Photon surfaces are timelike totally umbilic hypersurfaces of Lorentzian spacetimes. In the first part of this paper, we locally characterize all possible photon surfaces in a class of static spherically symmetric spacetimes that includes (exterior) Schwarzschild, Reissner–Nordström, and Schwarzschild–anti de Sitter in n + 1 dimensions. In the second part, we prove that any static, vacuum, and “asymptotically isotropic” n + 1-dimensional spacetime that possesses what we call an “equipotential” and “outward directed” photon surface is isometric to the Schwarzschild spacetime of the same (necessarily positive) mass using a uniqueness result obtained by the first author.

One of the cornerstone results in the theory of black holes (in 3 + 1 dimensions) is the static black hole uniqueness theorem first proposed by Israel1 for a single horizon and later by Bunting and Masood-ul-Alam2 for multiple horizons, which establishes the uniqueness of the Schwarzschild spacetime among all static asymptotically flat black hole solutions to the vacuum Einstein equations. Refer to the book by Heusler3 and the review article4 by Robinson for a (then) complete list of references on further contributions, Simon’s spinor proof recently described in the article by Raulot (Ref. 5, Appendix A), and the recent article by Agostiniani and Mazzieri6 for newer approaches in the case of a single horizon.

A well-known and intriguing feature of (positive mass) Schwarzschild spacetime is the existence of a photon sphere, namely, the timelike cylinder P over the {r=(nm)1n2,t=0}n1-sphere. P has the property of being null totally geodesic in the sense that any null geodesic tangent to P remains in P, i.e., P traps all light rays tangent to it.

In Ref. 7, the first author introduced and studied the notion of a photon sphere for general static spacetimes (see also Ref. 8 in the spherically symmetric case). Based on this study, in Ref. 9 she adapted Israel’s argument (which requires the static lapse function to have nonzero gradient) to obtain a photon sphere uniqueness result, thereby establishing the uniqueness of the Schwarzschild spacetime among all static asymptotically flat solutions to the vacuum Einstein equations, which admit a single photon sphere. Subsequent to that work, by adapting the argument of Bunting and Masood-ul-Alam,2 the authors10 were able to improve this result by particularly avoiding the gradient condition and allowing a priori multiple photon spheres. For further results on photon spheres, in particular uniqueness results in the electro-vacuum case and the case of other matter fields, see Refs. 11–20.

In this paper, we will be concerned with the notion of photon surfaces in spacetimes (see Refs. 8 and 21 for slightly more general versions of this notion). A photon surface in an n + 1-dimensional spacetime (Ln+1,g) is a timelike hypersurface Pn, which is null totally geodesic, as described above. As shown in Refs. 8 and 21, a timelike hypersurface Pn is a photon surface if and only if it is totally umbilic. By definition, a photon sphere in a static spacetime is a photon surface Pn along which the static lapse function N is constant (see Sec. II for details). While the Schwarzschild spacetime [in dimension n + 1, n ≥ 3, and with r>(2m)1n2, m > 0] admits a single photon sphere, it admits infinitely many photon surfaces of various types, as briefly described in Sec. III.

In Sec. III, we derive the relevant ordinary differential equation (ODE) describing spherically symmetric photon surfaces for a class of static spherically symmetric spacetimes, which includes (exterior) Schwarzschild, Reissner–Nordström, and Schwarzschild–AdS (anti–de Sitter). In the generic case, a photon surface in this setting is given by a formula for the derivative of the radius-time-profile curve r = r(t) (see Theorem 3.5). This formula is used in Ref. 22 to give a detailed qualitative description of all spherically symmetric photon surfaces in many (exterior) black hole spacetimes within the class S (defined in Sec. III), including the (positive mass) Schwarzschild spacetime. In addition, in Sec. III, we present a result that shows, for generic static, isotropic spacetimes, which include positive mass Schwarzschild and sub-extremal Reissner–Nordström, and that, apart from some (partial) timelike hyperplanes, all photon surfaces are isotropic (see Theorem 3.8 and Corollary 3.9). As a consequence, one obtains a complete characterization of all photon surfaces in Reissner–Nordström and Schwarzschild.

Finally, in Sec. IV, we obtain a new rigidity result pertaining to photon surfaces, rather than just to photon spheres. We prove that any static, vacuum, and asymptotically isotropic spacetime possessing an (possibly disconnected) “outward directed” photon surface inner boundary with the property that the static lapse function N is constant on each component of each time slice Σn−1(t) := Pn ∩ {t = const.} must necessarily be a Schwarzschild spacetime of positive mass, with the photon surface being one of the spherically symmetric photon surfaces in Schwarzschild classified in Sec. III. We call such photon surfaces equipotential. This generalizes static vacuum photon sphere uniqueness to certain photon surfaces and to higher dimensions.

The proof makes use of a new higher dimensional uniqueness result for the Schwarzschild spacetime due to the first author23 (see Sec. IV for a statement). This result generalizes in various directions the higher dimensional Schwarzschild uniqueness result of Gibbons et al.24 In particular, it does not a priori require the spacetime to be vacuum or static. A different proof of the result we use from Ref. 23 has since been given by Raulot,5 assuming that the manifolds under consideration are spin. These results rely on the rigidity case of a (low regularity version) of the Riemannian positive mass theorem.25–30 

The static spherically symmetric (n + 1)-dimensional Schwarzschild spacetime of massmR, with n ≥ 3, is given by (L̄n+1:=R×(Rn\Brm(0)̄),ḡ), where the Lorentzian metric ḡ is given by

ḡ=N̄2dt2+N̄2dr2+r2Ω,N̄=12mrn21/2,
(2.1)

with Ω denoting the standard metric on Sn1, and rm:=(2m)1n2 for m > 0 and rm := 0 for m ≤ 0 (see also the work of Tangherlini).31 For m > 0, the timelike, cylindrical hypersurface P̄n{r=(nm)1n2} is called the photon sphere of the Schwarzschild spacetime because any null geodesic (or “photon”) γ:RL̄n+1 that is tangent to P̄n for some parameter τ0R is necessarily tangent to it for all parameters τR. In particular, the Schwarzschild photon sphere is a timelike hypersurface ruled by null geodesics spiraling around the central black hole of mass m > 0 “at a fixed distance.”

