Self-Gravitating Matter in Stationary and Axisymmetric Black Hole Spacetimes
Prashant Kocherlakota Ramesh Narayan


Stationary Spacetimes]Self-Gravitating Matter in Stationary and Axisymmetric Black Hole Spacetimes




^1Black Hole Initiative at Harvard University, 20 Garden St., Cambridge, MA 02138, USA

^2Center for Astrophysics, Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA



All black holes (BHs) in nature are expected to be described by the Kerr vacuum solution of general relativity. However, the Kerr solution comes with several difficulties such as the existence of Cauchy horizons, curvature singularities, and causality-violating regions. Attempts to resolve some of these issues include phenomenological BH models, which typically contain nontrivial matter content. We introduce a simple framework here to examine the properties of matter in such phenomenological models for a broad class of stationary and axisymmetric spinning BH spacetimes, generated from nonspinning seed solutions via a metric ansatz. We apply this framework to a representative set of spinning BH spacetimes and the non-spinning seeds from which they are derived. The models span different types of matter – fluids, scalar fields, electromagnetic fields. For each model, we calculate the timelike four-velocity of the matter and thereby identify the rest frame of the matter, both outside and inside the horizon. We then examine the spatial distribution of the matter rest-frame energy density ϵ and the principal pressures. This provides a complete picture of how the matter moves, what its material properties are, and whether it obeys the classical energy conditions. Notably, at a horizon, the normal component of the pressure always satisfies p_n = -ϵ. We also investigate the expansions of the principal null congruences and explore the Hawking mass profiles of these spacetimes. These provide glimpses into the geometry of the stationary BH exterior as well as the nonstationary interior cosmology. The axisymmetric metric ansatz we work with can be used to generate new spinning solutions from a variety of nonspinning seeds. The matter in these models often satisfies the weak energy condition, at least in the BH exterior, and some models exhibit non-rigid, differential rotation. 




[
    
Prashant Kocherlakota^1,2 and
Ramesh Narayan^2,1

    Received 14 March 2024 / Accepted 23 April 2024
======================================================











§ INTRODUCTION
 


The discovery of the Kerr metric <cit.>, a stationary, vacuum solution to the Einstein equation, was a monumental achievement, realized nearly five decades after the discovery of general relativity (GR; <cit.>). Since the Kerr metric can be used to describe the spacetime geometry of a vacuum, spinning black hole (BH) in GR, it is hypothesized to describe all astrophysical BHs in nature. With the first detection of gravitational waves emitted from a binary BH merger <cit.> and the first horizon-scale images of supermassive BHs <cit.>, we have entered a new era of BH astrophysics, and the Kerr BH metric continues to be of fundamental importance in our examination of strong-gravity phenomena.

The interior geometry of a Kerr BH, however, contains several undesirable features such as a Cauchy horizon, a spacetime curvature singularity, as well as causality-violating regions permitting closed timelike curves (see, e.g., Ref. <cit.>). The instability of the Cauchy horizon (see, e.g., Ref. <cit.>) might render some of these issues physically irrelevant. To phenomenologically resolve some of these issues, alternative BH models have been also forwarded and studied extensively <cit.>. Such non-Kerr BH spacetimes typically have nontrivial Einstein tensors, and, therefore, due to the Einstein equation, can be seen to generically contain nontrivial matter content. Due to the complexity involved in obtaining stationary solutions to the coupled Einstein-matter field equations, solution-generating techniques are often employed instead to arrive at alternative BH models. For example, given a generic static and spherically-symmetric metric that solves appropriate field equations (Einstein-Klein-Gordon, Einstein-Maxwell, etc.), one attempts algorithmically, by leveraging mathematical tools such as coordinate transformations, to produce a legitimate stationary and axisymmetric counterpart, without having to resolve the field equations. 

Shortly after the discovery of the Kerr metric, Newman & Janis <cit.> proposed a novel algorithm, involving a complex coordinate transformation, that transforms the Schwarzschild metric <cit.>, which describes nonspinning vacuum BHs, to the Kerr metric. Shortly thereafter, this algorithm was applied to the Reissner-Nordström metric (see, e.g., Ref. <cit.>), a static and spherically-symmetric solution of the Einstein-Maxwell equations and used to describe nonspinning electrovacuum BHs, to successfully obtain the Kerr-Newman metric <cit.>, a legitimate stationary and axisymmetric solution to the Einstein-Maxwell system that is used to describe spinning electrovacuum BHs in GR. This algorithm <cit.> also converts the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) metric <cit.> into the Kerr-Sen metric <cit.>, both legitimate solutions of an Einstein-Maxwell-dilaton-axion theory, obtained in the low-energy effective limit of the heterotic string. Spurred by these successes, significant effort has been invested in developing and understanding such general relativistic solution-generating techniques <cit.>. 

Azreg-Aïnou has recently <cit.> fixed one crucial source of ambiguity in the Newman-Janis algorithm
[In the Newman-Janis algorithm, a complex coordinate transformation involving the spin parameter, a, and the angular coordinate, ϑ, is employed, after which a complex leg of the Newman-Penrose (NP; <cit.>) null tetrad “needs fixing” (cf. Sec. 5.1 of Ref. <cit.>). The inverse metric is then obtained using the fixed NP tetrad. Azreg-Aïnou achieves the decomplexification of the stationary metric functions by demanding that a Boyer-Lindquist <cit.> form of the metric should be attainable (cf. Sec. 7.1 of Ref. <cit.>). This removes an ambiguity in the Newman-Janis algorithm.] 
and has obtained a stationary and axisymmetric metric that ostensibly represents a legitimate stationary generalization of an arbitrary static “seed” metric. Since the class of spinning Azreg-Aïnou (AA) metrics generically admits a Carter constant <cit.> for null geodesics <cit.>, however, there is no guarantee that it will describe all stationary spacetimes. Furthermore, there is no concrete proof so far that this metric is the unique stationary generalization of an arbitrary static metric. Finally, the spinning AA metric is, in general, only conformally-related to its nonspinning seed metric <cit.>. Nevertheless, while several significant conceptual issues in this solution-generating procedure remain to be clarified, the AA algorithm has been extremely valuable in producing reasonable spinning generalizations of static regular BHs <cit.>. 

Our primary goal here is to investigate the properties of the self-gravitating matter that generates stationary spacetimes. The change in behavior of the background matter in spacetimes that differ only by their total angular momentum is, naturally, of particular interest. Towards this end, we introduce a new stationary metric ansatz, via a variation of the AA algorithm, which reduces to the static seed metric identically in the vanishing spin limit. This axisymmetric metric ansatz has several attractive qualities. For example, it is completely fixed by the seed metric (i.e., we can completely bypass the Einstein equations). It is also geodesically-integrable (i.e., all geodesics possess a Carter constant), and the corresponding Einstein tensor has only one off-diagonal component in Boyer-Lindquist coordinates (𝒢_tφ). Adopting this simple ansatz allows us to focus on exploring the qualitative features of the space of nonvacuum BH solutions. However, we expect our methods below will continue to apply for more general classes of stationary and axisymmetric spacetimes (see, e.g., Refs. <cit.>).

The outline of this paper is as follows. In Sec. <ref> we introduce the stationary and axisymmetric metric ansatz that we will employ here. In Sec. <ref>, we introduce a novel Boyer-Lindquist-like coordinate system to describe the BH interior. This greatly clarifies the analysis of the matter properties there. 

In Sec. <ref>, we identify the unique timelike comoving-frame, or rest-frame, of the background matter. This allows us to unambiguously introduce the rest-frame energy density, ϵ, and the three principal pressures, p_i. 
In Sec. <ref>, we demonstrate that matter on the horizon, including in spinning spacetimes, always satisfies p_n = -ϵ, where p_n is the principal pressure (or tension) normal to spherical surfaces. We point out that this result can be used to set up useful “no-go” theorems for BH solutions. An example of the latter is that static and spherically-symmetric BHs cannot be produced by a massless (real) scalar field that is minimally-coupled to gravity (Appendix <ref>; cf. also Ref. <cit.>). In Sec. <ref>, we examine a special class of “degenerate” nonspinning spacetimes as well as their spinning counterparts, whose matter content satisfies p_n = -ϵ everywhere. Electromagnetic fields are an example of such matter (Appendix <ref>). 

In Sec. <ref>, we obtain simple analytic expressions for the future expansions of the two principal null congruences (PNCs) of the reasonably broad class of spacetimes under consideration. Members of these congruences (null geodesics), along with the horizon null generators, together encode the global causal structure of the spacetime. Members of the ingoing and outgoing PNCs remain on conical surfaces (ϑ=const.), spiralling monotonically inwards to the center r=0 and outwards to r=∞, respectively (see, e.g., Sec. 33.6 of Ref. <cit.>). These expressions allow us to obtain a simple formula for the Hawking mass <cit.>, which is a useful measure of mass-energy in spacetime, and which determines the focusing of light. 

We apply the above results in Sec. <ref> to a study of the properties of matter – the rest-frame energy density and pressures as well as the Hawking mass distribution – in a variety of popular spherically-symmetric BH spacetimes containing different kinds of matter fields. We recover well-known results and also report new results on a few BH spacetimes. 

In Sec. <ref>, we turn to a similar study of axisymmetric BH spacetimes generated from the spherically symmetric metrics of Sec. <ref> via our variant of the AA algorithm. We comment on the the extent to which the classical energy conditions are satisfied by the underlying matter distributions in these spinning spacetimes. We also evaluate the Hawking mass as a function of Boyer-Lindquist radius r and explore the rotation characteristics of the matter, specifically the degree of differential rotation present.

Finally, we summarize our conclusions in Sec. <ref>.

Throughout we adopt geometrized units, in which G=c=1, and the metric signature is (-, +, +, +). We reserve M to denote the total Arnowitt-Deser-Misner (ADM; <cit.>) mass of the spacetime. Bracketed indices indicate that the corresponding tensor is projected onto a tetrad. Tensor components related to spherically-symmetric spacetimes are demarcated by hats. 






§ AXISYMMETRIC METRIC ANSATZ



In this section, we describe the ansatz adopted in this work to describe stationary and axisymmetric spacetimes. This spacetime metric is inspired by the modified version of the Newman-Janis algorithm <cit.> described by Azreg-Aïnou in Refs. <cit.>.

We begin with a general static and spherically-symmetric “seed” metric, ℊ̂_μν, whose line element, ds^2 = ℊ̂_μνdx̂^μdx̂^ν, is written in spherical-polar coordinates, x̂^μ = (t, r, ϑ, φ), as

    ds^2 = -fdt^2 + g/f dr^2 + R^2 dΩ_2^2 ,

where dΩ_2^2 = dϑ^2 + sin^2ϑ dφ^2 is the standard line-element on a unit 2-sphere, and the metric functions f, g, and R are functions of r alone. For the Schwarzschild metric, in particular, f(r) = 1-2M/r, g(r) = 1, and R(r) = r. 

The metric function R(r) measures the areal or curvature radius of a 2-sphere of coordinate radius r. It increases monotonically with r, and is required to be nonnegative for r>r_0 and permitted to (but not required to; cf. Refs. <cit.>) vanish at r=r_0. For most of the spacetimes considered here r_0 = 0. The only exception is the GMGHS spacetime (Table <ref>), for which r_0 = Q^2/M. Note, however, that by a simple change of coordinates, we can achieve r_0 = 0 for the GMGHS metric (see, e.g., Ref. <cit.>). The metric function g(r) is required to be positive-definite to preserve the Lorentzian nature of the metric everywhere: The determinant of the metric is given by det[ℊ̂_μν] = -g R^4sin^2ϑ. When horizons exist, they are located at the zeroes of ℊ̂^rr, i.e., at the roots of f(r). 

Since it is always possible to describe an arbitrary static and spherically-symmetric metric using just two metric functions (see, e.g., Ch. 14 of Ref. <cit.>), it is useful to move temporarily to a different radial-coordinate ρ in which the metric takes the form,

    ds^2 = -fdt^2 + 1/f dρ^2 + R^2 dΩ_2^2 ,

via the coordinate transformation dρ = √(g)dr. We note that sometimes this ordinary differential equation can be solved in terms of simple analytic functions, but in other cases, it is only possible to obtain numerical solutions for ρ(r) or, equivalently, for r(ρ). Of the six spacetimes considered in this paper (see Table <ref>), five can be cast analytically in the form (<ref>). For the Frolov BH metric <cit.>, however, it is unlikely that an analytic form of (<ref>) exists. In equation <ref>, the metric functions f and R are now functions of ρ alone, i.e., f=f(r(ρ)) and R=R(r(ρ)).

We now apply the Azreg-Aïnou (AA) algorithm <cit.> to the seed metric (<ref>) to obtain the line element, ds^2 = ℊ_μ^'ν^'dx^μ^'dx^ν^', of a stationary and axisymmetric metric, ℊ_μ^'ν^', in Boyer-Lindquist (BL; <cit.>) coordinates, x^μ^' = (t, ρ, ϑ, φ), as (see also Ref. <cit.>)

    ds^2
    =    X/Σ[-(1-2 F/Σ)dt^2 
    -22 F/Σasin^2ϑ dtdφ
    +Π/Σsin^2ϑ dφ^2. 
        .
    + Σ/Δdρ^2 + Σ dϑ^2] ,

where the spin parameter a corresponds to the specific angular momentum of the spacetime. In terms of two auxiliary functions A and B, which are related to the seed metric functions in eq. <ref> as

    A(ρ) = R^2(ρ) ; 
    B(ρ) = f(ρ) R^2(ρ) ,

the stationary metric functions, F(ρ), Δ(ρ), Σ(ρ, ϑ), and Π(ρ, ϑ), are given as

    3    F = (A - B)/2 ;        Δ = B + a^2 , 
        Σ = A + a^2cos^2ϑ ;        Π = (A + a^2)^2 - Δ a^2sin^2ϑ .

The remaining metric function X = X(ρ, ϑ) can be freely prescribed. For concreteness and simplicity, we adopt the choice X=Σ. As discussed in Ref. <cit.>, this choice has the attractive property that if the metric (<ref>) describes an asymptotically-flat spacetime, the metric (<ref>) is also assured to be asymptotically-flat. Furthermore, this class of spacetimes is geodesically-integrable, i.e., all geodesics possess a Carter constant <cit.>. Most importantly, this choice for the conformal factor ensures that the vanishing spin limit of the metric (<ref>) reproduces the metric (<ref>) exactly. 

We now move to a new radial coordinate r via dr = dρ/√(g) (this essentially inverts the previous r→ρ transformation) so that in the new set of BL coordinates, x^μ = (t, r, ϑ, φ), the axisymmetric line element (<ref>) becomes

    ds^2
    =    
    -(1-2 F/Σ)dt^2 
    -22 F/Σasin^2ϑ dtdφ
    +Π/Σsin^2ϑ dφ^2 
        
    + Σ/Δg dr^2 + Σ dϑ^2 .

