On the nature of cosmic strings in black hole spacetimes
NNoname manuscript No. (will be inserted by the editor)
On the nature of cosmic strings in black holespacetimes
David Kofroň
Received: date / Accepted: date
Abstract
A new model for cosmic strings (i.e. conical singularities) attachedto black holes is proposed. These string are obtained by a explicit constructionvia limiting process from the so – called Bonnor rocket. This reveals quite sur-prising nature of their stress – energy tensor which contains first derivative ofDirac δ distribution. Starting from the Bonnor rocket we explicitly constructthe Schwarzschild solution witch conical singularity and the C – metric. In thelatter case we show that there is a momentum flux through the cosmic string,causing the acceleration of the black hole and the amount of this momentumis in agreement with the momentum taken away by gravitational radiation. Keywords cosmic strings · Bonnor rocket · singularities · exact solutions Among the many solutions of Einstein’s equations with non-trivial topology,those possessing topological defects of various kind are of special interest. Be-side their specific geometrical properties, such spacetimes are often inspired byconsiderations in particle physics and cosmology. In particular, defects knownas cosmic strings typically arise as a result of spontaneous symmetry break-ing in Yang-Mills theories [1]. Phase transitions of this kind were conjecturedto happen in the early universe [2] and strings produced by this mechanismcould be sources of presently detectable gravitational radiation [3,4]. Anotherintriguing feature of cosmic strings is the possibility that their presence couldbe responsible for the actual existence of magnetic monopoles.In this paper we do not intend to study the microscopic origin of stringsand treat them from purely geometrical point of view. Topologically, the pres-ence of a cosmic string is reflected in the non-triviality of the first homotopy
David KofroňInstitute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University,V Holešovičkách 2, 180 00 Prague 8, Czech Republic E-mail: [email protected] a r X i v : . [ g r- q c ] S e p David Kofroň group of the spacetime, meaning that there exist closed curves surroundingthe string that cannot be continuously deformed to a point. Metrically, space-times exhibiting a cosmic string suffer from conical singularities in the planestransversal to the string [5].The cosmic string spacetimes — flat vacuum spacetimes with conical sin-gularity and cylindrical symmetry — are known and for a long time. Thestructure of the singularity was first studied by Sokoloff and Starobinskii [6]and the first attempt to model a cosmic string was due to Villenkin [7] in thelinearized theory. In the full general relativity Hiscock [8] reconstructed thesame results.Hiscock used the Weyl form [9] of static, cylindrically symmetric metricwhich is given by d s = − e ν d t + e λ (cid:0) d r + d z (cid:1) + e ψ d ϕ , (1)where ν , λ and ψ are functions of r only.He considered an infinite cylinder of matter with density (cid:37) = ε and negativepressure p = − ε in the direction of the axis and vacuum outside of this cylinder.Then the stress energy tensor reads T tt = T zz = − ε for r ≤ l , (2) T tt = T zz = 0 for r > l , (3)with all the other components trivially equal to zero. The constant l definesthe radius of the cylinder and the junction conditions by Israel [10,11] areimposed on the surface r = l .Then, in the limit when the radius of the cylinder is decreased and thedensity is increased simultaneously so that the ”mass per unit length” (definedas integral of T tt over surfaces of constant t, z ) remains constant a vacuumcosmic string spacetime is constructed, resulting in deficit angle around theaxis.In paper by Geroch and Trashen [12] which deals with distributional sourcesin general relativity is shown that the sources of the cosmic strings cannot befound unambiguously (in the sense that there is no relation in between thedeficit angle and the “mass per unit length”). Let us present here their exam-ple briefly. They consider the same model as Hiscock, this time in cylindricalcoordinates d s = − d t + d z + d r + β ( r ) d ϕ , (4)where β ( r ) = lγ sin (cid:0) γrl (cid:1) , r ≤ l , (cid:104) r − l + lγ tan γ (cid:105) cos γ , r > l . (5)This metric is C across r = l (the surface of the cylinder) and γ ∈ (0 , π ) is aconstant. The stress energy tensor corresponding to the metric (5) is identicallyzero for r > l (vacuum) and for r ≤ l T = β (cid:48)(cid:48) ( r ) β ( r ) (cid:2) d t − d z (cid:3) = − γ l (cid:2) d t − d z (cid:3) (6) n the nature of cosmic strings in black hole spacetimes 3 which allows us to calculate the mass per unit length of the cylinder µ l = 2 π (1 − cos γ ) . (7)The string (with a deficit angle given by πγ ) is obtained by taking the limit l → .But then they suggest the following modification of the metric (4) d s = e λf ( rl ) (cid:0) − d t + d z + d r + β ( r ) d ϕ (cid:1) , (8)where f is an arbitrary smooth nonnegative function with support on (cid:104) / , (cid:105) (this modifies the matter content of the outer half of the cylinder, keepingthe axis regular). The calculations of mass per unit length yield the followingvalue µ l = 2 π (1 − cos γ ) − πλ (cid:90) sin γ xx [ f (cid:48) ( x )] d x , (9)which is strictly less (but otherwise quite arbitrary up to the restrictions im-posed on the function f ( x ) above) than the value (7), even though the strongenergy condition is fulfilled (although the stress – energy tensor is more com-plicated).Aside from these ambiguities more sophisticated matter models of cosmicstrings have been proposed since then, see [13–15], but all of them consider cylindrical symmetry only.Now, the same property — deficit angle and, thus, the cosmic string — isinevitably found also in the C – metric spacetime and can be implemented inan arbitrary axisymmetric solution.The question we raise in this paper is whether it is possible to interpretthese strings (piercing the event horizon and extending to infinity) on the samelevel as a priori cylindrically symmetric cosmic strings? That is usually doneby attributing ”mass per unit length” to them.To our best knowledge the only attempt to resolve string like sources inGR in general has been done by Israel [16]. His pioneering work suggests toinvestigate the strings in the near string limit. In this paper Israel himselfconsidered, amongst other examples, static spacetime containing two blackholes endowed with cosmic string which keeps them apart in Weyl coordinates.But his near axis limit somehow pushes aside the black hole horizon (whichitself is degenerate in Weyl coordinates).In order to understand the origin of strings in the Schwarzschild solution orthe C – metric one should provide a constructive and well-controlled procedure.We will do so in the following text. We do not attempt to provide a generaltreatment of string-like sources.The Section 2 introduces relevant mathematical definitions employed laterin the text and fixes the notation.In Section 3 the starting point of our calculations — the so called Bonnorrocket solution of Einstein field equations — is reviewed. The Bonnor rocket[17] is a quite general black hole solution which contains an arbitrary axisym-metric and time dependent null dust along outgoing geodesic (being thus a David Kofroň generalization of Vaidya solution [18]). The Schwarzschild solution, resp. theC – metric, as a particular example of the Bonnor rocket is treated in Sub-section 3.1, resp. 3.2. Dynamical processes which lead to the string formationor to the smooth