Universal bounds on the size of a black hole

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$A_{\mathrm {ph,out}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">                   <mml:msub>                     <mml:mi>A</mml:mi>                     <mml:mrow>                       <mml:mi>ph</mml:mi>                       <mml:mo>,</mml:mo>                       <mml:mi>out</mml:mi>                     </mml:mrow>                   </mml:msub>                 </mml:math></jats:alternatives></jats:inline-formula>, the corresponding shadow area <jats:inline-formula><jats:alternatives><jats:tex-math>$$A_{\mathrm {sh,out}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">                   <mml:msub>                     <mml:mi>A</mml:mi>                     <mml:mrow>                       <mml:mi>sh</mml:mi>                       <mml:mo>,</mml:mo>                       <mml:mi>out</mml:mi>                     </mml:mrow>                   </mml:msub>                 </mml:math></jats:alternatives></jats:inline-formula> and the horizon area <jats:inline-formula><jats:alternatives><jats:tex-math>$$A_{{\mathcal {H}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">                   <mml:msub>                     <mml:mi>A</mml:mi>                     <mml:mi>H</mml:mi>                   </mml:msub>                 </mml:math></jats:alternatives></jats:inline-formula> satisfy a series of universal inequalities <jats:inline-formula><jats:alternatives><jats:tex-math>$$9A_{{\mathcal {H}}}/4\le A_{\mathrm {ph,out}}\le A_{\mathrm {sh,out}}/3\le 36\pi M^2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">                   <mml:mrow>                     <mml:mn>9</mml:mn>                     <mml:msub>                       <mml:mi>A</mml:mi>                       <mml:mi>H</mml:mi>                     </mml:msub>                     <mml:mo>/</mml:mo>                     <mml:mn>4</mml:mn>                     <mml:mo>≤</mml:mo>                     <mml:msub>                       <mml:mi>A</mml:mi>                       <mml:mrow>                         <mml:mi>ph</mml:mi>                         <mml:mo>,</mml:mo>                         <mml:mi>out</mml:mi>                       </mml:mrow>                     </mml:msub>                     <mml:mo>≤</mml:mo>                     <mml:msub>                       <mml:mi>A</mml:mi>                       <mml:mrow>                         <mml:mi>sh</mml:mi>                         <mml:mo>,</mml:mo>                         <mml:mi>out</mml:mi>                       </mml:mrow>                     </mml:msub>                     <mml:mo>/</mml:mo>                     <mml:mn>3</mml:mn>                     <mml:mo>≤</mml:mo>                     <mml:mn>36</mml:mn>                     <mml:mi>π</mml:mi>                     <mml:msup>                       <mml:mi>M</mml:mi>                       <mml:mn>2</mml:mn>                     </mml:msup>                   </mml:mrow>                 </mml:math></jats:alternatives></jats:inline-formula>, where <jats:italic>M</jats:italic> 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.</jats:p>

section_content: 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 black-hole 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. 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 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 + /2 ≤ r ph,out ≤ r sh,out / √ 3 ≤ 3M. 

(1. 2)

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 (1. 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: 9A H /4 ≤ A ph,out ≤ A sh,out /3 ≤ 36π M 2. 

(1. 3)

Although we do not have the full proof of Eq. (1. 

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.
section_name: Introduction

section_content: We first present the full proof of (1. 2) for spherically symmetric metrics in (3 + 1) dimensions, which read

(2. 1)

Here f (r + ) = 0, f (r ) is positive when r > r +. Einstein's equation reduces to the following three equations,

)

where N := 3 f -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. (2. 4)

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.
section_name: Spherically symmetric case

section_content: 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

The key is to show that W(r ) ≤ 1 when r ≥ r ph,out. 

