Quantifying closeness between black hole spacetimes: a superspace approach

The study of geometrodynamics championed by Wheeler wheeler1 ; wheeler2 ; wheeler3 aims to explore, in some precise sense, the configuration space of general relativity (GR). Remarkably, the collection of all metrics that can be placed on some manifold admits a rich geometric structure: the set $\text{Met}(M)$ of metrics that can be placed over $M$ can itself be viewed as an (infinite-dimensional) Riemannian manifold gil2 ; clark10 ; ebin ; gil1 . Considering a subspace of $\text{Met}(M)$ which includes only those metrics which are Einstein defines Wheeler’s superspace fishcer ; edwards ; guil09 , and provides a means to quantify the relationship between different spacetime structures in a purely geometrical setting. In particular, geodesics on superspace can be used to ‘measure’ a distance between distinct metrics. More ambitiously, understanding the configuration space of GR may provide a stepping stone to the building of a quantum path integral over spacetime histories hawk ; eucqg .

In GR, all stationary, vacuum, and asymptotically flat black holes (BHs) are locally isometric to the Kerr solution kerr63 ; mars1 . This result is related to the no-hair theorems chand1 , and implies that an appropriately defined configuration space of BHs consists only of various Kerr metrics with different mass and spin parameters (see also Ref. schconfig ). However, if GR is in some sense incomplete and provides but an inexact description for the gravitational field in the strong-field regime, it is possible that more general BHs, described by some finite set of generalised charges or ‘hairs’ $q^{i}$ (e.g. exact1 ; exact2 ; exact3 ; exact4 ), may exist in Nature. In this paper, we note that this vector of hairs $\bm{q}$ can be used to introduce a coordinate basis over a generalised BH superspace $\text{Met}_{\text{BH}}(M)$. Some technical considerations necessary to build the superspace, such as the extraction of a single Riemannian manifold $M$ from a collection of BH spacetimes, are resolved in detail. Distances on this space, essentially quantifying the difference between two different hairy BHs, can then be used for benchmarking ‘non-Kerrness’; see also Ref. bhag18 .

In general, the nonlinear and (often) higher-than-second order nature of non-GR field equations renders the task of finding exact solutions challenging, and only a handful of physically relevant ones are known (e.g. exact1 ; exact2 ; exact3 ; exact4 ). To address the dearth of exact solutions, various approaches to constructing metrics representing generic BHs in a theory-agnostic manner have been developed nkerr1 ; nkerr2 ; nkerr3 ; contfrac . These metrics are designed to represent parameterised departures from a Kerr description in some particular way, such as including deformations that still preserve the Killing tensor symmetry nkerr5 ; nkerr4 . Recently, Konoplya and Zhidenko (KZ hereafter) kz20 derived a metric which contains “the only parameters that matter”, in the sense that they found including higher-order terms beyond those in their parameterised metric affected electromagnetic observables very little. As an application of the formalism developed here, we further validate the claims made in the aforementioned study by showing that including additional, non-KZ parameters, leads to relatively small curvature differentials on the BH superspace.

This paper is organised as follows. In Section 2 we introduce the concept of a relativistic superspace, and describe how it may be applied to the case of BH geometries. Section 3 then details the necessary technical aspects of the construction (Secs. 3.1 and 3.2), and walks the reader through the various steps for the simple case of static, spherically symmetric BHs (Sec. 3.3). Section 4 is devoted to some worked examples, namely the GR case of the Kerr kerr63 metric (Sec. 4.1) and the theory-agnostic KZ kz20 solution (Sec. 4.2). Some general remarks about the applicability, and limitations, of the approach as a measure for BH closeness are detailed in Section 5. Some discussion is given in Section 6.

