 1 1 1 )
where F (η, cos θ) is given by Eq.( 82) and in going to the limit θ → θ we have again made use of Eq.(67). Now, using our series solution (83) for F (η, cos θ), we arrive at

[g rr ϕ ,r ϕ ,rθ ] r = κ 2 16π 2 M 2 α ∂ ∂θ A(cos θ) + κ 2 360π 2 M 2 (α 2 -1)(11α 2 + 1) cos θ α 4 sin 5 θ - κ 2 48π 2 M 2 (α 2 -1) α 2 cos θ sin 3 θ + O(η -1) (112)
Combining these results, we have

[∂ r T r θ ren ] = κ 2 16π 2 M 2 α ∂ ∂θ A(cos θ) - κ 2 48π 2 M 2 (α 2 -1) α 2 cos θ sin 3 θ . ( 113
)
There are no subtraction terms for this quantity so that this expression represents the renormalized expectation value ∂ r T r θ ren . Substituting Eqs.( 113), ( 100) and (101), for ξ = 0, into Eq.( 104), it is readily seen that ∇ a T a θ ren = 0 (114) as required. One can similarly, show that ∇ a T a r ren = 0. The equations for τ and φ are trivially satisfied.

The Trace Anomaly: The Conformally Coupled Case

In this short subsection, we check that we arrive at the standard trace anomaly for the stress tensor that we have derived. From Eqs.(98)-( 101), the trace of the renormalized stress tensor is easily seen to be

T a a ren = κ 2 32π 2 M 2 17 15 -6ξ + 3κ 2 4π 2 M 2 α (ξ -1 6 )A(cos θ) - κ 2 4π 2 M 2 (ξ -1 6 ) (α 2 -1) cos 2 θ α 2 sin 4 θ . (115)
In the conformally coupled case, ξ = 1 6 , we see that the trace is purely geometrical and given by T a a ren =

κ 2 240π 2 M 2 = 1 4π 2 v 1 (2M), (116)
where v 1 (x) is given by Eq.( 55). This is the standard trace anomaly for a conformally coupled scalar field [START_REF] Birrell | Quantum Fields in Curved Space[END_REF].

Conclusions

We have considered a massless, arbitrarily coupled quantum scalar field on a Schwarzschild black hole threaded by an infinite straight cosmic string. We have obtained simple expressions for the both the renormalized vacuum polarization and the renormalized stress tensor on the black hole horizon for this field in the Hartle-Hawking vacuum state, which is a two-fold generalization of the Candelas [1] result to encompass an arbitrary coupled field and a black hole threaded by a cosmic string. In order to do this, we have derived a very useful summation formula involving non-integer Legendre functions, which we have called a generalized Heine Identity given its analogy with the standard Heine Identity involving Legendre functions of integer degree. The derivation of this formula involved equating equivalent expressions for the same Green's function on a conveniently chosen space-time. Furthermore, the derivation did not rely on specific properties of the Legendre functions. This idea can be used to derive many useful summation formulae and addition theorems for non-integer Legendre functions, some of which are presented in Appendix B. Elsewhere [12], we have used this idea to obtain an expression for the vacuum polarization on the entire exterior region of the space-time. We believe this framework will prove useful in the other axially symmetric black hole calculations, most importantly, the Kerr-Newman black hole.

not true in the merely static Schwarzschild cosmic string space-time. We can also write this same Green's function in several other forms and in several coordinate systems, for example, performing the mode decomposition in cylindrical polar coordinates, ρ = r sin θ and z = r cos θ, we have

G 3 (x, x ) = 1 2π 2 α ∞ m=-∞ e im(φ-φ ) ∞ 0 cos k(z -z )I |m|/α (kρ < )K |m|/α (kρ > )dk.
(B.10)

The integral over k may further be performed to yield

G 3 (x, x ) = 1 4π 2 α(ρρ ) 1/2 ∞ m=-∞ e im(φ-φ ) Q |m|/α-1/2 (z -z ) 2 + ρ 2 + ρ 2 2ρρ . (B.11)
Another form of this Green's function is obtained from General Axi-Symmetric Potential Theory [START_REF] Weinstein | [END_REF]21]:

G 3 (x, x ) = 1 4π 2 α ∞ m=-∞ e im(φ-φ ) π 0 (ρρ ) |m|/α sin 2|m|/α Ψ [(z -z ) 2 + ρ 2 + ρ 2 -2ρρ cos Ψ] (|m|/α+1/2) dΨ.
(B.12)

These expressions are valid for all α ∈ R + ; equating any one of these expressions with another gives us a formula for the functions involved. Some of these results are well known and can be found in volumes such as Gradsteyn & Rhyzik [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF], nevertheless, we have non-obvious, important results, for example, equating (B.9) with (B.11) and taking r → r , we obtain

∞ l=|m| Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ ) = 1 π(sin θ sin θ ) 1/2 Q |m|/α-1/2 1 -cos θ cos θ sin θ sin θ . (B.13)
Also, equating Eq.(B.9) with Eq.(B.12) we arrive at the following formula