Here p T := T θ θ = T φ φ is the transverse pressure. As p r ≥ 0 and f ≤ 1, we see that

Thus the strong energy condition ensures that W(r ) is non-decreasing in the interval I + n. The maximal value of W(r ) at every interval I + n is theqrefore 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 → ∞, the asymptotical flatness requires matter fields to decay at least in following ways

Then we find

(2. 9)

Then we have [r 3 U (r )r 3 U (r )]| r →∞ = 0. Thus, we have

(2. 10)

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. (2. 2) and (2. 3) we find an identity

For r ≥ r ph,out, N ≥ 0, the null energy condition ensures M ≥ 0. This shows M(r ph,out ) ≤ M(∞) = M. At the photon sphere N = 0, we have M(r ph,out ) = r ph,out /3. Thus, we obtain r ph,out ≤ 3M. Compared to Hod's proof (1. 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. (2. 13) 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.
section_name: Proof of upper bounds

section_content: It follows from Eq. (2. 12) that m(r, ρ) ≥ r + /2 when ρ ≥ 0, then we see from Eq. (2. 11) 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 ) = ρ 0 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 find that the lower bound holds if p r and ρ satisfy an additional condition: there is a function (r ) such that, ∀r > r +, [r 2 (r )] ≥ 0 and -ρ ≤ (r ) ≤ p r (r ). (2. 15) This requirement is weak in the sense that we only need the existence of such function (r ). This condition admits the negative pressure. For example, in Einstein-Maxwell theory, ρ(r ) = -p(r ) ∝ Q 2 /r 4 and one we can take (r ) = p(r ). The proof is as follows. 

r 2 (r ) is a non-decreasing function outside the horizon and the null energy condition implies ρ ≥and p r ≥ and so M(r ) ≥ m(r, -) + 4πr 3 3. On the other hand, we find if r > r +. We thus prove the lower bound. This proof also applies to the stronger statement r ph,in ≥ 3r + /2. 

3 General static cases
section_name: Proof of lower bound

section_content: 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 ξ μ = (∂/∂t) μ outside the horizon and t is the time coordinate. The metric of spacetime has following 3+1 decomposition,

Here h ab is the metric of equal-t slice t and φ 2 = ξ μ ξ μ. Motivated by Ref. [22], we call a connected timelike co-dimensional 1 surface = {t} × S to be marginal transversely-trapping surface (MTTS), if S is topological 2sphere and any null geodesic that starts tangentially on will keep laying on. 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 | = 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, t is an equal-t slice and R is its scalar curvature, k μν is the extrinsic curvature of S embedded in t and its trace is k, l μ is the unit normal vector of t and (∂/∂t) μ = φl μ, r μ is the outward unit normal vector of S embedded in 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 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

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 (3. 2) implies φ -1 r μ D μ φk μν s μ s ν = 0. As the result we find Fig. 1 Notations in the MTTS and photon sphere. For convenience, we immerse all the vectors/tensors into the 3+1 dimensional spacetime and so their indexes become spacetime indexes. For example, h μν = e a μ e b ν h ab, where e a μ is the pull-black map from spacetime to the equal-t surface

Using the decomposition of the Einstein's tensor

where p r := T μν r μ r ν 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. (3. 3) is invariant under the conformal transformation {h μν → hμν = 2 h μν, φ → φ = φ}. Particularly, if we choose = φ -1, then hμν is just the "optical metric" [14] and the trace of extrinsic curvature is k| S = 0. Thus, photon sphere S is a minimal surface in the "optical metric". For a surface S, assume that dS is the surface element induced by original metric h μν. Then the surface element under the optical metric reads φ -2 dS. Thus, photon sphere is a critical surface which locally minimizes following functional

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 (3. 2) while keeping most of the essential properties. See e. g. Ref. [24] for an example. Nevertheless, we shall proceed with the assumption. 

3. 2 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. (3. 3) 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. (3. 7) 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 middle panel of Fig. 2. 

The S out may be disconnected and contain many connected branches S (i) out, i. e., S out = i S (i) out. See the right panel of Fig. 2 for example. 

We denote the area of

out to be A ph,out,i, i. e. 

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, A sh,out,i =

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

out. The inequalities in (2. 5) 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, 9 A H /4 ≤ 36π M 2 is simply the Penrose inequality and has been proven by several different methods [25]. In following we will offer the proofs on 9A H /4 ≤ A sh,out /3 and A ph,out ≤ A sh,out /3 ≤ 36π M 2 in the special case without the spherical symmetry.
section_name: Generalization of photon spheres

section_content: As the first step to consider the general case, we assume that the outmost horizon is connected, and always assume the weak and strong energy conditions. We also assume that outmost photon sphere is smooth at beginning. Finally, we will give argument to show that the smoothness of outmost photon sphere can be relaxed into piecewise smoothness. We will convert the inequalities into the problems of finding maximum/minimum, which can be solved by variational method. In general, to check if the variational problem will give us minimum or maximum, we may need to compute the second order variation. This in general will be complicated. However, if the on-shell value of variational problem has explicit expression, we have simpler way. 

