Periodic analogues of the Kerr solutions a numerical study Javier Peraza1 Mart n Reiris1 and Omar E. Ortiz2

2025-05-02 0 0 1.88MB 25 页 10玖币
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Periodic analogues of the Kerr solutions:
a numerical study
Javier Peraza ∗1, Mart´ın Reiris †1, and Omar E. Ortiz ‡2
1Universidad de la Rep´ublica, Uruguay
2FAMAF, Universidad Nacional de C´ordoba, Argentina
Abstract
In recent years black hole configurations with non standard topology or with non-
standard asymptotic have gained considerable attention. In this article we carry out
numerical investigations aimed to find periodic coaxial configurations of co-rotating
3+1 vacuum black holes, for which existence and uniqueness has not yet been theo-
retically proven. The aimed configurations would extend Myers/Korotkin-Nicolai’s
family of non-rotating (static) coaxial arrays of black holes. We find that numerical
solutions with a given value for the area Aand for the angular momentum Jof the
horizons appear to exist only when the separation between consecutive horizons is
larger than a certain critical value that depends only on Aand |J|. We also estab-
lish that the solutions have the same Lewis’s cylindrical asymptotic as Stockum’s
infinite rotating cylinders. Below the mentioned critical value the rotational energy
appears to be too big to sustain a global equilibrium and a singularity shows up at
a finite distance from the bulk. This phenomenon is a relative of Stockum’s asymp-
totic’s collapse, manifesting when the angular momentum (per unit of axial length)
reaches a critical value compared to the mass (per unit of axial length), and that
results from a transition in the Lewis’s class of the cylindrical exterior solution. This
remarkable phenomenon seems to be unexplored in the context of coaxial arrays of
black holes. Ergospheres and other global properties are also presented in detail.
1 Introduction
In recent years vacuum black hole configurations with non standard topology or with
non-standard asymptotic have gained considerable attention. In five dimensions, Em-
paran and Reall [1] have found asymptotically flat stationary solutions with ring-like
S1×S2horizon and Elvang and Figueras [2] have found asymptotically flat black Sat-
urns, where a black ring S1×S2rotates around a black sphere S3. More recently Khuri,
∗jperaza@cmat.edu.uy
†mreiris@cmat.edu.uy
‡omar.ortiz@unc.edu.ar
1
arXiv:2210.12898v1 [gr-qc] 24 Oct 2022
Weinstein and Yamada [3, 4], have found periodic static coaxial arrays of horizons ei-
ther with spherical S3or ring-like S1×S2topology. Rather than asymptotically flat,
these latter configurations have Levi-Civita/Kasner asymptotic and generalize to five
dimensions the important Myers/Korotkin-Nicolai (MKN) family of vacuum static 3 +1
solutions [5–7], referred sometimes as “periodic Schwarzschild”, as their are obtained as
“linear” superpositions of Schwarzschild’s solutions via Weyl’s method. The search for
new solutions has branched rapidly to higher dimensions and to different fields, giving
by now a rich landscape of topologies, [8]. However, the question whether “periodic
Kerr” configurations generalizing the “periodic Schwarzschild” exist, either in four or
higher dimensions, remains still open. In this article we aim to investigate this latter
problem in four dimensions numerically. Specifically, we carry out numerical investiga-
tions pointing at constructing periodic coaxial configurations of co-rotating 3 + 1 black
holes. The equally spaced horizons are all intended to have the same area A, angular
Figure 1: Periodic configuration of co-rotating coaxial black holes.
The horizons are equally spaced, have the same area Aand angular
momentum J.
momentum Jand angular velocity ΩH, (see Figure 1). This problem has been discussed
by Korotkin and Nicolai in [7] (section 4) and also by Mavrin in [9], however non conclu-
sively. Numerical investigations and an analysis of the solution space and of the global
properties of the solutions appear to be carried out here for the first time.
This problem poses a number of difficulties on the numerical side. The central, rele-
vant equation to solve is an harmonic map in two dimensions, whose solutions are found
as the stationary regime of a heat-type flow for the harmonic map. The numerical study
of the problem allowed us to find the correct formulation of the asymptotic boundary
conditions, which constituted the main difficulty of the problem.