∞ l=|m| Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )(r < ) λ (r > ) -(λ+1) = 1 π π 0 (ρρ ) |m|/α sin 2|m|/α Ψ [(z -z ) 2 + ρ 2 + ρ 2 -2ρρ cos Ψ] (|m|/α+1/2) dΨ, (B.14)
so that the integral expression on the RHS is a type of generating function for these non-integer Legendre functions. There are also a set of formulae that hold for a more restrictive set of α values. Linet [22] has shown that, when α > 1/2, the Green's function may be written as

G 3 (x, x ) = 1 4πσ + 1 8π 2 α ∞ 0 1 R F α (u, φ -φ )du (B. 15
)
where

σ = [ρ 2 + ρ 2 + (z -z ) 2 -2ρρ cos α(φ -φ )] 1/2 R = [ρ 2 + ρ 2 + (z -z ) 2 + 2ρρ cosh u] 1/2 (B.16)
and

F α (u, Ψ) = sin(Ψ -π/α) cosh(u/α) -cos(Ψ -π/α) - sin(Ψ + π/α) cosh(u/α) -cos(Ψ + π/α) . (B.17)
Equating this to (B.9) for α > 1/2 and taking r → r , we find

1 α ∞ m=-∞ e im(φ-φ ) ∞ l=|m| Γ(λ + |m|/α + 1) Γ(λ -|m|/α + 1) P -|m|/α λ (cos θ)P -|m|/α λ (cos θ ) = 1 [2(1 -cos αγ)] 1/2 + 1 2πα 1 (2 sin θ sin θ ) 1/2 ∞ 0 F α (u, φ -φ ) [cosh ξ + cosh u] 1/2 du (B. 18
)
where

cos αγ = cos θ cos θ + sin θ sin θ cos α(φ -φ ) cosh ξ = 1 -cos θ cos θ sin θ sin θ . (B.19)
This formula has the benefit of relating a 3D mode-sum expression to an expression that isolates the 3D Hadamard singularity structure (the first term on the RHS) together with a regular term that depends on the boundary conditions. We have used this result to prove the regularity of the mode-sum that arises in the calculation of the vacuum polarization on the exterior region of the spacetime [12]. Another form of the Green's function can be given in terms of the half-integer Legendre functions by considering the Green's function in toroidal coordinates, which are related to the cartesian coordinates by

x = sinh µ cos φ cosh µ -cos η y = sinh µ sin φ cosh µ -cos η z = sin η cosh µ -cos η , (B.20)
where the ranges of the coordinates are 0 ≤ µ < ∞, 0 ≤ η < 2π, 0 ≤ φ < 2π. The mode-sum form of the Green's function obtained by separating in these coordinates is

G 3 (x, x ) = 1 4π 2 α ∞ m=-∞ e im(φ-φ ) ∞ n=-∞ e in(η-η ) [(cosh µ -cos η)(cosh µ -cos η )] 1/2 e i|m|π/α Γ(n + |m|/α + 1/2) Γ(n -|m|/α + 1/2) P -|m|/α n-1/2 (cosh µ < )Q -|m|/α n-1/2 (cosh µ > ). (B.21)
Equating this to (B.11) gives us the following addition theorem The final coordinate system we shall investigate is prolate spheroidal coordinates which may trivially extended to the oblate spheroidal case. The relationship between prolate spheroidal coordinates and cartesian coordinates is given by x = sinh σ sin θ cos φ y = sinh σ sin θ sin φ z = cosh σ cos θ. (B.25)

∞ n=-∞ e in(η-η ) e i|m|π/α Γ(n + |m|/α + 1/2) Γ(n -|m|/α + 1/2) P -|m|/α n-1/2 (cosh µ < )Q -|m|/α n-1/2 (cosh µ > ) = 1 (sinh µ sinh µ ) 1/2 Q |m|/α-1/2 (χ). (B.22) where χ = (z -z ) 2 + ρ 2 + ρ 2 2ρρ (B.
where the ranges of the coordinates are 0 ≤ σ < ∞, 0 ≤ θ ≤ π, 0 ≤ φ < 2π. The mode form of the Green's function obtained by separating in these coordinates is

G 3 (x, x ) = 1 4πα ∞ m=-∞ e im(φ-φ ) ∞ l=|m| (2λ + 1)e i|m|π/α Γ(λ + |m|/α + 1) 2 Γ(λ -|m|/α + 1) 2 P -|m|/α λ (cos θ)P -|m|/α λ (cos θ )P -|m|/α λ (cosh σ < )Q -|m|/α λ (cosh σ > ) ( B . 2 6 )
We shall write down only one possible summation formula here, by equating the Green's function expression above to (B.11), we obtain a summation formula for a product of four Legendre functions: There are many more summation and addition formulae that can be derived in this way.

Those presented here were either of particular use to us elsewhere or are to the best of our knowledge previously unpublished.