Assume a functional is F : X → R, where X is the space spanned by all arguments of F. Mathematically, if F satisfies a few of very general restrictions 1 and bounded from above/below, then F must has maximum/minimum [32] and the maximum/minimum of F must be given by the maximal/minimal on-shell value of corresponding variational problem. Thus, to find the maximum/minium of F, we only need to find all on-shell values and pick out the maximal/minimal one. In the case that the on-shell value of variational problem has explicit expression, this is easy to perform.
section_name: Proofs without spherically symmetry

section_content: We will use Bondi-Scahs formalism [26, 27] in this section, which offers us simple way to study behaviors of metric in non-spherical case. Bondi-Scahs formalism foliates the spacetime by a series of null surfaces which are labeled by u =constant and the general metric has following form As the spacetime is asymptotically flat, we then fix the boundary conditions as follows

where d 2 is the metric of unit sphere. The photon spheres are geometrical structures of a static black hole, so they are independent of the choices of coordinates. To give readers clear visible pictures, we used static 3+1 decomposition. 

To study the inequalities proposed in this paper, the Bondi-Scahs coordinates gauge is more convenient. 

In Bondi-Scahs gauge, we have three gauge freedoms. Firstly, we have freedom to choose where the null hypersurface u = 0 locates. One convenient choice is that we use outgoing light rays of a photon sphere S to define the null hypersurface u = 0. See Fig. 3. 

The photon sphere S then locates at equal-t surface as well as locates at u = 0 null sheet. Thus, S is also defined by u = 0 and r = r S (x A ). After we obtain the first null sheet u = 0, all the other null sheets can be obtained by using the translation generated by static Killing vector ξ μ. Every null sheet is labeled by a constant u. In fact, besides photon sphere, for any closed surface S in the equal-t submanifold and outside horizon, we can always the immerse one closed surface S into its outgoing light rays. Using this initial null sheet and the translation generated by ξ μ, we can always obtain a local null foliation and S lays on one null sheet. 

Secondly, we have freedom to choose u-coordinates. The tangent vector of u-coordinate is (∂/∂u) μ. In Bondi-Scahs gauge, the normal covector of null sheets ∝ (du) μ but normal vector is not proportional to (∂/∂u) μ. In fact, the normal vector of equal-u surface satisfies

which lays on the equal-u null sheets. This defines the coordinate r and we find

which matches with the metric (4. 1). The u-coordinate is arbitrary timelike curve and passes through those null sheets, e. g. see Fig. 3. We can choose (∂/∂u) μ = ξ μ, i. e. the static Killing vector field. Then we have

Here φ is same as (3. 1). 

The third gauge freedom comes from the fact that the relationship (4. 3) does not fix the choice of coordinate r uniquely. We can choose r to replace r, where r satisfies (∂/∂r ) μ = ψ(∂/∂r ) μ with arbitrary nonzero scalar field ψ. To eliminate this freedom, we can impose gauge condition ∂ r h = 0. By this gauge choice, we see

and d is a surface element of unite sphere. The differential gauge ∂ r h = 0 leaves a freedom in choosing the initial condition of r. To fix this freedom, we can set the horizon radius r h to be constant, which leads to a simple formula for the area of horizon

Alternatively, we can choose a photon sphere S to be constant r = r ph, then the area of S ph has simple expression,

In general, we cannot require that both horizon and photon sphere have constant r. The coordinates {x A } lay on the null sheets. We consider the static case, so we can choose the transverse direction x A to be orthogonal to ξ μ, which means U A = 0. Then metric becomes
section_name: Bondi-Scahs formalism

section_content: In Appendix A we offer more detailed argument on Eq. (4. 8). Strictly speaking, in some spacetimes such coordinates may Fig. 3 Left: the outgoing null rays of photon sphere form the null surface u = 0. A photon sphere S locates at an equal-t spacelike surface as well as locates at a null sheet. On the null sheet, the photon sphere then is defined by r = r S (x A ). Right: the null foliation of spacetime. 

