Universal bounds on the size of a black hole

For static black holes in Einstein gravity, if matter fields satisfy a few general conditions, we conjecture that three characteristic parameters about the spatial size of black holes, namely the outermost photon sphere area $A_{\text{ph,out}}$, the corresponding shadow area $A_{\text{sh,out}}$ and the horizon area $A_{H}$ satisfy a series of universal inequalities $9A_{H}/4\leq A_{\text{ph,out}}\leq A_{\text{sh,out}}/3\leq 36\pi M^2$, where $M$ is the ADM mass. We present a complete proof in the spherically symmetric case and some pieces of evidence to support it in general static cases. We also discuss the properties of the photon spheres in general static spacetimes and show that, similar to horizon, photon spheres are also conformal invariant structures of the spacetimes.


I. INTRODUCTION
Black holes are fundamental objects in Einstein's general relativity. The spatial size of a black hole is usually characterized by its horizon; however, the horizon cannot be directly observed in classical theories either locally or from asymptotic infinity. A few of recent arguments (e.g. see Refs. [1,2]) suggest that quantum effects may render the horizon locally observable, but this topic remains controversial. There is another special surface named "photon sphere" where gravity is also so strong that photons are forced to travel in orbits [3][4][5]. Differing from the horizon, some photons can escape from the photon sphere, making it observable. The photon sphere plays a key role for gravitational lensing [6,7] or ringdown of waves around a black hole [8]. It is also related to the characteristic (quasinormal) resonances of blackhole spacetimes [5,[9][10][11]. For a Schwarzschild black hole of mass M , the radius of photon sphere is 3M . The outmost photon sphere is unstable and can cast a "shadow" for an observer at the asymptotic infinity. Recently, the first picture of a black hole shadow was taken [12], which gave us a direct impression of the appearance of the black hole size and shape.
Owing to the significance in astrophysical observations, it is important to study the photon spheres and their shadows. Although the classical properties of the horizon have been well studied, the photon spheres and shadows are still lack of extensive investigations. In a spherically symmetric black hole of mass M , Hod proved that for Einstein gravity coupled to matter satisfying the weak energy condition and negative trace energy condition, the innermost photon sphere radius r ph,in and total mass M satisfy [13] r ph,in ≤ 3M.
(1) By using the same energy condition, Ref. [14] proved an relationship between innermost photon sphere and its shadow radius: r sh,in ≥ √ 3r ph,in . A lower bound * mrhonglu@gmail.com r ph,in ≥ 2M was conjectured also by Hod [15] but counterexample was found by Ref. [14]. For the observational purpose, it is more relevant to consider the outermost photon sphere. The proof of Hod's does not apply to the outermost one when there are multiple photon spheres, which do exist in black holes satisfying the dominant energy condition [16]. Recently, a series of universal inequalities about outermost photon sphere was proposed [17,18] 3r Here r + is the radius of the horizon. Refs. [17,18] verify it in many different black holes. Its generalization to higher dimensions were discussed in [19].
In this paper, we will first prove the inequalities (2) for spherically symmetric and static black holes in Einstein gravity, for matter fields satisfying a few simple requirements. We then consider more general static configurations and define the corresponding "photon sphere" and "outermost" photon sphere. We conjecture that the area of outermost photon sphere A ph,out , the corresponding shadow area A sh,out and horizon area A H (if exists), also satisfy a series of universal inequalities, sandwiched within the Penrose inequality: Although we do not have the full proof of Eq. (3) yet, we will give some pieces of evidence to support it. We will also show that, similar to the horizon, photon spheres in static spacetimes are also conformal invariant structures.

II. SPHERICALLY SYMMETRIC CASE
We first present the full proof of (2) for spherically symmetric metrics in (3 + 1) dimensions, which read Here f (r + ) = 0, f (r) is positive when r > r + . Einstein's equation reduces to the following three equations, where N := 3f − 1 − 8πr 2 p r , and p r , ρ, T are radial pressure, energy density and the trace of stress tensor respectively. We require r 3 p r (r) → 0 and r 3 ρ(r) → 0 when r → ∞. Photon spheres are determined by U = 0, where U (r) := f (r)e −χ(r) /r 2 , and U = −N e −χ /r 3 . (7) It is clear that U must have an extremum. Multiple and odd numbers of extrema can also arise and the radii of photon spheres satisfy N = 0 [13]. Furthermore, we must have N > 0 and U < 0 if r > r ph,out .