The Schwarzschild photon sphere can be seen as a special case of what is called a “photon surface”8,21 in a general spacetime (or smooth Lorentzian manifold).

Definition 2.1

(Photon surface). A timelike embedded hypersurfacePnLn+1in a spacetime(Ln+1,g)is called a photon surface if any null geodesic initially tangent toPnremains tangent toPnas long as it exists or, in other words, ifPnis null totally geodesic.

The one-sheeted hyperboloids in the Minkowski spacetime (Schwarzschild spacetime with m = 0) are also examples of photon surfaces (see Sec. III). It will be useful to know that, by an algebraic observation, being a null totally geodesic timelike hypersurface is equivalent to being an umbilic timelike hypersurface.

Proposition 2.2

(See Theorem II.1 of Ref. 8 and Proposition 1 of Ref. 21). Let(Ln+1,g)be a spacetime andPnLn+1 be an embedded timelike hypersurface. Then,Pnis a photon surface if and only if it is totally umbilic, that is, if and only if its second fundamental form is pure trace.

As stated above, the Schwarzschild spacetime is “static” by the following definition.

Definition 2.3
(Static spacetime). A spacetime(Ln+1,g)is called (standard) static if it is a warped product of the form
Ln+1=R×Mn,g=N2dt2+g,
(2.2)
where (Mn, g) is a smooth Riemannian manifold andN:MnR+is a smooth function called the (static) lapse function of the spacetime.

Remark 2.4

[Static spacetime cont. and (canonical) time slices]. We will slightly abuse the standard terminology and also call a spacetime static if it is a subset (with boundary) of a warped product static spacetime(R×Mn,g=N2dt2+g),Ln+1R×Mn, to allow for inner boundaryLnot arising as a warped product. We will denote the (canonical) time slices {t = const.} of a static spacetime(Ln+1,g),Ln+1R×Mn, byMn(t) and continue to denote the induced metric and (restricted) lapse function onMn(t) bygandN, respectively.

In the context of static spacetimes, we will use the following definition of “photon spheres,” extending that of Ref. 7–9. Consistently, the Schwarzschild photon sphere clearly is a photon surface in the Schwarzschild spacetime in this sense.

Definition 2.5

(Photon sphere). Let(Ln+1,g)be a static spacetime andPnLn+1be a photon surface. Then,Pnis called a photon sphere if it is of the formPn=R×Σn1for some smooth hypersurfaceΣn1Ln+1and if the lapse functionNof the spacetime is constant along each connected component ofPn.

For our discussions in Secs. III, IV, we will make use of the following definitions.

Definition 2.6

(Equipotential photon surface). Let(Ln+1,g)be a static spacetime andPnLn+1 be a photon surface. Then,Pnis called equipotential if the lapse functionNof the spacetime is constant along each connected component of each time slice Σn−1(t) := PnMn(t) of the photon surface.

Definition 2.7

(Outward directed photon surface). Let(Ln+1,g)be a static spacetime andPnLn+1 be a photon surface arising as the inner boundary ofLn+1,Pn=L, and letηbe the “outward” unit normal toPn(i.e., the normal pointing intoLn+1). Then,Pnis called outward directed if theη-derivative of the lapse functionNof the spacetime is positive,η(N) > 0, alongPn.

As usual, a spacetime (Ln+1,g) is said to be vacuum or to satisfy the Einstein vacuum equation if

Ric=0
(2.3)

on Ln+1, where Ric denotes the Ricci curvature tensor of (Ln+1,g). For a static spacetime, the Einstein vacuum Eq. (2.3) is equivalent to the static vacuum equations,

NRic=2gN,
(2.4)
R=0,
(2.5)

on Mn, where Ric, R, and 2g denote the Ricci and scalar curvature and the covariant Hessian of (Mn, g), respectively. Combining the trace of (2.4) with (2.5), one obtains the covariant Laplace equation on Mn,

gN=0.
(2.6)

It is clear that, provided (2.4) holds, (2.5) and (2.6) can be interchanged without losing information. Of course, the Schwarzschild spacetime (R×M̄n,ḡ) is vacuum, and thus, (2.4) and (2.6) hold for the Schwarzschild spatial metric ḡ=N̄2dr2+r2Ω and lapse N̄ on its canonical time slice M̄n=Rn\Brm(0)̄.

Curvature quantities of a spacetime (Ln+1,g) such as the Riemann curvature endomorphism Rm, the Ricci curvature tensor Ric, and the scalar curvature R will be denoted in gothic print, and the corresponding covariant derivative will be denoted by g. The Lorentzian metric induced on a timelike embedded hypersurface PnLn+1 will be denoted by p, the (outward, see Definition 2.7) unit normal by η, and the corresponding second fundamental form and mean curvature by h and H=trph, respectively. With this notation, Proposition 2.2 can be restated to state that a photon surface is characterized by

h=Hnp.
(2.7)

Set sign conventions h(X,Y)=g(Xgη,Y) for vectors X, Y tangent to P.

If the spacetime (Ln+1,g) is static, its time slices Mn(t) have vanishing second fundamental form K = 0 by the warped product structure, or in other words, the time slices are totally geodesic. The time slices of a photon surfacePnLn+1 will be denoted by Σn−1(t) := PnMn(t), with induced metric σ = σ(t), second fundamental form h = h(t), and mean curvature H = H(t) = trσ(t)h(t) with respect to the outward pointing unit normal ν = ν(t). As an intersection of a totally geodesic time slice and a totally umbilic photon surfaces, Σn−1(t) is necessarily totally umbilic, and we have

h(t)=H(t)n1σ(t).
(2.8)

Our choice of sign of the mean curvature is such that the mean curvature of Sn1Rn is positive with respect to the outward unit normal in Euclidean space.

The following proposition will be useful to characterize photon surfaces in vacuum spacetimes.

Proposition 2.8
(Ref. 9, Proposition 3.3). Letn ≥ 2 and let(Ln+1,g)be a smooth semi-Riemannian manifold possessing a totally umbilic embedded semi-Riemannian hypersurfacePnLn+1. If the semi-Riemannian manifold(Ln+1,g)is Einstein, or, in other words, ifRic=Λgfor some constantΛR, then each connected component ofPnhas constant mean curvatureHand constant scalar curvature,
Rp=(n+12τ)Λ+τn1nH2,
(2.9)
whereτ:=g(η,η)denotes the causal character of the unit normalηtoPn.

