A New Class of Accelerating Black Hole Solutions
aa r X i v : . [ g r- q c ] J u l A New Class of Accelerating Black Hole Solutions
Joan Camps a , Roberto Emparan a,ba Departament de F´ısica Fonamental andInstitut de Ci`encies del Cosmos, Universitat de Barcelona,Mart´ı i Franqu`es 1, E-08028 Barcelona, Spain b Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA)Passeig Llu´ıs Companys 23, E-08010 Barcelona, Spain [email protected], [email protected]
Abstract
We construct several new families of vacuum solutions that describe black holes in uni-formly accelerated motion. They generalize the C-metric to the case where the energy densityand tension of the strings that pull (or push) on the black holes are independent parameters.These strings create large curvatures near their axis and when they have infinite length theymodify the asymptotic properties of the spacetime, but we discuss how these features can bedealt with physically, in particular in terms of ‘wiggly cosmic strings’. We comment on possi-ble extensions, and extract lessons for the problem of finding higher-dimensional acceleratingblack hole solutions.
Introduction
The C-metric is a solution of the Einstein equations that describes the spacetime of twoblack holes uniformly accelerating in opposite directions [1]. This solution and its variantshave been applied to a number of interesting problems, including gravitational radiation fromaccelerated sources [2], instantonic pair creation of black holes [3], black holes on branes [4],five-dimensional black rings [5], etc. It is remarkable that a simple, exact solution is availablefor the study of such a variety of problems, and so it seems desirable to investigate possibleextensions of it.The generalization that we study in this paper can be easily motivated. In the originalC-metric, conical singularities are unavoidable since they reflect the need of an external forceto accelerate the black holes. Conical deficit angles correspond to distributional string-likesources with linear energy density ε and tension T = ε . The black holes in the C-metric arethen accelerated under the pull of two such semi-infinite strings. These sources are physicallyappealing since they can be made sense of as cosmic strings (vortices) in the limit in whichtheir thickness is negligible. Note, however, that for the purpose of pulling on the black holesone merely needs a tensile string. In particular, it is not necessary that its energy densityequals its tension. Thus it appears that a one-parameter generalization of the C-metric wherethe energy density of the string is independent of its tension should be possible. Our purposeis to describe a solution where black holes are accelerated by such generic strings.String-like sources with ε = T arise as the zero-thickness limit of a variety of less singularsources, such as ‘wiggly’ cosmic strings or cylindrical shells. A main difference with the ε = T sources is that when ε = T the Newtonian gravitational potential does not vanish and thegravitational field has non-trivial local curvature. As a consequence the spacetime is notasymptotically flat, not even locally. Although this may appear as a serious drawback, itneed not be a problem in physical situations in which the string is not infinite but formsa (possibly long) loop, whose radius provides a natural cutoff for the geometry at largetransverse distance from the string (this is often assumed also for e.g., global strings). Onemight also take the view that, like e.g., for the Melvin universe, these strings define their ownasymptotic class. We shall accept that these solutions admit physical motivation.Interestingly, we can also have ‘struts’ that stretch between the black holes and pushthem apart. These solutions are locally inequivalent to those with pulling strings and sincethe struts have finite length, the asymptotic behavior at spatial infinity is expected to bebetter. The struts have negative tension, but in contrast to the C-metric, they do not violateany energy condition if the energy density on the struts is large enough. There is also thepossibility of solutions with both strings and struts. Although the physical import of thesemetrics is less clear, they provide the largest (five-parameter) family of solutions in this classand we present them explicitly for completeness.There are several interesting possible extensions of the solutions described in this paper.In fact one of the reasons leading us to their study has been the consideration of higher-1imensional generalizations of the C-metric. In the context of these elusive solutions, whichhave been sought for many of the applications mentioned in our opening paragraph, thepossibility of different kinds of string sources is potentially even more important than in fourdimensions. We shall address this point towards the end of the paper.In the next section we analyze the basic string solutions that later will be used to pull, orpush, on the black holes. In sec. 3 we present the new metrics for accelerating black holes,first with pulling strings, then with pushing struts, and we analyze their main properties.We also present the most general solution with both strings and struts. Sec. 4 discussesextensions of these solutions. The appendix contains an alternative form for the new metricsthat resembles more closely the conventional way of writing the C-metric. The Levi-Civita spacetime ds = − ρ m dt + ρ m ( m − ( dz + dρ ) + ρ − m ) dφ C , (2.1)is a long-known cylindrically-symmetric solution to the vacuum Einstein equations [6]. It isa Weyl metric of general Petrov type , and contains two parameters: m , which determinesthe local curvature, and C , which introduces a conical structure along the axis when thenormalization of φ is fixed by identifying φ ∼ φ + 2 π . When m = 0 , C = 1 itreproduces Minkowski and Rindler space, respectively, but for generic values of m and C thesolution is not asymptotically flat as ρ → ∞ and exhibits a curvature singularity at ρ = 0.As a Weyl solution, it corresponds to an infinite line source of the Newtonian potential withlinear density m/ G . We shall refer to it as the ‘Levi-Civita string’.It has been argued that, in order to admit an interpretation as the spacetime of a cylindri-cal source, one must have m < /
2, or possibly m < e.g., [7, 8] and references therein).The precise range will not concern us much, since later we shall be mostly interested in smallvalues of m , as well as C close to 1, for which the line source is readily interpreted. Nev-ertheless let us briefly discuss the general case. The singular behavior near the axis ρ = 0may be smoothed by replacing the region around it with an extended source, and a simpleexample is a cylindrical tubular shell [9, 10, 11, 12, 13, 14]. Cutting the metric at ρ = ρ s andreplacing the interior with flat Minkowski spacetime ( m = 0, C = 1), one can apply Israel’s The solution does not contain any length scale so the coordinates may be regarded as normalized relativeto an arbitrary length unit. Except for m = 0 , / , ± , T tt = ρ m − s πG (cid:16) (1 − m ) ρ − m s − C (cid:17) ,T zz = ρ m − s πG (cid:16) ρ − m s − C (cid:17) , (2.2) T φφ = ρ m − s πG m ρ − m s . In this manner, the problem of interpreting the strong curvature singularity at ρ = 0 isshifted to that of finding an adequate source that smoothens the milder singularity at theshell. Observe in (2.2) the presence of not only energy density and tension along z , butalso a hoop stress T φφ , which seems to be a necessary feature of any possible source of thesespacetimes. Typically the equation of state of the shell matter will impose a relationshipbetween m , C , and ρ s . We shall not dwell much on candidate shell sources, but merely notethat tubular structures with similar properties appear naturally in string theory in the form ofsupertubes and closely related helical strings (smeared along the z direction). It seems likelythat combinations or variants of these can provide adequate sources for these spacetimes.From (2.2) we can introduce the energy density per unit length [13], ε = − Z ρ = ρ s dφ √ g φφ T tt = 14 G (cid:18) − (1 − m ) Cρ m s (cid:19) (2.3)and tension T = − Z ρ = ρ s dφ √ g φφ T zz = 14 G (cid:18) − Cρ m s (cid:19) . (2.4)These are non-trivially equal only in the case of a conical defect spacetime, m = 0, C = 1.Let us now consider the Levi-Civita spacetime (2.1) expanded to linear order in m andin γ ≡ C − ds ≃ − (1 + 2 m log ρ ) dt + (1 − m log ρ ) ( dz + dρ )+ (1 − m log ρ ) (1 − γ ) ρ dφ . (2.5)In this linearized approximation we can also write m and γ in terms of the energy density(2.3) and tension (2.4) as m ≃ G ( ε − T ) , γ ≃ GT . (2.6)We will regard these simple relations as the basic interpretation of the parameters of theLevi-Civita string. Observe that ρ s does not appear in them. Relatedly, the hoop stress T φφ is O ( m ) and therefore it does not appear in the linearized approximation.Using (2.6), and performing the coordinate change(1 − G ( ε + T ) log r ) r = (1 − GT )(1 − G ( ε − T ) log ρ ) ρ (2.7)(to the required expansion order) the metric (2.5) is brought to the form ds ≃ − (1 + 4 G ( ε − T ) log r ) dt + (1 − G ( ε − T ) log r ) dz + (1 − G ( ε + T ) log r ) ( dr + r dφ ) . (2.8)3his can be recognized as the solution to the linearized Einstein equations, in transversegauge, (cid:3) (cid:18) h µν − h η µν (cid:19) = − πGT µν (2.9)with distributional stress tensor T µν = diag( ε, − T, , δ ( x ) δ ( y ) (2.10)which confirms our interpretation of ε and T .The solution (2.8) has been previously studied in the context of a different kind of stringsource, namely ‘wiggly strings’ (see e.g., [16]). Cosmic strings have Lorentz-invariant world-sheets so ε = T , but if they acquire a short-distance structure (wiggles), then when this isaveraged it produces an effective linear source with ε = T . This is an appealing physicalrealization of this linearized spacetime.Finally, observe that the tension of the string may be negative, and hence the string exertspressure, while satisfying the usual energy conditions if ε is large enough. We will refer tothis as the ‘Levi-Civita strut’. Let us discuss it in the case of small m and small C − C < γ < ε ≃ m − | γ | / G , ε − T ≃ m G , ε − | T | ≃ m − | γ | G . (2.11)Therefore the weak, strong, and dominant energy conditions are all satisfied if m > | γ | .While it does not seem possible to realize these struts in terms of wiggly cosmic strings,one may still obtain them from tubular shells. They might be elastically unstable due tothe negative tension, but presumably this depends on the specific shell matter (see [13] forenergy conditions on generic shells). We describe different families of solutions where the black holes are accelerated either bystrings that pull or by struts that push on them. They all contain the C-metric as a limit.