In the above, F and Δ are now functions of r alone and the functions Σ and Π are functions of r and ϑ, with all four being given via eq. <ref> in terms of the auxiliary metric functions,

    A(r) = R^2(r) ; 
    B(r) = f(r) R^2(r) .

Notice that the zero-spin limit (a=0) of the axisymmetric metric (<ref>) matches the spherically-symmetric metric (<ref>) exactly. We will use the metric (<ref>) as the ansatz for the spacetime geometry in this work.

We emphasize that while the metric (<ref>) is clearly inspired by the AA algorithm, it is fundamentally distinct from what we refer to as the “AA metric” below (<ref>). This can be seen from the zero-spin limit of the latter, which is, in general, only conformally-related to the spherically-symmetric seed metric (<ref>). We discuss this further in Sec. <ref>, after having introduced all the necessary technical ingredients. This becomes particularly transparent when considering the Kerr-Frolov spacetime metric introduced in Sec. <ref>.

Before proceeding further, we make one last change to the form of the general spherically-symmetric metric (<ref>). We trade the metric functions f and g for two new metric functions m and ψ so that the general spherically-symmetric metric takes the form

    ds^2 
    =    -e^2ψ(1-2m/R)dt^2 + (∂_r R)^2/(1-2m/R) dr^2 + R^2 dΩ_2^2 .

As we shall see, m(r) corresponds to the Hawking mass of the spacetime and ψ(r), the redshift function, is related to the matter pressure distribution. For the Schwarzschild metric, in particular, we have R(r) = r, m(r) = M, and ψ(r) = 0. The metric functions f and g are related to m and ψ as 

    f(r) = e^2ψ(1 - 2m/R) ;  
    g(r) = (e^ψ·∂_rR)^2 .

In the discussion below, for brevity, we will continue to use g(r) to denote the particular combination of ψ(r) and R(r) shown above.





 §.§ Properties of the Ansatz Metric



Ergosurfaces or static limit surfaces (cf., e.g., Ref. <cit.>) of the spacetime (<ref>) are defined as locations where the asymptotically timelike Killing vector 𝐓 = ∂_t changes character, i.e., where ℊ_tt = 0.:

    Ergosurfaces:   B + a^2cos^2ϑ = 0 .


When the metric (<ref>) is used to describe black hole (BH) spacetimes, its event horizon is a stationary, null hypersurface <cit.>, and is generated by the null Killing vector,

    ℓ_H^μ = [1, 0, 0, Ω_H] ;  Ω_H = a/A(r_H)+a^2 .

Here Ω_H denotes the angular velocity of the event horizon. Equivalently, since the horizon is a constant-r hypersurface, its normal, ∇_μ r, satisfies ℊ^μν(∂_μ r)(∂_ν r) = ℊ^rr = 0. Its location, r=r_H, therefore, corresponds to the largest positive root of Δ. Inner horizons correspond to other roots of Δ, hence

    Horizons:  Δ = B + a^2 = 0 .


Asymptotic flatness demands that lim_r→∞ℊ_tt = -1, i.e., lim_r→∞[-(B+a^2cos^2ϑ)/(R^2+a^2cos^2ϑ)] = -1. This implies that lim_r→∞B ≈ R^2. Thus, since B > 0 at asymptotic infinity, as we move closer to the BH, condition (<ref>) will be met before the condition (<ref>). This ensures that the outermost ergosurface always lies outside the event horizon. The region bounded by the two surfaces is referred to as the (exterior) ergoregion. For nonspinning BH spacetimes (a=0), the event horizon is a static (Ω_H = 0) null hypersurface and an ergoregion is absent.

The metric (<ref>) has a coordinate singularity at the location of the event horizon: Δ = 0 ⇒ℊ_rr→∞. However, if we assume that the metric components are real, analytic functions, it is possible to “extend” the spacetime by moving to better coordinates and adding new regions in the standard way <cit.> (see also Sec. <ref> below). 







We note that thus far we have referred to the metrics in eqs. <ref> and <ref> as being static and stationary, respectively. While this is true in the region of spacetime outside the event horizon (“region I”; See, e.g., Ch. 5 of Ref. <cit.>), it is not true for the maximally-extended spacetime. 

This is seen by considering the character of the general Killing vector, 𝐊. With 𝐓=∂_t the asymptotically-timelike Killing vector and Φ = ∂_φ the (asymptotically-spacelike) rotational Killing vector, the norm of the general Killing vector, 𝐊 = 𝐓 + ΩΦ (Ω=const.),

    N(Ω) := ℊ_μνK^μ K^ν = ℊ_tt + 2Ωℊ_tφ + Ω^2ℊ_φφ ,

is negative (timelike) in the exterior horizon geometry if Ω_- < Ω < Ω_+, and is positive (spacelike) otherwise. Here

    Ω_± = -ℊ_tφ±√(-Δ_tφ)/ℊ_φφ = 2aF ±√(Δ)Σϑ/Π

are solutions to N(Ω) = 0, and Δ_tφ := ℊ_ttℊ_φφ - ℊ_tφ^2 = -Δsin^2ϑ denotes the determinant of the tφ-sector of the metric. Thus, the Killing vectors 𝐊_± = 𝐓 + Ω_±Φ are null. For completeness, we note that the angular velocity of the zero angular momentum observer (ZAMO), Ω_Z, is given as 

    Ω_Z = -ℊ_tφ/ℊ_φφ = 2aF/Π .

At the event horizon, the critical null angular velocities Ω_± become ϑ-independent since Δ = 0, and we find Ω_+(r_H) = Ω_-(r_H) = Ω_Z(r_H) = Ω_H.

Immediately inside the (future) event horizon (see, e.g., region II of Fig. <ref>), however, Ω_± cease to be real since  Δ < 0. Thus, N(Ω) must have the same sign for all Ω, and is positive, in particular, since N(Ω=0) = (a^2sin^2ϑ - Δ)/Σ >0. Therefore, while there exist Killing vectors that are timelike (N<0) outside the event horizon, none remain timelike just inside it. This is to be expected since the event horizon is also a Killing horizon. 

Fig. <ref> shows four regions of the Penrose diagram <cit.> of a typical nonextremal BH spacetime, obtained by extending the metric in eq. <ref>. The set of BL coordinates, x^μ = (t, r, ϑ, φ), in (<ref>) cover region I, i.e., the principal region, corresponding to the exterior horizon geometry. We designate the region between the future event horizon, ℋ^+, and the first future inner horizon, ℋ_in^+, (this is located at the second largest positive root of Δ = 0) as region II. Similarly, we designate the region between the past event horizon, ℋ^-, and the first past inner horizon, ℋ_in^-, as region IV. The first inner horizon typically corresponds to a Cauchy horizon. Region III describes the exterior horizon geometry in a different asymptotic universe. The maximally-extended spacetime contains infinite such regions in general. For spacetimes containing a single horizon and a curvature singularity, such as the Schwarzschild BH spacetime, the Penrose diagram truncates at the red dotted lines. 

The absence of a timelike Killing vector in several regions (e.g., II, IV) of the BH interior implies that the spacetime is not stationary there. Instead, it corresponds to a cosmological spacetime (see, e.g., Refs. <cit.> for a discussion on the Schwarzschild BH interior as well as Refs. <cit.>). This is a generic feature of the maximally-extended spacetime: regions in which Δ > 0 are stationary and those in which Δ < 0 are not. These are separated by the horizons (Δ=0).

The positivity of g(r) for r ≥ r_0 implies that the determinant of the spacetime metric det[ℊ_μν] = -Σ^2 g sin^2ϑ is assuredly negative-definite, and, thus, that the spacetime (<ref>) is Lorentzian over r≥ r_0.

The denominators of the Ricci and Kretschmann scalars are proportional to ∝Σ^3 g^2 and ∝Σ^6 g^4, respectively, whereas their numerators are functions of the metric functions g, A, B, and their first and second derivatives. Assuming these metric functions are twice-differentiable, if the spacetime contains a curvature singularity, it must occur at Σ = 0. In spinning singular spacetimes, the singularity is located at r=r_0 in the equatorial (ϑ=π/2) plane. This is because the metric function Σ=R^2(r)+a^2cos^2ϑ inherits its properties from the seed metric function R(r), which is positive definite on r > r_0 and is permitted to vanish at r=r_0. 

Henceforth, including when referring to the maximally-extended spacetime, we will restrict our attention to regions of spacetime in which there are no closed timelike curves (CTCs; <cit.>), i.e., we focus on regions in which ℊ_φφ > 0 (see Footnote <ref>), or equivalently, Π > 0. Although possible to do, we do not extend the metric interior to the outermost surface r=r_CTC where Π(r_CTC, ϑ)=0. 

The statements above relating to the regions obtained by analytically extending the spacetime beyond the original region can be made precise by using better (e.g., Rindler <cit.>, ingoing null <cit.>, or double null <cit.>) coordinates. 

In Sec. <ref>, by performing a coordinate transformation, we introduce coordinates adapted to a congruence of radially-ingoing photons. These ingoing null (Kerr; <cit.>) coordinates allow us to extend the spacetime smoothly across the future event horizon. In addition to providing a clear interpretation of the character of the asymptotically-timelike Killing vector 𝐓 as well as the general Killing vector 𝐊, these coordinates also expose a crucial geometrical interpretation of the coordinate r as the affine parameter along such geodesics, throughout the maximally-extended spacetime.

In Sec. <ref>, by performing yet another coordinate transformation (starting now from the aforementioned ingoing null coordinates), we introduce a novel set of BL-like coordinates for the BH interior. These coordinates allow us to define a useful notion of “cosmological time” for the interior axisymmetric geometry. Furthermore, the minimal number of off-diagonal entries in the metric tensor, in these coordinates, buys us significant computational efficiency.
[This is further enhanced by moving to “rational polynomial” coordinates, in which ϑ is replaced by y=cosϑ (cf. Sec. 1.6 of Ref. <cit.>).] 
This series of coordinate transformations also makes it formally clear that the interior metric is simply given by the exterior one, with the timelike and spacelike coordinates legitimately switching character.





 §.§ Ingoing Null Coordinates



The outgoing (+) and ingoing (-) principal null congruences (PNCs; see Sec. 33.6 of Ref. <cit.>) are generated by the vector fields ℓ_+^μ and ℓ_-^μ respectively, which are given as (see also Ref. <cit.>)

    ℓ_±^μ = [(A + a^2)√(g)/Δ, ± 1, 0, a√(g)/Δ] .

At a horizon, these become identified and are given by the null generators of the horizon (see Sec. 33.6 of Ref. <cit.>), as given in eq. <ref>.

We can then introduce new coordinates, x^μ̅ = (v, r, ϑ, ϕ̅), adapted to the ingoing PNC as (to compare with the Kerr metric, see, e.g., Sec. 5.3.6 of Ref. <cit.>)

    v = t + r_⋆ ;  ϕ̅ = φ + r_⧫ ,

where
[The new basis-vectors, {∂_v, ∂_r, ∂_ϑ, ∂_ϕ̅} are given in terms of the old ones as {∂_v, ∂_r, ∂_ϑ, ∂_ϕ̅} = {∂_t, ∂_r - ℓ_-^t∂_t - ℓ_-^φ∂_φ, ∂_ϑ, ∂_φ}.]

    dr_⋆ = ℓ_-^t dr = (A + a^2)√(g)/Δdr ;   dr_⧫ = ℓ_-^φdr = a√(g)/Δdr .

From the PNC generators written in these coordinates,

    ℓ_-^μ̅ = [0, -1, 0, 0] ,  ℓ_+^μ̅ = [(A + a^2)√(g)/Δ, 1/2, 0, a√(g)/Δ] ,

we see that they are truly adapted to the ingoing PNC. For a closely related coordinate system adapted to a class of hypersurface-forming timelike observers, see Ref. <cit.>.

The above also reveals the geometric meaning of the radial coordinate r in both regions of the extended spacetime (in fact, throughout the maximally-extended spacetime): Since ℓ_- = -∂_r, it is clear that v, ϑ, and ϕ̅ are all constant for such ingoing null geodesics, and that -r is the affine parameter along them (see also Sec. 5.3.6 of Ref. <cit.>).

In these ingoing null coordinates,
[Outgoing null coordinates, x^μ̃ = (u, r, ϑ, ϕ̃), in which ℓ_+^μ̃ = +1, are arrived at by using u=t-r_⋆ and ϕ̃ = φ - r_⧫ (cf. Ref. <cit.>; To compare with the Kerr metric, see, e.g., Sec. 5.6 of Ref. <cit.>).
] 
the metric (<ref>) takes the form,

    ds^2
    =    
    -(1-2 F/Σ)dv^2 -22 F/Σasin^2ϑ dvdϕ̅
    +Π/Σsin^2ϑ dϕ̅^2
        
    + Σ dϑ^2 + 2√(g) dvdr - 2√(g)asin^2ϑ drdϕ̅ ,

and has no coordinate singularity at the future event horizon, and is therefore (future) horizon-penetrating. We can use this coordinate system to describe region I, the future event horizon ℋ^+, as well as all of region II.

The general Killing vector, 𝐊 = 𝐓 + ΩΦ, defined earliertransforms to 𝐊 = ∂_v + Ω∂_ϕ̅, whose norm in these coordinates, ℊ_μ̅ν̅K^μ̅K^ν̅, is precisely given by N(Ω) (<ref>). Thus, there exist Killing vectors that are timelike outside the horizon, one (Ω = Ω_H) that is null on the future horizon, but none that remain timelike in region II. 





 §.§ Interior Coordinates



In this section, we introduce a set of Boyer-Lindquist-like coordinates, x^μ̆ = (τ, z, ϑ, ϕ), for the BH interior. These will be valid specifically in region II of the maximally-extended spacetime, and are obtained from the ingoing null coordinates introduced above via the following coordinate transformation (effectively the inverse of equation <ref>),

    τ = r , z = v - r_⋆ , ϕ = ϕ̅ - r_⧫ .

The line element in these coordinates becomes,

    ds^2
    =    Σ/Δg dτ^2  + Σ dϑ^2 
        -(1-2 F/Σ)dz^2 
    -22 F/Σasin^2ϑ dzdϕ
    +Π/Σsin^2ϑ dϕ^2 .