In particular, connected components of photon surfaces in vacuum spacetimes (Λ = 0) have constant mean curvature and constant scalar curvature, related via

Rp=n1nH2.
(2.10)

We will now proceed to define and discuss the assumption of asymptotic flatness and asymptotic isotropy of static spacetimes.

Definition 2.9
(Asymptotic flatness). A smooth Riemannian manifold (Mn, g) withn ≥ 3 is called asymptotically flat if the manifoldMnis diffeomorphic to the union of a (possibly empty) compact set and an open endEn, which is diffeomorphic toRn\B̄,Φ=(xi):EnRn\B̄, whereBis some centered open ball inRn, and
(Φ*g)ijδij=Okr1n2ε,
(2.11)
Φ*R=O0(rnε)
(2.12)
fori, j = 1, , nonRn\B̄asr:=(x1)2++(xn)2for somekZ,k ≥ 2, andɛ > 0. Here,δdenotes the flat Euclidean metric, andδijdenotes its components in the Cartesian coordinates (xi).
A static spacetime(Ln+1=R×Mn,g=N2dt2+g)is called asymptotically flat if its Riemannian base (Mn, g) is asymptotically flat as a Riemannian manifold and, in addition, its lapse function satisfies
Φ*N1=Ok+1r1n2ε
(2.13)
onRn\B̄asrwith respect to the same coordinate chart Φ and numberskZ,k ≥ 2, and ɛ > 0. We will abuse language and callLn+1R×Mnasymptotically flat as long asLn+1has a timelike inner boundaryL.

Here and in the following, we say that a smooth functionf:Rn\B̄Rsatisfiesf=Ok(rp)forkN0andpRasrif there exists a constantC > 0 and a centered open ball centered open ballB1Bsuch that |Dαf| ≤ Crp−|α|inRn\B1̄for every multi-indexαsatisfying |α| ≤ k.

One can expect a well-known result by Kennefick and Murchadha32 to generalize to higher dimensions, which would assert that static vacuum asymptotically flat spacetimes are automatically “asymptotically isotropic” under suitable asymptotic coordinates. Here, we will resort to assuming asymptotic isotropy, leaving the higher dimensional generalization of this result to be dealt elsewhere.

Definition 2.10
(Asymptotic isotropy23). A smooth Riemannian manifold (Mn, g) of dimensionn ≥ 3 is called asymptotically isotropic (of massm) if the manifoldMnis diffeomorphic to the union of a (possibly empty) compact set and an open endEn, which is diffeomorphic toRn\B̄,Ψ=(yi):EnRn\B̄, whereBis some centered open ball inRn, and if there exists a constantmRsuch that
(Ψ*g)ij(g̃m)ij=O2(s1n),
(2.14)
fori, j = 1, , nonRn\B̄ass:=(y1)2++(yn)2, where
g̃m:=φm4n2(s)δ,
(2.15)
φm(s):=1+m2sn2
(2.16)
denotes the spatial Schwarzschild metric in isotropic coordinates.
A static spacetime(Ln+1=R×Mn,g=N2dt2+g)is called asymptotically isotropic (of mass m) if its Riemannian base (Mn, g) is asymptotically isotropic of massmRas a Riemannian manifold and, in addition, its lapse functionNsatisfies
NÑm=O2(s1n)
(2.17)
onRn\B̄asswith respect to the same coordinate chart Ψ and massm. Here,Ñmdenotes the Schwarzschild lapse function in isotropic coordinates, given by
Ñm(s):=1m2sn21+m2sn2.
(2.18)
As before, we will abuse language and callLn+1R×Mn asymptotically isotropic as long as it has a timelike inner boundary.

Here, we have rewritten the Schwarzschild spacetime, spatial metric, and lapse function in isotropic coordinates via the radial coordinate transformation,

r=:sφm2n2(s).
(2.19)

For m > 0, this transformation bijectively maps r ∈ (rm, ) ↦ s ∈ (sm, ), with sm:=m21n2. For m = 0, this transformation is the identity on R+, while for m < 0, it only provides a coordinate transformation for r suitably large, corresponding to s>|m|21n2.

Remark 2.11.

A simple computation shows that the parametermin Definition 2.10 equals the ADM-mass of the Riemannian manifold (Mn, g) defined in Refs.33 and 34 .

Remark 2.12.

One can analogously define asymptotically flat and asymptotically isotropic Riemannian manifolds and static spacetimes with multiple endsElnand associated massesml.

With these definitions at hand, let us point out that photon spheres are always outward directed in static, vacuum, asymptotically isotropic spacetimes, a fact that is a straightforward generalization to higher dimensions of Lemma 2.6 and Eq. (2.13) of Ref. 10.

Lemma 2.13.

LetPnLn+1be a photon sphere in a static vacuum asymptotically flat spacetime(Ln+1,g). Then,Pnis outwarddirected.

In this section, we will give a local characterization of photon surfaces in a certain class S of static spherically symmetric spacetimes (R×Mn,g), which includes the n + 1-dimensional (exterior) Schwarzschild spacetime. We will first locally characterize the spherically symmetric photon surfaces in (R×Mn,g)S in Theorem 3.5 and then show in Theorem 3.8 and particularly in Corollary 3.9 that there are essentially no other photon surfaces in spacetimes, (R×Mn,g)S. As mentioned above, these results have been used in Ref. 22 to give a detailed description of all photon surfaces in many spacetimes in class S, including the (positive mass) Schwarzschild spacetime.

The class S is defined as follows: Let (R×Mn,g) be a smooth Lorentzian spacetime such that

Mn=I×Sn1(r,ξ)
(3.1)

for an open interval I(0,), finite or infinite, so that there exists a smooth positive function f:IR for which we can express the spacetime metric g as

g=f(r)dt2+1f(r)dr2+r2Ω
(3.2)

in the global coordinates tR, (r,ξ)I×Sn1, where Ω denotes the canonical metric on Sn1 of area ωn−1. A Lorentzian spacetime (R×Mn,g)S is clearly spherically symmetric and moreover naturally (standard) static via the hypersurface orthogonal timelike Killing vector field t.

Remark 3.1.