We construct the metric using conventional integrability techniques for Weyl spacetimes (see e.g., [7, 17]). The rod structure for the solution is depicted in fig. 1. The metric reads ds = − e U dt + e ν ( dz + dρ ) + e − U ρ dφ C (3.1)with e U = ρ m µ − m µ µ , (3.2) I.e., the line sources for the Newtonian potential U . a a a Figure 1: Weyl rod structure for the solution with an accelerating black hole pulled by asemi-infinite Levi-Civita string. The rod at z < a has linear density m/ G . The rods at a < z < a and z > a have linear density 1 / G .and e ν = ρ m ( m − µ ( µ + ρ ) − m (cid:18) µ µ + ρ µ µ + ρ (cid:19) ! − m µ ( µ µ + ρ ) µ ( µ + ρ )( µ + ρ ) , (3.3)where µ i = a i − z + p ( a i − z ) + ρ . (3.4)The solution contains five parameters: m , C , a i ( i = 1 , , a i may beabsorbed by a shift in z so only their differences a i − a j are physical. The parameter C will be fixed presently by a regularity condition, so in the end we will be left with a three-parameter family of solutions. It contains the C-metric as the particular case m = 0 (seethe appendix), while for m = +1 we obtain a double Wick rotation of the Schwarzschildspacetime. When all the a i → + ∞ the solution reduces, after a rescaling of coordinates,to the Levi-Civita spacetime. Unlike the C-metric, for generic values of the parameters thesolution is not algebraically special.The rod structure allows a ready interpretation of the solution. Along the axis ρ = 0, weexpect to have: • A semi-infinite Levi-Civita string at z < a with rod density m/ G . • A black hole horizon at a < z < a . • An ‘exposed’ axis of rotation at a < z < a . • An acceleration (Rindler) horizon at a < z .We proceed to analyze the solution near each of these rods. Exposed axis.
