On geometry of deformed black holes:II. Schwarzschild hole surrounded by a Bach–Weyl ring

On geometry of deformed black holes: II. Schwarzschild hole surrounded by a Bach–Weyl ring

M. Basovník    O. Semerák Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic
July 16, 2019
Abstract

We continue to study the response of black-hole space-times on the presence of additional strong sources of gravity. Restricting ourselves to static and axially symmetric (electro-)vacuum exact solutions of Einstein’s equations, we first considered the Majumdar–Papapetrou solution for a binary of extreme black holes in a previous paper, while here we deal with a Schwarzschild black hole surrounded by a concentric thin ring described by the Bach–Weyl solution. The geometry is again revealed on the simplest invariants determined by the metric (lapse function) and its gradient (gravitational acceleration), and by curvature (Kretschmann scalar). Extending the metric inside the black hole along null geodesics tangent to the horizon, we mainly focus on the black-hole interior (specifically, on its sections at constant Killing time) where the quantities behave in a way indicating a surprisingly strong influence of the external source. Being already distinct on the level of potential and acceleration, this is still more pronounced on the level of curvature: for a sufficiently massive and/or nearby (small) ring, the Kretschmann scalar even becomes negative in certain toroidal regions mostly touching the horizon from inside. Such regions have been interpreted as those where magnetic-type curvature dominates, but here we deal with space-times which do not involve rotation and the negative value is achieved due to the electric-type components of the Riemann/Weyl tensor. The Kretschmann scalar also shapes rather non-trivial landscapes outside the horizon.

pacs:
0420Jb, 0440Nr, 0470Bw

I Introduction

Interaction of black holes with other gravitating sources is interesting for purely theoretical reasons (non-linear superposition in a strong-field regime) as well as within models of certain astrophysical sources. A black-hole near field is hard to modify significantly as regards potential and intensity, but its higher derivatives (curvature) may be affected by external sources considerably. Here we try to learn and visualize this effect on a Schwarzschild black hole subject to a presence of a concentric static and axially symmetric thin ring described by the Bach–Weyl solution. More specifically, we analyse the behaviour of the simplest invariants given by the metric and its first and second derivatives in dependence on parameters of the system, namely relative mass and radius of the ring. A special attention is given to the black-hole interior, including the vicinity of the central singularity.

In a previous paper Semerák and Basovník (2016), we tried to deform the black-hole field by another black hole and for that purpose we considered the Majumdar–Papapetrou binary system, made of two extremally charged black holes. Though “the other black hole” is a very strong source, we found that below the horizon the field is not much deformed within that class of space-times. This is connected with the extreme character of their horizons. Indeed, extreme charges are required as sources of the electrostatic field which just compensates the gravitational attraction; otherwise the holes would fall towards each other or would have to be kept static by an even more unphysical strut(s). Therefore, in the present paper we try to distort a black hole which is far from extreme state. Without the electrostatic repulsion, the external source has to be supported by pressure (hoop stresses) or by centrifugal force. The simplest configuration of this kind involves a thin ring or dics surrounding the hole in a static and axially symmetric, concentric manner. Such a setting may capture at least some features of the accreting black holes studied in astrophysics, while still allowing for an exact analytical treatment.

In section II, we first compose the total metric and analyse its behaviour at the horizon. Then in section III we extend the metric to the black-hole interior by solving Einstein’s equations numerically along null geodesics starting tangentially to the horizon. In section IV, we compute and visualize on contours the behaviour of the basic invariants in dependence on parameters of the system, namely relative mass of the Bach–Weyl ring and its radius. Some more attention is devoted to the Kretschmann scalar and to the regions where it turns negative, in particular to their relation with the Gauss curvature of the horizon (subsection IV.1). Final section V concludes with a summary, a brief scan of similar literature, a remark concerning visualization and some further plans. More details on the null geodesics important for extension of the metric inside the black hole are shifted to Appendix A and the question of extension of the Weyl coordinates is treated in Appendix B. Let us stress that when speaking of “black hole”, we everywhere have in mind a section of the 3D horizon given by constant Killing time ().

Note on notation: equations/values valid on the horizon will be denoted by the index ‘H’, , while expansions valid there will be denoted by an asterisk, . The black-hole mass is called , while the ring mass and its Weyl radius . The Weyl-radius coordinate will be denoted by ; below horizon where it is pure imaginary, we will introduce by . We use geometrized units in which , , index-posed comma/semicolon indicates partial/covariant derivative and usual summation rule is employed. Signature of the space-time metric is (+++), Riemann tensor is defined according to and Ricci tensor by . Cosmological constant is set zero.