transition of Schwarzschild black hole to the C – metric arediscussed.The dynamical situations are difficult to treat, thus, the following Section4 contains the detailed calculations of the structure of the string in a sequenceof static spacetimes. S ( x ) S ( x ) = for x < x , f (cid:16) x − x x − x (cid:17) f (cid:16) x − x x − x (cid:17) + f (cid:16) − x − x x − x (cid:17) for x ∈ (cid:104) x , x (cid:105) , for x > x , (10)where f ( x ) = e − / x , which is a step in between x and x .The another class of step functions, so called smooth-step functions, arepolynomials of order n +1 with boundary conditions prescribed by f ( x ) = 0 , f ( x ) = 1 and d j f ( x ) / d x j = d j f ( x ) / d x j = 0 for j = 1 , , . . . n which aresimple to construct, manage analytically and are smooth up to the order n .In general we will have a step up function (cid:19) S ( a,b ) ( x ) which vanishes for x < a , has a desired interpolation in between 0 and 1 for x ∈ (cid:104) a, b (cid:105) and isequal 1 for x > b . The step down is then simply (cid:83) S ( a,b ) ( x ) = 1 − (cid:19) S ( a,b ) ( x ) .Also the “table” function T ( a,b,c,d ) ( x ) = for x < a , (cid:19) S ( a,b ) ( x ) for x ∈ (cid:104) a, b (cid:105) , for x ∈ (cid:104) b, c (cid:105) , (cid:83) S ( c,d ) ( x ) for x ∈ (cid:104) c, d (cid:105) , for x > d , (11)will be of use.2.2 Fourier – Legendre series L functions on the interval (cid:104)− , (cid:105) can be expanded in the basis of Legendrepolynomials as f ( x ) = ∞ (cid:88) n =0 a n P n ( x ) , (12) n the nature of cosmic strings in black hole spacetimes 5 where, due to the normalization of Legendre polynomials, the coefficients a n are a n = 2 n + 12 (cid:90) − f ( x ) P n ( x ) d x . (13)The expansion of the Dirac δ distribution to this basis is given by ([19](1.17.22)) δ ( x − a ) = ∞ (cid:88) n =0 ( n + / ) P n ( x ) P n ( a ) . (14)Legendre polynomials arise as the result of Gramm – Schmidt orthogonal-ization of monomials { x j , j = 0 . . . ∞} . In the following calculation the inverserelation is necessary x k = k (cid:88) j =0 a ( k ) j P j ( x ) . (15)This will allow us to rewrite polynomial series in term of Legendre polynomialsand sum them up.The explicit formulae (which differ for even and odd powers of x ) we foundto be x k = √ π Γ (2 k + 1)2 k +1 k (cid:88) j =0 ˜ a (2 k )2 j P j ( x ) , ˜ a (2 k )2 j = 4 j + 1 Γ ( k + j + / ) Γ ( k − j + 1) , (16)for even powers of x and x k +1 = √ π Γ (2 k + 2)2 k +2 k (cid:88) j =0 ˜ a (2 k +1)2 j +1 P j +1 ( x ) , ˜ a (2 k +1)2 j +1 = 4 j + 3 Γ ( k + j + / ) Γ ( k − j + 1) , (17)for odd powers. In 1996 Bonnor [17] found an explicit solution of Einstein field equations withnull dust in which the central black hole can radiate the null dust with anarbitrary axisymmetric pattern and time profile. This solution belongs to theRobinson – Trautman class and thus posses an expanding null geodesic con-gruence which is shear free and twist free. The modern version of this metric
David Kofroň can be found in [20] and reads as follows d s = − (cid:18) − G ,xx − m ( u ) r − r ( bG ) ,x − b Gr (cid:19) d u − u d r + 2 br d u d x + r (cid:18) d x G + G d ϕ (cid:19) , (18)where b ( x, u ) = − A ( u ) − (cid:90) G ,u ( x, u ) G ( x, u ) d x , (19) G ( x, u ) = (cid:0) − x (cid:1) (cid:104) (cid:0) − x (cid:1) ˜ h ( x, u ) (cid:105) , (20)with A ( u ) an arbitrary function of u and ˜ h ( x, u ) > − an arbitrary smoothbounded function. This represents a Bonnor rocket, a particle emitting nulldust (pure radiation) with the angular (as the x = cos θ ) dependence π n ( x, u ) = −