Assuming that φ ∼ φ + 2 π , the absence of conical singularities at theexposed axis at { ρ = 0 , a < z < a } requires that C = lim ρ → e − ( U + ν ) (cid:12)(cid:12)(cid:12) a Near the rod at a < z < a , to leading order in ρ the metric is ds ≃ (2( z − a )) m − a − za − z − ρ dt + (cid:18) a − a a − a (cid:19) − m ) ( dz + dρ ) ! + a − za − z dφ /C (2( z − a )) m − . (3.6)5he horizon at ρ = 0 is a regular surface away from the Levi-Civita singularity at its pole z = a . The horizon area is A BH = Z π dφ Z a a dz √ g zz g φφ (cid:12)(cid:12) ρ =0 = 2 πC ( a − a ) (cid:18) a − a a − a (cid:19) − m = 2 m π ( a − a ) ( a − a ) − m ( a − a ) − m . (3.7)We can compute the surface gravity at the horizon of the Killing vector ∂ t , κ BH = lim ρ → ∂ ρ √− g tt √ g ρρ (cid:12)(cid:12)(cid:12)(cid:12) a Near ρ = 0 and a < z we find ds ≃ (2( z − a )) m − z − a z − a (cid:0) − ρ dt + dz + dρ (cid:1) + z − a z − a dφ /C (2( z − a )) m − . (3.10)There is an infinite Killing horizon (Rindler) at ρ = 0 generated by ∂ t . The apparentsingularity at z = a is just a coordinate artifact and the horizon is regular everywhere.We compute the surface gravity as was done for the black hole horizon, κ R = 1 . (3.11)The acceleration of the black hole is ambiguous in that it depends on the normalization of ∂ t ,which for a spacetime with a Levi-Civita string is unclear, and also because the black hole is anextended object. In the case of the C-metric, when the black hole is small ( a − a ≪ a − a )its acceleration relative to static asymptotic observers can be unambiguously identified toleading order as A ≃ (2 a − ( a + a )) − .The ambiguities in the normalization of κ cancel when we consider the quotient κ R κ BH = (cid:18) a − a a − a (cid:19) − m . (3.12) See [18] for a definition of ‘boost mass’. T BH,R = κ BH,R / π .Since T R < T BH the two temperatures are never equal. Thus, even if the Levi-Civita singular-ities at the string and infinity could be disposed of, an otherwise regular Euclidean instantoncould not be constructed. Levi-Civita string. For small ρ and z < a we have e U ≃ ρ m − m ( a − z ) − m ( a − z ) a − z ,e ν ≃ ρ m ( m − a − za − z (cid:18) m − ( a − z ) m − ( a − z ) ( a − z ) (cid:19) − m . (3.13)The radial dependence is like in (2.1), but there is a dependence on z as well. However, awayfrom the string endpoint at z = a these functions vary slowly with z . Thus let us introduce,at any given z along the string, the functions ˆ U ( z ) and ˆ ν ( z ) by e U ( z ) = 2 − m ( a − z ) − m ( a − z ) a − z ,e ν ( z ) = a − za − z (cid:18) m − ( a − z ) m − ( a − z ) ( a − z ) (cid:19) − m . (3.14)These are approximately constant in a neighbourhood of a given z not close to a , so we maylocally absorb them through a change of coordinates in such a way that the geometry is wellapproximated by a metric of the form (2.1) with a z -dependent C parameter ˆ C ( z ) = Ce ˆ U ( z ) e ˆ ν ( z ) − mm − m +1 . (3.15)This is a monotonically decreasing function of z . In this sense, we may say that the stringtension increases along the string from infinity towards the black hole. Note that for theC-metric with m = 0 this z -dependence cancels out. For small m ˆ C ( z ) = a − a a − a (cid:18) − m log ( a − z )( a − a )( a − z )( a − z ) + O ( m ) (cid:19) . (3.16)It is tempting to interpret this effect as saying that the wiggles in the string get stretchedwhen this pulls on the black hole, but we have not pursued this interpretation further.While it does not seem feasible to identify in a unique manner the mass and accelerationof the black hole, we note that when the black hole-rod length a − a is small (much smallerthan a − a ) and m is small, on general grounds we might expect to identify the ratio ofsurface gravities as κ R κ BH ≃ GM A . (3.17)In this limit C is close to 1 so we can identify the string tension from (2.6). If ε − T ≪ T wefind that Newton’s second law T ≈ M A (3.18)is recovered. It is interesting to observe that for the C-metric ( m = 0) the identity 1 − C − = κ R /κ BH is exactly satisfied. We may equivalently say that we are matching the metrics induced on a surface at constant z < a . a a a Figure 2: Weyl rod structure for the solution with a finite Levi-Civita strut pushing the blackhole. The rod at a < z < a has linear density m/ G . The rods at a < z < a and z > a have linear density 1 / G . Now we place a finite Levi-Civita rod along a < z < a while leaving the semi-infiniteaxis z < a exposed. If the latter is non-singular, then, as we will see below, the segment a < z < a must support a conical