We see immediately that ∂_τ is timelike in this region since ℊ_μ̆ν̆(∂_τ)^μ̆(∂_τ)^ν̆ = ℊ_μ̆ν̆δ^μ̆_τδ^ν̆_τ = Σ g/Δ, and Δ< 0. We can similarly see that ∂_ϑ is spacelike. Furthermore, the two Killing vectors are given, in region II, as 𝐓= ∂_z and Φ = ∂_ϕ. The first, 𝐓, is spacelike since ℊ_zz = (2F-Σ)/Σ, and 2F > Σ in this region.
[Since 2F = A - B and Σ = A + a^2cos^2ϑ, this inequality is B + a^2cos^2ϑ < 0, which is true in such regions, where Δ = B+a^2 < 0.] 
We reiterate that here our attention is restricted to the regions of spacetime in which no causality-violation occurs, where ℊ_φφ > 0.
[Since ∂_t and ∂_τ are timelike in the stationary and nonstationary regions of spacetime respectively, regions for which additionally ℊ_φφ < 0 are regions in which ∂_φ is also timelike. Since the integral orbits of ∂_φ are closed, these correspond to the regions that permit closed timelike curves, which we ignore in this work. See also the related discussion in Appendix <ref>.
] 
Thus, the general Killing vector 𝐊 = ∂_z + Ω∂_ϕ (Ω=const.) is everywhere spacelike in this region, consistent with the earlier discussion.

Furthermore, the inverse metric is given as,

    g^μ̆ν̆ = 
    [           Δ/Σ1/g                0                0                0;                0            -Π/ΣΔ                0          -2aF/ΣΔ;                0                0              1/Σ                0;                0          -2aF/ΣΔ                0 -(2F-Σ)/ΣΔsin^2ϑ;                  ] ,

from which we see that the normal to constant-τ hypersurfaces, ∇_μ̆τ, is timelike: ℊ^μ̆ν̆(∂_μ̆τ)(∂_μ̆τ) = ℊ^ττ = Δ/(Σ g). Therefore, these are spacelike hypersurfaces. Similarly, the one-forms ∇_μ̆z, ∇_μ̆ϑ, and ∇_μ̆ϕ are all seen to be spacelike.

We can express the above equivalently in terms of the metric functions of the spacetime in region I as,

    ds^2
    =    ℊ_rrdτ^2 + ℊ_ϑϑdϑ^2 + ℊ_ttdz^2 + 2ℊ_tφdzdϕ + ℊ_φφdϕ^2 ,

to see that, unsurprisingly, the metric in region II (<ref>) is simply the metric in region I (<ref>) with the time and “radial” coordinates legitimately exchanging character, r ↔τ and t ↔ z. The interior angular coordinates ϕ and ϑ match the exterior ones φ and ϑ precisely, by construction.

Note that the topology of spacelike-hypersurfaces in the nonstationary region II of these spacetimes is radically different from that in the stationary region I (in these coordinates). In particular, z is not a radial coordinate because it takes values in (-∞, +∞)  (cf. Refs. <cit.>). Nevertheless, its level sets in the hypersurfaces are still spheres. On the other hand, τ takes values from r_H to r_IH, decreasing in the future, with r=r_IH < r_H the location of the first inner horizon. We understand this by remembering that -r parameterizes the ingoing PNC, as discussed below eq. <ref>.
[In region III, the timelike coordinate, say τ̅, for the interior metric must increase from r_IH to r_H so that τ̅ increases to the future. Again, this can be understood by remembering that +r parameterized the outgoing PNC (see Footnote <ref>).]

The above is to be kept in mind when interpreting the generators of the PNCs in these coordinates,

    ℓ_±^μ̆ = [-1, ∓ (A + a^2)√(g)/Δ, 0, ∓ a√(g)/Δ] .

With this useful coordinate system for the interior in hand, we will consider below the properties of matter both in the BH exterior as well as in its interior but only work with coordinate-invariant quantities. 






§ MATTER COMOVING FRAME



We have thus far discussed the properties of the axisymmetric metric ansatz (<ref>). We now turn to a systematic consideration of the properties of the matter that generates the spacetime metric, beginning with the determination of the matter rest-frame.

We define the comoving or rest frame of the self-gravitating matter that generates the stationary spacetime as being given by an orthonormal tetrad, {e_(a)^μ (a = 0-3)}, adapted to a timelike congruence in which there are no energy or momentum fluxes. That is, we will orient the tetrad legs such that the matter energy-momentum-stress (EMS) tensor, 𝒯_μν, appears diagonal in this frame. If such a frame exists, then it can be obtained by a local boost along with a local SO(3)-rotation from the coordinate basis frame, {δ^μ_α/√(|ℊ_αα|) (α = 0-3)}.

In the comoving frame, we identify the rest-frame energy density, ϵ, and principal pressures, p_i (i = 1-3), of the background matter content as

    ϵ =    𝒯_(t)(t) := 𝒯_μνe^μ_(t)e^ν_(t) , 
    
    p_i =    𝒯_(i)(i) := 𝒯_μνe^μ_(i)e^ν_(i) .

In the above, e^μ_(t) denotes the timelike leg of the tetrad, corresponding to the normalized 4-velocity of the matter, and e^μ_(i) denote the three normalized spacelike legs that are orthogonal to it.

Matter that is minimally-coupled to Einstein-Hilbert gravity satisfies the Einstein equations (see, e.g., Ref. <cit.>), which relate the curvature of spacetime to the matter EMS tensor as 𝒢_μν = 8π𝒯_μν. In such theories, the properties of the matter given in eq. <ref> are equivalently written as

    ϵ =    𝒢_μνe^μ_(t)e^ν_(t)/(8π) , 
    
    p_i =    𝒢_μνe^μ_(i)e^ν_(i)/(8π) .

Here 𝒢_μν is the Einstein tensor, which is given in terms of the Ricci curvature tensor, ℛ_μν, and its trace, ℛ = ℊ^μνℛ_μν, as 𝒢_μν = ℛ_μν - ℊ_μνℛ/2. 

Equations <ref> and <ref> reveal how the orthonormalized eigenvectors of the Einstein tensor of the stationary spacetime define the matter comoving frame.
[Note that the orthonormalization condition implies that these are also eigenvectors of the metric tensor.] 
Furthermore, the rest-frame energy density is the eigenvalue of the Einstein tensor corresponding to the timelike eigenvector. The remaining eigenvalues yield the rest-frame principal pressures.

When the matter content has several different components (e.g., a scalar field, an electromagnetic field, fluid matter), no restrictions are imposed on the form of the field-couplings between them, i.e., they can either be minimally- or nonminimally-coupled to each other. The EMS tensor in the Einstein equations then is the “total” EMS tensor, envisioned as having contributions from all matter components. We interpret the energy density and the principal pressures in eq. (<ref>) then as the total rest-frame values.

In stationary regions of the spacetime (where Δ > 0; e.g., region I), the Einstein tensor for the stationary metric (<ref>) has only one independent off-diagonal component, 𝒢_tφ.
[This is ensured by the choice of the conformal factor X = Σ. As an aside, we note that for the metric (<ref>) with general X≠Σ, one also obtains 𝒢_rϑ≠ 0. In this case, two of the eigenvectors of the rest-frame do not, in general, lie precisely along E_1 and E_2 (<ref>) but lie in their span instead.
] 
This enables finding, straightforwardly, its eigenvectors to be, in general,

    3    E^μ_0 ∝ [1, 0 ,0, Ω] ;       E^μ_3 ∝ [1, 0 ,0, χ] , 
        E^μ_1 ∝ [0, 1, 0, 0] ;       E^μ_2 ∝ [0, 0, 1, 0] .

In particular, we will require, reasonably, that the timelike eigenvector of the above corresponds to the matter 4-velocity.

Since in stationary regions E_1 and E_2 are spacelike, we see that the matter configuration obeys the underlying symmetries of the spacetime geometry, i.e., its 4-velocity is always described by (a quasi-Killing congruence <cit.>)

    e_(t)^μ∝ E_0^μ∝[1, 0 ,0, Ω(r, ϑ)] .

As indicated above, in general, Ω need not be constant on spherical (constant-r) surfaces, i.e., the matter can be differentially-rotating. However, as we will see, Ω is independent of ϑ in a large class of special spacetimes.

The complete rest frame is then defined concretely as being 

    3    e^μ_(t) = 1/√(-N(Ω))[1, 0 ,0, Ω] ;       e^μ_(φ) = 1/√(+N(χ))[1, 0 ,0, χ] ;
        e^μ_(r) = 1/√(+ℊ_rr)[0, 1, 0, 0] ;       e^μ_(ϑ) = 1/√(+ℊ_ϑϑ)[0, 0, 1, 0] .

In the above, we have made explicit the requirement that e_(t) and e_(φ) be timelike (N(Ω) < 0) and spacelike (N(χ) > 0) respectively. This simply implies that Ω_- < Ω < Ω_+ as well as that either χ < Ω_- or χ > Ω_+. For the special case of Ω = Ω_Z, we can simply set e^μ_(φ) = δ^μ_φ/√(ℊ_φφ) (cf. Ref. <cit.>).

Furthermore, due to orthonormality (ℊ_μνe^μ_(t)e^ν_(φ) = 0), χ is fixed by Ω as

    χ(Ω) =    - ℊ_tt + Ωℊ_tφ/ℊ_tφ + Ωℊ_φφ = Ω_- - (Ω_+ - Ω_-)(Ω - Ω_-)/2(Ω_Z-Ω)
    
    =    Ω_+ + (Ω_+ - Ω_-)(Ω_+ - Ω)/2(Ω-Ω_Z) .

Clearly, if Ω_- < Ω < Ω_Z, we automatically have χ < Ω_-. Similarly, if Ω_Z < Ω < Ω_+, we automatically have χ > Ω_+. 

Thus, finding the matter comoving frame in region I reduces to finding a real root Ω_- < Ω < Ω_+ to the quadratic equation 𝒢_(t)(φ) = 0,

    [ℊ_tφ𝒢_φφ - ℊ_φφ𝒢_tφ]Ω^2 
    + [ℊ_tt𝒢_φφ - ℊ_φφ𝒢_tt]Ω
        + [ℊ_tt𝒢_tφ - ℊ_tφ𝒢_tt] = 0 .

In case such a root does not exist, we can conclude that matter described by the metric above (<ref>) flows on achronal orbits (those that do not contain any points connected by a timelike path; cf. Ref. <cit.>). 

In nonstationary regions of the maximally-extended spacetime (e.g., region II), we can perform an identical analysis, using the interior BL-like coordinates introduced above (Sec. <ref>). In this way, we find that the matter orthonormal rest-frame is given as,

    3    e^μ̆_(τ) = 1/√(-ℊ_rr)[-1, 0, 0, 0] ;       e^μ̆_(ϕ) = 1/√(+N(χ))[0, 1, 0, χ] ;
        e^μ̆_(z) = 1/√(+N(Ω))[0, 1, 0, Ω] ;       e^μ̆_(ϑ) = 1/√(+ℊ_ϑϑ)[0, 0, 1, 0] ,

where e_(τ) is the matter timelike 4-velocity and e_(ϑ) is a spacelike leg. Moreover, e_(z) and e_(ϕ) are assuredly spacelike in this region since they are both proportional to the general Killing vector 𝐊 (see discussion below eq. <ref>). In the above, we have used the interior metric in the form (<ref>), so that ℊ_rr and ℊ_ϑϑ continue to denote the BL metric in region I, i.e., ℊ_rr=(Σ/Δ)g and ℊ_ϑϑ=Σ. Similarly, N(Ω) is also given in terms of the BL metric functions in region I via eq. <ref>.

Putting everything together, the rest-frame energy density and principal pressures are given everywhere in the maximally-extended spacetime as

    ϵ =    
    +𝒢_(t)(t)/(8π) ,    Δ > 0 
    
    -𝒢_(r)(r)/(8π) ,    Δ < 0
     , 
    
    p_n =    
    +𝒢_(r)(r)/(8π) ,    Δ > 0 
    
    -𝒢_(t)(t)/(8π) ,    Δ < 0
     , 
    
    p_ϑ =    𝒢_(ϑ)(ϑ)/(8π) , 
    
    p_φ =    𝒢_(φ)(φ)/(8π) .

In the above, on the right-hand side, 𝒢_(a)(b) always denotes the rest-frame projected Einstein tensor in region I. This is because we have used the interior metric in the form (<ref>). We note that p_n denotes the principal pressure in directions normal to 2-spheres everywhere in the spacetime. Therefore, we will henceforth refer to it as the normal pressure. In spherically-symmetric spacetimes, by symmetry, the two tangential pressures, p_ϑ and p_φ, are always equal.






 §.§ Classical Energy Conditions



The null energy condition (NEC) is the requirement that the energy density measured by an arbitrary null observer is always non-negative (see Sec. 2.1 of Ref. <cit.>), and is satisfied when ϵ + p_i ≥ 0 for each principal direction (i). 

The weak energy condition (WEC) requires the energy density measured by an arbitrary timelike observer to be non-negative, and is met when ϵ≥0 and ϵ + p_i ≥ 0. When the WEC is satisfied, the NEC is also automatically satisfied. As noted in Sec. 4.3 of Ref. <cit.>, the WEC is satisfied by all experimentally detected fields. Discussed there are also the adverse consequences of WEC-violating matter.

The dominant energy condition (DEC) demands that, as measured by an arbitrary timelike observer, the energy density be non-negative, and that the flow of energy be in non-spacelike directions  (Sec. 4.3 of Ref. <cit.>). This is met when ϵ≥ 0 and ϵ≥ |p_i|. Note that the DEC implies the WEC (and therefore the NEC). It is clear then that if the DEC is violated but the WEC is not, there exist timelike observers that view the flow of energy to be in spacelike directions. 

Finally, for completeness, we note that the strong energy condition (SEC) is satisfied if ϵ + p_i ≥ 0 and ϵ+Σ_ip_i ≥ 0. This last energy condition is really a statement regarding the Ricci tensor (see Sec. 2.1 of Ref. <cit.>), i.e., that it satisfy R_μνv^μ v^ν≥ 0 for an arbitrary timelike vector v^μ. As discussed in Sec. 4.3 of Ref. <cit.>, this ensures the “timelike convergence condition,” i.e., the expansion of a timelike geodesic congruence with zero vorticity will monotonically decrease. Violation of the NEC automatically implies that of the SEC. An extensive survey of various energy conditions can be found in Ref. <cit.>.

In regions of spacetime where the rest-frame energy-densitive is non-negative ϵ≥ 0, the statements of the various energy conditions can be reduced into simpler conditions on the normal (ω_n), polar (ω_ϑ), and azimuthal (ω_φ) equations-of-state (EoSs),

    ω_n = p_n/ϵ ; ω_ϑ = p_ϑ/ϵ ; ω_φ = p_φ/ϵ .

For example, the NEC and the WEC are satisfied if ω_i ≥ -1. The DEC is satisfied if |ω_i| ≤ 1. The SEC is satisfied if ω_i ≥ -1 as well as Σ_i ω_i ≥ -1.






 §.§ Matter on Horizons



We now consider the properties of matter at the future horizon. First, we restrict ourselves to spherically-symmetric spacetimes (<ref>). The components of the Einstein tensor for these spacetimes are given explicitly in Appendix <ref>. 