Ii Weyl metric for Schwarzschild plus ring

All vacuum static and axially symmetric space-times can be described by the Weyl-type metric

 ds2=−e2νdt2+ρ2e−2νdϕ2+e2λ−2ν(dρ2+dz2), (1)

where and are Killing time and azimuthal coordinates, and the unknown functions and depend only on cylindrical-type radius and the “vertical” linear coordinate which cover the meridional planes (orthogonal to both Killing directions) in an isotropic manner. Einstein’s equations reduce to

 ν,ρρ+ν,ρρ+ν,zz=0, (2) λ,ρ=ρ(ν,ρ)2−ρ(ν,z)2,λ,z=2ρν,ρν,z, (3)

i.e. to the Laplace equation and a simple line integral (which is however only rarely solvable explicitely). Hence, the potential behaves like in Newtonian theory and adds linearly, whereas the second function does not “superpose” that simply. For two sources, with and denoting their individual potentials, one can write , where and describe the first and the second source alone (i.e., they satisfy the above equations with just and , respectively) and is the interaction term which is given by

 λint,ρ =2ρ(ν1,ρν2,ρ−ν1,zν2,z), (4) λint,z =2ρ(ν1,ρν2,z+ν1,zν2,ρ). (5)

Typically, the potential scales linearly with the source mass, hence scales with the mass square.

We are specifically interested in space-time generated by a Schwarzschild-type black hole surrounded by a thin ring described by the Bach–Weyl solution. The Schwarzschild solution appears, respectively in the Weyl and Schwarzschild coordinates, as

 νSchw =12lnd1+d2−2Md1+d2+2M (6) =12ln(1−2Mr), (7) λSchw =12ln(d1+d2)2−4M24d1d2 (8) =12lnr(r−2M)(r−M)2−M2cos2θ, (9)

where

 d1,2:=√ρ2+(z∓M)2=r−M∓Mcosθ.

 ρ=√r(r−2M)sinθ, z=(r−M)cosθ; (10) r−M=d2+d12, Mcosθ=d2−d12. (11)

Let us stress that these relations can only be safely used above the horizon (see Appendix B).

It is worth noting that in the case of a Schwarzschild-type centre () the field equations for appear quite simple in Schwarzschild coordinates when expressed in terms of . Actually, after transforming ( is some quantity)

 X,r =X,ρρ,r+X,zz,r= =X,ρr−M√r(r−2M)sinθ+X,zcosθ, X,θ =X,ρρ,θ+X,zz,θ= =X,ρ√r(r−2M)cosθ−X,z(r−M)sinθ, νSchw,ρ =(d1+d2)[4M2−(d2−d1)2]8Mρd1d2 (12) =M(r−M)sinθ[(r−M)2−M2cos2θ]√r(r−2M), (13) νSchw,z =d2−d12d1d2=Mcosθ(r−M)2−M2cos2θ, (14)

 λint,r=2Mν2,ρρsin2θ,λint,θ=−2Mν2,zsinθ. (15)

Therefore, if depends linearly on the “external”-source mass (we will call it ), then is linear in it, too, while is quadratic. Hence, in the decomposition of the parameter appears as

 λ=λSchw+λint+λ2=λSchw+M~λint+M2~λ2, (16)

where the pure-Schwarzschild term as well as the tilded functions and do not depend on .

Our “second” source is a thin ring with Weyl radius and mass , described by the Bach–Weyl solution

 νBW =−2MK(k)πl2,l1,2:=√(ρ∓b)2+z2, (17) λBW =−M24π2b2ρ× ×[(ρ+b)(E−K)2+(ρ−b)(E−k′2K)2k′2], (18)

where

 K≡K(k):=∫π/20dα√1−k2sin2α, E≡E(k):=∫π/20√1−k2sin2αdα

are complete elliptic integrals of the 1st and the 2nd kind, with modulus and complementary modulus

 k2:=1−(l1)2(l2)2=4bρ(l2)2,k′2:=1−k2=(l1)2(l2)2.

Especially on the axis , one has , , so and (the latter must actually hold for any Weyl solution should the axis be regular). The solution was derived by Bach and Weyl (1922) and more recently studied e.g. by Hoenselaers (1995); O. et al. (1999); D’Afonseca et al. (2005).111We thank our colleague Pavel Čížek for pointing out that we did not give properly in O. et al. (1999) and for suggesting a correct form.