Note that we do not assume that spacetimes(R×Mn,g)Ssatisfy any kind of Einstein equations or have any special type of asymptotic behavior toward the boundary of the radial intervalI, such as being asymptotically flat or asymptotically hyperbolic asrsupI, or such as forming a regular minimal surface asrinfI.

Remark 3.2.

Astis a Killing vector field, the time translation of any photon surface in a spacetime(R×Mn,g)Swill also be a photon surface in(R×Mn,g). As all spacetimes(R×Mn,g)Sare also time-reflection symmetric (i.e.,t → −tis an isometry), the time reflection of any photon surface in(R×Mn,g)will also be a photon surface in(R×Mn,g).

While the form of the metric (3.2) is certainly non-generic even among static spherically symmetric spacetimes, the class S contains many important examples of spacetimes, such as Minkowski spacetime and the regions exterior to the black holes and white holes in Schwarzschild spacetime, Reissner–Nordström spacetime, and Schwarzschild–anti de Sitter spacetime (in n + 1 dimensions), each for a specific choice of f.

Before we proceed with characterizing photon surfaces in spacetimes in this class S, let us first make the following natural definition.

Definition 3.3.
Let(R×Mn,g)S. A connected timelike hypersurfacePn(R×Mn,g)will be called spherically symmetric if, for eacht0Rfor which the intersectionMn(t0) ≔ Pn ∩ {t = t0} ≠ ∅, there exists a radiusr0I(whereMn=I×Sn1) such that
Mn(t0)={t0}×{r0}×Sn1{t0}×Mn.
(3.3)
A future timelike curveγ: IPn, parametrized by the arclength on some open intervalIR, is called a radial profile ofPnifγspan{t,r}Tγ(R×Mn)onIand if the orbit ofγunder the rotation generatesPn.

With this definition at hand, we will now prove the following lemma, which will be used in the Proof of Theorem 3.5.

Lemma 3.4.
Let(R×Mn,g)Sand letPn(R×Mn,g)be a spherically symmetric timelike hypersurface. Assume thatPn(R×Mn,g)has a radial profileγ: IPn, which may be written asγ(s)=(t(s),r(s),ξ*)R×I×Sn1for some fixedξ*Sn1. IfPn(R×Mn,g)is a photon surface, i.e., is totally umbilic with the umbilicity factorλ, then the following first order ODEs hold onI:
ṫ=λrf(r),
(3.4)
(ṙ)2=λ2r2f(r),
(3.5)
whereλis constant (also where=dds). Conversely, providedṙ0, if the ODEs(3.4)and(3.5)hold, withλbeing constant, thenPis a photon surface with the umbilicity factorλ.

Proof.

To simplify the notation, we write P for Pn(R×Mn,g) and f for f(r). As in Sec. II, let p and h denote the induced metric and second fundamental form of P, respectively.

Set e0=γ̇, and extend it to all of P by making it invariant under the rotational symmetries. Thus, e0 is the future directed unit tangent vector field to P orthogonal to each time slice {t(s) = const.}, sI. In terms of coordinates, we have
e0=ṫt+ṙr.
(3.6)
Let η be the outward pointing unit normal field to P. From (3.2) and (3.6), we obtain
η=ṙft+ṫfr.
(3.7)

Claim.
P is a photon suface, with umbilicity factor λ=frṫ, if and only if e0 satisfies
e0ge0=λη.
(3.8)

Proof of the claim.
Extend e0 to an orthonormal basis {e0, e1, …, en−1} in a neighborhood of an arbitrary point in P. Thus, each eI, I = 1, …, n − 1, where defined, is tangent to the time slices. A simple computation then gives
eIgη=ṙfeIgt+ṫfeIgr=ṫfreI,
(3.9)
from which it follows that
h(eI,eJ)=g(eIgη,eJ)=λδIJ,I,J=1,,n1,
(3.10)
where δIJ is the Kronecker delta and
λ=frṫ.
(3.11)
Similarly,
h(e0,eI)=h(eI,e0)=g(eIgη,e0))=0.
(3.12)
Hence,
h(eI,eJ)I,J=0,,n1
(3.13)
is a diagonal matrix with h(eI,eI)=λ for I = 1, …, n − 1. It remains to consider h(e0,e0).
The profile curve γ and its rotational translates are “longitudes” in the “surface of revolution” P. As such, each is a unit speed geodesic in P, from which it follows that
e0ge0=η
(3.14)
for some scalar . This implies that
h(e0,e0)=g(e0gη,e0)=g(e0ge0,η)=.
(3.15)
From this and (3.13), we conclude that P is a photon surface if and only if =λ=frṫ, which establishes the claim.
Using the coordinate expressions for e0, η, and λ, a straightforward computation shows that (3.8), with λ=frṫ, is equivalent to the following system of second order ODEs in the coordinate functions t = t(s) and r = r(s):
ẗ+ffṙṫ=ṙrṫ,
(3.16)
r̈+ff2ṫ2f2fṙ2=(fṫ)2r.
(3.17)
Now, assume that P is a photon sphere with the umbilicity factor λ so that, particularly, (3.16) and (3.11) hold. Treating (3.16) as a first order linear equation in ṫ, we have
ẗ+ffṙṙrṫ=0,
which, multiplying through by the integrating factor fr, gives ddsfrṫ=0 so that (3.4) holds, with λ=frṫ>0 being a constant on P. The assumption that γ is parameterized with respect to arc length gives
f(ṫ)21f(ṙ)2=1.
(3.18)
Together with (3.4), we see that (3.5) also holds.
Conversely, now assume that (3.4) and (3.5) hold, with λ=frṫ= being constant, and, in addition, that ṙ0. Differentiating (3.4) with respect to s and then using (3.11) easily implies (3.16). Differentiating (3.5) with respect to s, then using (3.11), and dividing out by ṙ gives
r̈+f2=(fṫ)2r.
(3.19)
Together with (3.18) [which follows from (3.4) and (3.5)], this implies (3.17). We have shown that (3.16) and (3.17) hold, from which it follows that (3.8) holds with λ=frṫ. Invoking the claim then completes the Proof of Lemma 3.4.□

From Lemma 3.4, we obtain the following.