excess angle instead of a deficit angle, and hence we finda Levi-Civita strut pushing on the black holes. This configuration has the advantage that,since the strut has finite length, we expect the metric to be asymptotically flat at spatialinfinity. Furthermore, as discussed at the end of the previous section, the positive pressurealong the strut need not imply a violation of energy conditions as long as m is sufficientlylarge. This is unlike in the C-metric with a strut, which always violates positivity of energyand therefore makes the solutions manifestly unphysical.The rod structure is as in fig. 2. The metric functions are e U = µ (cid:18) µ µ (cid:19) − m , (3.19)and e ν = µ µ (cid:18) µ µ + ρ µ µ + ρ (cid:19) (cid:18) ( µ µ + ρ ) ( µ + ρ )( µ + ρ ) (cid:19) − m ! − m µ µ + ρ . (3.20)In contrast to the C-metric, when m = 0 the geometry is locally inequivalent to our previoussolution where the strings run to infinity. Along ρ = 0 we now have an exposed axis at z < a , a black hole horizon at a < z < a , a Levi-Civita strut with rod density m/ G at a < z < a and an acceleration horizon at a < z .The area and surface gravities of the horizons take the same form as in eqs. (3.8), (3.9),(3.11), but now regularity at the exposed axis z < a requires C = 1 . (3.21)We can then expect an excess angle along the Levi-Civita rod. Indeed, for small ρ and a < z < a we find e U ≃ ρ m − m (( a − z )( z − a )) − m z − a ,e ν ≃ ρ m ( m − − m − m ( z − a ) m − (cid:18) ( a − a ) − m (( a − z )( z − a )) − m ( a − a ) (cid:19) − m (3.22)8nd we can define a z -dependent C parameter along this rod like we have done above. Forsmall m we findˆ C ( z ) = a − a a − a (cid:18) m log ( a − a )( a − z )( z − a )( z − a )( a − a ) + O ( m ) (cid:19) , (3.23)which is smaller than 1, reflecting the need of a negative tension (pressure) to push the blackholes. The rest of the analysis can be carried out as in the previous solution and we omit it. Clearly, one can construct a larger class of metrics with a Levi-Civita rod at z < a withdensity m L / G and another rod at a < z < a with density m R / G . The construction ofthese solutions is straightforward, and the metric functions are e U = ρ m L µ − m L (cid:18) µ µ (cid:19) − m R (3.24)and e ν = (cid:18) µ µ (cid:19) − m R (cid:18) ( µ µ + ρ ) ( µ + ρ )( µ + ρ ) (cid:19) (1 − m R ) µ ρ − m L ( µ + ρ ) − m L (cid:18) µ µ + ρ µ µ + ρ (cid:19) − m R ) ! − m L . (3.25)Since there is no exposed axis there does not seem to be any preferred value for the parameter C . This is then a five-parameter family of solutions. Their analysis does not introduce anyother important novelties so we shall not dwell on it. We have exhibited several new families of explicit solutions that describe black holes accelerat-ing under the pull or push of a string-like object. Their construction is fairly straightforwardand our aim has been to underscore that these solutions can have physical significance, inparticular when strings pull on the black holes. One important feature is that, even if theLevi-Civita string (or strut) is strongly singular, it can end on the black hole without de-stroying the regularity of the horizon (away from the touchpoint). This feature was not apriori obvious, but it follows essentially from the properties of Weyl rod structures: close toa ‘horizon rod’ the geometry is always of Rindler type (as we have explicitly exhibited). Thisis indeed the reason that, while we have not performed a detailed analysis of the extensionof the solutions across the horizons, we do expect that this poses no difficulty. When theself-gravity of the string is weak (and hence the acceleration is small) it can be regardedas the zero-thickness limit of a wiggly cosmic string, but it may also correspond to othernon-singular sources. The main difficulties in interpreting this solution and identifying itsphysical parameters stem from its unconventional asymptotics. But this is a problem only ifwe consider the string to be infinitely long, and if the solution is taken to approximate only9 portion of a closed loop of string then the asymptotic behavior will be improved. On theother hand, the solutions with finite struts are presumably spatially asymptotically flat.The existence and properties of these solutions raise a number of suggestions for futurework: String sources and cylindrical shells in the accelerating black hole solution. Wehave not investigated the regularization of the Levi-Civita string in the accelerating blackhole spacetime, but there are reasons to expect that this should not be