The results of this paper can be summarized as follows. Periodic configurations
having a given value of Jand A(>8π|J|), appear to exist only when the separation
2
between the horizons is larger than a certain critical value that depends only on Aand
|J|. The smaller the separation the bigger the angular velocity |ΩH|and the bigger
the rotational energy and the total energy. The asymptotic (suitably defined) matches
always a Lewis’s solution [10] as in the Stockum’s rotating cylinders [11]. Furthermore,
as the separation between horizons gets smaller than the critical value, no asymptotic
can hold the given amount of energy and we evidence a singularity formation at a
finite distance from the bulk. Roughly speaking, the asymptotic “closes up” due to too
much rotational energy. This phenomenon is similar to Stockum’s asymptotic collapse,
studied in [11], and manifesting when the angular momentum (per unit of axial length)
increases past a critical value compared to the mass (per unit of axial length). When
the rotation increases, the exterior Lewis solution transits from one extending to infinity
to one blowing up at a finite distance from the material cylinder. There is a change in
the Lewis’s class. This change of class, that prevents black holes from getting too close,
appears to be entirely novel in our context and is in sharp contrast to what occurs
for the periodic Schwarzschild configurations, where horizons can get arbitrarily near.
The ergo-regions are always bounded and disjoint. We observe though that below the
critical separation the ergospheres can indeed merge, but such solutions do not extend
to infinity. The Komar mass Mper black hole satisfies the relevant inequality,
rA
16π+4πJ2
A≤M, (1)
and equality is approached as the separation between the black holes grows unboundedly
and the geometry near the horizons approaches that of Kerr.
The paper is organized as follows. In section 2 we overview the theoretical and
numerical problem commenting on the difficulties and strategies. We present the main
equations to be solved and recall the Lewis’s classes. We also discuss the boundary
condition for the harmonic map heat flow and give the precise set up for the numerical
study. In section 3 we discuss the numerical techniques used to solve the equations. In
section 4 we present our results, discussing several features of the numerical solutions
obtained, and in section 5 we contextualize our work and give some possible future
directions to our study.
2 The setup
2.1 Overview of the mathematical and numerical problem
The configurations that we are looking for have three degrees of freedom that we will
take to be the area A, the angular momentum J, and the period L, with Llinked to
the physical separation between consecutive black holes. These parameters need to be
incorporated into the boundary conditions of the main equations. The stationary and
axisymmetric metrics are assumed to be in Weyl-Papapetrou form and are recalled in
(2), (see for instance [12]). In this expression ρ∈R+and z∈Rare the Weyl-Papapetrou
cylindrical coordinates and play a role in what follow. For Weyl-Papapetrou metrics,
3
the Einstein equations reduce to find an axisymmetric harmonic map (ω(ρ, z), η(ρ, z)) :
R3→H2from R3into the hyperbolic space H2. Here ηis the norm squared of the
axisymmetric Killing field and ωis the twist potential. All the metric components can
be recovered from ηand ωafter line-integrations. The full set of reduced equations
are presented in section 2.2. The harmonic map equations are (3)-(4) and the metric
components are (5) after the line integrations of (6) and (7).
Figure 2: The numerical domain with the horizon marked as a thick
interval.
As we look for metrics periodic in z, we restrict the analysis to the region {ρ≥
0,−L/2≤z≤L/2}keeping appropriate periodicity conditions on the top and bottom
lines {ρ≥0, z =±L/2}. Here Lis the period mentioned earlier and is a free parameter.
The harmonic map equations for ωand ηneed to be supplied with boundary data on
the border {ρ= 0,−L/2≤z≤L/2}. This boundary contains the horizon H={ρ=
0,−m≤z≤m}, and the two components of the axis Awhich is the complement of H.