18 ( GG ,xxx ) ,x + 32 m ( bG ) ,x − m ,u . (21)Corresponding stress energy tensor reads T ab = (cid:37) l a l b , (cid:37) = n r , l a = (cid:18) ∂∂r (cid:19) a . (22)The form of G ( x, u ) given by (20) is not the most general one. It hadbeen chosen by Bonnor so that the axis is regular. And, clearly, it does notguarantee that the quantity n defined in (21) is positive.Let us relax these restrictions, first of all we will consider the function G ( x, u ) = (cid:0) − x (cid:1) (1 + h ( x, u )) (23)and then we will omit the second power in the definition of n ( x ) .Investigating the regularity condition [21] of the axis ( x = ± ) given interms of the norm of the axial Killing vector ξ ( ϕ )
14 lim x →± F ,a F ,a F = 1 + h ( ± , u ) , (24)where F = ξ ( ϕ ) · ξ ( ϕ ) , it is clear the function h ( x, u ) determines the regularityof the axis.The Bonnor rocket (18) – (20) was fine tuned and thus in its original formdoes not contain conical singularities. We can introduce them by rescaling G → KG which is equivalent to the choice h ( x, u ) = const.Then, scaling the coordinates and parameters as ˜ u = √ K u , ˜ r = r/ √ K , (25) ˜ m = m/K √ K , ˜ A = A/ √ K , (26)leads to the same form (18) the metric, except the term K d ϕ which showsthe presence of conical singularity as we consider ϕ to run form to π strictly.This “relaxed” class of Bonnor rockets contains as a special cases Schwarzschildsolution with conical singularities and the C – metric. n the nature of cosmic strings in black hole spacetimes 7 u = u u = u (a) Schwarzschild u = u u = u (b) Schwarzschild −→ C – metric
Fig. 1: Space-time diagram of dynamical formation of the cosmic string inthe Schwarzschild solution, Fig. (a), and the smooth transition (nonuniformacceleration) of the Schwarzschild black hole to the C – metric, Fig. (b), ac-companied with the formation of cosmic strings with different stress energytensors along the north and south poles.The black hole horizon is depicted by bold red line. At u = u the radiationphase starts, the gray scale represents the time evolution of the intensity ofthe radiation, and at u = u the spacetime becomes static again, with conicalsingularities present.3.1 The Schwarzschild solutionThe metric (18) with b = 0 and G = (1 − x ) is the Schwarzschild solu-tion. For b = 0 and G = K (1 − x ) and after the aforementioned rescalingthe Schwarzschild black hole with the horizon pierced by cosmic string is ob-tained. (For regular Schwarzschild solution x = cos θ where θ is standard polarcoordinate on sphere.)Using G ( x, u ) = (cid:0) − x (cid:1) (cid:32) w (cid:19) S ( u ,u ) ( u ) e − (cid:65) S ( u ,u u )1 − x (cid:33) ,A ( u ) = 0 . (27)we get a transition between Schwarzschild for ( u < u ) through a radiatingphase u ∈ (cid:104) u , u (cid:105) during which the axis is still regular, to a Schwarzschildpierced by cosmic string with K = 1 + 2 w which appears at u = u and thereis no evolution later, see Figure 1 (a) for a schematic picture.Investigating the radiation pattern (21) we can see that the radiation getsmore and more focused (see Fig. 3 for an example for the C – metricor the Fig.4 for the Schwarzschil solution – the latter one is in polar coordinates and thussome of the properties are better readable from the picture) until the stringappears and propagates to the infinity along a null world-line. David Kofroň d s = − u d r + A r G (cid:18) x − Ar (cid:19) d u − Ar d u d x + r (cid:18) d x G ( x ) + G ( x ) d ϕ (cid:19) , (28)with G ( x ) = (cid:0) − x (cid:1) (1 + 2 Amx ) . (29)Clearly, the metric element (18) of the Bonnor rocket contains the C – metric(28) as a special case, we simply have to set m ( u ) = m , (30) b ( x, u ) = − A , (31) G ( x, u ) = (cid:0) − x (cid:1) (1 + 2 Amx ) . (32)Choosing the functions G ( x, u ) and A ( u ) in general Bonnor rocket metric (18)as G ( x, u ) = (cid:0) − x (cid:1) (cid:32) A ( u ) mx e − (cid:65) S ( u ,u u )1 − x (cid:33) ,A ( u ) = A (cid:19) S ( u ,u ) ( u ) , (33)we get a smooth transition from the static Schwarzschild solution for u < u ,through a dynamic radiation