problematic. In termsof wiggly strings, looking sufficiently close to the black hole one may resolve the wiggles anduse the analysis of [19] to conclude that the vortex string can pierce the black hole. It wouldremain to solve the problem of how the wiggly structure extends to all the length of thestring, possibly with z -dependent effective parameters as suggested by our analysis above.One may also replace the Levi-Civita string with a tubular shell. It does seem possible to cutthe solution at some ρ = ρ ( z ) in a region z ≤ z s , with z s < a , and replace the interior witha smooth spacetime, so the Levi-Civita string is replaced by an empty cylindrical shell thatends on the black hole. Israel’s construction will yield the shell stress tensor. Stationaritydemands that it be orthogonal to the null generator of the horizon k , i.e., k µ k ν T µν = 0.Other than this, in the absence of a specific model for the shell there do not seem to be anyrestrictions on its stress tensor. Black hole charge and pair creation. The impossibility of matching the black holeand acceleration temperatures prevents the construction of a Euclidean instanton that wouldmediate the snapping of the string by spontaneous formation of a pair of black holes at itsendpoints. Extending our solution to include black hole charge should allow one to lower theblack hole temperature to match the acceleration temperature, as in [3], and then study thisprocess. For black holes in Kaluza-Klein theory the construction of this solution should berather straightforward given the integrability of the five-dimensional equations. AdS and black holes on branes. There does not seem to exist any obstacle of principleto extending our solutions to include a (negative) cosmological constant, even if in practicefinding exact solutions might not be feasible (for instance, inverse scattering techniques areunavailable for this case). At any rate, with these solutions one could investigate extensionsof the construction of [4] of black holes localized on a Randall-Sundrum two-brane. Note,however, that the existence of solutions to Israel’s junction conditions for a vacuum brane, i.e., one with extrinsic curvature proportional to its induced metric, is not guaranteed. Also,if the additional parameter in the solutions allowed one to construct a continuous familyof black holes localized on a two-brane, this might seem to entail a continuous violation ofuniqueness of black holes on the brane. However, although we are not aware of any theoremsagainst this, it is unlikely to be realized in this manner since the Levi-Civita string (‘hiddenbehind the brane’) presumably makes it impossible to have flat asymptotics along the brane10irections. Global structure and gravitational radiation. In this paper we have not attempted tostudy the maximal analytic extension and global structure of these solutions, but there maybe more to this than a point of mathematical rigour. In particular it should be interestingto study the extension beyond the Rindler horizon of the solution with pulling strings todescribe the ‘roof’ in the Penrose diagram, where the radiative properties of the spacetimebecome apparent (see the second reference in [2]). The Levi-Civita string is absent from thisregion and so it may be interesting to study whether the asymptotic geometry at null infinityis better behaved. If radiation at infinity can be suitably characterized, this may provide aninteresting extension of the class of boost-rotation symmetric radiative spacetimes. On theother hand, the solutions with struts probably have worse asymptotic behavior in the ‘roof’. Non-uniform rod density. The only parameter that must be fixed in order to avoidsingularities on the exposed axis is C , which amounts to a simple rescaling of φ , and whichin the linearized limit corresponds to the string tension. Thus it would seem possible toconstruct Weyl solutions analogous to the ones we have studied, with naked singularitiesonly at the pulling string, where the rod at z < a would have z -dependent density m ( z )(varying in the range (0 , / 2) or possibly (0 , C remains constant. Obviously thesame could be done with finite struts. In general, explicit solutions could be found presumablyonly up to quadratures, but their properties might perhaps still be analyzable. This wouldgive a one-function family of accelerating black holes. It is conceivable that if m approacheszero sufficiently fast as z → −∞ the asymptotic behavior might be as in the C-metric. Accelerating black holes in higher dimensions. No exact