Appendix C: Determining the β λ Coefficients

We briefly describe the procedure for determining the β λ coefficients occurring in the definition of A(cos θ). The method described is completely analogous to the Candelas [1] method except that we replace l by λ = l -|m| + |m|/α everywhere it occurs. We begin with the Wronskian relation between the n = ±1 radial functions of the first and second kind,

p 1λ (η) d dη q 1λ (η) -q 1λ (η) d dη p 1λ (η) = - 2 η 2 -1 . (C.29)
Dividing across by (p 1λ ) 2 and integrating we obtain the following integral expression for q 1λ (η)

q 1λ (η) = 2p 1λ (η) ∞ η dξ (ξ 2 -1)[p 1λ (ξ)] 2 , (C.30)
where we have used the fact that q 1λ (η) → 0 as η → ∞. One can easily verify that the series solution to p 1λ (ξ) about ξ = 1 is p 1λ (ξ) = (ξ -1) 1/2 + 1 4 λ(λ + 1)(ξ -1) ) that the first integral is convergent as η → 1, and the second integral can be evaluated explicitly. We thus obtain the following asymptotic expansion of q 1λ (η) as η → 1, q 1λ (η) = 1 (η -1) 1/2 + 1 2 [1 + λ(λ + 1)](η -1) 1/2 ln(η -1) + 2I λ + 1 4 λ(λ + 1) -1 2 [1 + λ(λ + 1)] ln 2 (η -1) 1/2 + O((η -1) 3/2 ln(η -1)) , (C.34) where

I λ = ∞ 1 dξ (ξ 2 -1) 1 [p 1λ (ξ)] 2 - 1 (ξ -1) + λ(λ + 1) 2 . (C.35)
Recall that we also had an alternate expression for q 1λ (η) (76) which we expand about η = 1 to get q 1λ (η) = 1 (η -1) 1/2 + 1 2 [1 + λ(λ + 1)](η -1) 1/2 ln(η -1)

+ [1 + λ(λ + 1)][Ψ(λ + 1) + γ -1 2 ln 2 -1 2 ] + 1 4 + β λ (η -1) 1/2 +O((η -1) 3/2 ln(η -1)) , (C.36)

where Ψ(z) = Γ (z)/Γ(z) and γ is Euler's constant. Finally, comparing our two equivalent expressions for q 1λ (η) given by (C.34) and (C.36) yields an expression for β λ that is numerically tractable:

β λ = 2I λ -[1 + λ(λ + 1)][Ψ(λ + 1) + γ -3 4 ] -1 2 .
(C.37)

That the derivation of the β λ coefficients depends on the integral (C.30) underlines the fact these coefficients are truly global in nature since the dependence on the boundary conditions is explicit here. If we considered the case where we place the black hole in a spherical box, then the upper limit of the integration above would be the radius of the box and we would obtain different β λ coefficients.

The integral I λ is most effectively calculated by splitting the range of integration into the regions 1 ≤ ξ ≤ 2 and 2 ≤ ξ < ∞. In the former range, one can use a series solution to p 1λ (η) in order to cancel the divergences in the integrand near η = 1, and the integral can now be performed without difficulty. In the latter range, the first term cuts off exponentially as η → ∞ since p 1λ (η) → ∞ as η → ∞. The remaining part of the integral can be performed explicitly. We have tabulated above some β λ coefficients for α = 0.95 and α = 0.6 for l up to l = 5. We note that for m = 0, we have λ = l an integer, and we retrieve the Candelas [1] β l values.

1 - 1 P

 11 -|m|/α l-|m|+|m|/α (cos θ)P -|m|/α l -|m|+|m|/α (cos θ)d(cos θ) = 2 (2λ + 1) Γ(λ -|m|/α + 1) Γ(λ + |m|/α + 1)δ ll .

)

  Figure 1. φ2 horizon ren

  23) which are related to the toroidal coordinates by ρ = sinh µ cosh µcos η z = sin η cosh µcos η . (B.24)

  1)e i|m|π/α Γ(λ+ |m|/α + 1) 2 Γ(λ -|m|/α + 1) 2 P -|m|/α λ (cos θ)P -|m|/α λ (cos θ ) P -|m|/α λ (cosh σ < )Q -|m|/α λ (cosh σ > ) = 1 π Q |m|/α-1/2 (χ) √ sinh σ sinh σ sin θ sin θ (B.27)where χ = cosh 2 σ + cosh 2 σsin 2 θsin 2 θ -2 cosh σ cosh σ cos θ cos θ 2 sinh σ sinh σ sin θ sin θ (B.28)

  3/2 + O(ξ -1) 5/2 (C.31) and therefore we have 1 [p 1λ (ξ)] and subtract the first two terms on the right-hand side above in the integral expression (C.30), q 1λ (η) = 2p 1λ (η) from Eqs.(C.31) and (C.32

Table C1 .

 C1 The coefficients β λ for α = 0.95.

	m= 0	1	2	3	4	5
	l=0 0.119377314
	l=1 0.002779252 0.002318785
	l=2 0.000144497 0.000127179 0.000112214
	l=3 0.000018348 0.000016749 0.000015310 0.000014014
	l=4 0.000003975 0.000003705 0.000003456 0.000003226 0.000003013
	l=5 0.000001181 0.000001115 0.000001053 0.000000995 0.000000941 0.000000890

Table C2 .