Theorem 3.5.
Let(R×Mn,g)Sand letPn(R×Mn,g)be a spherically symmetric timelike hypersurface. Assume thatPn(R×Mn,g)is a photon surface, with the umbilicity factorλ, i.e.,
h=λp,
wherepandhare the induced metric and second fundamental form induced onPnbyPn(R×Mn,g), respectively.
Letγ: IPnbe a radial profile forPnand writeγ(s)=(t(s),r(s),ξ*)R×I×Sn1for someξ*Sn1. Then,λis a positive constant and either rr*alongγfor somer*Iat which the photon sphere condition
f(r*)r*=2f(r*)
(3.20)
holds,λ=f(r*)r*, and(Pn,p)=(R×Sn1,f(r*)dt2+r*2Ω)is a cylinder and thus a photon sphere or r = r(t) can globally be written as a smooth non-constant function oftin the range ofγandr = r(t) satisfying the photon surface ODE,
drdt2=f(r)2(λ2r2f(r))λ2r2.
(3.21)

Conversely, whenever the photon sphere conditionf′(r*)r* = 2f(r*) holds for somer*I, then the cylinder(Pn,p)=(R×Sn1,f(r*)dt2+r*2Ω)is a photon sphere in(R×Mn,g)with the umbilicity factorλ=f(r*)r*. In addition, any smooth non-constant solutionr = r(t) of (3.21) for some constantλ > 0 gives rise to a photon surface in(R×Mn,g)with the umbilicity factorλ.

Proof.

From Lemma 3.4, we know that λ is a positive constant. Moreover, we know that t = t(s) and r = r(s) satisfy Eqs. (3.4) and (3.5).

In the case when rr* for some constant r*, these equations immediately imply
ṫ=1f(r*),λ=f(r*)r*.
(3.22)
Furthermore, (3.17) implies
f(r*)r*=2f(r*).
(3.23)

In the general case, Eqs. (3.4) and (3.5) clearly imply (3.21). The converse statements are easily obtained from (3.21) and the unit speed condition (3.18).□

Remark 3.6.

In view of Remark 3.2, note that in the “either” case, the photon sphere is time-translation and time-reflection invariant in itself. In the “or” case, note that the photon surface ODE(3.21)is time-translation and time-reflection invariant and will thus allow for time-translated and time-reflected solutions corresponding to the sameλ > 0.

Example 3.7.
Choosing(R×Mn,g)=(R1,n,m), wheremis the Minkowski metric andf:(0,)R:r1, the photon sphere condition cannot be satisfied for anyr* ∈ (0, ) so that every spherically symmetric photon surface in the Minkowski spacetime must satisfy ODE(3.21), which reduces to
drdt2=λ2r21λ2r2r(t)=λ2+(tt0)2for some t0R
and describes the rotational one-sheeted hyperboloids of radiiλ−1for any 0 < λ < .

This is, of course, consistent with the well-known fact that the only timelike totally umbilic hypersurfaces in the Minkowski spacetime are, apart from (parts of) timelike hyperplanes, precisely (parts of) these hyperboloids and their spatial translates, the formula for which explicitly displays the time-translation and time-reflection invariance of the photon surface characterization problem.

Note that the photon sphere condition is satisfied precisely at the well-known photon sphere radius r*=(nm)1n2 in the n + 1-dimensional Schwarzschild spacetime, where f(r)=12mrn2 for m > 0 and r>rH=(2m)1n2 and there is no photon sphere radius for m ≤ 0 and r > 0. While there are no other photon spheres, there are many non-cylindrical photon surfaces in the Schwarzschild spacetime. The analysis in Ref. 22, based on Theorem 3.5, shows that, up to time translation and time reflection (cf. Remarks 3.2 and 3.6), there are five classes of non-cylindrical spherically symmetric photon surfaces in the (exterior) positive mass Schwarzschild spacetime [as well as in many other (exterior) black hole spacetimes in class S]; the profile curves for representatives from each class are depicted in Fig. 1.

FIG. 1.

Profile curves for all types of spherically symmetric photon surfaces in Schwarzschild spacetime grouped according to the umbilicity factor (see Ref. 22 for details).

FIG. 1.

Profile curves for all types of spherically symmetric photon surfaces in Schwarzschild spacetime grouped according to the umbilicity factor (see Ref. 22 for details).

Close modal

In each case, they approach asymptotically the event horizon r = rH and/or the photon sphere r = r* and/or become asymptotically null at infinity in (t, r)-coordinates. An analysis of the behavior of the asymptotics of the non-cylindrical spherically symmetric photon is performed for both Schwarzschild and many other (exterior) black hole spacetimes in class S in (generalized) Kruskal–Szekeres coordinates in Ref. 35. There, it is found that the photon surfaces appearing to approach r = rH in (t, r)-coordinates, in fact, cross the event horizon, while those approaching the photon sphere r = r* or asymptotically becoming null in (t, r)-coordinates do so in (generalized) Kruskal–Szekeres coordinates too.

Using quite different methods, in Ref. 36, the same types of photon surfaces are found in a 2 + 1-dimensional spacetime obtained by dropping an angle coordinate from 3 + 1-dimensional Schwarzschild of positive mass.

The following question naturally arises: What about non-spherically symmetric photon surfaces? This is addressed in the following theorem (see Corollary 3.9).

Theorem 3.8.
Letn ≥ 3,IR+ be an open interval, andDn:={yRn||y|=sI}, and letÑ,ψ:IR+be smooth positive functions. Consider the static isotropic spacetime
R×Dn,g=Ñ2dt2+ψ2δ
(3.24)
of lapseÑ=Ñ(s)and conformal factorψ = ψ(s), withs := |y| foryDnhere and in the following. We writeg̃:=ψ2δ. A timelike hypersurfacePnin(R×Dn,g)is called isotropic ifPn{t=const.}=Ss(t)n1(0)Dnfor some radiuss(t) ∈ Ifor everytfor whichPn ∩{t = const. } ≠ ∅. A (partial) centered vertical hyperplane in(R×Dn,g)is the restriction of a timelike hyperplane in the Minkowski spacetime containing thet-axis toR×Dn, i.e., a set of the form
{(t,y)R×Dn|yu=0}
(3.25)
for some fixed Euclidean unit vectoruRn, where · denotes the Euclidean inner product. Centered vertical hyperplanes are totally geodesic in(R×Dn,g).
Furthermore, assume that the functionsÑandψsatisfy
Ñ(s)Ñ(s)ψ(s)ψ(s)
(3.26)
for allsSfor some dense subsetSI. Then, any photon surface in(R×Dn,g)is either (part of) an isotropic photon surface or (part of) a centered vertical hyperplane.