phase for u ∈ (cid:104) u , u (cid:105) during which the radia-tion gets more a more focused along the still regular axis (but this time thisradiation pattern is not reflection symmetric) to the C – metric for u > u . SeeFigure 1 (b) for schematic picture. The axis start to posses a conical singularityat u = u when the radiation is completely focused into an infinitely narrowbeam and this singularity propagates along the null direction to infinity.This dynamically obtained C – metric is for u > u diffeomorphic to theC – metric but, clearly, cannot be analytically extended and does not containthe second black hole accelerated in the opposite direction.The term (cid:90) G ( x, u ) ,u G ( x, u ) d x (34)hidden in the definition of the function b ( x, u ) and thus in the radiation pattern n ( x, u ) , see (21), is difficult, even impossible, to threat analytically. Therefore,in the next section, we will investigate these spacetimes as a sequence of dif-ferent static spacetimes parameterized either by continuous parameter ε orinteger N . n the nature of cosmic strings in black hole spacetimes 9 h x uu u Fig. 2: The function h = (cid:19) S ( u ,u ) ( u ) e − (cid:65) S ( u ,u u )1 − x which enters the structurefunction G ( x, u ) and changes it dynamically in between u and u . The Bonnor rocket emits null radiation and thus loses its mass, given by theenergy outflow − d m ( u )d u = (cid:73) r (cid:37) d Ω = (cid:73) n ( x ) d Ω . (35)Even if there is no time evolution there can be pure radiation, therefore ourdemand for staticity requires (cid:73) n ( x ) d Ω = 0 , m ( u ) = m . (36)From this follows that n ( x ) cannot be positive for ∀ x ∈ (cid:104)− , (cid:105) .The arbitrariness in the choice of function h ( x ) is almost infinite. Let usconsider a sequence of spacetimes labelled by N ∈ ( N ∪ given by G ( x, u ) = (cid:0) − x (cid:1) (cid:0) w (cid:0) − x N (cid:1)(cid:1) ,A ( u ) = 0 . (37)which can be treated completely analytically.For N = 0 we get a standard Schwarzschild solution while in the limit N → ∞ the Schwarzschild solution with a cosmic string is obtained. Duringthe limiting process the axis is regular all the time. Evaluating the radiation pattern (21) for the structure function (37) leadstrivially to π n N ( x ) = − w (4 N + 1)(2 N + 1)( N + 1) N x N + 4 w (2 N + 1)(4 N − N x N − − w (4 N − N − N ( N − x N − + (2 w + 1) w (2 N + 1) ( N + 1) N x N − w + 1) w (2 N + 1)(2 N − N x N − + (2 w + 1) w (2 N − N − N ( N − x N − . (38)In this explicit and exact form the monomials x j can be expressed (orexpanded) in the basis of Legendre polynomials as shown in the Section 2.2.Then the limit N → ∞ of n N ( x ) leads to π n ( x ) = (cid:0) w + w (cid:1) ∞ (cid:88) n =0 n (2 n + / ) (2 n + 1) P n ( x ) . (39)This series can be summed up using the expansion of Dirac δ distribution (14)in the basis of Legendre polynomials from which we get ∆ + ≡ δ ( x + 1) + δ ( x −
1) = 2 ∞ (cid:88) n =0 (2 n + / ) P n ( x ) . (40)Now, employing the standard properties of Legendre polynomials and applyingthe following differential operator we get
12 dd x (cid:20)(cid:0) − x (cid:1) dd x ∆ + (cid:21) = − dd x δ ( x + 1) + dd x δ ( x − − ∞ (cid:88) n =0 n (2 n + / ) (2 n + 1) P n ( x ) , (41)in which we recognize the right hand side of (39) and thus the final radiationpattern is π n ( x ) = − ( w + w ) (cid:20) dd x δ ( x + 1) − dd x δ ( x − (cid:21) . (42)This leads us to one of the main results of this paper – to the explicit formof stress energy tensor for the cosmic string piercing the Schwarzschild blackhole T ab = − ( w + w ) (cid:20) dd x δ ( x + 1) − dd x δ ( x − (cid:21) l a l b r . (43) n the nature of cosmic strings in black hole spacetimes 11 Analogously, the C – metric can be obtained as a limiting case of the fol-lowing sequence of spacetimes G ( x, u ) = (cid:0) − x (cid:1) (cid:0) Amx (cid:0) − x N (cid:1)(cid:1) ,A ( u ) = (cid:18) − N + 1 (cid:19) A, (44)for which the condition (36) of zero mass flux through an arbitrary sphereholds. Evaluating the radiation pattern is straightforward (but not short enoughto be presented). Expressing monomials in the basis of Legendre polynomialsand taking the limit N → ∞ yields π n ( x ) = A m ∞ (cid:88) n =0 n (2 n + / ) (2 n + 1) P n ( x )+ Am ∞ (cid:88) n =0 (2 n + 1) (2 n + / ) (2 n + 2) P n +1 ( x ) . (45)After some rearrangement of the expansion of the Dirac δ distribution in Leg-endre polynomials, ∆ − ≡ δ ( x + 1) − δ ( x −
1) = 2 ∞ (cid:88) n =0 (2 n + / ) P n +1 ( x ) , (46)and employing the properties of Legendre polynomials again we get
12 dd x (cid:20)(cid:0) − x (cid:1) dd x ∆ − (cid:21) = − dd x δ ( x + 1) − dd x δ ( x − − ∞ (cid:88) n =0 (2 n + 1) (2 n + / ) (2 n + 2) P n +1 ( x ) . (47)As a result, we can recognize (45) to be π n ( x ) = − Am ( Am + 1) dd x δ ( x + 1) + Am ( Am −
1) dd x δ ( x − , (48)with the stress energy tensor given again by T ab = (cid:37) l a l b .A different profile whose advantages lie in the fact that for x ∈ (cid:104)− ε, − ε (cid:105) the spacetime is locally Schwarzschild or the C – metric can be found G ( x, u ) = (cid:0) − x (cid:1) (cid:0) w (cid:19) S ( − , ( − ε ) T ( − , − ε, − ε, ( x ) (cid:1) , (49)for the Schwarzschild or G ( x, u ) = (cid:0) − x (cid:1) (cid:0) Amx (cid:19) S ( − , ( − ε ) T ( − , − ε, − ε, ( x ) (cid:1) , (50)for the C – metric. Step functions are now the polynomial smooth-step of order7 or higher. In these cases we can calculate n ε ( x ) and then, using computer n ε ( x ) x − (a) n ε ( x ) on x ∈ (cid:104)− , (cid:105) n ε ( x ) + 2 log / ε x . ε =2 ε =1 ε = / ε = / (b) n ε ( x ) on x ∈ (cid:104) . , (cid:105) Fig. 3: An example of radiation pattern n ε ( x ) and its focusing properties.These particular profiles are calculated for the structure function G ( x ) givenby (52) for ε = (2 , , / , / ) . In figure ( a ) the plot for the whole angle is shown,whereas in ( b ) the grey patch of ( a ) is zoomed. Also, for the sake of clarity,there is an offset in the y axis for every value of ε . The maxima of a every curveis depicted and the enveloping curve of these maxima is shown by dashed line,which is continuous function of ε (with zero offset in the y axis; thus it in thispicture passes just through the maxima of the curve for ε = / ).algebra systems, its Fourier – Legendre expansion. In the next step — in thelimit ε → + we recover the results (39) and (45).A completely different approach, which shows that these results are robust,is to use the functions G ε ( x, u ) = (cid:0) − x (cid:1) (cid:16) we − ε − x (cid:83) S (0 , ( ε ) (cid:17) ,A ( u ) = 0 , (51)for Schwarzschild G ε ( x, u ) = (cid:0) − x (cid:1) (cid:16) Amxe − ε − x (cid:83) S (0 , ( ε ) (cid:17) ,A ( u ) = A (cid:83) S (0 , ( ε ) , (52)for C – metric.Evaluating the radiation pattern n ε ( x ) is straightforward but it is impossi-ble to express this function in the Fourier – Legendre series due to the integrals— they consist of rational function multiplied by e − ε / (1 − x .For the Schwarzschild solution we get π n ε ( x ) = − w (cid:83) S , ( ε ) p ( x )( x − ( x + 1) ε e − ε − x − w (cid:83) S (0 , ( ε ) q ( x )( x − ( x + 1) ε e − ε − x , (53)where p ( x ) and q ( x ) are polynomials p ( x ) = 8 ε x − x (cid:0) − x (cid:1) (cid:0) x + 9 (cid:1) ε + 3 (cid:0) x + 8 x + 1 (cid:1) (cid:0) − x (cid:1) , (54) n the nature of cosmic strings in black hole spacetimes 13 zθ(cid:37)c − c (a) ε = / z(cid:37) (b) ε = / z(cid:37) (c) ε = / z(cid:37) (d) ε = / Fig. 4: The radiating pattern n ε ( x = cos θ ) for the structure function (51)with w = − / depicted in polar plot. As the function n ε ( x ) is not strictlypositive for all x ∈ (cid:104)− , (cid:105) an