 C2 The coefficients β λ for α = 0.6.

	m= 0	1	2	3	4	5
	l=0 0.119377314
	l=1 0.002779252 0.000344996
	l=2 0.000144497 0.000033858 0.000010533
	l=3 0.000018348 0.000006342 0.000002579 0.000001181
	l=4 0.000003975 0.000001723 0.000000828 0.000000431 0.000000239
	l=5 0.000001181 0.000000592 0.000000319 0.000000182 0.000000109 0.000000068

Acknowledgements

PT is supported by the Irish Research Council for Science, Engineering and Technology, funded by the National Development Plan.

The authors would like to thank Cormac Breen for many helpful conversations, and Stuart Dowker for bringing MacRobert's analysis of generalized Legendre functions to our attention.

Appendix A: Normalization Condition

In this Appendix, we shall prove the normalization condition Eq. [START_REF] Christensen | [END_REF] for the noninteger Legendre functions that arise on separation of the wave-equation on cosmic string spacetimes. We begin by noting the following relationship between the Legendre function and the Gegenbauer polynomial [START_REF] Oberhettinger | Higher Transcendental Functions[END_REF],

where C ν n (z) is the Gegenbauer polynomial and n is an integer. We also have a normalization for these polynomials [START_REF] Oberhettinger | Higher Transcendental Functions[END_REF] given by

Choosing n = l -|m| and ν = 1/2 + |m|/α in Eq.(A.1) gives us the appropriate Legendre function in terms of the Gegenbauer polynomials, and with this choice of n and ν, Eq.(A.2) becomes

We now use the so-called doubling formula for the Gamma function [START_REF] Gradshteyn | Table of Integrals, Series and Products[END_REF] Γ

Finally, substituting this into Eq.(A.3) gives the desired result, viz.,

Appendix B: Summation Formulae

In this Appendix, we shall derive some interesting summation formulae for non-integer Legendre functions based on considering equivalent forms of the 3D Green's function on flat space threaded by a cosmic string

The 3D Green's function for Laplace's equation satisfying

can be written in spherical polar coordinates as the mode sum:

(B.9)

Note that this corresponds to the n = 0 term of the 4D Green function in the corresponding ultrastatic (g tt = -1) space-time, but that the corresponding statement is


Sven Zschocke
email: sven.zschocke@tu-dresden.de

Sergei A Klioner Lohrmann

On the efficient computation of the quadrupole light deflection

Keywords: numbers: 95.10.-a, 95.10.Jk, 98.62.Tc, 98.62.Sb

Although the formulas for the light deflection due to quadrupole gravitational field of deflecting bodies are well known, the formulas are rather complicated, so that massive computations of quadrupole light deflection (e.g., in the framework of astrometric survey missions like Gaia) are time-consuming. Considering an observer situated within a few million kilometers from the Earth (clearly the most practical case), we derive the simplest possible form of the relevant formulas still having numerical accuracy of 1 µas. This form leads to simple upper estimates for the quadrupole light deflection in various cases allowing one to relate the magnitude of the actual quadrupole deflection with the corresponding monopole deflection due to the same body. These upper estimates can be used to decide if, for a given configuration, the actual quadrupole deflection should be computed for a given accuracy goal.

Introduction

In the nearest future astrometric observations will reach an accuracy of 1 microarcsecond (µas) level. This level of accuracy requires a precise modelling of light propagation. In particular, the light deflection due to quadrupole gravitational field of deflecting bodies should be taken into account [START_REF] Klioner | [END_REF]. On the other hand, the accuracy of radio and laser radar links of future missions like BepiColombo or Juno require modelling of the light travel time at the level of millimeters. Therefore, also the Shapiro delay due to quadrupole fields is of practical interest.

Analytical formulas for quadrupole light deflection are well known. Analytical solutions of light deflection in a quadrupole gravitational field have been investigated by many authors [START_REF] Ivanitskaya | Lorentz Basis and Gravitational Effects in Einstein's Theory of Gravitation[END_REF]2,14,1,6,7,[START_REF] Klioner | [END_REF][START_REF] Klioner | Gaia report[END_REF]. For the first time the full analytical solution for the light trajectory in a quadrupole field has been obtained in [6]. These results were confirmed by a different approach in [13]. Various generalization (higher-order multipole moments, time-dependence, etc.) were derived in [10,11,12]. The formulas suitable for high-accuracy data reduction are given e.g. in [START_REF] Klioner | [END_REF]. These formulas are rather complicated, so that massive computations of quadrupole light deflection are time-consuming. This represents a problem for data processing of astrometric surveys. For instance, the ESA mission Gaia (to be launched 2012) will have to process about 10 12 individual observations of about 10 9 distinct celestial objects. It is, therefore, obvious that efficient analytical algorithms to compute the quadrupole light deflection are mandatory. Furthermore, for observations in solar system, the quadrupole light deflection reaches the microarcsecond level only for objects at a relatively small angular distance from giant planets. Accordingly, it is highly useful to find simple analytical formulas by means of which one can decide whether or not the quadrupole field needs to be taken into account for a given accuracy and a given geometrical configuration.