solution for acceleratingblack holes in D > D > D ≥ ∼ /r D − . Near the string, however, there are significant differencesbetween sources. Requiring the symmetry R t × R z × SO ( D − D = 5 by uplifting the solutions in [22]) to find ds = − f ( D − ε − Tµ dt + f ( D − T − εµ dz + f − D − D − − ε + Tµ dr + f D − − ε + Tµ r d Ω ( D − (4.1)11ith f = 1 − πG ( D − D − ( D − µr D − ,µ = s ( D − D − (cid:18) ε + T − D − εT (cid:19) . (4.2)We may regard these as the D -dimensional versions of the Levi-Civita strings. The twoparameters ε , T are the energy density and tension measured at asymptotic infinity. Theycoincide with the energy density and tension of the sources for the linearized approximationto the solutions. When T = ε/ ( D − 3) we recover the black string , but in all other casesthe solutions present naked singularities where f = 0, including in particular the strings withLorentz-invariant worldsheet, ε = T [23] . One might nevertheless expect that, like in fourdimensions, all of these strings with T > .This suggests that in D > 4, as in four dimensions, a family of accelerating black holesolutions should exist with at least three independent parameters, for the black hole massand the string energy density and tension, with an open set of their values being potentiallyuseful for physical applications. But, unlike in four dimensions, the asymptotic behaviordoes not seem to single out any specific solution, so in this respect they all appear to be ona similar footing and different pulling strings may be relevant to different problems. It iseven possible that strings with non-uniform density need to be considered, e.g., in order tosatisfy the junction conditions on the brane as suggested by the results of [20]. One may alsoconsider struts pushing on the black holes, but they are always nakedly singular since thereare no ‘black struts’. Acknowledgments Work supported by DURSI 2009 SGR 168, MEC FPA 2007-66665-C02 and CPAN CSD2007-00042 Consolider-Ingenio 2010. JC was also supported in part by FPU grant AP2005-3120. A The solutions in ( x, y ) coordinates The C-metric is customarily written not in Weyl coordinates but in a set of coordinates ( x, y )adapted to uniformly accelerated motion. In order to write our solutions in these coordinates, Observe that this contains the conical-defect strings for D = 4. The case ε = T / ( D − 3) has conical singularities when −∞ < z < ∞ . Note also that only when ε > ( D − T or ε < T / ( D − 3) does the angular S D − shrink to zero at the singularity. The methods of [24] should be of help here. 12e perform the change ρ = 2 A α ( x − y ) p (1 − x )( y − νx )(1 + νy ) ,z = (1 − xy )(2 + ν ( x + y )) A α ( x − y ) . (A.1)and a = − νA α , a = νA α , a = 1 A α , (A.2)which can be generically applied to any Weyl solution with two finite rods. The parameter A fixes the overall scale, and its exponent is α = 21 + m for solutions with strings ,α = 2 for solutions with struts . (A.3)We take y ≤ − − ≤ x ≤ < ν < µ i in (3.4) then become µ = 2 ( x − y )(1 + νy ) A α ( x − y ) ,µ = 2 ( x − y )(1 + νx ) A α ( x − y ) , (A.4) µ = 2 ( y − νx ) A α ( x − y ) . Defining G ( ξ ) = (1 − ξ )(1 + νξ ), the metric with strings of sec. 3.1 reads ds = 2 A ( x − y ) " G ( y ) (cid:18) G ( x )( x − y ) − y − x (cid:19) m dt + G ( x ) (cid:18) G ( x )( x − y ) − y − x (cid:19) − m dφ ¯ C + Υ A (cid:18) − dy G ( y ) + dx G ( x ) (cid:19) , (A.5)whereΥ = 2 (cid:18) − νx − y (cid:19) (cid:20) (1 − x )(1 − y )(2 + ν (1 + x + y − xy )) − m G ( x ) (cid:18) G ( x )(1 − y )(1 − x )( x − y ) (cid:19) m (cid:21) m , (A.6)and ¯ C = 12 m (1 + ν ) − m − ν . (A.7)For the solution with struts of sec. 3.2 we get ds = 2 A ( x − y ) " G ( y ) (cid:18) − x − y (cid:19) m dt + G ( x ) (cid:18) − x − y (cid:19) − m dφ + Υ A (cid:18) − dy G ( y ) + dx G ( x ) (cid:19) , (A.8)now withΥ = 2 (cid:18) − νx − y (cid:19) [ − ( x + y + ν (1 + xy ))(2 + ν ( − x + y + xy ))] − m − ν ) − m ) ((1 − x )(1 − y )) − m ! m . (A.9)When m = 0 both solutions reduce, up to a constant rescaling of coordinates, to theuncharged C-metric with the factorized form for G ( ξ ) first given in [25] (see also [26]).13 eferences [1] T. Levi-Civita, “ ds einsteiniani in campi newtoniani,” Rend. Acc. Lincei , 343 (1918).H. Weyl, “Bemerkung uber die axisymmetrischen losungen der Einsteinschen Gravita-tionsgleinchungen,” Ann. Phys. (Germany) , 185 (1919).W. Kinnersley and M. Walker, “Uniformly accelerating charged mass in general relativ-ity,” Phys. Rev. D (1970) 1359.[2] For instance see:J. 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