Due to linearity of the Laplace equation, the partial potentials and can simply be added, while the total function has to be found from the total by quadrature. In Schwarzschild coordinates, the total metric reads O. et al. (1999)

 ds2= −e2νdt2+r(r−2M)e−2νsin2θdϕ2+ +[(r−M)2−M2cos2θ]e2λ−2ν[dr2r(r−2M)+dθ2] = −(1−2Mr)e2νextdt2+e2λext−2νext1−2Mrdr2+ +r2e−2νext(e2λextdθ2+sin2θdϕ2), (19)

where in our case , while . Regarding that

 νBW=−2MπMK(k)l2/M,∂νBW∂ρ=1M∂νBW∂(ρ/M)

(and similarly for derivatives with respect to and ), we can now add to the decomposition (16), on the basis of equations (15), that scales with and as

 λint(ρM,zM;bM;M,M)= =MMλint(ρM,zM;bM;M=1,M=1). (20)

Thanks to this property, one can find the -field for a given system (given , , ) by simple scaling of its form obtained for and (and the given ).

ii.1 Behaviour on the horizon

Our main interest is to learn how the external source affects the geometry inside the black hole, which requires to extend the metric below the horizon. It will thus be useful to know how the metric functions behave on the horizon. In the Weyl coordinates, the horizon is given by , . The black-hole potential has there a logarithmic divergence while the exterior potential is regular,222Asterisk / index ‘H’ denote expansions/values valid at the horizon.

 νSchw\lx@stackrel∗=lnρ2√M2−z2+O(ρ2),νBW\lx@stackrelH=−M√z2+b2,

so the total potential expands there as

 ν\lx@stackrel∗=lnρ2√M2−z2−M√z2+b2+O(ρ2), (21)

which implies, for example,

 ρ2e−2ν \lx@stackrelH=4(M2−z2)exp(2M√z2+b2), (22) λ,ρ−ν,ρ =ρ(ν,ρ)2−ρ(ν,z)2−ν,ρ\lx@stackrel∗=O(ρ). (23)

On any static (in fact even stationary) horizon, (see e.g. Will (1974), eq. (24)), therefore, applying this for the total as well as pure-Schwarzschild metric, one finds

 λ−ν \lx@stackrelH=λSchw−νSchw+νBW(z)−2νBW(z=M)\lx@stackrelH= \lx@stackrelH=ln2M√M2−z2−M√z2+b2+2M√M2+b2. (24)

Using (23) and (24),

 λ−ν (λ−ν)H+∫ρ0(λ,ρ−ν,ρ)dρ\lx@stackrel∗= \lx@stackrel∗=ln2M√M2−z2−M√z2+b2+2M√M2+b2+O(ρ2) (25)

and, by subtraction of (21) from (25),

 λ−2ν\lx@stackrel∗=ln2M√M2−z2−ρ4M+O(ρ2). (26)

Iii Extension of the metric below horizon

Interior of a black hole deformed by an external source is known to remain regular, except for the central singularity which however keeps its point-like character Geroch and Hartle (1982). In order to extend the metric explicitely, let us first allow the spheroidal radius to go below . The Schwarzschild potential involves imaginary part there, because the lapse squared is negative below horizon. More seriously, the potential induced by the external source has to be continued there since it is not at all defined at that region originally.

iii.1 External potential inside the black hole

For , the Weyl radius turns pure imaginary, which makes the distances and the modulus of the integral complex. However, this need not lead to complex since the latter is even in , as seen, for example, from the known identity

 K(k)=21+k′K(1−k′1+k′)

which in our case () implies

 −π2MνBW=K(k)l2=2l2+l1K(l2−l1l2+l1). (27)

This is symmetrical with respect to the exchange . But such an exchange is equivalent to the change of the sign of , so is even.

Now, if is even in , it should remain real when becomes pure imaginary. However, the behaviour of for complex involves a feature which leaves this conclusion only partially valid. Let be pure imaginary, , where . From the explicit form of the modulus

 k2=4bρ(l2)2=4ibϱ−ϱ2+b2+z2+2ibϱ (28)

it is seen that inside black hole there is a surface where is pure real, . But has a branch cut along the real axis at , so it is discontinuous on the above surface. More specifically, when crossing the cut from to side (which means from to side of the surface), the integral jumps from to , hence in our case it jumps from to the complex conjugate . In addition, the same surface also marks the location where , with on its side and on its side. Due to these two circumstances, the expression changes from pure real to pure imaginary when crossing the surface from to .

A possible solution of this issue is offered by the above formula (27). Actually, when writing the potential as

 νBW=−4Mπ(l2+l1)K(l2−l1l2+l1) (29)

rather than in the usual form , it is real for both real and imaginary , it smoothly crosses the horizon and coincides with the original form in the outer region.