In this paper we assume that the observer is located within a few million kilometers from the Earth's orbit which is clearly the most practical case (e.g. Gaia will have a Lissajous-like orbit around Lagrange point L 2 of the system Earth-Sun). This allows one to simplify the formulas for the quadrupole light deflection considerably. Besides these simplified formulas we derive simple analytical estimates of the quadrupole deflection allowing one to decide if the effect should be computed and taken into account for a given accuracy. We also give a strict upper estimate for the quadrupole Shapiro delay.

The paper is organized as follows. In Section 2 we summarize some basics about light deflection and introduce the notation. In Section 3 the full quadrupole formula in post-Newtonian order, the simplest possible expression for stars and quasars still having an accuracy of 1 µas and simple upper estimates of the latter are presented. In Section 4 the full quadrupole formula in post-Newtonian order, the simplest possible expression for solar system objects still having an accuracy of 1 µas and criteria are presented. An improved estimation of quadrupole Shapiro effect is given in Section 5. The efficiency and correctness of the upper estimates and simplified quadrupole formulas have been investigated numerically and analytically, and the results are discussed in Section 6. A summary of the findings is given in Section 7.

Some basic formulas of light propagation

Let us summarize some basic formulas of light propagation in post-Newtonian approximation. The geodetic equations in post-Newtonian order is linear with respect to the metric components and, therefore, the coordinates of a photon and the derivative with respect to coordinate time t are given by [START_REF] Klioner | [END_REF] x

(t) = x(t 0 ) + c σ (t -t 0 ) + i ∆x i (t) + O(c -4 ), (1)
ẋ(t) = c σ + i ∆ ẋi (t) + O(c -4 ). (2)
The sum runs over individual terms in the metric of various physical origins (e.g. monopole gravitational field of various bodies, quadrupole fields, higher-order multipole fields, etc.). Here, t 0 is the moment of emission, x 0 = x(t 0 ) is the position of source and

σ = lim t→-∞ ẋ(t)
c is the unit tangent vector of light path at infinitely past. The position of observer is x 1 = x(t 1 ) and t 1 is the moment of observation. The unit coordinate direction of the light propagation at the moment of observation reads n = ẋ(t 1 ) | ẋ(t 1 )| . In post-Newtonian order the transformation σ to n reads [6,[START_REF] Klioner | [END_REF]

n = σ + i δσ i + O(c -4 ) , δσ i = σ × c -1 ∆ ẋi (t 1 ) × σ . (3)
The spherical symmetric part (monopole contribution) due to one massive body A and its absolute value are given by (cf. Eq. (102) in [START_REF] Klioner | [END_REF])

δσ A pN = -(1 + γ) G c 2 M A d A d 2 A 1 + σ • r A 1 r A 1 , δσ A pN = (1 + γ) G c 2 M A d A 1 + σ • r A 1 r A 1 . ( 4
)
Here γ is the PPN parameter, M A is the mass of body A, c is the speed of light, G is the gravitational constant, and

d A = σ × r A 1 × σ is the impact parameter, d A = |d A |, r A 1 = x(t 1 ) -x A , r A 1 = r A 1
, and x A is the position of massive body A. In order to consider light propagation between two given points x 0 and x 1 (as it is needed for the data processing for solar system objects) let us define vectors

r A 0 = x(t 0 )-x A and R = x 1 -x 0 = r A 1 -r A 0 with absolute values R = |R| and r A 0 = |r A 0 |, and unit vector k = R/R. In post-Newtonian approximation, the transformation k to n reads [5] n = k + i δk i + O(c -4 ) , δk i = k × c -1 ∆ ẋi (t 1 ) -R -1 ∆x i (t 1 ) × k . (5)
The impact parameter d A can be computed as

d A = k × r A 1 × k + O(c -2 ) = k × r A 0 × k + O(c -2
). The spherical symmetric part (monopole contribution) due to one massive body A and its absolute value are given by (cf. Eq. (70) in [START_REF] Klioner | [END_REF] or Table 1. Numerical parameters of the giant planets taken from [START_REF] Weissman P R, Mcfadden | Encyclopedia of the Solar System[END_REF]. In this table the values of r A min 1 are given under assumption that the observer is in the vicinity of Earth's orbit. The value J 2 for the Sun is taken from [3].

Parameter

Sun Jupiter Saturn Uranus Neptune Eq. ( 24) in [START_REF] Klioner | [END_REF])

GM A /c 2 [
δk A pN = -(1 + γ) G c 2 M A r A 1 k × r A 0 × r A 1 r A 0 r A 1 + r A 0 • r A 1 , δk A pN = (1 + γ) G c 2 M A r A 1 r A 0 × r A 1 r A 0 r A 1 + r A 0 • r A 1 . (6)

The quadrupole light deflection for stars and quasars

Using the expression ∆ ẋQ (t 1 ) given by Eq. (44) of [START_REF] Klioner | [END_REF] one gets [START_REF] Klioner | Gaia report[END_REF][START_REF] Zschocke | [END_REF]

δσ Q = A δσ A Q , δσ A Q = 1 + γ 2 G c 2 α A UA c + β A ĖA c + γ A ḞA c + δ A VA c . (7)
The scalar functions and vectorial coefficients are given by Eqs. (A.1) -(A.8) of Appendix A.1. The last three terms in Eq. ( 7) can be estimated as

T A 1 = 1 + γ 2 G c 2 β A ĖA c + γ A ḞA c + δ A VA c ≤ 13 G c 2 M A J A 2 P 2 A (r A min 1 ) 3 . (8)
Here, P A is the equatorial radius, J A 2 is the second zonal harmonics of massive body A, and r A min 1 is the minimal distance between massive body and observer. The proof of this estimation is given in [START_REF] Zschocke | [END_REF].