Other null sheets are obtained by using translation generated by static Killing vector ξ μ. The u-coordinate is free and we can choose arbitrary timelike curve which passes through these null sheets only cover a finite neighborhood. Here we assume that it can cover the whole region outside horizon H. 

In above coordinates gauge, the Einstein's equation shows [27]

and

Here (2) R and D A are the scalar curvature and covariant derivative operator of h AB. T rr := T μν (∂/∂r ) μ (∂/∂r ) ν is the null-null component of energy momentum tensor T μν, ρ := T μν ξ μ ξ ν /(ξ μ ξ μ ). The weak energy condition then insures

Then Eq. (4. 9) implies

In following we will define some target functionals on some surfaces S, which will depend on the spacetime geometry and choices of S. We always assume the Einstein's equation, so the spacetime geometry is determined by distribution of matters. As the result, the functionals will depend on distribution of matters and choices of S. It is necessary to know how many degrees of freedoms the target functionals have. The energy momentum tensor has 10 components, 4 of which are constrained by Bianch identity (or the conserved law ∇ μ T μν = 0). The requirement U A = 0 also supplies 2 constraints according to Einstein's equation. Thus, we In this subsection, we will use variational method to prove: in all black holes of which outmost horizon areas are A H, if photon sphere exists, then the minimum value of A sh,out is 27A H /4. To do that, we introduce an auxiliary functional

Here S is arbitrary surface which encloses outmost horizon H and lays on the equal-t surface, A S is the area of S. The Cauchy-Schwartz inequality shows that

Then for outmost photon sphere S sh,out, we have

In Appendix B we show that, if we fix the horizon area and strong energy condition is satisfied, then B is bounded from above. Thus, functional B has maximum. To find the this maximum, we need to find all extreme values and pick up the maximal one. For every surface S lay on the equal-t submanifold, we can immerse it into the null sheet u = 0 and parameterize it by {u = 0, r = r S (x A )} in the Bondi-Scahs coordinates gauge. In the Appendix C, we have shown that: if a surface S makes B extreme among all surfaces which lay on equal-t surface, it will also makes B extreme among all surfaces which lay on u = 0 null sheet. Then we have following relationship

≤ (maximal extreme value of B on null sheet u = 0). (4. 16)

Assume that B m is the the maximal critical value of B on the null sheet u = 0, 2 then we see that max B in Eq. (4. 15) satisfies max B ≤ B m. In the null sheet u = 0, if we can show B m = 4/(27A H ), then A sh,out ≥ 27A H /4 is a corollary. This is what we will do in this subsection. 

In the null sheet = 0, the value of B depends on the function r S (x A ) as well as the metric and matter distributions. In following we will use variational method to find the maximum of B on the null sheet u = 0. As we have analyzed at the end of Sec. 4. 1, we can choose at most 4 bulk variables and at most 8 surface variables as the independent variables when we use variational method. We will first do not specify what are independent variables but only use Einstein's equation to rewrite B into a different form (see Eq. (4. 26)) and then specify what are independent variables, by which we can easy to perform variation and find the on-shell values. 

In Appendix D we use a concrete function as an example to explain the main idea of following steps. Though what we will do in following is more complicated than the example of Appendix D, the mathematical essence has no difference. Referring to the metric (4. 8), our target functional reads

We take the gauge that the horizon has constant radius r = r h. Then we find

(4. 18)

2 It does not mean that B m is the maximal value of B on the null sheet u = 0. Though B on the equal-t hypersurface is shown to be bounded from above, it may have no upper bound on the null sheet u = 0. 

In general V e 2β will depend on r 0 and x A both. The induced metric on S reads ds 2 S = r 2 S h AB dx A dx B, so we find dS = r 2 S d. Eqs. (4. 18) and Eq. (4. 17) then show

and area of S is

We define,

and

which take the constraints β ≤ 0, ρ ≥ 0 and V ≥ 0 into account. Then we find

It needs to note that r S (x A ) in general may not be constant. 