The boundary conditions arise from two sides: from the natural conditions that ηand
ωmust verify on horizons and axes, and from fixing the values of the area Aand of the
angular momentum J. The angular momentum Jis set by specifying Dirichlet data for
ωon the axis, whereas the area Ais set by specifying the limit values of η/ρ2at the
poles z=±m. The fact that one can incorporate Aand Jinto the boundary conditions
in this simple way is a great advantage of the formulation. The parameter mthat may
seem to be a free parameter, is equal to the surface gravity κ, (κ2:= −h∇∂t,∇∂ti/2), of
the horizons times A/4πand therefore can be fixed using the freedom in the definition
of the stationary Killing field ∂t, or the definition of time in (2). In this article we chose
κ=κ(A, J) equal to the surface gravity of the Kerr black holes for the given Aand J.
The full set of boundary conditions on {ρ= 0,−L/2≤z≤L/2}is discussed in
section 2.4.
In practice, the harmonic map equations need to be solved numerically on a finite
rectangle [0, ρMAX]×[−L/2, L/2], and this adds the extra difficulty of finding natural
boundary conditions also at ρ=ρMAX, for ωand for η. On physical grounds one expects
4
the solutions to become asymptotically independent of zand indeed approaching the
Lewis models of Stockum’s rotating cylinders that we recall in section 2.3. But the
problem is that one does not know a priori which Lewis solution shows up for the given
A,Jand L. If that information were known, then one could easily supply appropriate
boundary conditions at ρMAX. Now, all Lewis solutions have ω=wz, so we make
ω(ρMAX, z) = wz. For ηhowever not such single choice is possible. We set a Neumann
type of boundary condition for ηand for that we use that, on actual solutions, the
Komar mass expression M(ρ), in (34), is constant, so that M(ρMAX) = M(ρ) easily
relates ∂ρη(ρMAX) to ηand ωat any ρ < ρMAX. Then, to define the condition for η
we make use of this relation, equating ∂ρη(ρMAX) to the average of the Komar mass
expression in the bulk 0 < ρ < ρMAX. We discuss this condition in section 2.4, and state
it in (34). This peculiar Neumann condition for ηat ρMAX is what gave us the best
numerical results in terms of the speed and the stability of our code.
To find solutions for the harmonic map system, we run a harmonic map heat flow with
certain initial data, and look for the stationary solutions in the long time. We can show
that the numerical solutions for two different values of ρ, say 1 ρ1
MAX < ρ2
MAX, but
the same values of A,Jand L, overlap on the smaller rectangle [0, ρ1
MAX]×[−L/2, L/2].
This shows that the solutions constructed indeed depend on A,Jand Lonly and the
boundary condition at ρMAX does not introduce any spurious new degree of freedom,
but just drives the harmonic map heat flow to settle with the right asymptotic for the
given A,Jand L.
2.2 The reduced stationary equations
The black hole configurations that we are looking for will be in Weyl-Papapetrou form,
g=−V dt2+ 2W dtdφ +ηdφ2+e2γ
η(dρ2+dz2),(2)
where (t, ρ, z, φ)∈R×R+×R×S1and where the components V, W, η and γdepend
only on (ρ, z). We require V, W, η and γto be z-periodic with period L, and, to prevent
struts on the axis, we demand in addition V, W, η and γto be symmetric with respect
to the reflection z→ −z+L.
If we denote by ωthe twist potential of ∂ϕ, then η(ρ, z) and ω(ρ, z) satisfy the
harmonic map equations (Ernst equations),
∆ρη=|∂η|2− |∂ω|2
η,(3)
∆ρω= 2h∂ω, ∂η
ηi,(4)
where the inner product h,iis with respect to the flat metric dρ2+dz2, and ∆ρ=
∂2
ρ+1
ρ∂ρ+∂2
zis the 3-dim Laplacian in cylindrical coordinates (ρ, z, φ) under axial
symmetry. The metric components Wand Vfollow from the identities,
W=ηΩ, V =ρ2−W2
η,(5)
5
摘要：

PeriodicanaloguesoftheKerrsolutions:anumericalstudyJavierPeraza*1,MartnReiris„1,andOmarE.Ortiz…21UniversidaddelaRepublica,Uruguay2FAMAF,UniversidadNacionaldeCordoba,ArgentinaAbstractInrecentyearsblackholecon gurationswithnonstandardtopologyorwithnon-standardasymptotichavegainedconsiderableattent...

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