The collection of all metrics definable over a Riemannian manifold $M$ can be shown to admit enough structure that it itself defines an (infinite-dimensional) Riemannian manifold ebin . Points on $\text{Met}(M)$ are Riemannian metrics on $M$, i.e. each $p\in\text{Met}(M)$ corresponds to a symmetric, positive-definite $(0,2)$-tensor over $M$. We take the base-space $M$ to be 3-dimensional (though a generalisation to the higher-dimensional case is straightforward), as later it will be identified with the leaves of a spacelike foliation of a 4-dimensional BH spacetime.

A metric $G$ can be introduced over $\text{Met}(M)$, in the $L^{2}$-topology clark10 , as ebin ; gil1

$$G(\mu,\nu)=\int_{M}d^{3}x\sqrt{h}\operatorname{Tr}\left(h^{-1}\mu h^{-1}\nu\right),$$

(1)

where $\mu$ and $\nu$ are tangent vectors to the space of metrics at the ‘reference point’ $h$, provided that the integral converges (see Sec. 3). It is important to note that other metrics exist on $\text{Met}(M)$, all of which fall under the class of the so-called $\alpha$-metrics osmo which include (1) and the DeWitt gil2 metric as special cases. However, the choice (1) is invariant under the action of the diffeomorphism group $\text{Diff}(M)$ on $\text{Met}(M)$ gil1 , and is therefore ‘canonical’ in some appropriate sense. Nevertheless, using a different metric (e.g. DeWitt gil2 ), does not qualitatively change the picture discussed herein.

As it stands, the metric (1) is defined over the entire manifold $\text{Met}(M)$. In this work we are exclusively interested in those metrics that could describe BHs, so we restrict our consideration to only those $\mu$ and $\nu$ which are tangent vectors to some appropriately defined (see below) space of black hole metrics, $\text{Met}_{\text{BH}}(M)$. The submanifold $\text{Met}_{\text{BH}}(M)\subset\text{Met}(M)$ inherits a metric from its parent space via pullback, which we also call $G$ through a slight abuse of notation. In theories other than GR, it is difficult to define $\text{Met}_{\text{BH}}(M)$ in total generality since, depending on the validity of the no-hair theorem chand1 , there may be an arbitrarily large (but finite) number of parameters (‘hairs’) which characterise the BH (cf. Refs. pappas15 ; suvmel16 ). Nevertheless, suppose that a BH can be described by $N$ macroscopic hairs $q^{1},\ldots,q^{N}$. The parameters $\bm{q}$ can be used to define a natural coordinate basis for the $N$-dimensional submanifold $\text{Met}_{\text{BH}}(M)$. In vacuum GR, Kerr uniqueness mars1 implies that $\text{Met}_{\text{GR-BH}}(M)$ is a two-dimensional space, spanned by appropriate mass ($q^{1}$, say) and spin ($q^{2}$, say) vectors. Electric chand1 or more general Yang-Mills ymbh1 ; ymbh2 charges giving rise to additional hairs may also be relevant for electrovacuum BHs in GR.

With respect to the $\bm{q}$ basis, the tensor components of (1) read suv20

$$G_{ij}=\int_{M}d^{3}x\sqrt{h}h^{nk}\frac{\partial h_{mn}}{\partial q^{i}}h^{\ell m}\frac{\partial h_{\ell k}}{\partial q^{j}},$$

(2)

where $1\leq i,j\leq N$. Assuming a (Levi-Civita) connection, all relevant geometric quantities on $\text{Met}_{\text{BH}}(M)$ can be defined from (2), including the Christoffel symbols $\Gamma$. The distance between two metrics $\ell$ and $k$, described by parameter vectors $\bm{q}_{\ell}$ and $\bm{q}_{k}$, respectively, is then given by the length of a geodesic $\gamma:[0,1]\mapsto\text{Met}_{\text{BH}}(M)$ connecting these points, viz.

$$d(\ell,k)=\int^{1}_{0}d\lambda\sqrt{G_{ij}\frac{d\gamma^{i}}{d\lambda}\frac{d\gamma^{j}}{d\lambda}},$$