A. Proof of upper bounds
Here we show that if both the weak and strong energy conditions are satisfied for r ≥ r ph,out , we have The radius of shadow r sh,out is related to the outermost photon sphere by r sh,out = 1/ U (r ph,out ) [14]. We introduce an auxiliary function W(r) := e −χ(r) [1 + 8πr 2 p r (r)], which has following relations U (r ph,out ) = W(r ph,out )/(3r 2 ph,out ), (r 3 U ) = W . (9) The key is to show that W(r) ≤ 1 when r ≥ r ph,out . The Einstein's equation and null energy condition imply χ ≥ 0. The weak energy condition tells us ρ ≥ 0. Using Eq. (5), we see max f = 1−8πr 2 ρ| f =0 ≤ 1. We now split the interval [r ph,out , ∞) into two groups: {I + 1 , I + 2 , · · · } where p r ≥ 0 and {I − 1 , I − 2 , · · · } where p r ≤ 0. If r ∈ I − n (here n = 1, 2, · · · ), we see W(r) ≤ 1. If r ∈ I + n , i.e., p r ≥ 0, we find that the derivative of W(r) is Thus the strong energy condition ensures that W(r) is nondecreasing in the interval I + n . The maximal value of W(r) at every interval I + n is therefore at the endpoint, where p r = 0. Thus, we also have W(r) ≤ 1 at the interval I + n . We can now immediately see r sh,out ≥ √ 3r ph,out . To prove the shadow upper bound, we introduce U = (1 − 2M/r)/r 2 for the Schwarzschild black hole of the same mass. It is clear that (r 3 U ) = 1 ≥ W = (r 3 U ) . At r → ∞, we have [r 3 U (r) − r 3 U (r)]| r→∞ = 0. Thus, we have U (r) ≤ U (r) when r ≥ r ph,out . As r ph,out ≤ 3M , in the interval [r ph,out , ∞), we see the upper bound In addition, we have the rigidity: for all spherically symmetric static spacetimes of mass M , r ph,out = 3M or r sh,out = 3 √ 3M arises if and only if the exterior of photon sphere is the Schwarzschild.
If we only focus on the photon sphere, the requirement can be much relaxed and we only need the null energy condition outside the outermost photon sphere. Our main tool is a new mass function Here m(r, ρ) is the Hawking-Geroch mass [20,21], given by Applying Eqs. (5) and (6) we find an identity For r ≥ r ph,out , N ≥ 0, the null energy condition ensures Thus, we obtain r ph,out ≤ 3M . Compared to Hod's proof (1), our condition is much weaker but the conclusion is stronger (as r ph,out ≥ r ph,in ). This result also implies a new positive mass theorem and entropy bound for static spherically symmetric black holes: if the null energy condition outside the black hole is satisfied, the mass M must be positive and the horizon radius must be smaller than 3M (cannot saturate this bound). In fact, the condition can be even weaker: There exists a photon sphere outside which the null energy condition is satisfied. This is remarkable since even in the spherical case, the previous proofs often require the weak energy condition. Stronger inequalities may be obtained for some particular matters. For example, if the matters contain the standard Maxwell field with charge Q and all the other matters satisfy null energy condition, then Eq. (14) implies M ≥ 2Q 2 /(3r 2 ) and so M (r ph,out ) ≤ M − 2Q 2 /(3r ph,out ), which leads to a tighter bound in the charged case r ph,out ≤ 3M 2 (1 + 1 − 8Q 2 /(9M 2 )). Thus, in all spherically symmetric black holes of same mass and charge, Reissner-Norström (RN) black hole has largest photon sphere.