Corollary 3.9.

Letn ≥ 3 andm > 0, and consider then + 1-dimensional Schwarzschild spacetime of massm. Then, any connected photon surface is either (part of) a centered vertical hyperplane, as described above, or (part of) a spherically symmetric photon surface, as described in Theorem 3.5.

Proof of Corollary 3.9.

Recall the isotropic form of the Schwarzschild spacetime (2.15), (2.16), and (2.18), with I = (sm, ), and note that (3.26) corresponds to sn2m2(n1) (which can be quickly seen when exploiting Ñ=2φφ, ψ=φ2n2). This, however, is automatic as (m2(n1))1n2<sm=(m2)1n2<s holds for all sI = (sm, ).□

Remark 3.10.
Condition(3.26)has the following geometric interpretation: If
Ñ(s)Ñ(s)=ψ(s)ψ(s)
holds for allsO, withOIbeing an open subset, then, on each connected componentJO, there exists a positive constantA > 0 such thatÑ(s)=Aψ(s)for allsJ, where we used thatÑ,ψ>0. This shows thatg=Ñ2dt2+ψ2δ=ψ2(A2dt2+δ)or, in other words, the static isotropic spacetime(R×(O×Sn1),g)is locally conformally flat and hence possesses additional photon surfaces corresponding to the totally geodesic timelike hyperplanes that do not contain thet-axis and to the spatially translated totally umbilic rotational one-sheeted hyperboloids of the (time-rescaled) Minkowski spacetime (see Example 3.7).

Hence, Theorem (3.8) gives a full characterization of photon surfaces in nowhere locally conformally flat static isotropic spacetimes.

Remark 3.11.

A static isotropic spacetime(R×Dn,Ñ2dt2+ψ2δ)can be globally rewritten as a spacetime of classSif and only ifÑ2(s)=(1+sψ(s)ψ(s))2>0for allsIby settingr(s) := (s) andf(r):=Ñ(s(r)), wheres = s(r) denotes the inverse function ofr = r(s). In this case, the photon sphere and photon surface conditions on the isotropic radius profiles = S*ands = S(t) [3.57and (3.60)] reduce to the much simpler photon sphere and photon surface conditions for the area radius profiler = r*andr = r(t) [3.20and (3.21)], respectively.

Conversely, a spacetime of classScan always be locally rewritten in isotropic form by picking a suitabler0I(orr0Ī) and setting
s=s(r):=expr0rρ(f(ρ)1dρ,
ψ(s):=r(s)s, andÑ(s):=f(r(s)), wherer = r(s) denotes the inverse ofs = s(r).

The main reason for switching into the isotropic picture lies in the spatial conformal flatness allowing us to easily describe centered vertical hyperplanes and to exclude photon surfaces that are neither centered vertical hyperplanes nor isotropic.

Proof of Theorem 3.8.
Let Pn be a connected photon surface in a static isotropic spacetime (R×Dn,g=Ñ2dt2+ψ2δ). As before, set Mn(t) := {t = const.}. Let T:={tR|PnMn(t)} and note that T is an open, possibly infinite, interval. Set Σn−1(t) := PnMn(t) for tT. As timelike and spacelike submanifolds are always transversal, Σn−1(t) is a smooth surface. Furthermore, Σn−1(t) is umbilic in Mn(t) by the time symmetry of Mn(t), or, in other words, because the second fundamental form of Mn(t) in a static spacetime vanishes. As (Mn(t),g̃) is conformally flat, exploiting the conformal invariance of umbilicity, the only umbilic hypersurfaces in (Mn(t),g̃) are the conformal images of pieces of Euclidean round spheres and pieces of Euclidean hyperplanes. Slightly abusing the notation and denoting points in the spacetime by their isotropic coordinates, by continuity, and by connectedness of Pn, Σn1(t)tT is thus either a family of pieces of spheres,
Σn1(t){yRn||yc(t)|=S(t)},
(3.27)
with centers c(t)Rn and radii S(t) > 0 for all tT, or a family of pieces of hyperplanes,
Σn1(t){yRn|yu(t)=a(t)}
(3.28)
for some δ-unit normal vectors u(t)Rn and altitudes a(t)R for all tT, where |⋅| and · denote the Euclidean norm and inner product, respectively. The (outward, where appropriate) unit normal η to Pn can be written as η = αν + β∂t, with α > 0, recalling that ν denotes the (outward, where appropriate) unit normal to Σn−1(t) in Mn(t). By time symmetry of Mn(t), the second fundamental form h of Pn in the spacetime restricted to Σn−1(t) can be expressed in terms of the second fundamental form h of Σn−1(t) in Mn(t) via h|TΣn1(t)×TΣn1(t)=αh. By umbilicity, h=λp, with p denoting the induced metric on Pn, and this implies that
h=λασ,
(3.29)
where σ is the induced metric on Σn−1(t). We will treat the planar and the spherical cases separately. We will denote t-derivatives by and s-derivatives by ′.

Let Σn−1(t) be as in (3.28) for all tT. We will show that a(t) = 0 and u̇(t)=0 for all tT and, moreover, that λ = 0 along Pn. This then implies that Pn is contained in a centered vertical hyperplane with unit normal η=ν=ψ1(s)uiyi and, moreover, that all centered vertical hyperplanes are totally geodesic as they can be written in the form (3.28) with u̇(t)=0 and a(t) = 0 for all tT.