offset has been introduced as is clear from Fig. ( a ) — the value c is at the outermost dash-dotted circle, zero is represented bythe middle circle and − c is represented as the innermost circle (the origin ofcoordinates is in the centre). In this case the value of c = − . The focusationof radiation along the z axis is clearly visible. In Fig. ( d ) the scaling had to bereadjusted so the circles are closer to each other. The visualisation of n ( x, u ) ,Eq. (21), with G ( x, u ) as in (27) is visually indistinguishable.and q ( x ) = p ( x ) − ε x + 2 x (cid:0) − x (cid:1) (cid:0) x + 6 (cid:1) ε . (55)Therefore, for now, consider the n ε ( x ) as a distribution and let it act on testfunctions. We anticipate the result, of course. The behavior of n ε ( x ) is governedby the term e − ε / (1 − x , for x ∈ ( − , the limit ε → + tends to 0.In the radiation pattern n ε ( x ) we can interpolate for small εe − ε − x ∼ e − εx +1 , for x ∈ (cid:104)− , denoted by n − ε (56) e − ε − x ∼ e εx − , for x ∈ ( − , (cid:105) ; denoted by n + ε and similarly for e − ε / (1 − x .Using computer algebra systems it can be analytically calculated how thisdistribution acts on basis of polynomials, i.e. evaluate the integral n ε ( x ) (cid:2) x N (cid:3) = (cid:82) x x N n ε ( x ) . In the limit the result, independent on x ∈ ( − , , is lim ε → + (cid:90) x x N n + ε ( x ) d x = ( w + w ) N (57) = − ( w + w ) δ (cid:48) ( x + 1) (cid:2) x N (cid:3) , lim ε → + (cid:90) x − x N n − ε ( x ) d x = ( − N N ( w + w ) (58) = − ( w + w ) δ (cid:48) ( x − (cid:2) x N (cid:3) , and thus it acts as derivative of Dirac δ distribution as in (42). z ε = / (cid:27) ε =1 (cid:88)(cid:88)(cid:121) ε →∞ (cid:107) ε → + (cid:63) ε = / (cid:63) (cid:37) Fig. 5: The embedding of surface of constant u and r for the structure function(51) for various values of the parameter ε ∈ (0 + , / , / , , ∞ ) and w = − / .Basically the transition from a sphere ( ε → ∞ ) — a dashed halfcircle —through a cigar shaped surfaces, ε ∈ (1 , / , / ) , with a regular axis — in red— into a sphere with cut-out angle ( ε → + ) and thus a singularity aroundpoles is seen.The same procedure can be repeated for the C – metric with results as in(48), of course.So far we have calculated the stress energy tensor for the strings attachedto the Schwarzschild black hole and the C – metric using three different reg-ularization schemes with the same results. This shows that the procedure isrobust.Moreover, another interesting conclusion is at hand: in the case of theC – metric the null dust is not radiated away in a symmetric manner and thuscarries the momentum away, in the rest frame of the black hole we find P z = (cid:90) − n ( x ) x d x = Am , (59)where x is spherical harmonics Y ( x, ϕ ) and actually should be replaced bythe solution of eigenfunctions on two sphere t = const and r = const as wehave done in [22,23]. Unfortunately the solution can be found only in terms ofHeun general function and cannot be normalized. Yet, this solution for small Am tend to x and the corrections to P z given by (59) would be of order A m .Finally, let us investigate the geometry of surfaces (with spherical topology)of constant u and r as embedded surfaces into R . Assume its embedding inspherical coordinates d s = d r + r (d θ + r d ϕ ) is of the form r = R ( θ ( x )) (we apply the coordinate transformation θ = θ ( x ) at the simultaneously).Then by comparison of the induced metrics d s = (cid:0) R ,x + R θ ,x (cid:1) d x + R sin θ d ϕ , (60) d s = 1 G ( x ) d x + G ( x ) d ϕ , (61)we get the following differential equation for a newly introduced function f ( x ) R ( x ) = (cid:112) f ( x ) + G ( x ) , f ,x = (cid:113) − G (cid:0) G ,x − (cid:1) G . (62) n the nature of cosmic strings in black hole spacetimes 15