B. Proof of lower bound
It follows from Eq. (13) that m(r, ρ) ≥ r + /2 when ρ ≥ 0, then we see from Eq. (12) that the lower bound 3r + /2 ≤ r ph,out holds if weak energy condition holds and p r ≥ 0 at r = r ph . This is generally satisfied by astronomical black holes. In theory, many important solutions such as the RN black holes have negative p r . In these cases, the weak energy condition alone is not enough to ensure 3r + /2 ≤ r ph,out . As an example, we consider χ = 0 and for which ρ(r) = −p r (r) = ρ0 8πrr+ e −r/(2r+) with ρ 0 > 0, satisfying the weak energy condition. After specifying ρ 0 = 1, we find r ph,out /r + ≈ 1.417 < 3/2.
However, we define Ξ(r) := αp r (r) − (1 − α)ρ(r), and find that the lower bound holds if p r and ρ satisfy an additional condition: there is at least one α ∈ [0, 1] such that, This requirement is weak in the sense that we only need the existence of one such α. The proof is as follows. r 2 Ξ(r) is a non-decreasing function outside the horizon and the null energy condition implies ρ ≥ −Ξ and p ≥ Ξ and so M (r) ≥ m(r, −Ξ) + 4πr 3 3 Ξ. On the other hand, we find which gives us Substituting r = r ph,out and M (r ph,out ) = r ph,out /3 into the above, we obtain As r + is the outermost horizon, we have f (r + ) ≥ 0. Eq. (5) implies 1 − 8πr 2 + ρ(r + ) ≥ 0 and therefore if r > r + . We thus prove the lower bound. This proof also applies to the stronger statement r ph,in ≥ 3r + /2.

A. Generalization of photon spheres
In this part we consider the general static spacetimes, where it is more instructive to study the "photon sphere" in the spacetime rather than only to focus on its spatial projection. In static spacetime, there is a timelike Killing vector (∂/∂t) µ outside the horizon and t is the time coordinate. Motivated by Ref. [22], we call a connected timelike co-dimensional 1 surface Γ = {t} × S to be marginal transversely-trapping surface (MTTS), if S is topological 2-sphere and any null geodesic that starts tangentially on Γ will keep laying on Γ. Let n µ be its outward unit normal covector, T µ be the tangent vector of a null geodesic. We see T µ n µ | Γ = 0 and so T µ ∇ µ (n ν T ν ) = 0. We thus have the condition for an MTTS [23]: Here K µν is extrinsic curvature of the MTTS. Then an equal-t cross-section S is a photon sphere.
We can give a more explicit expression to find a photon sphere in the static spacetime. Assume that R is the scalar curvature of S, Σ t is an equal-t slice and R is its scalar curvature, k µν is the extrinsic curvature of S embedded in Σ t and its trace is k, l µ is the unit normal vector of Σ t and (∂/∂t) µ = φl µ ,r µ is the outward unit normal vector of S embedded in Σ t . (γ µν , D µ ) and (h µν , D µ ) are the induced metrics and covariant derivatives of the photon sphere S and static slice Σ t , respectively. See Fig. 1 for schematic explanations on these notations. In static cases, one can find that n µ | S =r µ | S and so we have the decomposition K µν = −l µ l ν φ −1rτ D τ φ+k µν . Assuming that s µ is an arbitrary unit tangent vector field of S, then T µ = l µ + s µ is a null vector tangent to MTTS. The requirement (21) implies φ −1rµ D µ φ − k µν s µ s ν = 0. As the result we find Using the decomposition of the Einstein's tensor where p r := T µνr µrν is the pressure on S, we can obtain It reduces to N = 0 in the spherical case. Similar to the horizon, the photon sphere is also a conformal invariant structure. This can be understood by the fact that the null geodesics are conformal invariant, or that Eq. (22) is invariant under the conformal transformation {h µν →h µν = Ω 2 h µν , φ →φ = Ωφ}. Particularly, if we choose Ω = φ −1 , thenh µν is just the "optical metric" [14] and the trace of extrinsic curvature isk| S = 0. Thus, photon sphere S is a minimal surface in the "optical metric". It should be pointed out that we generalized the photon sphere concept directly from the well-defined spherical case. However, whether such thin-shell photon sphere exists in general remains to be further investigated. It may be necessary for us to introduce some "weak photon spheres" by relaxing requirement (21) while keeping most of the essential properties. See e.g. Ref. [24] for an example. Nevertheless, we shall proceed with the assumption.