For each tT, extend u(t) to a δ-orthonormal basis {e1(t) = u(t), e2(t), …, en(t)} of Rn so that eI(t) is smooth in t, and constant for each t, for all I = 2, …, n. Then, clearly, XI(t,y):=eIk(t)yk is tangent to Pn for all I = 2, …, n and {XI(t,)}I=2n is an orthogonal frame for Σn−1(t) with respect to g̃ by conformal flatness. To find the missing (spacetime-) orthogonal tangent vector to Pn, consider a curve μ(t) = (t, y(t)) in Pn with tangent vector μ̇(t)=t+ẏi(t)yi. Let the capital Latin indices run from 2, …, n. Now decompose ẏ(t)=ρ(t)u(t)+ξI(t)eI(t)Rn. By (3.28), we find that ρ(t)=ẏ(t)u(t)=ȧ(t)y(t)u̇(t). Hence, a future pointing tangent vector to Pn orthogonal in the spacetime to all XI is given by

X1(t,y):=t+ȧ(t)yu̇(t)ui(t)yi
(3.30)

so that we have constructed a smooth orthogonal tangent frame {Xi}i=1n for Pn. Hence, we can compute the (spacetime) unit normal to Pn to be

η(t,y)=ψ(s)Ñ(s)ȧ(t)yu̇(t)t+Ñ(s)ψ(s)ui(t)yiÑ2(s)ψ2(s)ȧ(t)yu̇(t)2.
(3.31)

In other words, using that ν(t,y)=ψ1(s)ui(t)yi, we have

α(t,y)=Ñ(s)Ñ2(s)ψ2(s)ȧ(t)yu̇(t)2,
(3.32)
β(t,y)=ψ(s)ȧ(t)yu̇(t)Ñ(s)Ñ2(s)ψ2(s)ȧ(t)yu̇(t)2.
(3.33)

We are now in a position to compute the second fundamental forms explicitly and take advantage of the umbilicity of Pn. Using u(t)ėJ(t)=u̇(t)eJ(t) for all tT, the condition h(X1,XJ)=0 gives

u̇(t)eJ(t)+1sψ(s)ψ(s)Ñ(s)Ñ(s)ȧ(t)yu̇(t)yeJ(t)=0
(3.34)

for J = 2, …, n. As {u(t),eJ(t)}J=2n is a δ-orthonormal frame, this is equivalent to

1sψ(s)ψ(s)Ñ(s)Ñ(s)ȧ(t)yu̇(t)ya(t)u(t)=u̇(t).
(3.35)

As Σn−1(t) has the dimension n − 1, (3.35) tells us that u̇(t)=0 for all tT by linear dependence considerations (otherwise, if the term in braces {…} vanishes). Hence, (3.35) simplifies to

ψ(s)ψ(s)Ñ(s)Ñ(s)ȧ(t)ya(t)u=0
(3.36)

so that, for a given tT, again using that Σn−1(t) has dimension n − 1 and linear dependence considerations, we find ȧ(t)=0 if the term in braces {…} does not vanish along Pn, i.e., when assuming (3.26).

Let us now compute the umbilicity factor λ, exploiting that u and a are constant. Note that eI is also constant, (3.32) and (3.33) reduce to α = 1 and β = 0, and using (3.31) and (3.30), we get η = ν and X1 = t. As eI is independent of y, we find

h(XI,XJ)=ψ(s)saδIJ
(3.37)

so that by (3.29), the photon surface umbilicity factor λ satisfies

λ(t,y)=λ(y)=ψ(s)sψ2(s)a
(3.38)

and is particularly independent of t. From h(X1,X1)=λp(X1,X1), we find

λ(y)=Ñ(s)sÑ(s)ψ(s)a.
(3.39)

Thus, (3.38) and (3.39) combine to

λ(y)=ψ(s)sψ2(s)a=Ñ(s)sÑ(s)ψ(s)a,
(3.40)

which implies that a = 0 and, indeed, λ(y) = λ = 0 is also independent of y when assuming that (3.26) holds along Pn. This shows that centered vertical hyperplanes are totally geodesic and that any photon surface Pn as in (3.28) along which (3.26) holds is (part of) a centered vertical hyperplane.

Let Σn−1(t) be as in (3.27) for all tT. We will show that c(t) = 0 for all tT. This then implies that Pn is contained in an isotropic photon surface as desired, namely, in a photon sphere with isotropic radius s = S* satisfying (3.57) or with isotropic radius profile s = S(t) as in (3.60). We will use the abbreviation

u(t,y):=yc(t)S(t)
(3.41)

to reduce the notational complexity.

The outward unit normal ν to Σn−1(t) in Mn(t) is given by

ν=ψ1(s)ui(t,y)yi.
(3.42)

Now, choose a smooth δ-orthonormal system of vectors eI(t, y) locally along Pn such that eI(t, y) · u(t, y) = 0 for all (t, y) ∈ Pn and set XI(t,y):=eIk(t,y)yk for all (t, y) ∈ Pn and all I = 2, …, n so that {XI(t,)}I=2n is an orthogonal frame for Σn−1(t) with respect to g̃ by conformal flatness. To find the missing (spacetime-) orthogonal tangent vector to Pn, consider a curve μ(t) = (t, y(t)) in Pn with the tangent vector μ̇(t)=t+ẏi(t)yi. Let capital Latin indices again run from 2, …, n. Now, decompose

ẏ(t)=ρ(t)u(t,y(t))+ξI(t)eI(t,y(t))Rn.
(3.43)

By (3.27) and the fact that ddtu(t,y(t))δ2=0 for all tT, we find

ρ(t)=ẏ(t)u(t,y(t))=ċ(t)u(t,y(t))+Ṡ(t).
(3.44)

Hence, a future pointing tangent vector to Pn orthogonal in the spacetime to all XI is given by

X1(t,y):=t+ċ(t)u(t,y)+Ṡ(t)ui(t,y)yi
(3.45)

so that we have constructed a smooth orthogonal tangent frame {Xi}i=1n for Pn. Hence, we can compute the outward (spacetime) unit normal to Pn to be

η(t,y)=ψ(s)Ñ(s)ċ(t)u(t,y)+Ṡ(t)t+Ñ(s)ψ(s)ui(t,y)yiÑ2(s)ψ2(s)ċ(t)u(t,y)+Ṡ(t)2.
(3.46)

In other words, using that ν(t,y)=ψ1(s)ui(t,y)yi, we have

α(t,y)=Ñ(s)Ñ2(s)ψ2(s)ċ(t)u(t,y)+Ṡ(t)2,
(3.47)
β(t,y)=ψ(s)ċ(t)u(t,y)+Ṡ(t)Ñ(s)Ñ2(s)ψ2(s)ċ(t)u(t,y)+Ṡ(t)2.
(3.48)