To relax the dependence of upper limit r S in the second integration of Eq. (4. 23), we can introduce step-function (x) so that
section_name: .8)

section_content: As h AB is metric of 2-dimensional space, we can always find a coordinate transformation x A → y A such that

where γ AB is the standard metric of unit sphere. By using this conformal metric, we find that B becomes

Here DA is the covariant derivative operator corresponding to γ AB. From Eqs. (4. 13) to (4. 26), we just made identical deformations according to Einstein's equation. In Eq. (4. 26), we find that {β, r S, S, N 1, N 2, } has 4 independent bulk variables and only 2 independent surface variables, so we can treat all them as independent variables. These are independent arguments of functional B. 

We solve the variational problem by following steps. The variation with respective to is easy computed from Eq. (4. 23) and we obtain and so N 2 = N 1 = 0 when r h < r < r S. Then by using Eq. (4. 26), we find that variation with respective to gives us

This equation shows that (r Sr )r S must be independent of y A. This is because

Thus Eq. (4. 30) shows r S to be constant. Then the variation with respective to β shows

which shows that β = β(r ). Thus, the variation with respective to r S shows If this is true, we then have

(4. 37)

To prove Eq. (4. 36), our tool is variational method combining with Lagrangian multipliers. The target functional now is

According to the conclusion in Appendix C, this can be converted into the task of finding critical surface among the surfaces laying on null sheet u = 0. Similar to (4. 16), we have following relationship

≤ (maximal extreme value of A S on null sheet u = 0). 

As we want to find the upper bound of A S when S is a photon sphere, there are a few of restrictions on surface S. Firstly, in the equal-t surface, we have know that a photon sphere is a critical surface which makes the integration φ -2 dS to be extremal. In the null sheet u = 0, Appendix C shows that a photon sphere should also be a critical surface which makes the integration φ -2 dS to be extremal on the null sheet u = 0. On the null sheet u = 0, we have φ -2 dS = r 3 S e -4β /(V e -2β )d. The extreme condition means that S on the null sheet u = 0 should satisfy following equation

As we now fix the value of functional B = B 0, we have following constraint

Thus, we construct following target functional on the null sheet u = 0 with two Lagrangian multipliers

Here {λ 1 (x A ) λ 2 } are two Lagrangian multipliers and λ 2 is constant. The critical values of A ph,out in equal-t surface are given by the critical values of F in the null sheet u = 0. 

To solve the variational problem, we separate functional F into two parts

where

and

The ∂ r (V e -2β ) and ∂ r β in F bd are given by Eq. (4. 9). The V e -2β in F nd is given by Eq. (4. 18). Using Eqs. (4. 9), (4. 10) and (4. 18) we find

and

(4. 45)

Here β S = β| S, N 2 1S and N 2 2S are defined by Eq. ( 4. 22) but restricted on surface S. We use the lower index "S" to explicitly show that they are surface quantities. Then using the conformal transformation (4. 25), we have

and Eq. (4. 18) shows

Functional F bd only involves quantities at the surface S, but functional F bulk involves quantities at surface S as well as in bulk region between S and H. The functional F now depends on 6 surface variables {N 1S, N 2S, β S, S, W S, r S } and 4 bulk variables {N 1, N 2, β, }. We can treat all these variables as independent variables. The S, W S, N 1S and N 2S appear only in boundary part F bd. Using Eq. (4. 46), we find that the variations with respective to W S, N 1S and N 2S show

This shows that λ 1 (y A ) is a constant. The variation with respective to β S involves both F bulk and F bd. The result reads,

As λ 1 is constant, we integrate Eq. (4. 49) on S and find

As V > 0, we find λ 2 = 0, which implies D2 β S = 0 3 and so β S is constant. The variables {N 1, N 2 } appear only in bulk part F bulk. The variation with respective to N 1 and N 2 shows N 1 = N 2 = 0. Now we have

Then we variation with respective to shows D2 [ (r Sr )r -1 S ] = 0 and so r S is constant. The variation of β shows D2 β = 0 and so β = β(r ) when r < r h < r S. Thus, we have following results at the surface