The following combination of the energy density and the normal pressure is particularly instructive,

    ϵ + p_n = ±𝒢̂_(t)(t) + 𝒢̂_(r)(r)/8π = ±1/4π(1 - 2m/R)∂_r ψ/R∂_r R ,

where the upper (lower) sign applies in regions where 𝐓 = ∂_t is timelike (spacelike).

Since the horizons for the spherically-symmetric spacetimes always occur at R(r) = 2m(r), eq. <ref> immediately reveals that matter in the limit of approach to the horizon always satisfies (see also Refs. <cit.>)

    ϵ + p_n = 0,   ω_n=-1.

The equation above is quite remarkable. It is a statement about the properties of matter in its own rest frame, and can, therefore, be used to set up several “no-go” theorems. Matter content whose normal EoS (<ref>) does not satisfy ω_n = -1 cannot form static BH spacetimes. 

As we show in appendix <ref>, a massless minimally-coupled scalar field in a spherically-symmetric spacetime has ω_n = +1. Thus, it cannot describe a static BH solution (cf. also Ref. <cit.> for a related no-go theorem). Indeed, the static and spherically symmetric solution to the Einstein-Klein-Gordon system is given by the Wyman-Janis-Newman-Winicour metric <cit.>, which describes a naked singularity and not a black hole. On the other hand, we discuss in appendix <ref> how ω_n = -1 for arbitrary spherically symmetric configurations of electromagnetic fields. Indeed, the unique electrovacuum solution to the Einstein-Maxwell system, namely the Reissner-Nordström metric (cf., e.g., Ref. <cit.>), can describe BH solutions. Finally, we note that the Joshi-Malafarina-Narayan-1 naked singularity metric <cit.> describes a spherically symmetric configuration of anisotropic fluid matter with ω_n = 0. Equation <ref> then indicates that this matter model cannot be used to form static BH solutions. 

Importantly, as can be seen from eq. <ref>, we note that eq. <ref> also holds for matter on the horizons of the broad class of stationary and axisymmetric metric that we consider here (<ref>). See also the related discussion in Sec. 5.4.2 of Ref. <cit.>. Thus, similar no-go theorems can be set up even in stationary and axisymmetric spacetimes. It is interesting to consider the implications of such no-go theorems for the weak cosmic censorship conjecture <cit.>.

We emphasize that these no-go theorems apply only to stationary (equilibrium) solutions. This becomes immediately clear upon considering the spacetime generated by the continual collapse <cit.> or continual expansion <cit.> of a spherically-symmetric configuration of self-gravitating dust (ω_i = 0). 






 §.§ Degenerate Spacetimes



The line element for a very special class of spherically-symmetric spacetimes, whose metric functions take values ψ(r) = 0 and R(r) = r,
[For spacetimes with ψ = 0, we can trivially set R = r.] 
is given as,

    ds^2 =-(1-2m/r)dt^2 + (1-2m/r)^-1 dr^2 + r^2 dΩ_2^2 .

Since these spacetimes are described by a single metric function m(r), we will refer to them as degenerate spacetimes. These spacetimes satisfy ℊ̂_ttℊ̂_rr = - 1 in areal-radial coordinates, and have been analyzed in Ref. <cit.>. 

Well-known examples of degenerate spacetimes include the Schwarzschild and Reissner-Nordström BHs of GR as well as several “regular” BH models that have been proposed to address the issue of the curvature singularity (cf. Refs. <cit.>).

Since ψ(r) = 0, we can see from eq. <ref> that degenerate spacetimes contain matter that satisfies p_n = -ϵ everywhere, not just at the event horizon. More completely, in region I, we can obtain 

    𝒯̂_(t)(t) = -𝒯̂_(r)(r) = ∂_r m/4π r^2 ; 𝒯̂_(ϑ)(ϑ) = 𝒯̂_(φ)(φ) = -∂_r^2 m/8π r .

All of the energy conditions reduce to conditions on the metric function m and its first two derivatives. For spacetimes with ∂_r m > 0, the validity of the energy conditions can be determined by simply examining the behaviour of ω_ϑ = -r∂_r^2m/(2∂_r m). All of the above applies equally well in region II with the formal replacement r ↔τ.

As an illustrative example, let us consider the class of regular Hayward-like BH spacetimes (cf. Refs. <cit.>), whose mass functions are given as

    m(r) = M r^p/r^p + α L^p-q ,

where L, p, q, and α are some positive constants. It can be checked that the rest-frame energy density for these spacetimes is assuredly positive-definite, ϵ > 0. The tangential equation of state for these BH models is given as,

    ω_ϑ(r) = 1 + p/2 - p α L^p-q/r^p + α L^p-q ,

so that, at the center (r=0), we can write

    ω_ϑ(0) = (1-p)/2 .

Thus, degenerate Hayward-like BH spacetimes with p>3 generically violate all of the classical energy conditions. For p≤ 3, energy condition violating regions may still persist (<ref>). For the original Hayward BH metric <cit.> in particular, for which (p, q, α) = (3, 1, 2M), as we shall see in Sec. <ref>, all energy conditions barring the DEC are satisfied everywhere.

Furthermore, the rest-frame angular velocity of the matter in the spinning counterparts (<ref>) of degenerate spacetimes (<ref>), can be obtained, in the stationary regions of spacetime, by solving eq. <ref>. It is found to be (cf. Refs. <cit.>)

    Ω(r) = a/A(r)+a^2 = Ω_PNC(r) ,

which is independent of ϑ. In the above, we have introduced Ω_PNC to denote the angular frequency of the principal null congruences in stationary regions of spacetime (see equation <ref>),

    Ω_PNC = ℓ_±^φ/ℓ_±^t .

We can also check that the matter 4-velocity (<ref>) is assuredly timelike in region I since N(Ω_PNC) = -ΔΣ/(A+a^2)^2.






 §.§ Comparison to the Azreg-Aïnou Algorithm



Applying the Azreg-Aïnou (AA) algorithm <cit.> directly on the general spherically-symmetric seed metric (<ref>) leads to an axisymmetric metric, which we refer to as the AA metric, of the form (cf. eq. 2 of Ref. <cit.>),

    ds^2
    =    X/Σ[-(1-2 F/Σ)dt^2 
    -22 F/Σasin^2ϑ dtdφ. 
        . +Π/Σsin^2ϑ dφ^2 + Σ/Δdr^2 + Σ dϑ^2] ,

where the metric functions, F, Δ, Σ, Π, are expressed in terms of auxiliary functions A and B, as in eq. <ref>. However, it is important to note that the auxiliary functions for the AA metric (see eq. 3 of Ref. <cit.>),

    A(r) = R^2/√(g) ; B(r) = (f/g)R^2 ,

differ from the A(r), B(r) we define in equation (<ref>) for the ansatz metric employed here. The two metrics (<ref> and <ref>) also differ, as written, additionally in the form of their rr-components. Finally, the conformal factor X (denoted by Ψ in Refs. <cit.>) in the AA metric (<ref>) is, in general, an unknown.

The AA algorithm <cit.> provides a prescription to fix the conformal factor X by solving a nonlinear hyperbolic partial differential equation (PDE) and a linear PDE. These equations arise from demanding that the rest-frame of the matter generating the spacetime (<ref>) be of the form (<ref>), with Ω = Ω_PNC = a/(A+a^2). Tetrad orthonormality (see eq. <ref>) automatically fixes χ = 1/(asin^2ϑ).

The first of the two PDEs is a concomitant of demanding that the rϑ-component of the Einstein tensor of the AA metric (<ref>) vanish, i.e., 𝒢_rϑ = 0,
[Note that demanding that e_(r)∝∂_r and e_(ϑ)∝∂_ϑ when X≠Σ is significantly stronger than requiring that 𝒢_(r)(ϑ) = 0 (cf. Footnote <ref>).

] 
and the second is a consequence of demanding that 𝒢_(t)(φ) = 0 for Ω=Ω_PNC specifically. These PDEs are given respectively as (with y=cosϑ; cf. eq. 4 and 7. of Ref. <cit.>),

    0 =    √(X)∂_r∂_y(1/√(X)) - √(Σ)∂_r∂_y(1/√(Σ)) 
    
    0 =    X[-Σ∂_r^2A + (∂_r A)^2 + 2(A - a^2y^2)] 
        + Σ[2y∂_yX - ∂_r A ·∂_r X] .

While X=Σ is clearly a (trivial) solution of (<ref>), there is no guarantee that it also solves (<ref>). This is of critical importance since for a solution X ≠Σ, the zero-spin limit of the AA metric (<ref>) is, in general, only conformally-related (by a factor of √(g(r))) to the spherically-symmetric seed metric (<ref>) that was used as an input (see Sec. 4 of Ref. <cit.>). 

Remembering that Σ = A+a^2y^2, we recognize that both PDEs involve only the metric functions X and A. Setting X=Σ in eq. <ref> yields an ordinary differential equation for A(r): ∂_r^2 A = 2. Thus, the choice X=Σ yields a legitimate spinning AA counterpart for seed metrics with A(r) = c_2 r^2 + c_1 r + c_0, where c_i are constants (cf. eq. 8 of Ref. <cit.>). Since A=R^2/√(g) for the AA metric (<ref>), the equation above does not imply specific forms for either R or g. However, for the special case when the seed metric is of the form

    R^2(r) = c_2 r^2 + c_1 r + c_0 ;   g(r) = 1 ,

we find that the zero-spin limit of the AA metric identically matches the seed metric. For other choices of R and g (even when X=Σ), the former is, in general, only conformally-related to the latter. The GMGHS metric <cit.> is an example of a seed metric that satisfies eq. <ref> (see Table <ref>). Note that the above contains the important class of degenerate spacetimes (Sec. <ref>), i.e., c_2 = 1, c_1 = c_0 = 0 (cf. Sec. 3 of Ref. <cit.>). 

We have already alluded to the differences between the ansatz metric described here (<ref>) and the AA metric (<ref>), as well as between the methods used to obtain the two metrics.  We now highlight two key points. First, we apply the AA algorithm to a general spherically-symmetric metric in specific coordinates (<ref>). Second, we impose X=Σ, ensuring that we recover the spherically-symmetric seed metric in the zero-spin limit of its stationary generalization (<ref>), and then solve for the matter rest-frame via the procedure described in Sec. <ref> instead. As noted above, the latter reduces to solving a single quadratic equation (<ref>) for Ω, allowing us to bypass the nontrivial PDE system (<ref>, <ref>). In particular, we do not require the matter rest frame to satisfy Ω = Ω_PNC.

In conclusion, we emphasize that both methods can be used to obtain legitimate spinning counterparts when starting from the same seed metric. For spherically-symmetric seed metrics with metric functions (<ref>), both methods output the same axisymmetric metric. This is because (i) when g(r)=1 both auxiliary metric functions, A and B, in both methods (<ref>, <ref>) match identically, and (ii) we find Ω = Ω_PNC (which the AA method imposes on its solution) to be the unique legitimate solution (<ref>) to the quadratic equation for the matter rest-frame angular velocity in our framework, and X=Σ (which we impose right from the start) is a legitimate solution to the PDE system in the AA framework. However, since we do not impose the requirement Ω = Ω_PNC in our framework, it is possible for matter in spacetimes described by (<ref>) to be differentially-rotating on a coordinate 2-sphere (see the Kerr-Frolov spacetime below; Sec. <ref>), which is not permitted in the Azreg-Aïnou framework (those solutions have matter that is always rotating rigidly in spherical shells). Therefore, while our method relies crucially on the AA algorithm, it provides an alternative implementation of, and a distinct new perspective on, this thread of solution-generating techniques.






§ HAWKING MASS



Thus far we have described a method to obtain the angular velocity of the background matter that generates the ansatz spacetime metric (<ref>), as well as its rest-frame properties. In this section, we consider the distribution of mass in the spacetime.

Finding appropriate notions for mass and angular momentum in GR has proven, thus far, to be difficult <cit.>. Ref. <cit.> provides a clear description of the various requirements that a candidate for the mass should meet. Of the several promising and useful suggestions, the quasilocal Hawking mass <cit.> meets all but one of these requirements: It is not necessarily positive-definite nor is it necessarily monotonically decreasing with decreasing distance from the center of space. Nevertheless, since it fundamentally measures the strength of light-focussing in any spacetime, and is ubiquitously used, we elect to use it as a measure of mass distribution here. An analysis of the spatial distribution of angular momentum in spinning spacetimes is beyond the scope of this work.

For convenience, we will obtain below the Hawking mass for spheres of constant coordinate radius r. This measure of mass, which can be defined in arbitrary spacetimes, is constructed from the null expansions, Θ_±, of the outgoing (+) and ingoing (-) principal null congruences as (see also Ref. <cit.>)

    𝓂_Haw = √(𝒜/16π)[1 + 1/2π∫_𝒮Θ_+Θ_-dS] ,

where 𝒜 = ∫_𝒮dS is the area of the sphere 𝒮. The area element, dS, is given in terms of the determinant of the induced metric on the sphere

    σ = ℊ_ϑϑℊ_φφ = Πsin^2ϑ ,

as dS = √(σ)dϑdφ. We then obtain

    𝒜(r) = 
    2π[A+a^2 + Π(π/2)/a√(Δ)arcsinh(a√(Δ)/√(Π(π/2)))] .

where Π(π/2) := Π(r, ϑ = π/2) = (A+a^2)^2 - Δ a^2. Equation <ref> is written above for both the stationary regions of spacetime (Δ > 0, e.g., region I) as well as the nonstationary regions of spacetime (e.g., region II) where Π > 0. In the latter case, r is a time coordinate. It is more intuitive to think of r uniformly as the affine parameter for ingoing or outgoing null geodesics that are members of the principal null congruences (see discussion below eq. <ref>) everywhere in spacetime.

Note also that eq. <ref> reveals the area of the surface Π=0 to be,

    𝒜_CTC = 2π(A(r_CTC) + a^2) .

This corresponds to the area of the coordinate sphere r=r_CTC inside which closed timelike curves are permitted (since Π = 0 ⇒ℊ_φφ = 0). For all of the specific spacetimes considered here (barring the Kerr-Newman BH spacetime), we find A(r_CTC) = 0. Thus, the area of the Π=0 surface is given as 𝒜_CTC = 2π a^2. See also Appendix <ref> for related discussion.

The area of the event horizon, in particular, is given as,

    𝒜_H = 4π(A_H+a^2) ,

where the subscript “H" denotes that the function is evaluated at r=r_H. 

To obtain the null expansions, it is useful to work with an orthonormal complex null tetrad <cit.>. This Newman-Penrose (NP) tetrad, {ℓ_+^†μ, ℓ_-^†μ, ζ^μ, ζ̅^μ}, has two real legs, ℓ_+^†μ and ℓ_-^†μ, and two complex ones, with ζ̅^μ denoting the complex-conjugate of ζ^μ. The tetrad orthonormality relations can be read off from

    ℊ^μν = -2ℓ_+^(†μℓ_-^†ν) + 2ζ^(μζ̅^ν) .