B. Outermost photon sphere and conjectures about its size
For general static spacetimes, the photon spheres may intersect with each others and have many inequivalent homology classes. See the left panel of Fig. (2) for example. The meaning of the "outermost" photon sphere needs to be clarified. We propose a proper definition about the "outermost" should satisfy the following four requirements: (1) it satisfies Eq. (22) piecewise and no tangentially null geodesic can escape outside; (2) it is closed; (3) no any part of photon spheres is outside it; and (4) ∀ topological 2-sphere X outside the "outermost" photon sphere, we have where k, D 2 and R are the trace of extrinsic curvature, Laplace operator and scalar curvature of X respectively, and p r is the pressure normal to X. In the spherical case, Eq. (25) recovers the condition N > 0. Based on these considerations, we define the outermost photon sphere S out as the enveloping surface of outermost segments of all photon spheres, illustrated in the right panel of Fig. 2. The S out may be disconnected and contain many connected branches S We denote the area of S (i) out to be A ph,out,i . The S (i) out will cast a shadow at the observer's sky. In the spherical case, the shadow is a disk, of which the radius is independent of the angle of view. In general cases, the shadow may have complicated shapes and depend on the angle of view. It is more convenient to study the apparent area of photon sphere measured at infinity, which is given by following integration, Here dS is the surface element induced by original metric h µν . We can use A sh,out,i to characterize the size of shadow. In the spherically symmetric case r sh,out = A sh,out,i /(4π). Assume A H,i to be the area of horizon inside S (i) out . The inequalities in (8) have a naturally generalization: We also conjecture a global version which involves the union of all the connected branches: Here A H = i A H,i and the same for others. Although we do not have a complete proof beyond the spherical case, we can already prove some parts now in special situations. For example, 9A H /4 ≤ 36πM 2 is simply the Penrose inequality and has been proven by several different methods [25]. We can prove the lower bounds 9A H /4 ≤ A ph,out,i ≤ A ph,out by using the "inverse mean curvature flow" (IMCF) [21,26,27] if a connected smooth branch S (i) ph,out and horizon H (if exists) can be connected by an IMCF (See appendices A and B) and the energy momentum tensor satisfies the conditions similar to the spherical case. In addition, if the outermost photon sphere S ph,out is connected, has positive mean curvature and satisfies k −1rµ ∂ µ p r ≥ −3(ρ + p r )/2 under the IMCF (In the spherically symmetric case, this can be guaranteed by the null energy condition), then we can prove A ph,out ≤ 36πM 2 . (See appendix C.) This upper bound is also obtained in Ref. [22] by assuming weak energy condition and p r ≤ 0 at S (i) ph,out . It is interesting to study the proofs about Eqs. (28) and (27) in more general cases.

IV. CONCLUSION
In this paper, we conjectured a series of universal inequalities about the size of a static black hole in Einstein gravity. We gave a complete proof in the spherically symmetric case. We studied the properties of the photon spheres in general static spacetimes and proved that photon spheres are conformal invariant structures of the spacetime. Our results strongly suggest that black holes photon spheres may have rich physical contents and mathematical structures.
Our conjecture gives us a simple way to estimate the size of the horizon and black hole mass. For the spherically symmetric case, though we assume the spacetime is static outside the horizon (if exists), we in fact only need it being static outside the photon sphere due to Birkhoff theorem. It needs to emphasize that the upper bound in (2) do not require the existence of a black hole. This has significance in astronomy. Birkhoff theorem implies the interior of photon sphere may not contain a black hole. For example, a neutron star can also form a photon sphere and the corresponding shadow. However, our inequality (2) implies if the radius of photon sphere is larger than 2.25M or the radius of shadow is larger than 3.89M , then the interior of the photon sphere cannot be a neutron star. If we find a larger size photon sphere or shadow, the interior must be a black hole or it is to form a black hole.