Note that S = 0 ⇒ (2) R| S = 2. Take these into Eqs. (4. 38) and (4. 40) we have

Thus, we find that F| on-shell = e 4β S /(3B 0 ). The on-shell values of target functional are not unique. We then have max

This proves Eq. (4. 36) and so we obtain A ph,out ≤ A sh,out /3. It needs to note that above proof is still true for every connected photon spheres, i. e. A ph,out,i ≤ A sh,out,i /3. Thus, we prove one part of our stronger conjecture (3. 11).
section_name: .24)

section_content: The solution of Eq. (4. 10) can be written as

This shows 3 The other solution is λ 1 = 0, which should be abandoned as it leads to the variation with respective to r S has no solution. 

In the gauge that horizon has constant r = r h, we find

(4. 57)

This shows the Penrose inequality 9A H /4 ≤ 36π M 2. 

To show A sh,out /3 ≤ 36π M 2, we immerse the photon sphere into the null sheet u = 0 and take the gauge that the radius of outmost photon sphere r S is constant. The size of shadow reads

(4. 58)

In general, the horizon radius r h is no longer constant. Eq. (4. 56) becomes

In following we will use variational method to show that, in all black holes of which the size of shadow is A sh,out = 12πa 2 0, the minimal value of mass is a 0 /3. The main idea is as follow. 

We treat M as a functional, the positive mass theorem shows M ≥ 0. Thus, the functional M is bounded from below. Then the minimum value can be found according to variational problem. As S out is a photon sphere, so constraint (4. 38) should also satisfied. We construct the target functional

Here λ 1 (x A ) and λ 2 are two Lagrangian multipliers and λ 2 is constant. The λ 2 insures that the size of shadow is fixed to be 12πa 2 0 and the λ 1 insures that condition (4. 38) is satisfied. 

We use the variable transformations (4. 22) and find

The functional M contains the boundary part and bulk part. 

In principle, we can follow the similar step of Sect. 4. 3 to solve the variational problem. However, in this case, we have simpler method. We first choose β, N 1, N 2 and K := √ V e -2β as four independent bulk variables. The bulk variations with respective to β, N 1 and N 2 only involves the third line of Eq. (4. 61). The result shows

Thus we see h AB = h AB | r =∞. This shows (2) R = (2) R| r →∞ = 2. Then we find V e -2β = r -2M. 

Thus, the on-shell geometry outside photon sphere S out becomes spherically symmetric. We then find that min M = min M| on-shell = min M| spherically symmetric. 

(4. 62)

In spherically symmetric case, we have shown that min M = A ph,out /(108π) = a 0 /3. Thus Eq. (4. 62) shows min M = a 0 /3. Then we finish the proof of A sh,out /3 ≤ 36π M 2 in the case without spherical symmetry. Note that above proof also implies that A sh,out /3 = 36π M 2 is true only if the geometry outside S out is Schwarzschild. Now we make a short summary on this section. We use variational method to prove 9A H /4 ≤ A sh,out /3 and A ph,out ≤ A sh,out /3 ≤ 36π M 2 under two assumptions: (1) weak and strong energy condition are satisfied, and (2) outmost horizon and outmost photon sphere are connected and smooth. Our proof also implies a rigidity theorem: A ph,out = 36π M 2 only if the geometry outside outmost photon sphere is Schwarzschild. Our proof on A ph,out ≤ A sh,out /3 is also true when outmost photon sphere is not connected. As the result, we obtain A ph,out,i ≤ A sh,out,i /3 for every connected branch and so prove a part of our stronger conjecture (3. 11). The lower bound 9A H /4 ≤ A ph,out in spherically symmetric case involves some non-typical energy condition (2. 15). It is not clear how to generalize it into the case without spherical symmetry. We leave the study of this inequality for the future. 