For region I, in BL coordinates, we can write the NP tetrad explicitly as, 

    ℓ_±^†μ =    √(Π/2ΔΣ)[1, ±1/√(Π)Δ/√(g), 0, 2aF/Π] , 
    ζ^μ =    (√(A) - iacosϑ)/√(2Σ)[0, 0, 1/√(Σ), i/√(Π/Σ)sinϑ] .

The vector fields ℓ_±^†μ are indeed the generators of the PNCs and are related to the ones defined earlier in eq. <ref>, ℓ^μ_±, through a series of null tetrad rotations (see Sec. 8 (g) of Ref. <cit.>). 

With the above, the expansions of the outgoing and ingoing PNCs are defined as,
[Denoted in Ref. <cit.> by ρ and μ, up to signs.]

    Θ_+ = (ℓ_+^†)_μ; νζ^μζ̅^ν ;  Θ_- = (ℓ_-^†)_μ; νζ̅^μζ^ν ,

where the semicolons denote covariant derivatives. In the relevant ϑφ-sector, we find (ℓ_+^†)_[μ; ν] = (ℓ_-^†)_[μ; ν] = 0, so that the above wonderfully becomes

    Θ_± = (ℓ_±^†)_(μ; ν)ζ^(μζ̅^ν) ,

from which we can explicitly see that each null expansion is the covariant derivative of the congruence projected onto the 2-sphere (see eq. <ref>; See also, e.g., Sec. 2.4 of Ref. <cit.>). Furthermore, for each component we have |(ℓ_+^†)_μ; ν| = |(ℓ_-^†)_μ; ν|. Therefore, the null expansions are equal in magnitude Θ_+ = -Θ_-, and can be read off from (region I)

    Θ_+Θ_- = -Δ/32Σ g(∂_r Π/Π)^2 =  -ℊ^rr/8(∂_rln[√(σ)])^2 .

For the Schwarzschild metric, we find Θ_± = ±√(r-2M)/√(2r^3). Thus, for r > 2M, we can see that the outgoing radial PNC is expanding and the ingoing radial PNC is converging. We also find that the Hawking mass (<ref>) does not vary with radius for the Schwarzschild metric, i.e., 𝓂_Haw(r) = M. For completeness, we note that, in spherically-symmetric spacetimes, H = √(-8Θ_+Θ_-) is independent of ϑ. It corresponds to the mean curvature, H, of the coordinate sphere.

For region II, using the interior BL coordinates (Sec. <ref>), we can write the NP tetrad explicitly as, 

    ℓ^†μ̆_± =    √(Π/-2ΔΣ)[1/√(Π)Δ/√(g), ± 1, 0, ±2aF/Π] , 
    ζ^μ̆ =    (√(A) - iacosϑ)/√(2Σ)[0, 0, 1/√(Σ), i/√(Π/Σ)sinϑ] ,

and the null expansions are equal, Θ_+ = Θ_-, and negative. This indicates that the outgoing and the ingoing principal null congruences are both converging with time. Remarkably, 

    Θ_+Θ_- = -Δ/32Σ g(∂_r Π/Π)^2 =  -ℊ^rr/8(∂_rln[√(σ)])^2 ,

which is the same as eq. <ref>. For the Schwarzschild metric, we find Θ_± = -√(2M-r)/√(2r^3). 

In this way, we find the Hawking mass to be given everywhere in our axisymmetric spacetimes (<ref>) as,

    𝓂_Haw = √(𝒜/16π)[1 - ∫_0^πℊ^rr/8(∂_rln[√(σ)])^2√(σ)dϑ] .

The nonpositivity and the nonmonotonicity of the Hawking mass with radius are both immediately evident from the above. While the first term (square root of area) is positive-definite everywhere and decreases monotonically with decreasing radius, the sign of the second term is determined by the sign of ℊ^rr, which changes at a horizon. Therefore, the nonmonotonicity of the Hawking mass or its derivative is tied to the change in the sign of the outgoing principal null expansion as well as the magnitudes of the two expansions, a consequence of the interior (collapsing and expanding) cosmology behind the future horizon. Fortunately, it is reasonable to expect, at least, that the Hawking mass decreases monotonically from its asymptotic value as we approach the event horizon. In asymptotically-flat spacetimes, the asymptotic value of the Hawking mass always produces the Arnowitt-Deser-Misner (ADM; <cit.>) mass M of the spacetime <cit.>. 

Clearly, at a horizon, where ℊ^rr = 0, the Hawking mass is given by the area. In particular, for the event horizon, r=r_H,

    𝓂_Haw; H = √(𝒜_H/(16π)) = √(A_H + a^2)/2 ,

where the subscript “H” denotes that the function is evaluated at the event horizon, i.e., 𝓂_Haw; H := 𝓂_Haw(r_H). The event horizon Hawking mass, therefore, corresponds to the irreducible mass of the BH. 

Furthermore, if we introduce the areal-radius of a sphere as

    ℛ_𝒜 = √(𝒜/4π) ,

then the compactness, 𝒞(r) := 𝓂_Haw(r)/ℛ_𝒜(r), of the event horizon is identically 𝒞_H = 1/2, even in stationary and axisymmetric spacetimes. 

We can also write the horizon angular velocity (cf. eq. <ref>) in terms of the horizon areal radius as

    Ω_H = a/A_H+a^2 = a/4𝓂_H^2 = a/ℛ_𝒜; H^2 ,

to set up a loose comparison between the event horizon and a uniform spherical solid that is rigidly-rotating about its axis in Newtonian mechanics. The angular momentum, J, mass, M, angular frequency, Ω, and radius, ℛ, of the latter are related as J/M = (2/5)ℛ^2Ω. For a rigidly-rotating spherical surface instead, we have J/M = (2/3)ℛ^2Ω.

In nonspinning spacetimes (<ref>), the Hawking mass 𝓂_Haw is simply given by the metric function m, i.e., (see also, e.g., Refs. <cit.>)

    𝓂_Haw = m = R/2[1-ℊ̂^αβ∂_α R·∂_β R] .


For spinning spacetimes, equation (<ref>) gives the general formula for computing m(r) vs r. For the class of metrics considered this paper (<ref>), the Hawking mass can be calculated analytically. The somewhat lengthy expression is relegated to Appendix <ref>.

For stationary counterparts of degenerate static BH spacetimes we can expand the Hawking mass, for small spins (a ≪ 1), to leading order in a to find,

    𝓂_Haw(r) =    m - m^2/r^3a^2 - [(r-2m)∂_rm/3r^2]a^2 + 𝒪(a^3) 
    ≈     m - m^2/r^3a^2 - (r-2m)4πϵ/3a^2 .

For the slowly-spinning Kerr metric, in particular, we have

    𝓂_Haw(r) ≈     M - J^2/r^3 .

Since the Kerr metric contains no matter, it is tempting to speculate on the distribution of “spin-energy” stored in the spacetime from the equation above. If appropriately formalized, the latter can be used to understand energy extraction from black hole ergoregions via Penrose processes <cit.> in greater detail. The Blandford-Znajek process <cit.>, which is an electromagnetic Penrose process <cit.>, is particularly interesting due to its role in the production of astrophysical jets.






 §.§ Linking the Hawking Mass to Matter Properties



A fundamentally distinct measure of the mass-energy is provided by the Misner-Sharp-Hernandez (MSH; <cit.>) mass. The MSH mass was originally interpreted as a volume integral of a quantity that is closely linked to the rest-frame energy density (see eq. 6.9 of Ref. <cit.>), in the context of spherical-symmetric gravitational collapse. It can also be seen, from eq. 6.7 there, as a simpler improper volume integral of the rest-frame energy density (cf. eq. <ref> below). The MSH mass can be defined in arbitrary spherically-symmetric spacetimes (including cosmological spacetimes) and has been shown to always coincide precisely with the Hawking mass (cf. Ref. <cit.>; See also Sec. 4.2.1 of Ref. <cit.>). 

In Appendix <ref>, we find the relation between the Hawking/MSH mass and the material properties of the matter, now in spherically-symmetric BH spacetimes (<ref>), to be given by

    m =
    ∫_𝒱^r (+ϵ)[+(1 - 2m/R)]^1/2 dV^r , 
       if ℊ̂_tt < 0 , 
    ∫_𝒱^τ (-p_n)[-(1 - 2m/R)]^1/2 dV^τ ,
       if ℊ̂_tt > 0 .

In the above, 𝒱^r and 𝒱^τ are spacelike and timelike volumes bounded by the surfaces r=r and τ=τ in the static (ℊ̂_tt < 0) and nonstatic (ℊ̂_tt > 0) regions of spacetime (Sec. <ref>) respectively. Their respective proper volume elements are denoted by dV^r and dV^τ. Finally, we remind the reader that m=m(r) and m=m(τ) respectively in such regions.

In the BH exterior (as well as in the static regions of spacetime), we find from the above that the expression for the Hawking/MSH matches with eq. 6.9 of Ref. <cit.>. Therefore, the same physical interpretation of the Hawking/MSH mass holds true: it receives contributions from the kinetic energy as well as the gravitational potential energy. This interpretation is seen to apply equally well in the interior cosmological regions of the spherically-symmetric BH spacetime.

In this way, eq. <ref> reveals the connection between the Hawking/MSH mass and the particular aspect of the matter distribution that they measure, albeit in spherically symmetric spacetimes.

For the special case of degenerate spacetimes (Sec. <ref>), since the spacetime is filled with special matter content which satisfies p_n = -ϵ everywhere, eq. <ref> simplifies to 

    m(r) = ∫_𝒱 (+ϵ)[1 - 2m/r]^1/2 dV ,

In the above, r can be interpreted as the affine parameter along a radial null geodesic everywhere in the spacetime. 






§ PROPERTIES OF MATTER IN NONSPINNING BLACK HOLE SPACETIMES















We are finally in a position to use the concepts and methods described in the previous sections to study the properties of matter in black hole (BH) spacetimes. In this section, we consider spherically-symmetric nonspinning BHs, and in Sec. <ref> we analyze the more general case of spinning BHs.

To guide the discussion, we focus on the six representative spherically-symmetric black hole (BH) spacetimes listed in Table <ref>. This set of BH metrics allows us to cover several different aspects of the problem.

The Schwarzschild and Reissner-Nordström (RN) spacetimes are solutions to the vacuum and electrovacuum equations of GR respectively (see, e.g., Ref. <cit.>). The Hayward metric <cit.> describes a regular BH spacetime, and can be interpreted as containing an anisotropic fluid with no heat fluxes or viscosity. These three models are examples of degenerate spacetimes (Sec. <ref>), and are described by their Hawking mass distribution, m(r). Modified-Hayward metrics, with different mass functions, were studied in Ref. <cit.>. We will refer to the n=4 modified-Hayward metric of Ref. <cit.> as the Zhou-Modesto-4 (ZM4) metric. This metric contains no curvature singularity and remains geodesically-complete even when extended to negative r. Note the minor change in the denominator of the mass function (Table <ref>) from eq. 13 there.

The Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS; <cit.>) solution describes a BH spacetime containing a scalar (dilaton) and an electromagnetic field. These fields arise naturally in the low-energy limit of the heterotic string with nonminimal field-couplings, with the gravitational Lagrangian still given by the Einstein-Hilbert term. The GMGHS metric is not degenerate due to a nontrivial R(r), as shown in Table <ref>. However, as with the previous metrics, it is described in coordinates in which g(r) = (e^ψ ∂_r R)^2 = 1. For further discussion on these spacetimes, see also Refs. <cit.>. 

Finally, yet another nontrivial modification to the Hayward metric, the regular Frolov BH metric <cit.>, modifies the redshift function, ψ(r), as shown in Table <ref>. We note that this metric is expressed in “g(r)≠ 1 coordinates” as (see eqs. 2.47, 2.50 of Ref. <cit.>),

    ds^2 =    -[r^6 + r_-^6/r^6 + r_+^4 r_-^2]^2(1-2Mr^2/r^3 + 2ML^2)dt^2 
        + (1-2Mr^2/r^3 + 2ML^2)^-1dr^2 + r^2dΩ_2^2 ,

where r_±(L) are the locations of the two horizons, i.e., the real positive roots of the cubic r^3 - 2Mr^2 + 2L^2. Putting the above (<ref>) into coordinates in which g(r) = 1 requires numerically solving a differential equation (see discussion below eq. <ref>). This is not convenient, so we leave the metric in the form given in Table <ref>.

The left panel of figure <ref> shows the (Hawking) mass distribution for the six spacetimes as a function of the areal-radius of 2-spheres in the spacetime. We emphasize that the areal-radius ensures meaningful comparisons between different spacetime geometries since it is a gauge-invariant quantity, unlike the 2-sphere coordinate radius r. The horizons in these BH spacetimes can be located by finding the intersection of their metric/Hawking mass function with the (red dot-dashed) line m(r)=R(r)/2. This is because the compactness of event horizons, 𝒞_H = 𝓂_Haw; H/R_H, is always 1/2 (see below eq. <ref>). Thus, it can be seen visually that all BHs barring the Schwarzschild BH and the GMGHS BH have inner horizons. 

For the singular RN BH, the mass decreases rapidly with decreasing r and becomes negative at the center, due to a divergent electromagnetic self-energy. On the other hand, for the three regular BHs (Hayward, ZM4, Frolov), the mass remains positive at all R before vanishing at the center. 

As discussed in Appendix <ref>, in the exterior geometry of a BH spacetime, the Hawking/MSH mass m(R) can be seen as an exact (improper) volume-integral over the matter rest-frame energy density ϵ (<ref>). Thus, the monotonicity (increasing with increasing R) of m(R) implies the positivity of ϵ. 

However, in the nonstatic regions of spacetime (e.g., between the outer and the first inner horizon), the Hawking mass 𝓂_Haw corresponds to the (improper) volume-integral over the rest-frame energy density only in degenerate spacetimes, i.e., only for matter content that satisfies p_n = - ϵ. In the nonstatic regions of nondegenerate spacetimes, the Hawking mass corresponds to the (improper) volume-integral over the rest-frame normal pressure instead (<ref>). 

This explains why the Hawking mass is nonmonotonic for the GMGHS metric, despite, as we will see in Fig. <ref>, its rest-frame energy density remaining positive-definite and increasing monotonically. The nonmonotonicity of the Hawking mass is an unfortunate but well-known feature <cit.>. Note, however, that the Hawking mass still accurately measures the bending of light (and in particular the expansion of the principal null congruences) in spacetime. Finally, we emphasize also that the non-monotonic behavior of the Hawking mass function does not immediately mean that the matter generating the spacetime violates the classical energy conditions. We will see this explicitly for the case of the GMGHS metric when discussing Fig. <ref>.