(3)

for affine parameter $\lambda$, where $\gamma^{j}(0)=q_{\ell}^{j}$ and $\gamma^{j}(1)=q_{k}^{j}$, and $\gamma$ satisfies the geodesic equation,

$$0=\frac{d^{2}\gamma^{i}}{d\lambda^{2}}+\Gamma^{i}_{jk}\frac{d\gamma^{j}}{d\lambda}\frac{d\gamma^{k}}{d\lambda}.$$

(4)

From the Christoffel symbols $\Gamma$, we can also calculate the Ricci curvature

$$R_{ij}=\Gamma^{k}_{ij,k}-\Gamma^{k}_{ik,j}+\Gamma^{k}_{ij}\Gamma^{\ell}_{\ell k}-\Gamma^{k}_{i\ell}\Gamma^{\ell}_{jk},$$

(5)

whose contraction, $R=G^{ij}R_{ij}$, contains additional information about the structure of $\text{Met}_{\text{BH}}(M)$; see Sec. 4.

In GR and its extended variants, dynamics traditionally take place on a Lorentzian spacetime $(\mathcal{M},g)$ rather than on a Riemannian manifold, i.e. the metric $g$ is not positive-definite. To construct the space $\text{Met}_{\text{BH}}(M)$ we must first extract a Riemannian $(\text{sub-})$manifold $M$ from the 4-dimensional pseudo-Riemannian manifold $\mathcal{M}$. The standard approach to achieving this is through a $3+1$ Arnowitt-Deser-Misner (ADM) split admsplit ; for a globally hyperbolic spacetime, a foliation of $\mathcal{M}$ by spacelike (Cauchy) hypersurfaces $\Sigma_{t}$ parameterised by some global time coordinate $t$ is always possible ger70 . For stationary spacetimes, the underlying leaves of the foliation are necessarily time-independent in some appropriate sense and we need only consider one such $\Sigma_{t}$, which will ultimately play the role of our $M$ (see Sec. 3.1).

Although we will not present a detailed account of the rich theory of ADM decompositions (which can be found in Refs. gour06 ; gour07 , for example), we note that the 4-dimensional line element, in adapted coordinates $(t,x^{i})$ where $t$ is such that each spatial $x^{i}$ is constant along its field lines, can be written as

$$ds^{2}_{g}=-n^{2}dt^{2}+h_{ij}\left(dx^{i}+\beta^{i}dt\right)\left(dx^{j}+\beta^{j}dt\right),$$

(6)

where $n$ and $\bm{\beta}$ are known as the lapse function and shift vectors, respectively, and $h$ defines a Riemannian metric on each $\Sigma_{t}$. The metric $h$ is just the restriction of $g$ to $\Sigma_{t}\subset\mathcal{M}$, and will serve as the natural Riemannian metric uniquely associated to any given BH.

Here we present some explicit examples of the mathematical machinery developed in the previous sections. We focus on the case of vacuum GR (i.e. Kerr superspace; Sec 4.1), and on KZ static spacetimes (Sec. 4.2).

It is important to stress that the localised superspace metric (2) is tailored to measure distances between various members of a parameterised family of metrics. Under these circumstances, distances (3) are relatively straightforward to compute and yield physically reasonable results, at least for the Kerr and KZ cases considered in Sec. 4. However, the general idea may break down when considering theories of gravity that admit separate families of BH solutions with mutually exclusive hairs. In this sense, there may not be an obvious parameterisation $h$ that interpolates between two disjoint solution branches. For example, there are no slowly rotating BHs that are simultaneously solutions in the Hořova-Lifshitz and Einstein-aether theories horlif , even though the two are related in the infrared limit and admit the Schwarzschild solution as a limiting case. When considering sets of solutions with mutually exclusive hairs, the interpretation of distances on $\text{Met}_{\text{BH}}(M)$ as measuring BH similarity becomes less clear.