Comment on smoothness: In above proofs, we assumed that the outmost photon sphere S out is smooth. This requirement can be be replaced by piecewise smoothness. The reason is that, in variational method, the target functionals and constraints are defined in integrations. For a outmost photon sphere S (i) ph,out, we can always find a smooth surface S (i) out,ε which satisfies: (1) ∀ε > 0, the maximal volume enclosed by S (i) out and S (i) out,ε is smaller than εM 3, i. e. where V is arbitrary spacelike 3-dimensional surface and satisfies

out,ε, and (2) A ph,out,i and A sh,out,i are approximated in arbitrary accuracy A ph,out,i -

and (3) the constraint (4. 38) is broken arbitrarily small

(4. 65)

We can use the smooth surface S (i) out,ε to replace S (i) out and obtain the same conclusion. Because of this reason, the conclusions in our above proofs are still valid if assume the outmost photon sphere is piecewise smooth.
section_name: Proof of

section_content: 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 (1. 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 (1. 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. In astrophysical observations, to verify whether or not a gravitational system has photon sphere can give us useful information about the gravitational system. For example, if the spacetime contains a naked singularity rather than a black hole, we can classify naked singularities into two kinds based on whether or not a naked singularity is covered within a photon sphere [28]. For the case which contains a photon sphere, the gravitational lensing effects of naked singularity is very similar to Schwarzschild black hole; however, for the case without photon sphere, gravitational lensing effects is qualitatively different from the case of a Schwarzschild black hole. The combinational observations between the shadow and other gravitational lensing effects in principle can gives us some physical information of gravitational source such as the charge/mass ration, distance to the gravitational source and so on [29] [30] [31]. Thus, we can study how to read these useful physical information from the observations on shadows of black holes in the future. or the data will not be deposited. [Authors' comment: This is a purely theoretical study, and all the necessary results and analyses are included in the manuscript. ] Open Access This article is licensed under a Creative Commons Attribution 4. 0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons. org/licenses/by/4. 0/. Funded by SCOAP 3.
section_name: Conclusion

section_content: The momentum constraint and evolutional equation of induced metric are trivial due to extrinsic curvature of t is identical zero. The evolutional equation of extrinsic curvature then shows The strong energy condition implies ρ+T ≥ 0. Thus, outside horizon H we have

Then the maximum principle shows that the maximum of φ must be at the boundaries, i. e. the spital infinity or outmost horizon H. This show φ < 1. 

The proof is as follow. Outside horizon, we can take the orbit of static Killing vector as the time and so metric has a form ds 2 = -φ 2 dt 2 + ds 2 t. We can take a special gauge for the induced metric in t such that ds 2 t = dr 2 +q AB dx A dx B. This can be achieved locally for arbitrary static black hole. Then the spacetime metric has following 3+1 decomposition form ds 2 = -φ 2 dt 2 + dr 2 + q AB dx A dx B. 

(C. 2)

A general 3-dimensional submanifold is locally parameterized by t = t (r, x A ). A surface S ⊂ then is parameterized by r = r S (x A ) on. As the result, S is determined by following two functions:

and the induced metric of S reads

Here qAB := q AB + ∂ A r S ∂ B r S, which is independent of t S. It needs to note that t S depends on the function r S (x A ) and so

Then the target function (C. 1) becomes

q AB and q are the inverse and determinate of qAB, respectively. If S is the critical surface on, then we have following equation This shows that S 0 is also one solution of Eq. (C. 5). Thus, S 0 also makes integration S φ -2 dS to be extreme among all surfaces laying on if S 0 also lays on. 

In our above discussion, the 3-dimensional submanifold can be spacelike, timelike or null. Assume that the null sheet u = 0 is formed by the outgoing light rays of surface S 0. Above conclusion shows that the task of finding critical surface on equal-t surface can be converted into the task of finding critical surface on the null sheet u = 0. 

Above result can be generalized. Consider a general diffeomorphically invariant functional F[S, φ i ] in static spacetime, which is defined on a 2-dimensional surface S and depends on some static fields φ i (i. e. L ξ φ i = 0 and φ i is invariant under the time reversal transformation). Assume that is arbitrary 3-dimensional hypersurface and surface S 0 ⊂ ∩ t. Using the similar step, we can prove following conclusion: if S 0 is a critical surface of F[S, φ i ] among the surfaces laying on t, it is also a critical surface of F[S, φ i ] among the surfaces laying on.
section_name: gauge (B.2). The Hamiltonian constraint reads

section_content: To explain our logics clearly when we use variational method to prove the inequalities of photon spheres, we will use a concrete function as an example. This example, though is much simpler than the cases in the main content, can exhibit all the necessary ideas. 