In the right panel of figure <ref>, we show the Hawking mass of the BH horizon for all the spacetimes (except Schwarzschild, which is trivial) listed in Table <ref>, with varying relevant physical parameter. This plot shows that with increasing physical charge (e.g., Q or L), an increasing amount of energy density is stored in the field outside the event horizon. Thus, the mass of the event horizon (as a fraction of the asymptotic ADM mass) generically decreases. The stars mark the locations of the extremal BHs, which have finite masses. The horizon of the GMGHS BH approaches its curvature singularity in the near-extremal limit, eventually meeting the latter. For this reason, the GMHGS BH has no extremal limit. 

Fig. <ref> shows the energy-density ϵ (left panel), the normal equation of state (EoS; cf. eq. <ref>) ω_n = p_n/ϵ (top-right panel), and the tangential EoS ω_ϑ = p_ϑ/ϵ (which is equal to ω_φ by spherical symmetry) for all the spacetimes considered above, in the appropriate timelike comoving-frame of the matter. The matter content in all cases has positive-definite energy density. Furthermore, for all the degenerate spacetimes (RN, Hayward, ZM4), we see, as predicted (Sec. <ref>), that ω_r = -1 identically. In addition, all the models (except Schwarzschild) contain matter with anisotropic pressure, ω_n ≠ω_ϑ.

Furthermore, for the singular BH models (RN, GMGHS), it can be seen then that their matter satisfies the dominant energy condition (DEC) since |ω_i| ≤ 1 (see Sec. <ref>). This automatically implies the validity of the weak energy condition (WEC) as well as of the null energy condition (NEC). Additionally, since ω_r ≥ -1 and ω_ϑ = ω_φ > 0, the strong energy condition (SEC) is also satisfied (since ω_r + 2ω_ϑ≥ -1). Thus, the matter content in these spacetimes satisfies all classical energy conditions. The variations in the GMGHS EoSs from the RN ones are not clearly visible due to the scale. Note, in particular, that our conclusion regarding the GMGHS spacetime differs from that of Ref. <cit.>, where it was found that the energy conditions are violated. The reason is that our definition of the rest-frame energy density corresponds correctly to the eigenvalue of the Einstein tensor corresponding to its timelike eigenvector (<ref>).

The matter content in the regular Hayward BH spacetime satisfies the WEC, and therefore also the NEC, everywhere but not the DEC (since ω_ϑ > 1). This matter also satisfies the SEC outside the horizon (at the center, see that ω_r + 2ω_ϑ = -3).

On the other hand, the matter content in the regular ZM4 BH spacetime violates all of the energy conditions, albeit only inside its inner horizon (ω_ϑ < -1). Therefore, we see that even when the Hawking mass decreases monotonically with radius, and the rest-frame energy-density is positive-definite, it is still possible for all energy conditions to be violated. Outside the event horizon, it only violates the DEC. 

Finally, the matter content in the regular (non-degenerate) Frolov BH spacetime again violates all of the energy conditions. Here, the energy-condition-violating region is not restricted to the interior of the inner horizon though it is still contained within the event horizon. In the exterior horizon geometry, however, only the DEC is violated.









The recent Event Horizon Telescope images of the supermassive ultracompact objects M87^⋆ <cit.> and Sgr A^⋆ <cit.> provide strong observational evidence for the existence of a photon shell in their spacetimes <cit.>. Black hole spacetimes generically admit photon shells <cit.>, which in spherically-symmetric spacetimes is referred to as the photon sphere. The photon sphere is a 2-sphere in space where photons are permitted to move on circular orbits due to extreme light-bending, i.e., it is a location, r=r_PS, where massless particles have zero radial acceleration. This last condition determines this radius as a solution of (in regions where 𝐊 is timelike)

    m/(4/3)π R^3 + p_n - 1/4π R^2 = 0 .

The equation above relates the areal-radius of the photon sphere to the material properties of the background matter at the photon sphere. It is tempting to think of the first term above as a quasilocal energy density and the last term as a “curvature energy density” or “curvature pressure.”

In Fig. <ref>, we show the compactness of the photon sphere, 𝒞_PS = 𝓂_PS/R(r_PS), for the nonspinning BH spacetimes considered here. The minimum compactness of the photon sphere is seen to be ≈ 1/3, i.e., well below 1/2, which corresponds to the compactness of the event horizon. This suggests that non-BHs of sufficient compactness, 𝒞≳ 1/3, could bend light sufficiently to generically possess photon spheres (see, e.g., Ref. <cit.>).






§ PROPERTIES OF MATTER IN SPINNING BLACK HOLE SPACETIMES



Thus far we have considered in detail the properties of the equilibrium matter that generates various popular nonspinning spacetimes. In this section, we present an analysis of the matter content that generates their spinning counterparts. In all cases here, the spinning spacetimes are described by the metric ansatz (<ref>). For a specific axisymmetric spacetime, one requires as inputs the spherically-symmetric “seed” metric functions given in Table <ref> above.

The spinning spacetimes we will consider here are as follows. The Kerr metric is the stationary generalization of the Schwarzschild metric and is a vacuum solution of the Einstein equations, used to describe the vacuum, spinning BHs of general relativity (GR). The Kerr-Newman (KN) metric <cit.> is the spinning counterpart of the RN metric and is a solution to the Einstein-Maxwell equations, used to describe electromagnetically charged, spinning BHs in GR. The KN metric was originally obtained via the Newman-Janis (NJ) algorithm <cit.>.

The Kerr-Hayward (KH) metric was obtained from the Hayward metric in Ref. <cit.>, via the original NJ algorithm. An analysis of the energy conditions for this spacetime was also presented there, with which our results are consistent. This metric can be used to describe spinning regular BHs containing an anisotropic fluid. The Azreg-Aïnou (AA) algorithm produces the same metric <cit.>. 

We introduce the Kerr-Zhou-Modesto-4 (KZM4) metric here, which can also similarly be obtained via either the NJ or AA algorithm, using the ZM4 metric as the seed metric. We remind the reader that all of the aforementioned spacetimes are special, and belong to the class of degenerate spacetimes discussed in Sec. <ref>.

The Kerr-Sen (KS) spacetime <cit.> describes the stationary generalization of the GMGHS metric, and contains a scalar (dilaton), an electromagnetic, and a pseudoscalar (axion) field. It was shown in Ref. <cit.> that this metric can be obtained as a product of the original NJ algorithm. It can also be obtained via the AA algorithm. It is interesting to note that while the existence of a Carter constant for null geodesics for this metric <cit.> leads to the expectation that it is of Petrov-Pirani type D, it has been shown in Ref. <cit.> that it is actually of type I. 

Finally, we introduce the Kerr-Frolov (KF) spacetime here as being given by the metric (<ref>), with the seed metric functions given in Table <ref>. Note that neither the KS nor the KF spacetimes are degenerate spacetimes. Furthermore, the nonspinning seed metrics for all but the KF spacetime are written in “g=1” coordinates. The Frolov metric does not admit a simple analytic form in coordinates in which g(r)=1. Strikingly, the KF BH metric, obtained in this way, contains a curvature singularity. We remind the reader that the seed metric was perfectly regular.

The simplicity in the construction of the KF metric in particular highlights the potential value of our method, which is inspired by the AA algorithm but is distinct from it. We will first discuss the properties of matter in the set of spacetimes for which the rest-frame angular velocity of the matter is given as Ω =Ω_PNC and return later to the KF metric, for which Ω≠Ω_PNC.

In the discussion below, we will exclusively employ the areal-radii ℛ_𝒜(r) of 2-spheres of coordinate radius r, to enable meaningful comparisons of spatially-varying quantities across different spacetimes. This is because the areal-radius is a gauge-invariant (coordinate-invariant) quantity (<ref>).

Furthermore, the areal-radius is defined only in regions that are devoid of closed timelike curves (CTCs), where Π > 0 (see Footnote <ref>), since ℛ_𝒜 involves an integral over √(Π). We identify the causality-violating region to lie within the largest surface r=r_CTC on which Π(r_CTC, ϑ) = 0 for some ϑ. As a reminder, we note that when the spacetime contains a curvature singularity, it is located in the equatorial plane at r=r_0 such that Σ(r_0, π/2) = 0 or equivalently, R(r_0) = 0.

For the KN spacetime, the equatorial ring singularity is located at r_0=0, whereas its causality-violating region extends to positive-r, i.e., r_CTC > r_0 (see Sec. 4.2 of Ref. <cit.>; See also Ref. <cit.>). Note, however, that these regions are constrained to the interior of the KN inner horizon. For the regular BH spacetimes here (KH, KZM4), we still find causality-violating regions, with r_CTC = 0. For the KS spacetime, we find r_0 = r_CTC = Q^2/M (note that R(Q^2/M)=0; Table <ref>). Finally, for the KF spacetime, one obtains r_0 = r_CTC = 0. Therefore, for all barring the KN metric, we find R(r_CTC) = 0 ⇒ A(r_CTC) = 0, leading to the Π=0 surface having an areal-radius of (see eq. (<ref>))

    ℛ_𝒜(r_CTC) = a/√(2) .

The left panel of Fig. <ref> in Appendix <ref> shows the variation of the areal radius as a function of the coordinate radius in all spinning spacetimes considered here. The right panel there confirms that the regions of spacetime considered below are indeed CTC-free. In short, by using the areal-radius, we automatically discount the causality-violating regions of these spacetimes.






 §.§ Rigidly Rotating Models















Fig. <ref> shows the matter angular velocity for the KN, KH, KZM4, and KS BH spacetimes, all of which satisfy the special requirement given in equation (<ref>), as a function of the areal-radii ℛ_𝒜(r) of 2-spheres. The plot covers both the exterior and the interior regions of these BHs. Since Ω_PNC is a function of r alone, we find that the matter in these spacetimes is rigidly-rotating on any given coordinate 2-sphere. 
The vertical line corresponds to the surface R(r) = 0 for all barring the KN spacetime,
on which (see eq. <ref>),

    Ω = 1/a ,

as verified by the upper horizontal line. For the KS and KF spacetimes, this corresponds to the angular velocity of the equatorial timelike ring singularity. For the KN spacetime since r=r_CTC > 0, R=R(r_CTC) > 0 (Table <ref>).

Since the matter angular velocity defines the matter rest-frame completely (<ref>), we can project the Einstein tensor onto this tetrad to obtain the rest-frame energy density ϵ and principal pressures p_i (<ref>). The top-left panel of Fig. <ref> shows ϵ as a function of the areal-radius in the equatorial plane as well as along the spin-axis in the four BH spacetimes discussed above. It is clear to see that the rest-frame energy density is nonnegative everywhere. It diverges in the equatorial plane for the KN and KS spacetimes, due to the presence of the curvature singularity, and remains regular everywhere for the regular BH spacetimes. For the KZM4 metric in particular, the energy density vanishes at r=0.

The remaining panels of Fig. <ref> show the spatial-variation of the three equations of state (EoSs) ω_i = p_i/ϵ. The top-right panel shows that the normal EoS for all the degenerate spacetimes (KN, KH, KZM4) is identically equal to -1. For the KS metric, which is a nondegenerate spacetime, the normal EoS shows nontrivial variation. Nonetheless, we see that it converges to -1 at both its horizons, as anticipated by eq. <ref>. This is remarkable since, even though the spacetime contains three different matter fields (including a massless scalar), with nonminimal couplings between them (in the Einstein frame), their pressure distributions manage to conspire to produce a total normal EoS of ω_n = -1 at the two horizons.  

Combining all the four panels of Fig. <ref>, we see that the rest-frame energy-momentum-stress (EMS) tensor for the KN metric, which contains only an electromagnetic field, is precisely 𝒯_(μ)(μ) = diag.[ϵ, -ϵ, ϵ, ϵ], as derivable from first-principles (see, e.g., Appendix <ref>).

Furthermore, while the polar and azimuthal EoSs for the two regular BH spacetimes (KH, KZM4) display nontrivial variation with radius and polar angle, the inset makes clear that the two tangential EoSs are identically equal to each other everywhere in these degenerate spacetimes. This is striking due to the absence of spherical-symmetry in such spinning BH spacetimes. 

The KS metric again displays nontrivial variation of the tangential EoS with r and θ, but with the additional feature that the two tangential EoSs, ω_ϑ and ω_φ, differ significantly from each other in the equatorial plane (inset).

With the nonnegativity of the energy density and the spatial variation of the principal EoSs for each of these spacetimes in hand, we can finally comment on the validity of the classical energy conditions in these spacetimes. 

The KN BH spacetime satisfies all energy conditions (Sec. <ref>), as can be seen from its rest-frame EMS tensor above. 

Since the tangential EoSs for the two regular degenerate BH spacetimes enter the red regions in the bottom panels (ω_ϑ = ω_φ < -1, or, equivalently, p_ϑ + ϵ = p_φ + ϵ < 0), it becomes clear that they violate the null energy condition (NEC), albeit deep inside the inner horizon (see also Ref. <cit.>). Therefore, they also violate the weak energy condition (WEC), the dominant energy condition (DEC), as well as the strong energy condition (SEC). It is interesting to note that the KH metric does not violate the NEC or the WEC in the equatorial plane anywhere, underscoring the importance of carefully examining the properties of matter throughout the entire spacetime. In the BH exterior, however, the NEC, WEC, and SEC are satisfied but the DEC remains violated (ω_ϑ = ω_φ > 1).

Finally, the KS spacetime satisfies the NEC, WEC, and DEC everywhere. However, the SEC is violated inside the inner horizon. The latter is seen from the value of the EoSs at equatorial singularity, ω_r < 0, ω_ϑ = 0, and ω_φ = -1, implying ω_r + ω_ϑ + ω_φ < -1. 

Finally, in Fig. <ref>, we show the variation of the Hawking mass (left) and of the expansions of both the principal null congruences (right) for various BH spacetimes. It is clear to see that the Hawking mass always asymptotes to the ADM mass, M. The red dot-dashed line locates the horizons of the BH spacetimes since it traces 2-spheres having compactnesses 𝒞 = 1/2. The outermost (largest areal-radius) intersection of the Hawking mass function with the dot-dashed red line for any given BH spacetime locates its event horizon and measures its irreducible mass. 

Notice that while the Kerr metric is a vacuum solution to the Einstein equations, unlike the Schwarzschild metric, it has a nontrivial Hawking mass profile. It is tempting to link this to the distribution of “spin energy” stored in the spacetime. Furthermore, It is clear to see that the Hawking mass distribution for any of the nonvacuum BH spacetimes is smaller than that of the vacuum Kerr BH for the same spin parameter at any given 2-sphere of the same areal-radius, indicating that a portion of the mass-energy is stored in the background matter outside the sphere. 