ACKNOWLEDGMENTS
We are grateful to Zhong-Ying Fan and Jun-Bao Wu for useful discussions. This work is supported in part by NSFC (National Natural Science Foundation of China) Grant No. 11875200 and No. 11935009. In this appendix, we will give a very brief introduction on inverse mean curvature flow (IMCF). This will be a crucial tool in our analysis. The IMCF is a very powerful tool in proving some geometrical inequalities such as Penrose inequality, positive mass theorem and so on. This method was initiated by Geroch [21], extended by Wald and Jang [26] and completed by Ilmanen and Huisken [27].
We consider a family of 2-dimensional closed level sets S y which are smoothly immersed in a 3-dimensional spacelike hypersurface Σ. The metrics of S y and Σ are γ µν and h µν , respectively. The extrinsic curvature of S y is k µν and its trace is k. The scalar curvatures of S y and Σ are R and R, respectively.r µ is the unit outward normal vector of S y embedded in Σ. For a given initial level set S 0 , the other level sets are generated by a flow, of which the direction is along the outward normal vector and the speed is the inverse of mean curvature, i.e., the flow vector is Here y is the parameter of integral curve of v µ . See Fig. 4 for an schematic explanation. Assume the area of S y to be A(S y ). Under the IMCF, the area of S y has following growth rate The Hawking-Geroch mass of surface S y is defined as [20,21] m(S y ) := A(y) Geroch, Wald and Jang found that the growth rate of the Hawking-Geroch mass satisfies (A4) Herek µν is the traceless part of k µν and D µ is the covariant derivative operator of 2-surface S y . We see that the Hawking-Geroch mass is non-decreasing under the IMCF if R is nonnegative.
If Σ is a maximal slice of asymptotic spacetime, the Hawking-Geroch mass will approach the ADM mass, i.e. m(S ∞ ) = M . If we choose a minimal surface as the initial surface, for example, the horizon H, then we have We thus have the Penrose inequality: M ≥ A(H)/π/4. IMCF may meet singularities before it reaches the infinity and so has to stop. G. Huisken and T. Ilmanen developed a "weak inverse mean curvature flow" in geometric analysis and geometric measure theory [27], which treats (or at least improves) such sensitive part. Roughly speaking, it admit that the flow "jumps" over the singular point but keep all the integrating results invariant, see Ref. [27] for more details.

Appendix B: A proof on the lower bound
To simplify the discussion, we assume that there is only one connected horizon H with the topology S 2 . We choose that horizon H as the initial surface of IMCF and denote the other surfaces under the IMCF to be H y , see Fig. 5. We can prove our lower bound in a class of special cases. We will show that: if (a) weak energy condition is satisfied, (b) ∃y 0 > 0, such that the one connected branch S (i) ph,out coincides with H y0 except for some zero-measured sets, and (c) the normal pressure p r and energy density ρ satisfy This provides one part of Eqs. (27) and (28). The condition (b) is automatically satisfied for the spherical case, but it is an additional assumption in general. The requirements (B1) is the generalizations of p r ≥ 0 for the non-spherical case. We first summarize the basic idea of the proof. We introduce M (H y ) and show (1) M (H y0 ) = A ph,out,i /π/6, We define the mass of H y to be Here It reduces to Eq. (12) in the spherical case. As H y0 coincides with S This is true if condition (II) is satisfied. This is the conclusion reported in Ref. [22]. However, the requirement p r | S0 ≤ 0 is not satisfied usually by astronomical black holes.
Let us turn to the condition (I), which admits positive pressure in general. We define an auxiliary function J(y) = We can thus reach a conclusion: if condition (I) is satisfied, we also have A ph,out,i ≤ 36πM 2 .

(C6)
Thus, we obtain the previous conclusion in the spherically symmetric case: A ph,out,i ≤ 36πM 2 if the null energy condition is satisfied outside the photon sphere.
In non-spherical cases, ther µ ∂ µ p r can be rewritten in terms of geometrical quantities and Einstein's equation by Eq. (23). It remains to be investigated whether this will lead to the same conclusion as that in the spherically symmetric case.
To use the smooth IMCF, we have to assume also that the outermost photon sphere is smooth. However, this assumption is excessive due to (1) the outermost photon sphere must be piecewise smooth by our definition and (2) we can obtain same results for piecewise smooth surface by using the weak IMCF developed by G. Huisken and T. Ilmanen [27].