In general the function f ( x ) can be found in term of integral, the embeddingin R is not possible if G ,x > .The embedding is the a surface of revolution defined by a curve in Euclideancoordinates [ z, (cid:37) ] = (cid:104) f ( x ) , (cid:112) G ( x ) (cid:105) . (63)We numerically evaluate and plot these surfaces for G ( x ) given by (51) in Fig.5. And it can be nicely seen how the axis remains regular until the very lastmoment of the procedure. Let us briefly discuss the approach proposed by Israel [16] applied to our case.This approach relies on explicit construction of coordinates in the vicinity ofthe axis such that the metric is of the form d s = d (cid:37) + A ( z, t )d z + B (cid:37) d ϕ − C ( z, t )d t , (64)and investigating The extrinsic curvature K ab (and its densitized form K ba )of cylinders of constant (cid:37)K ab = 12 ∂g ab ∂(cid:37) , K ba = (cid:112) − det g K ba , (65)and its limit C ba = lim (cid:37) → + K ba . (66)Let us have a Schwarzschild solution endowed with cosmic string in Weylcoordinates d s = − e ψ d t + e λ − ψ ) (cid:0) d r + d z (cid:1) + e − ψ r d ϕ , (67)where ψ = 12 ln (cid:20) R + + R − − mR + + R − + 2 m (cid:21) , λ = 12 ln (cid:34) ( R + + R − ) − m R + R − (cid:35) + K , (68)with R ± = (cid:113) r + ( z ± m ) . The parameter K controls the regularity of theaxis — regularity condition reads λ (0 , z ) = 0 .We can find the transformation from Weyl coordinates ( r, z ) to approxi-mate coordinates ( (cid:37), ζ ) in which (cid:37) is the affine parameter of geodesic connect-ing the axis with the point in its vicinity to an arbitrary order of precision (for ζ > m ) r = 0 + e − K (cid:115) ζ − mζ + m (cid:37) + e − K (cid:112) ζ − m m ( ζ + m ) (cid:37) − e − K (cid:112) ( ζ − m ) m (2 m + 9 ζ )( ζ + m ) (cid:37) + . . . , (69) z = ζ − e − K m ( ζ + m ) (cid:37) − e − K m ( m − ζ )( ζ + m ) (cid:37) + e − K m (cid:0) m + 6 mζ − ζ (cid:1) ( ζ + m ) (cid:37) + . . . , (70)and then the Schwarzschild metrics reads d s = − ζ − mζ + m (cid:32) me − K ζ + m ) (cid:37) − me − K
12 13 m − ζ ( ζ + m ) (cid:37) + o ( (cid:37) ) (cid:33) d t + (cid:0) o ( (cid:37) ) (cid:1) d (cid:37) + o ( (cid:37) ) d (cid:37) d ζ + ζ + mζ − m (cid:32) e K + m ( ζ + m ) (cid:37) + me − K
12 7 m − ζ ( ζ + m ) (cid:37) + o ( (cid:37) ) (cid:33) d ζ + e − K (cid:37) (cid:32) − e − K ( ζ + m ) (cid:37) + o ( (cid:37) ) (cid:33) d ϕ . (71)The limit of densitized external curvature tensor is simply C ba = , (72)in coordinates ( t, ζ, ϕ ) . But this tensor has null eigenvectors; and therefore theIsrael’s approach does not have to provide an decisive answer, as “Condition(vi) excludes “lightlike” sources which (like null surface layers) require specialtreatment” . Although the axis itself is not a null hypersurface, the stressenergy tensor is composed of null dust.We have already seen in Section 3.1 and 3.2 that the nature of singularitiesis lightlike.This shows that the conical defects are a very subtle subject which has tobe treated carefully. Citation from [16].n the nature of cosmic strings in black hole spacetimes 17
We have proposed a new model of cosmic strings attached to black holes andrevealed their corresponding stress energy tensor in the case of Schwarzschildblack hole and the C – metric. The strings are made of null dust.Our explicit construction proved to be quite regularization independent(we used three different schemes).For the C – metric the deficit angle is different on the north pole and on thesouth pole. This asymmetry suggests that there is a momentum flux throughthe cosmic strings. This flux has been calculated.
Acknowledgements
D.K. acknowledges the support from the Czech Science Foundation,Grant 17-16260Y. Moreover, D.K. would like to thank Dr. M. Scholtz for inspiring discussionsand comments on the manuscript.
References
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