Interior solution – in particular in the region where direct extension of the original exterior potential to imaginary did not bring a real result – can also be checked by returning to the field equations and by solving them once again for . The equations then read

 ν,ϱϱ+ν,ϱϱ−ν,zz=0, (30) λ,ϱ=ϱ(ν,ϱ)2+ϱ(ν,z)2,λ,z=2ϱν,ϱν,z, (31)

so in comparison with (3) there appear sign changes in the first two equations. In particular, the first equation is the wave equation in the “interior meridional plane” . Its solution, appropriate for our situation, is given by infinite series involving the Legendre functions :

 νinBW= −M√b2+ϱ2× ×∞∑n=0(−1)n(2n)!22n(n!)2Pn−12(b2−ϱ2b2+ϱ2)z2n(b2+ϱ2)n.

This sum is really an expansion of (29) valid inside the horizon.333However, it only converges uniformly within , elsewhere the convergence is just point-wise and slow. In particular, on the horizon (more precisely, on all the axis ) it yields correctly

 νinBW(ϱ=0) =−M√b2+z2=νBW(ρ=0), (32) =0=∂νBW∂ρ∣∣∣ρ=0, (33) ∂2νinBW∂ϱ2∣∣ ∣∣ϱ=0 =M2b2−2z2(b2+z2)5/2=−∂2νBW∂ρ2∣∣∣ρ=0. (34)

An example of the ring-potential behaviour inside the black hole is given in figure 2.

iii.2 Function λ on the axis and at the horizon

The last function needed in order to complete the metric (19) is . Its extension below the horizon is given by field equations (31) which can be rewritten for as

 λext,ϱ ≡λ,ϱ−λSchw,ϱ=ϱ(ν,ϱ)2+ϱ(ν,z)2−λSchw,ϱ= =ϱ[(νBW,ϱ)2+(νBW,z)2 +2νSchw,ϱνBW,ϱ+2νSchw,zνBW,z], (35) λext,z ≡λ,z−λSchw,z=2ϱν,ϱν,z−λSchw,z= =2ϱ(νSchw,ϱνBW,z+νSchw,zνBW,ϱ+νBW,ϱνBW,z). (36)

Transforming to the Schwarzschild-type coordinates,

 ϱ=√r(2M−r)sinθ,z=(r−M)cosθ,

while now using

 X,r =X,ϱϱ,r+X,zz,r =X,ϱM−r√r(2M−r)sinθ+X,zcosθ, X,θ =X,ϱϱ,θ+X,zz,θ =X,ϱ√r(2M−r)cosθ+X,z(M−r)sinθ, νSchw,ϱ =(d1+d2)[4M2−(d2−d1)2]8Mϱd1d2 (37) =M(r−M)sinθ[(r−M)2−M2cos2θ]√r(2M−r), (38) νSchw,z =d2−d12d1d2=Mcosθ(r−M)2−M2cos2θ (39)

(these formulas are the same as (12)–(14) valid outside, only is changed for ), the equations assume the form

 λext,r =2νBW,ϱsinθ√r(2M−r)[r(2M−r)νBW,zcosθ−M] +(M−r)[(νBW,ϱ)2+(νBW,z)2]sin2θ, (40) λext,θ =2νBW,zsinθ× ×[(M−r)√r(2M−r)νBW,ϱsinθ−M] +r(2M−r)[(νBW,ϱ)2+(νBW,z)2]sinθcosθ. (41)

Note that in Schwarzschild coordinates all the expressions are “ready to use”, whereas if using Weyl coordinates (below horizon), one has to choose the signs of and properly (“by hand”) – see Appendix B.

The first of these reduces, for , just to

 (λext,r)sinθ=0=0, (42)

hence the function is constant along the axis. Regarding that on the Weyl axis (, ) one has ( surfaces are required to be regular there), one thus finds that

 (λext)sinθ=0=(λSchw)sinθ=0=0 (43)

holds everywhere on the (Schwarzschild) axis, including the black-hole interior.

Notice now that the second equation for reduces to the same relation at the singularity and on the horizon ,

 (λext,θ)r=0 =−2M(νBW,z)r=0sinθ, (44) (λext,θ)r=2M =−2M(νBW,z)r=2Msinθ. (45)

But , and is even in (hence is odd in ), so we have

 (νBW)r=0=(νBW)r=2M,(νBW,z)r=0=−(νBW,z)r=2M

and, therefore,

 (λext,θ)r=0=−(λext,θ)r=2M, (46)

namely the latitudinal dependence of is just opposite at the singularity and on the horizon. However, on the horizon we have for the total metric as well as for pure Schwarzschild, so the same must also hold for , hence

 (λext)r=0=−(λext)r=2M= =−2νBW(r=2M)+2νBW(r=2M,θ=0)= =2M√M2cos2θ+b2−2M√M2+b2. (47)