It is well known (see Table 1 of [START_REF] Klioner | [END_REF]) that the quadrupole light deflection in solar system can achieve the level of 1 µas only for the giant planets (and, possibly, the Sun). Using the parameters in Table 1 we obtain from (8) that T A 1 ≤ 10 -6 µas for all these bodies. Therefore, these terms in [START_REF] Klioner | Gaia report[END_REF] can safely be neglected at the level of a microacrsecond. Accordingly, the simplest possible expression of quadrupole light deflection for stars and quasars still having an accuracy of 1 µas and valid for an observer situated within a few million kilometers of the Earth orbit reads:

δσ A Q = 1 + γ 2 G c 2 α A UA c . (9)
The simplified formula of quadrupole light deflection ( 9) is still a complicated expression.

In order to avoid evaluation of this term for each object in the data reduction, a simple criterion is needed allowing one to decide whether or not it is necessary to compute the quadrupole light deflection for a source. The absolute value of light deflection due to the quadrupole field of objects A can be estimated as [START_REF] Zschocke | [END_REF] |

δσ A Q | ≤ 9 4 1 + γ 2 GM A c 2 J A 2 P 2 A d 3 A 1 + σ • r A 1 r A 1 . (10)
A comparison of (10) with the absolute value of the monopole deflection given by ( 4) gives

δσ A Q ≤ 9 8 J A 2 P 2 A d 2 A δσ A pN . (11)
This estimate relates the quadrupole light deflection for stars and quasars to the corresponding monopole deflection. The latter is relatively large, defined by a simple formula and usually computed for each source and each deflecting body. In this case the estimate (11) can be computed at cost of two multiplications (note that d A is known since it is used for δσ A pN ). In case when | δσ pN | is not readily available, one can use [START_REF] Zschocke | [END_REF] δσ

A Q ≤ 2 (1 + γ) GM A c 2 J A 2 P 2 A d 3 A ( 12
) ≤ 2 (1 + γ) GM A c 2 J A 2 1 P A , (13)
where we use d A ≥ P A and (A.1) is estimated by |2 + 3 cos α -cos 3 α| ≤ 4 for α being the angle between vectors σ and r A 1 . Estimate (13) coincides with Eq. (41) of [6].

The quadrupole light deflection for solar system objects

The quadrupole light deflection for solar system objects δk Q is defined by Eqs. ( 36)-( 47) and (69) of [START_REF] Klioner | [END_REF] and can be written as [START_REF] Klioner | Gaia report[END_REF][START_REF] Zschocke | [END_REF]:

δk Q = A δk A Q , δk A Q = 1 + γ 2 G c 2 A α A A A c + β A B A c + γ A C A c + δ A D A c . ( 14
)
The scalar functions and vectorial coefficients are given in Eqs. (A.10) -(A.17) in Appendix A.2. In [START_REF] Zschocke | [END_REF] it has been shown that the last three terms in ( 14) can be estimated by

T A 2 = 1 + γ 2 G c 2 β A B A c + γ A C A c + δ A D A c ≤ 9 2 1 P 2 A r A min 1 + 1 P A (r A min 1 ) 2 + 19 2 1 (r A min 1 ) 3 GM A c 2 J A 2 P 2 A . ( 15
)
Using the parameters given in Table 1 we obtain the numerical estimates of T A 2 given in Table 2. In view of these numerical values, the simplest possible form of quadrupole light deflection (14) for solar system objects with an accuracy of 1 µas and valid for an observer situated within a few million kilometers of the Earth orbit is given by

δk A Q = 1 + γ 2 G c 2 α A A A c . (16)
Again, in order to avoid unnecessary computations, one needs an efficient way to estimate the magnitude of δk A Q . This can be done using the following inequality [START_REF] Zschocke | [END_REF]

| δk A Q | ≤ 3 (1 + γ) 2 GM A c 2 1 d 2 A 1 r A 1 J A 2 P 2 A r A 0 × r A 1 r A 0 r A 1 + r A 0 • r A 1 . (17)
A comparison with (6) yields

δk A Q ≤ 3 2 P 2 A d 2 A J A 2 δk A pN . (18)
This estimate allows one to estimate the magnitude of the quadrupole light deflection in observations of solar system objects using the monopole deflection. As it was mentioned above, the monopole deflection is significantly larger and should be usually calculated for each source and each gravitating body (at least for those bodies, for which the quadrupole deflection could be sufficiently large). If | δk pN | has been calculated, the magnitude of δk A Q can be estimated at cost of three multiplications (we note that d A is required to compute | δk pN | and can be considered as known). If | δk pN | is not readily available, we can use [START_REF] Zschocke | [END_REF]

δk A Q ≤ 2 (1 + γ) GM A c 2 J A 2 P 2 A d 3 A (19) ≤ 2 (1 + γ) GM A c 2 J A 2 1 P A , ( 20
)
where in the last estimate we have used P A ≤ d A .