The superspace of metrics also cannot account for non-metric hairs, and may thus fail to capture certain facets of ‘closeness’. As an extreme example, consider two scalar-tensor theories that differ only by the dynamics in their respective scalar sectors. These theories may admit BH solutions described by the same metric, but with different scalar field profiles (due to, e.g., spontaneous scalarization dam93 ; don18 ). These two BH families are not to be thought of as identical, and yet the distance (3) between them on $\text{Met}_{\text{BH}}(M)$ for fixed metric parameters would be zero. In the same way, the metric (2) may also not reasonably quantify differences between solutions in bi- or multi-metric theories bimetric .

Our ability to experimentally validate GR (or otherwise) in the strong-field regime is continually improving, with now over a dozen measurements of gravitational waves from events involving BHs abb . For this reason, there is interest in developing generic, parameterised BH spacetimes nkerr1 ; nkerr2 ; nkerr3 , with the goal of using these to test for a variety of hypothetical deviations from the Kerr metric in a theory-agnostic way. On the other hand, a high level of generality implies that many of the proposed metrics can be mapped to one another nkerr5 ; nkerr4 , though it is not always obvious how this can be achieved in practice. It is therefore useful to have a tool which can quantitatively distinguish between various parameterised metrics, or indeed between non-Kerr metrics arising as exact solutions in beyond-Einstein theories of gravity. A similar approach for neutron star geometries was recently developed in Ref. suv20 .

Borrowing ideas from the study of geometrodynamics wheeler1 ; wheeler2 ; wheeler3 , we show in this paper how one can build a configuration manifold $\text{Met}_{\text{BH}}(M)$ of BHs; points on this non-flat manifold (see Figs. 2–4) correspond to spatial $3$-metrics associated to ADM-split BH spacetimes. Distances on this manifold can be computed by solving the geodesic equations (4). The construction of a geometric distance between two distinct metrics allows for a natural way to quantify the ‘closeness’ of two BH geometries and could be used, for example, as a general measure for ‘non-Kerrness’; see also Ref. bhag18 . For the particular case of the Kerr configuration manifold (Sec. 4.1), which is 2-dimensional and is spanned by the mass $m$ and spin $a$, we found that the Ricci curvature, though remaining finite, develops stronger gradients as the extremal limit $a/m\rightarrow 1$ is approached (see Fig. 2). The superspace approach therefore reaffirms the expectation that greater spacetime distortions accompany more rapidly rotating compact objects, as distances between BHs with greater $a$ values are larger than distances between slowly rotating holes.

For static BHs, we further validate the claims made by Konoplya and Zhidenko kz20 about which parameters are most important (that is, those which modulate astrophysical observables the most) when constructing a parameterised BH metric. This is achieved by noting that distances between (generalised) KZ metrics with an additional free parameter and those used in Ref. kz20 are comparatively small (see Sec. 4.2), which means that these spaces are geometrically similar.

Although we have only considered asymptotically flat BHs here, it is likely that the approach can be generalised to the asymptotically de Sitter case without much difficulty, since the conformal factor $\psi$ can be used to tune the convergence of (2) even when $h$ grows as $r\rightarrow\infty$. The inclusion of non-stationary black holes, where time can be thought of as an additional hair of sorts, could also be achieved given a clear choice for a single $M$ among the family of metrics under consideration. Finally, as discussed in Sec. 5, we note that the approach presented here may not capture all features that might be expected of BH ‘closeness’. Importantly, the localised metric (2) on the superspace $\text{Met}_{\text{BH}}(M)$ requires a parameterised family of BH metrics $h$. It is not obvious how to construct such a parameterisation for theories admitting disjoint branches of solutions with mutually exclusive hairs horlif ; don18 . Purely non-metric charges are also not accounted for in expression (2), as non-metric (e.g. scalar) fields do not appear.

We thank Prof. Bill Moran for hospitality shown during a research visit to Melbourne, where some of this work was completed. We are grateful to Sebastian Völkel for discussions. We thank the anonymous referees for providing feedback which helped improve the clarity of this manuscript. This work was supported by the Alexander von Humboldt Foundation.