The nonmonotonicity of the Hawking mass, or its derivative, in the BH interior, is also evident in Fig. <ref>. To better understand this, we show in the panel on the right the (absolute) future expansions of both the ingoing (-) and outgoing (+) principal null congruences (PNCs) for each BH spacetime (see also the discussion below eq. <ref>). We emphasize that our choice of null rotations ensures that each component of the two PNC generators differs only in sign and not in magnitude both in the stationary (<ref>) as well as the nonstationary (<ref>) regions of spacetime. This enables introducing a gauge-invariant definition for each individual null expansion as |Θ_±| = √(|Θ_+Θ_-|). As described in Sec. <ref>, the ingoing null expansion is always negative whereas the outgoing one is positive in the stationary regions of spacetime but negative in the nonstationary regions of spacetime. Both null expansions vanish at a horizon since both PNCs are given by the horizon generators. The main panel shows the null expansions in the equatorial plane whereas the inset shows the same on the BH spin axis. The mismatch in their magnitudes shows differential rates of expansion over any particular coordinate 2-sphere. 





 §.§ The Kerr-Frolov Spacetime



In the previous section, we considered a variety of BH models whose matter was seen to be rigidly-rotating over each coordinate 2-sphere. In this section, we discuss the properties of the novel Kerr-Frolov (KF) BH spacetime, for which this is not the case. The KF model is different from the other models because we do not have an analytical spherically symmetric seed metric with g(r)=1. We therefore retain the non-trivial g(r), which propagates into the spinning metric (see ℊ_rr) via our ansatz equation <ref>. An immediate consequence is that we no longer have Ω = Ω_ PNC. In addition, as we describe below, Ω at a given r now varies with θ.









In the top row of Fig. <ref>, the solid line shows the angular velocity of the background matter, Ω, in a non-extremal KF BH spacetime (with a=0.5M and L=0.4M), in the equatorial plane (panel [a.]) and on the spin axis ([b.]), as a function of the areal radius ℛ_A(r) of a 2-sphere of coordinate radius r. The solid vertical line in all panels shows the location of the curvature ring singularity. The rest-frame angular velocity is obtained by solving the quadratic equation eq. (<ref>) for a real root that lies between the critical null angular velocities Ω_+ (dashed line) and Ω_- (dotted). As a reminder, the condition, Ω_- < Ω < Ω_+ ensures that the matter 4-velocity is timelike (<ref>). 

Interestingly, we find regions inside the inner horizon (vertical dashed line), where no real solution to eq. <ref> is obtained. We interpret this to mean that the matter flows on achronal orbits in such regions. We also note that ℊ_φφ is positive-definite everywhere outside the ring singularity r > 0 (Appendix <ref>). The structure of these regions is nontrivial, as can be seen by comparing the panels [a.] and [b.]. We see that the KF matter flows on timelike orbits everywhere outside an areal-radius of ℛ_A≈ 1.1M, corresponding, equivalently, to a BL coordinate radius of r ≈ 1M. Inside this radius, the solution is pathological.

Such comparison also reveals that the matter is differentially-rotating, i.e., Ω = Ω(r, ϑ), on each coordinate 2-sphere, except at the event horizon (vertical dotted line) and the inner horizon, where the matter angular velocity matches that of the horizon. We note that, although not visible here, outside the inner-horizon, the angular velocity at the equator differs from that on the spin-axis by less than 0.1%.

In all of the BH models considered previously, we saw that the matter was not only rigidly-rotating on each coordinate 2-sphere, but that it was rotating at a special frequency, namely that of the principal null congruences (PNCs), Ω_PNC(r) = a/(A(r)+a^2). The dot-dashed line in panels [a.] and [b.] show Ω_PNC for the KF metric. The insets in each panel confirm that we have a nonzero (but mostly moderate) fractional deviation of the matter angular velocity Ω from Ω_PNC (except at the horizons). 

Panel [c.] in Fig. <ref> shows the matter rest-frame energy density profile and panel [d.] shows the principal equations of state. In the region ℛ_A≳ 1.1M, we find the energy density to be positive-definite, and all classical energy conditions, except for the dominant energy condition, to be satisfied. This includes the nonstationary region between the two horizons, which is interesting because the matter in the nonspinning Frolov metric contained energy-condition-violating regions between the horizons. 

For completeness, we note that as we move further inwards along the pole, approaching the surface where Ω first ceases to be real-valued, ℛ_A≈ 0.5M, we find the energy density to take negative values (not shown here due to scale). Furthermore, the null energy condition is clearly violated close by since ω_r < -1 (panel [d.]). 

Panel [e.] in Fig. <ref> shows the variation of the Hawking mass throughout the KF BH spacetime. For comparison, we also show the same in a Kerr BH spacetime of the same spin parameter. Due to the additional “charge,” L, it is clear to see that the two horizons of the KF BH are significantly closer to each other than those of the Kerr BH. While the nonmonotonicity of the derivative of the Hawking mass for the Kerr metric is clearly visible, the same for the KF metric is suppressed. 

Panel [f.] shows the absolute values of the expansions of the two PNCs of the Kerr KF metrics throughout the spacetime. As discussed above, it is the nonmonotonicity of the null expansions that leads to the nonmonotonicity in the Hawking mass or of its derivative (see the discussion below eq. <ref>).






§ SUMMARY & CONCLUSIONS



Given a general nonspinning and spherically-symmetric metric of the form

    ds^2 
    = -e^2ψ(1-2m/R)dt^2 + (∂_r R)^2/(1-2m/R) dr^2 + R^2 dΩ_2^2 ,

where the metric functions R(r) and m(r) denote the areal-radius of a 2-sphere of coordinate radius r and its Hawking mass respectively, and ψ(r) is the redshift function that measures aspects of the pressure profile of the background matter, we have proposed in this paper a unique spinning and axisymmetric generalization metric, inspired by the Azreg-Aïnou algorithm <cit.>, whose form is given as

    ds^2
    =    
    -(1-2 F/Σ)dt^2 
    -22 F/Σasin^2ϑ dtdφ
    +Π/Σsin^2ϑ dφ^2 
        
    + Σ/Δg dr^2 + Σ dϑ^2 .

In the above, by uniqueness, we simply mean that the spinning metric functions F, Δ, Σ, and Π are completely fixed by the nonspinning ones:

    3    F(r) = (A - B)/2 ;        Δ(r) = B + a^2 , 
        Σ(r, ϑ) = A + a^2cos^2ϑ ;        Π(r, ϑ) = (A + a^2)^2 - Δ a^2sin^2ϑ ,

with 

    A(r) = R^2 ;  B(r) = e^2ψ(R^2 - 2mR) ; g(r) = (e^ψ·∂_rR)^2 .

The axisymmetric metric (<ref>) is asymptotically-flat, permits a Carter constant for all geodesics, and can be used to describe several popular black hole (BH) spacetimes (Sec. <ref>).

In general, the metric (<ref>) possesses a nontrivial Einstein tensor, indicating that it describes a nonempty spacetime. We describe a simple method to identify the timelike 4-velocity of the axially-spinning background matter, 

    e_(t)^μ∝ [1, 0, 0, Ω] ,

which involves solving a single quadratic equation (<ref>) for a real-valued angular frequency Ω. This allows us to unambiguously introduce the rest-frame or comoving-frame of the matter distribution, both in the BH exterior as well as in its interior (Sec. <ref>). We then obtain the rest-frame energy density ϵ and principal pressures p_i of the matter by projecting the Einstein tensor onto the matter rest-frame (<ref>). 

With the above, it is found that at a BH horizon of the axisymmetric metric (<ref>), the matter rest-frame energy density and normal pressure p_n always satisfy,

    ϵ + p_n = 0 .

We highlight that the equation above yields powerful “no-go” theorems for the type of permissible matter content in stationary BH spacetimes. For example, for a massless-scalar field minimally-coupled to Einstein-Hilbert gravity in a spherically-symmetric configuration, p_n = ϵ in the matter rest-frame (Appendix <ref>). Therefore, such matter by itself cannot form either a static or a stationary BH solution (cf. also Ref. <cit.>). This has interesting consequences for the solution space of general relativity minimally-coupled to matter, as well as for the cosmic censorship hypothesis <cit.>.

Spherically-symmetric spacetimes filled with matter content that satisfies (<ref>) everywhere are special. Such spacetimes are described by a single metric function, i.e., their Hawking mass function m(r) (with ψ = 0 and R = r). Various interesting properties of these “degenerate” spacetimes have been discussed in Ref. <cit.>. In Sec. <ref>, we review how degenerate regular BH spacetimes (no curvature singularity at the center; cf. Ref. <cit.>), typically violate all energy conditions (cf. also Ref. <cit.>), close to the center r=0 (<ref>). 

While there is no generally accepted expression for a measure of mass or angular momentum in GR, the Hawking mass <cit.> is widely used and and is a useful measure of the (quasilocal) mass of a closed surface. The key ingredients needed to define this mass for a coordinate 2-sphere are the area of the sphere and the expansions of the principal null congruences (PNCs) of the spacetime over its surface. In Sec. <ref>, we obtain simple analytic expressions for the area (<ref>) and for the (gauge-invariant) null expansions (<ref>), as well as for the Hawking mass (<ref>), both in the BH exterior as well as in its interior. 

In Sec. <ref>, we apply the general framework we have developed in the previous sections to examine the properties of matter and spacetime in six repesentative nonspinning BH spacetimes (Table <ref>). Of these, four are degenerate BH spacetimes. The first of these is the Schwarzschild BH metric <cit.>, which describes a vacuum spacetime. Naturally, its Hawking mass matches the Arnowitt-Deser-Misner (ADM) mass M everywhere. We review how the Reissner-Nordström (RN) solution (see, e.g., Ref. <cit.>) satisfies all classical energy conditions. This is to be expected since the spacetime contains an electromagnetic field (see Appendix <ref>). Its Hawking mass decreases monotonically with decreasing radius, from its asymptotic value M, before diverging (→ -∞) at r=0 due to the field's self-energy. The Hayward regular BH metric <cit.> is an interesting exception to the aforementioned rule for degenerate regular BH spacetimes. Its matter content only violates the dominant energy condition (DEC) since its rest-frame tangential pressure p_ϑ exceeds its rest-frame energy density in magnitude, i.e., p_ϑ > ϵ. The Zhou-Modesto-4 (ZM4) regular BH metric <cit.> violates all energy conditions but only inside its inner horizon. Outside the inner horizon, it only violates the DEC (p_ϑ > ϵ). The Hawking mass for these two regular spacetimes decreases monotonically from infinity before vanishing at the center.

We also re-examine the matter content in the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) BH metric <cit.>, which is a solution of a particular Einstein-Maxwell-dilaton/scalar theory. Since the GMHGS spacetime does not belong to the class of degenerate spacetimes (Table <ref>), we find a nontrivial normal pressure profile, i.e., p_n ≠ - ϵ everywhere. However, precisely at the horizon, we do find p_n=-ϵ, as required by equation <ref>. As an aside we note that this BH does not possess an inner (Cauchy) horizon, and that its surface gravity as well as its causal structure are identical to that of a Schwarzschild BH. We find that its matter fields combined satisfy all classical energy conditions. Its Hawking mass displays nonmonotonic behavior, which we explore further in Appendix <ref>.

Finally, we consider the Frolov regular BH metric <cit.>, which was proposed as a novel modification of the Hayward metric. While its Hawking mass function is identical to that of the Hayward metric, its redshift function ψ is nontrivial. We find that the matter content violates all energy conditions well inside the event horizon. In the exterior horizon geometry, only the DEC (p_ϑ > ϵ) is violated. In each of these BH spacetimes, the matter content obeys eq. <ref> at every horizon (Fig. <ref>).

In Sec. <ref>, we consider the properties of matter and spacetime in the spinning counterparts of the nonspinning spacetimes discussed in Sec. <ref>, as described by the metric (<ref>). More specifically, the zero-spin (a=0) limits of the six BH metrics, Kerr <cit.>, Kerr-Newman (KN; <cit.>), Kerr-Hayward (KH; <cit.>), Kerr-ZM4 (KZM4), Kerr-Sen (KS; <cit.>), and Kerr-Frolov (KF), correspond to the nonspinning BH metrics described in Sec. <ref> and listed in Table <ref>. Strikingly, we find that while the nonspinning Frolov BH is everywhere regular, the KF BH picks up a curvature singularity in this solution-generating technique.

In all of these nonvacuum spinning spacetimes, the self-gravitating matter flows on circular orbits about the BH spin axis (<ref>) in the stationary regions of spacetime (i.e., outside the event horizon and inside the inner horizon; See Sec. <ref>). In the nonstationary-region of spacetime (i.e., between the two horizons), the matter comoves with the interior cosmology (<ref>), i.e., it exhibits no spatial motion in the interior Boyer-Lindquist coordinates introduced in Sec. <ref>.

Furthermore, in the spinning counterparts of degenerate nonspinning spacetimes specifically, we find the matter angular velocity Ω to be constant over each coordinate 2-sphere in the stationary regions of spacetime. Additionally, it is given by a special value there (see Refs. <cit.>),

    Ω(r) = a/R^2(r) + a^2 = Ω_PNC(r) .

This is precisely the angular velocity of both the ingoing and the outgoing PNCs, Ω_PNC (<ref>). Therefore, of the six spacetimes considered here, the background matter in the stationary regions of the KN, KH, and KZM4 spacetimes is rigidly-rotating on each coordinate 2-sphere. This property is also found to hold for the KS metric (see Sec. <ref>), consistent with the prediction in Sec. 3 of Ref. <cit.>. 









For the KF metric alone we find the matter to be differentially-rotating on each coordinate 2-sphere outside the BH, i.e., Ω = Ω(r, ϑ). At the event horizon itself, the matter rotates rigidly at the horizon angular velocity. This is similarly true for the matter at the inner horizon. We note, however, that the angular velocity in the equatorial plane differs from that on the spin axis by ≲ 0.1% everywhere outside the inner horizon. Interestingly, inside the inner horizon, we find regions where the quadratic equation (<ref>) for Ω returns complex values. We interpret this as implying that, in such regions, the background matter flows on achronal orbits. In this way, the framework developed in this paper introduces a new “sanity check” on the physical reasonability of self-gravitating matter in spinning spacetimes.

We find the rest-frame energy-density of the matter to be positive-definite everywhere in the physically-reasonable regions of each of the spinning BH spacetimes we consider (i.e., when the matter flows on timelike orbits and in regions containing no closed timelike curves). The pure electromagnetic field in the singular RN and KN spacetimes is the only matter that satisfies all of the energy conditions in such regions. The regular BHs (Hayward, KH, ZM4, KZM4) still permit closed timelike curves inside the surface r=0, and all violate the dominant energy condition (DEC). However, they satisfy the null energy condition (NEC), weak energy condition (WEC), and strong energy condition (SEC) in the BH exterior. The KS metric satisfies the NEC, WEC, and DEC outside the ring singularity but the strong energy condition (SEC) is violated inside the inner horizon. Its nonspinning limit, the GMGHS BH spacetime, satisfies all of the energy conditions everywhere. Finally, the matter in the KF BH metric satisfies the NEC, WEC, and SEC everywhere outside the event horizon. A summary of the important similarities and differences between these BH spacetimes and their matter content is presented in Table <ref> for easy access.