Shapiro effect for solar system objects

According to Eq. ( 1), the propagation time c τ = c (t 1 -t 0 ) is given by [6]:

c τ = R + c i δτ i + O c -4 , c δτ i = -k • ∆x i (t 1 ) . ( 21
)
The formula for the Shapiro delay due to one mass monopole with mass M A is well known:

c δτ A pN = (1 + γ) GM A c 2 log r A 0 + r A 1 + R r A 0 + r A 1 -R . ( 22
)
This monopole Shapiro delay becomes unboundedly large for growing distance R between the points of emission and observations (although it is growing logarithmically with R). The quadrupole Shapiro effect c δτ

Q = -k • ∆x Q (t 1
) is given by [6]:

c δτ Q = A c δτ A Q , c δτ A Q = 1 + γ 2 G c 2 (δ A V A + γ A F A + β A E A ) , (23)
G c 2 | δ A V A | ≤ G M A c 2 J A 2 P 2 A d 2 A , (24)
G c 2 | γ A F A + β A E A | ≤ G M A c 2 J A 2 P 2 A (r A 0 ) 2 + P 2 A (r A 1 ) 2 . ( 25
)
Now since d A ≥ P A , r A 0 ≥ P A , and r A 1 ≥ P A , we conclude that

c δτ A Q ≤ 3 J A 2 GM A c 2 , (26)
which represents a strict upper bound of quadrupole Shapiro delay and slightly improves the estimate given in Eq. ( 47) in [6]. This estimate implies that the quadrupole Shapiro delay has an upper bound that depends only on physical parameters of the massive body. Table 3 gives maximal possible quadrupole effects in the Shapiro delay for any positions of the source and observer.

Efficiency of the upper estimates

As explained above, the principal merit of the simple upper estimates for the quadrupole light deflection (Eqs. ( 11), ( 12) and ( 13) for stars and quasars and Eqs. ( 18), ( 19) and (20) for solar system objects) is the possibility to use them, at very low computational cost, as criteria to decide if the quadrupole deflection should be calculated or not for a given configuration and a given numerical accuracy. In this Section we investigate the numerical efficiency of the criteria for two situations: (1) purely random homogeneous distribution of sources and the position of observer with respect to the deflecting body, and (2) light rays grazing the surface of the deflecting body, but with directions still randomly distributed with respect to the body (note that the body is not spherically symmetric and the orientation does play a role). For both of these situations we compute minimal, maximal and mean values of the ratio between the quadrupole deflection and its upper estimate. The higher is the mean value the more efficient is the corresponding estimate as a criterion.

For stars and quasars, starting from ( 11), ( 12) and ( 13), we consider the following ratios

r 1 = δσ A Q 9 8 P 2 A d 2 A |J A 2 | δσ A pN , r 2 = δσ A Q 4 G M A c 2 |J A 2 | P 2 A d 3 A , r 3 = δσ A Q 4 G M A c 2 |J A 2 | 1 P A , (27)
where δσ A Q and δσ A pN are determined by Eqs. ( 9) and (4), respectively. For solar system objects, starting from Eqs. ( 18), ( 19) and (20), we consider the ratios

r 4 = δk A Q 3 2 P 2 A d 2 A |J A 2 | δk A pN , r 5 = δk A Q 4 G M A c 2 |J A 2 | P 2 A d 3 A , r 6 = δk A Q 4 G M A c 2 |J A 2 | 1 P A , (28)
where δk A Q and δk A pN is determined by Eqs. ( 16) and ( 6), respectively. For all six ratios it is easy to compute the minimal and maximal values analytically. Besides that for both distributions of sources and observers it is possible to compute analytically the mathematical expectations (i.e. the mean values) of each of six ratios r i . These values are given in Table 4. The analytical calculations have been also confirmed by direct numerical simulations in which the ratios r i were computed for correspondingly distributed sources and positions of the observer and statistically analyzed.

The minimal values of all r i is zero. The maximal values of the ratios are 1 except for r 1 and r 4 for grazing rays. In the latter case the maximal values are less than 1. This reflects the fact that for grazing rays the numerical coefficients in (11) and ( 18) can be improved (but only for grazing rays and not for arbitrary situation). The fact that no maximal values are greater than 1 confirms the validity of the estimates. The efficiency of the criteria are characterized by the mean values of the ratios. Considering that the random distribution of sources is much more realistic situation than the grazing rays we can conclude that the trivial estimates (13) and (20) leading to r 3 and r 6 are extremely inefficient: the value of quadrupole deflection is typically five orders of magnitude lower than "predicted" by those estimates. The criteria (12) and (19) are already better: the quadrupole deflection is typically only 3 times lower than "predicted" (the mean value of both r 2 and r 5 for random sources is 0.33). It is clear, however, than the most efficient criteria are given by ( 11) and (18). For stars and quasars the value of the quadrupole deflection "predicted" by ( 11) is only two times larger than the real value. It means that only 50% of the computations based on (11) lead to values lower than the desired numerical cut-off value and could be saved. Estimates (11) and (18) will be used for the Gaia data processing. In the numerical simulations, the expressions T 1 in (8) and T 2 in (15) have also been calculated and have shown, independently from the analytical calculations, the correctness of simplified quadrupole formula (9) for stars and quasars and (16) for solar system objects.