Note that the “duality” between the horizon and the singularity was already observed by Frolov and Shoom (2007).

iii.3 Function λ inside the black hole

It has thus been possible to find along the axis and on the horizon. One would however like to know its behaviour everywhere inside the black hole. For such a purpose, it has proved advantageous to subtract equations (35), (36) and rewrite the result

 λext,ϱ∓λext,z=ϱ(ν,ϱ∓ν,z)2−ϱ(νSchw,ϱ∓νSchw,z)2

in terms of the derivatives

 ∂∂η∓:=∂∂ϱ∓∂∂z:
 λext,η∓ =ϱ[(ν,η∓)2−(νSchw,η∓)2]= =ϱ[(νSchw,η∓+νext,η∓)2−(νSchw,η∓)2]= =ϱνext,η∓(2νSchw,η∓+νext,η∓), (48)

where, from (38) and (39),

 νSchw,η∓=2Md1[ϱ∓(z+M)]+d2[ϱ∓(z−M)]d1d2[(d1+d2)2−4M2]= =−M2r(2M−r)[ϱ∓(z+M)d2+ϱ∓(z−M)d1]. (49)

Regarding that

 d1,2 =√(z∓M)2+ρ2=√(z∓M)2−ϱ2= =√(z∓M+ϱ)(z∓M−ϱ),

the above can also be written

 νSchw,η∓= =±M2r(2M−r)(√z+M∓ϱz+M±ϱ+√z−M∓ϱz−M±ϱ).

Equation (48) has now to be integrated toward the black-hole interior. This is best performed along the family of curves given by

 d1,2=0⇔ϱ=|z∓M|⇔r=M(1±cosθ),

namely

 r=M[1±cos(θ−θ0)],θ∈⟨θ0,θ0+π⟩, (50) whereθ0=const∈⟨0,π⟩.

These curves are null geodesics starting tangentially to the horizon and descending toward the central singularity (see figure 1 and appendix A); they represent characteristics of the Einstein equations. Multiplying equations (48) by the tangent vector of the respective curves, where is some parameter, one obtains an ordinary differential equation suitable for integration,

 dλextdσ=ϱνext,η∓d(2νSchw+νext)dσ. (51)

The main benefit of the latter is that it no more contains (which does not behave nicely below the horizon).

However, the formulation we have found the most advantageous still requires one more transformation.

iii.4 Horizon angles fixed by characteristics and a trapezoid rule

It is seen on figure (1) that any two null geodesics which “counter-inspiral” (with respect to each other) from the horizon to the singularity intersect at a certain point inside the black hole. Let us denote by and the angles on the horizon from where these geodesics start, assuming , and make the transformation

 r=M(1+cosθ+−θ−2),θ=θ++θ−2, ϱ=M2(cosθ−−cosθ+),z=M2(cosθ−+cosθ+).

In terms of these angles, the metric reads (notice that it is no longer diagonal)

 ds2= −(1−2Mr)e2νextdt2 +r2e−2νext(e2λextdθ−dθ++sin2θdϕ2), (52)

where , and Einstein equations have the form

 2(cosθ−−cosθ+)∂2νext∂θ−∂θ+=∂νext∂θ+sinθ−−∂νext∂θ−sinθ+, (53) ∂λext∂θ−sinθ−=[2sinθ−(cosθ−−cosθ+)∂νext∂θ−]∂νext∂θ−, (54) ∂λext∂θ+sinθ+=[2sinθ+(cosθ−−cosθ+)∂νext∂θ+]∂νext∂θ+. (55)

To solve the first equation, it is sufficient to know the axis values ,

 νext(ϱ,z)=1ππ∫0νext(0,z+ϱcosα)dα. (56)

This integral can be calculated using a simple trapezoid rule. Actually, for a function having the same odd derivatives with respect to the integration variable at the end points of the integration interval (which is the case of our ), the error of this scheme falls exponentially with the number of discretization points (see e.g. Press et al. (2007), chapter 4).

In order to find , we have solved, instead of equations (54) and (55) themselves, their integrability condition

 ∂2λext∂θ−∂θ+=M(νext,θ+−νext,θ−)2√r(2M−r)−νext,θ+νext,θ−. (57)

Using a reversible discretization scheme which respects propagation of the boundary conditions along characteristics (like in numerical treatment of the wave equation), one obtains very precise results, mainly thanks to a regular behaviour everywhere inside the black hole (including the shells where ).

The language of and angles is also advantageous for the Kretschmann scalar: in a vacuum, the Riemann tensor has 3 independent components which satisfy

 Rtθ−tθ−=Rt