Finally, the spatial-profile, i.e., as a function of the areal-radius of coordinate 2-spheres, of the Hawking mass or its derivative generically exhibits nonmonotonic behavior these spacetimes, behind the event horizon. We understand this to be due to the nonmonotonicity of the null expansions.

The analysis of the KZM4 and the KF BH spacetimes here demonstrates how when using the metric (<ref>) as a “solution-generating metric” (i.e., to find a spinning, axisymmetric generalization for a given nonspinning, spherically-symmetric metric), the theoretical framework outlined here can be employed to systematically investigate the material properties of the self-gravitating matter content of the newly-generated spinning spacetime. While we restricted our attention to a specific class of axisymmetric spacetimes (<ref>), we expect that the general principles underlying this framework can be extended to cover more general classes of axisymmetric spacetimes (e.g., Ref. <cit.>). 

As a final remark, we note that for non-fluid matter models in general, additional equations of motion for the matter field must also be solved to obtain a consistent solution to both the gravitational as well as the matter field equations. While we have not attempted this here, we direct the reader to Ref. <cit.> for the status and successes of the original Newman-Janis solution-generating technique.








We thank Rahul Kumar Walia for stimulating discussions. Support comes from grants from the Gordon and Betty Moore Foundation (GBMF-8273) and the John Templeton Foundation (#62286) to the Black Hole Initiative at Harvard University, and from NSF award OISE-1743747. 












§ SPHERICALLY-SYMMETRIC SPACETIMES: THE EINSTEIN TENSOR AND THE BIANCHI IDENTITIES



The Einstein tensor 𝒢̂ corresponding to the static and spherically-symmetric metric (<ref>) is diagonal and its (mixed) components are given as,

    𝒢̂^t_ t =    -2∂_r m/R^2∂_r R , 
    𝒢̂^r_ r-𝒢̂^t_ t =    (1-2m/R)2∂_r ψ/R∂_r R .

The only other nontrivial component of the Einstein tensor 𝒢̂^ϑ_ ϑ = 𝒢̂^φ_ φ can be read off more conveniently from the (r-)Bianchi identity.
Of the Bianchi identities, ∇_μ𝒢̂^μ_ ν = 0, only the ν=r one, 0=∇_μ𝒢̂^μ_ r = ∂_r𝒢̂^r_ r + Γ^μ_μ r(𝒢̂^r_ r - 𝒢̂^μ_ μ), is nontrivial. Here Γ^μ_αβ are the Christoffel symbols and are given as 2Γ^μ_αβ:=ℊ̂^μγ[-∂_γℊ̂_αβ + ∂_αℊ̂_γβ + ∂_βℊ̂_αγ]. The r-Bianchi identity takes the form,

    0 =    ∂_r𝒢̂^r_ r + ∂_r[ln(R^2√(-ℊ̂_tt))](𝒢̂^r_ r-𝒢̂^t_ t) 
        - ∂_r[ln(R^2)](𝒢̂^ϑ_ ϑ - 𝒢̂^t_ t) .







§ COMOVING FRAME STRESS-ENERGY TENSOR FOR SCALAR AND ELECTROMAGNETIC FIELDS



We turn now to the specific form of the matter stress-energy tensor for various types of matter fields, that are minimally-coupled to gravity, in their comoving frame. 





 §.§ Scalar Field



The stress-energy tensor for a scalar field Φ with mass m_Φ that is minimally coupled to gravity is given as (see Ch. 4 of Ref. <cit.>),

    𝒯_μν = ∇_μΦ ∇_νΦ - ℊ_μν/2[∇^μΦ∇_μΦ + m_Φ^2Φ^2] .

In a static and spherically-symmetric spacetime, in particular, it is reasonable to expect that the scalar field is purely a function of r, i.e., Φ = Φ(r). Thus,

    𝒯̂_tt =    -ℊ̂_tt/2[ℊ̂^rr(∂_rΦ)^2 + m_Φ^2Φ^2] ,
    𝒯̂_rr =    (∂_rΦ)^2 - ℊ̂_rr/2[ℊ̂^rr(∂_rΦ)^2 + m_Φ^2Φ^2] ,
    𝒯̂_ϑϑ =    -ℊ̂_ϑϑ/2[ℊ̂^rr(∂_rΦ)^2 + m_Φ^2Φ^2] ,
    𝒯̂_φφ =    -ℊ̂_φφ/2[ℊ̂^rr(∂_rΦ)^2 + m_Φ^2Φ^2] .

In its comoving frame (<ref>), we can write,

    𝒯̂_(t)(t) =    
    -𝒯̂_(ϑ)(ϑ) = 
    -𝒯̂_(φ)(φ) = 
    1/2[ℊ̂^rr(∂_rΦ)^2 + m_Φ^2Φ^2] ,
    
    𝒯̂_(r)(r) =    1/2[ℊ̂^rr(∂_rΦ)^2 - m_Φ^2Φ^2] .

Finally, for a massless scalar m_Φ = 0, we find that the stress-energy tensor in its comoving frame is given as, 𝒯̂_(a)(b) = diag.[ϵ, ϵ, -ϵ, -ϵ]. 

Indeed, it can explicitly be checked that the comoving stress-energy tensor for the Wyman-Janis-Newman-Winicour <cit.> solution, which describes a static and spherically-symmetric configuration of a massless scalar field, is of this form.





 §.§ Electromagnetic Field



The stress-energy tensor for an electromagnetic field with vector potential 𝒜_μ and field strength ℱ_μν = 2∇_[μA_ν] that is minimally coupled to gravity is given as (see Ch. 4 of Ref. <cit.>),

    𝒯_μν = 1/4π[ℊ^σρℱ_μσℱ_νρ - ℊ_μν/4ℱ_σρℱ^σρ] .

For an electric field in a static and spherically-symmetric spacetime, in particular, we can write Â_μ = [Φ(r), 0, 0, 0]. Here Φ(r) is the Coulomb potential for this configuration. Thus, ℱ̂_μν = 2(∂_rΦ) δ_[μ^rδ_ν]^t and ℱ̂_μνℱ̂^μν = 2(∂_rΦ)^2ℊ̂^ttℊ̂^rr. With this, we can write the stress-energy tensor as,

    𝒯̂_μν = (∂_rΦ)^2/4π[ℊ̂^rrδ^t_μδ^t_ν + ℊ̂^ttδ^r_μδ^r_ν - ℊ̂_μν/2ℊ̂^ttℊ̂^rr] .

In its comoving frame (<ref>), we can write,

    𝒯̂_(t)(t) = -𝒯̂_(r)(r) = 𝒯̂_(ϑ)(ϑ) = 𝒯̂_(φ)(φ) = -(∂_rΦ)^2/8πℊ̂^ttℊ̂^rr .

Thus, the comoving frame stress-energy tensor for a static and spherically-symmetric configuration of electric field is of the form 𝒯̂_(a)(b) = diag.[ϵ, -ϵ, ϵ, ϵ]. 

Indeed, the vector potential for the electrically-charged  Reissner-Nordström (RN; <cit.>) solution is given as Â_μ = (-Q/r, 0, 0, 0) so that 𝒯̂_(a)(b) = Q^2/(8π r^4)diag.[1, -1, 1, 1]

For a magnetic field in a static and spherically-symmetric spacetime, in particular, we can write Â_μ = [0, 0, 0, Ψ(ϑ)]. Here Ψ(ϑ) is related to the poloidal magnetic flux function (see, e.g., Ref. <cit.>) for this configuration. Thus, ℱ̂_μν = 2(∂_ϑΨ) δ_[μ^ϑδ_ν]^φ and ℱ̂_μνℱ̂^μν = 2(∂_ϑΨ)^2ℊ̂^ϑϑℊ̂^φφ. With this, we can write the stress-energy tensor as,

    𝒯̂_μν = (∂_ϑΨ)^2/4π[ℊ̂^φφδ^ϑ_μδ^ϑ_ν + ℊ̂^ϑϑδ^φ_μδ^φ_ν - ℊ̂_μν/2ℊ̂^ϑϑℊ̂^φφ] .

In its comoving frame (<ref>), we can write,

    𝒯̂_(t)(t) = -𝒯̂_(r)(r) = 𝒯̂_(ϑ)(ϑ) = 𝒯̂_(φ)(φ) = +(∂_ϑΨ)^2/8πℊ̂^ϑϑℊ̂^φφ .

Thus, the comoving frame stress-energy tensor for a static and spherically-symmetric configuration of a magnetic field is also of the form 𝒯̂_(a)(b) = diag.[ϵ, -ϵ, ϵ, ϵ]. 

Indeed, the vector potential for the magnetically-charged RN solution is given as Â_μ = (0, 0, 0, Qcosϑ). 






§ MASS IN SPHERICALLY-SYMMETRIC SPACETIMES



To examine the link between the Hawking/MSH mass m with material properties in spherically-symmetric BH spacetimes, we begin by considering the mixed tt-component of the Einstein tensor in static regions (ℊ̂_tt < 0) of the spacetime (Appendix <ref>), and its mixed zz-component in nonstatic regions (ℊ̂_tt > 0) of spacetimes, where the metric is given by eq. <ref>. These are

    𝒢̂^t_ t/8π = -1/4π∂_r m/R^2∂_r R , 
       if ℊ̂_tt < 0 , 
    𝒢̂^z_ z/8π = -1/4π∂_τ m/R^2∂_τ R ,
       if ℊ̂_tt > 0 .

The integral version of the above reads

    m =
    ∫_r (+𝒢̂_(t)(t)/8π)4π R^2∂_r R dr ,
       if ℊ̂_tt < 0 , 
    ∫_τ(-𝒢̂_(z)(z)/8π)4π R^2∂_τ R dτ ,
       if ℊ̂_tt > 0 .

In the above, we have used the appropriate rest frames in the respective regions, i.e., ê^μ_(t) = δ̂^μ_t/√(-ℊ̂_tt) and ê^μ̆_(z) = δ̂^μ̆_z/√(+ℊ̂_zz) to relate the rest-frame projected Einstein tensors 𝒢̂_(a)(b) to the mixed coordinate-frame Einstein tensors 𝒢̂^μ_ ν and 𝒢̂^μ̆_ ν̆ respectively.









Eq. <ref> can be restated in terms of volume integrals of the appropriate matter rest-frame energy density ϵ and normal pressure p_n (<ref>), as

    m =
    ∫_𝒱^r (+ϵ) dV^r_imp. , 
       if ℊ̂_tt < 0 , 
    ∫_𝒱^τ (-p_n) dV^τ_imp. ,
       if ℊ̂_tt > 0 .

In the above, note that the top row is an integral over a spacelike-volume 𝒱^r whereas the bottom is an integral over a timelike-volume. We have also introduced dV^r_imp. := R^2sinϑ∂_rR drdϑdφ regions and dV^τ_imp. := R^2sinϑ∂_τ Rdτdϑdφ.

It is clear that neither of these volume elements correspond to the proper volume elements, which are given instead as dV^r = √(ℊ̂_rrℊ̂_ϑϑℊ̂_φφ)drdϑdφ = (1-2m/R)^-1/2dV^r_imp. and dV^τ = √(-ℊ̂_ττℊ̂_ϑϑℊ̂_φφ)dτdϑdφ= (2m/R-1)^-1/2dV^τ_imp..

Thus, from eq. <ref>, we finally arrive at

    m =
    ∫_𝒱^r (+ϵ)[+(1 - 2m/R)]^1/2 dV^r , 
       if ℊ̂_tt < 0 , 
    ∫_𝒱^τ (-p_n)[-(1 - 2m/R)]^1/2 dV^τ ,
       if ℊ̂_tt > 0 .

In the static regions of spacetime (ℊ̂_tt < 0), the above matches with eq. 6.9 of Ref. <cit.>. Therefore, the original physical interpretation of the MSH mass there holds true. That is, the MSH mass receives contributions from the kinetic energy as well as the gravitational potential energy. As stated, this interpretation applies equally well in the interior cosmological regions of the spherically-symmetric BH spacetime.

For the special case of degenerate spacetimes (Sec. <ref>), since the spacetime is filled with special matter content which satisfies p_n = -ϵ everywhere, we can simplify the above to write,

    m = ∫_𝒱 (+ϵ)[1 - 2m/R]^1/2 dV .

In the above, we have suppressed the superscripts for 𝒱 and dV but we emphasize that they have the same physical meaning as in eq. <ref>.






§ HAWKING MASS IN AXISYMMETRIC SPACETIMES



In this section, we report an analytic expression for the Hawking mass (<ref>) for the general class of axisymmetric metrics (<ref>), which applies both outside the black hole event horizon as well as inside it. If we re-express the Hawking mass (<ref>) as

    𝓂_Haw 
    =    √(𝒜/16π)[1 + 1/2π∫_𝒮Θ_+Θ_-dS] , 
    
    =    √(𝒜/16π)[1 - Δ/32gH̃^2] ,

then the nontrivial piece (with y=cosϑ)

    H̃^2 = ∫_-1^+1(∂_r Π)^2/ΣΠ^3/2dy ,

is given as (with Π(π/2) := Π(r, π/2))

    H̃^2 =    -[(A+a^2)∂_rΔ -2Δ∂_rA]^2/F ΔΠ(π/2)
        + 2/a√(A+a^2/A)(2∂_r A - ∂_r B)^2/(2F)^3/2arctan[a√(2F)/√(A(A+a^2))]
        + (∂_r Δ)^2/aΔ^3/2log[A+a^2 + a√(Δ)/A+a^2 - a√(Δ)] .







§ AREAL-RADIUS IN AXISYMMETRIC SPACETIMES AND CLOSED TIMELIKE CURVES



The left panel of Fig. <ref> shows the areal radius of coordinate 2-spheres in all of the spinning black hole (BH) spacetimes considered here. The blue-shaded regions show the overall variation in the sizes of their event horizons and the green-shaded regions show that for their inner horizons. 

The horizontal black line on which these curves terminate bounds the region containing closed timelike curves (CTCs), which we do not consider here. 

These pathological regions of spacetime (see, e.g., <cit.>) occur when φφ-component of the metric tensor vanishes, i.e., when Π = 0. We see from Π = (A+a^2)^2 - Δ a^2sin^2ϑ immediately that in regions where Δ < 0, i.e., in the nonstationary cosmological regions of spacetime, Π > 0. Such regions are generically devoid of CTCs. Furthermore, it is also clear that Π = 0 must first occur in the equatorial plane. 

The right panel of Fig. <ref> shows the variation of this metric component in the equatorial plane and on the pole, for the regions we consider here. It is verified then that no CTCs exist in these regions.