We consider following 4-variables function

This is the analogy of functional B. The four variables of f are not independent. They satisfy following constraint

This plays the role of Einstein's equation. It is clear that f has no upper bound if 4 variables are all free. However, only 3 of them are independent due to constraint (D. 2). This is the analogy of the analysis at the end of Sec. 4. 1, where we argued that target functionals involves 10 bulk variables and 11 surface variables but only 4 of bulk variables and 8 of surface variables are independent. In following we will prove that f ≤ 2. 

Firstly, by using constraint (D. 2), we can prove f = (1 + cos 3 z+sin 2 z)/(w 2 +1) < 3. Thus, we find that f is bounded from above. This step is the analogy of Appendix B, where we use Einstein's equation to prove that B is bounded from above. 

Secondly, we use constraint (D. 2) to rewrite f into a form f = 1 + cos 3 z + sin 2 z w 2 + 1 and treat w, z as the independent variables. This is the analogy of the steps between Eqs. (4. 17) and (4. 26), where we used Einstein's equation to rewrite the formula of target functional and chose the independent variables suitably. Thirdly, we make the variation with respective to w and z, and find all critical values (including when w = ∞ and z = ∞). The variation with respective to w shows that w = 0, or w = ∞. 

When w = ∞, we find that on-shell value of f is 0. When w = 0, the variation with respective to z shows that sin z = 0, or cos 2 z = 2/3, which shows f | on-shell = 2, 4/3 ± (2/3) 3/2 = 4/3 ± 0. 54432 • • •. Thus, we find that max f = max{0, 2, 4/3 ± (2/3) 3/2 } = 2 so we prove f ≤ 2. This is the analogy of steps between Eqs. (4. 27) and (4. 35). 

To conclude, our basic ideas are as follows. By using constrained equations (in our main text, they are Einstein's equation and requirements of photon sphere), we (1) analyze how many independent variables the target function/functional has, (2) prove target function/functional is bounded from above/blow, (3) rewrite the mathematical expression of target function/functional into a suitable form and choose independent variables suitably, and (4) compute all critical values and find the maximal/minimal one.
section_name: Appendix D: An intuition about finding upper bound

section_name: Acknowledgements

section_content: 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.
section_name: Acknowledgements

section_name: Data Availability Statement This manuscript has no associated data

section_name: Appendix B: Boundedness of functional B

section_content: where (∂/∂u) μ = ξ μ is the static Killing vector and |U | := r 2 h AB U A U B. For the same static spacetime, we can also choose the infalling null time v, which satisfies (∂/∂v) μ = ξ μ and keeps {r, x A } coordinates unchanged. As the result, the metric becomes

Here all components are independent of v and

As vectors (dx A ) μ and (dr ) ν are unchanged in two different coordinates, we have 4

and

Define ṽ = -v and we obtain

This metric and metric (A. 1) describe same geometry at every point only if Ũ A = U A and β = β. Combine it with Eq. (A. 3) and we find U A = 0.

section_content: There is a simple way to prove that B is bounded from above. Firstly, the area of any closed surface outside horizon is given by

If we can prove 0 < φ < 1, we can find

Thus, in all black holes of fixed horizon area, the functional B is bounded from above. 

In order to prove 0 < φ < 1, we take a 3+1 decomposition

Here h ab is the metric of equal-t slice t. We write down the decomposition of static Einstein's equation in the coordinates Appendix C: Critical surfaces on equal-t submanifold and u = 0 null sheet

In this appendix, we will explain why the task of finding critical surface of a target functional on equal-t plane can be converted into the task of finding critical surface of the same target functional on the null sheet u = 0. We first consider a concrete example. Assume that t is equal-t submanifold and define a target functional

Assume that S 0 is a critical surface (i. e. the surface makes P extreme) among all surfaces laying on t. In following, we will show that, for arbitrary 3-dimensional manifold, if S 0 ⊂, then S 0 also makes integration S φ -2 dS to be extreme among all surfaces laying on. 5
section_name: Appendix B: Boundedness of functional B