Summary

In this paper we have developed efficient numerical algorithms allowing one to compute the quadrupole light deflection with minimal computational efforts. These algorithms 0.00 0.72 × 10 -5 1.00

will be used for data processing of the ESA astrometric survey mission Gaia and can be useful in other cases. In this work we assume that the observer is situated within a few million kilometers from the Earth orbit. This is clearly the most practical case. Other situations can be analyzed along the lines of our reasoning. The main results which are valid with an accuracy of at least 1 µas are as follows:

1. Quadrupole light deflection for stars and quasars can be computed as (9).

2. Eqs. ( 11), ( 12) and ( 13) can be used as an a priori criterion if the quadrupole light deflection (9) has to be computed for a given source.

3. Quadrupole light deflection for solar system sources can be computed as [START_REF] Weissman P R, Mcfadden | Encyclopedia of the Solar System[END_REF].

4. Eqs. (18), ( 19) and (20) can be used as an a priori criterion if the quadrupole light deflection [START_REF] Weissman P R, Mcfadden | Encyclopedia of the Solar System[END_REF] has to be computed for a given solar system object.

The efficiency of the upper estimates has been investigated numerically and analytically, and the results are shown in Table 4. They demonstrate high efficiency and correctness of the upper estimates, both for randomly distributed sources and sources which generate grazing rays. According to these investigation, the most efficient upper estimate of quadrupole light deflection is (11) for stars and quasars and (18) for solar system objects. The correctness of simplified quadrupole formulas has also been shown by numerical simulations.

Additionally, we give a strict upper bound (26) for the quadrupole effect in the Shapiro delay. This upper bound can be used to decide whether or not the quadrupole Shapiro delay should be taken into account if high-accuracy ranging measurements are to be modelled; e.g. in the framework of missions like BepiColombo [15] or Juno.

Table 2 .

 2 Maximal numerical values of the neglected terms as given by (8) and(15)

		m]		1476. 1.40987 0.42215 0.064473 0.076067
	J A 2 [10 -3 ]		0.0002	14.697	16.331	3.516	3.538
	P A [10 6 m]		696.	71.492	60.268	25.559	24.764
	r A min 1	[10 12 m]		0.147	0.59	1.20	2.59	4.31
	GM A J A 2 P 2 A /c 2 [10 15 m 3 ]	0.143	0.106	0.025 0.000148 0.000165
	Parameter	Sun	Jupiter	Saturn	Uranus	Neptune
	T A 1 [µas]		< 10 -6 for Sun and giant planets
	T A

2 [µas] 1.86 × 10 -3 3.26 × 10 -2 5.32 × 10 -3 8.11 × 10 -4 5.79 × 10 -5

Table 3 .

 3 Numerical values of estimate (26).

	Parameter	Sun Jupiter Saturn Uranus Neptune
	3 J A 2	G M A c 2 [mm] 0.89	62.16	20.68	0.68	0.81
	where the scalar functions and scalar coefficients are given in Eqs. (A.18) -(A.23) in
	Appendix A.3. This expression for the quadrupole Shapiro delay cannot be reasonably
	simplified. However, one can give a strict upper bound for the quadrupole effect in the
	Shapiro delay. One can demonstrate [17] that

Table 4 .

 4 Statistical properties of the ratios r i for two distributions of sources (see text for further explanations).

	ratio	random		grazing
		min mean	max	min mean max
	r 1	0.00 0.49	1.00	0.00 0.59	0.89
	r 2 r 3	0.00 0.33 0.00 0.72 × 10 -5 1.00 1.00	0.00 0.67	1.00
	r 4	0.00 0.38	1.00	0.00 0.44	0.67
	r 5 r 6	0.00 0.33	1.00	0.00 0.67	1.00

Acknowledgments

This work was partially supported by the BMWi grants 50 QG 0601 and 50 QG 0901 awarded by the Deutsche Zentrum für Luft-und Raumfahrt e.V. (DLR).

Appendix A. Explicit formulas for the quadrupole terms

We give the full expressions of coefficients and scalar functions of quadrupole light deflection and quadrupole Shapiro effect in post-Newtonian order, because so far they were not being presented in a refereed journal.

Appendix A.1. Light deflection for stars and quasars

The functions in (7) read

3 , (A.4)

Here M A ij is the symmetric and trace-free quadrupole moment of body A. For an axial symmetric body (this approximation is sufficient for the solar system and the accuracy of 1 µas) one has

where R A is the rotational matrix giving the orientation of the figure axis of body A (see Eq. ( 48)-(53) of [START_REF] Klioner | [END_REF]).

Appendix A.2. Light deflection for solar system objects

The functions in (14) read

(A.10)

5 , (A.12)

3 , (A.13)

)

The functions in (23) read

3 , (A.18)

(A.20) .23)