Extending the fields eJ trivially in the radial direction, let us first collect the following explicit formulas arising from differentiating eJ in direction u:

eIi(t,y)eJ,yi(t,y)u(t,y)=eIi(t,y)eJ(t,y)y,yiS(t)=δIJS(t),
(3.49)
ui(t,y)eJ,yi(t,y)u(t,y)=ui(t,y)eJ(t,y)y,yiS(t)=0.
(3.50)

Now, let us compute the second fundamental forms explicitly. Using (3.50), we find that the umbilicity condition h(X1,XJ)=0 gives

0=u(t,y)ėJ(t,y)+ċ(t)u(t,y)+Ṡ(t)ui(t,y)uk(t,y)×eJ,yki(t,y)+ψ(s)sψ(s)ykeJi(t,y)+yeJ(t,y)δkiyi(eJ)k(t,y)ċ(t)u(t,y)+Ṡ(t)Ñ(s)sÑ(s)yeJ(t,y)=ċ(t)S(t)eJ(t,y)+ċ(t)u(t,y)+Ṡ(t)1sψ(s)ψ(s)Ñ(s)Ñ(s)yeJ(t,y)

for all J = 2, …, n. As {u(t,y),eJ(t,y)}J=2n is a δ-orthonormal frame, this turns out to be equivalent to

ċ(t)S(t)=ċ(t)u(t,y)+Ṡ(t)1sψ(s)ψ(s)Ñ(s)Ñ(s)yċ(t)u(t,y)+Ṡ(t)1sψ(s)ψ(s)Ñ(s)Ñ(s)yu(t,y)+ċ(t)u(t,y)S(t)u(t,y).
(3.51)

As Σn−1(t) has the dimension n − 1 for all tT, (3.51) tells us by linear dependence considerations that ċ(t)=0 for all tT. Consequently, (3.51) simplifies to

0=Ṡ(t)ψ(s)ψ(s)Ñ(s)Ñ(s)y(yu(t,y))u(t,y).
(3.52)

Assuming (3.26), the term in braces {…} does not vanish and (3.52) implies that, for a fixed tT, Ṡ(t)=0 or y = (y · u(t, y))u(t, y) for all y ∈Σn−1(t). However, y = (y · u(t, y))u(t, y) for all y ∈ Σn−1(t) is equivalent to c = 0 and S(t) = s for all y ∈Σn−1(t), again by linear dependence considerations and as Σn−1(t) has dimension n − 1. In other words, assuming (3.26), we now know that c = 0 unless S is constant along Pn, the case in which a constant center c ≠ 0 is potentially possible.

Let us now continue with our computation of the conformal factor λ using the simplification ċ(t)=0 for all tT. We first treat the caseṠ(t)=0for alltT: We know that S(t) = S = s. Moreover, (3.47) and (3.48) give α = 1 and β = 0, and by (3.46) and (3.45), η = ν and X1 = t. Moreover, u(t, y) = u(y) and eJ(t, y) and, hence, XJ are independent of t. By (3.49) and y · u(y) = c · u(y) + S via (3.41), we find

h(XI,XJ)=ψ(S)S1+ψ(S)ψ(S)cu(y)+SδIJ,
(3.53)

so that, by (3.29), the photon surface umbilicity factor λ satisfies

λ(t,y)=1Sψ(S)1+ψ(S)ψ(S)cu(y)+S,
(3.54)

and thus, particularly, λ(t, y) = λ(y) is independent of t. Similarly, from h(X1,X1)=λp(X1,X1), we find

λ(y)=Ñ(S)SÑ(S)ψ(S)cu(y)+S,
(3.55)

and hence,

1+ψ(S)ψ(S)Ñ(S)Ñ(S)cu(y)+S=0
(3.56)

for all y ∈ Σn−1(t) and all tT. As Σn−1(t) has the dimension n − 1 and S is constant, we conclude that c · u(y) is constant and, hence, by (3.41) that c · y must be constant along Pn. Using again that Σn−1(t) has the dimension n − 1, this leads to c = 0 as desired. Hence, the isotropic radii s = S* for which this photon sphere can occur are the solutions of the implicit photon sphere equation,

1+ψ(S*)ψ(S*)Ñ(S*)Ñ(S*)S*=0,
(3.57)

provided that such solutions exist.

Let us now treat the other casec = 0: We find X1=t+Ṡ(t)ui(t,y)yi by (3.45), and indeed, eJ(t, y) = eJ(y) is independent of t as u(t,y)=yS(t). Moreover, s = S(t) holds for all (t, y) ∈ Pn. Thus, using (3.49), we can compute

h(XI,XJ)=ψ(S(t))S(t)1+ψ(S(t))ψ(S(t))S(t)δIJ,
(3.58)

so that, by (3.29), the photon surface umbilicity factor λ satisfies

λ(t,y)=Ñ(S(t))1+ψ(S(t))ψ(S(t))S(t)S(t)ψ(S(t))Ñ2(S(t))ψ2(S(t))Ṡ2(t)
(3.59)

from which we see that λ(t, y) = λ(t) only depends on t. From the remaining umbilicity condition h(X1,X1)=λp(X1,X1), we obtain

λ(t)=Ñ(S(t))ψ(S(t))Ñ2(S(t))ψ2(S(t))Ṡ2(t)3×Ñ(S(t))Ñ(S(t))ψ2(S(t))+S̈(t)+ψ(S(t))ψ(S(t))2Ñ(S(t))Ñ(S(t))Ṡ2(t)

and can conclude that the implicit equation,

1+ψ(S(t))ψ(S(t))S(t)Ñ2(S(t))ψ2(S(t))Ṡ2(t)=S(t)Ñ(S(t))Ñ(S(t))+S(t)ψ2(S(t))S̈(t)+ψ(S(t))ψ(S(t))2Ñ(S(t))Ñ(S(t))Ṡ2(t),
(3.60)

holds for the isotropic radius profile s = S(t).□

Now that we know that all photon surfaces except some vertical partial hyperplanes in “most” spacetimes of class S are spherically symmetric, let us gain a complementary perspective on these by relating spherically symmetric photon surfaces to null geodesics. The following notion of maximality will be useful for this endeavor.

Definition 3.12

(Maximal photon surface). Let(R×Mn,g)Sand letPn(R×Mn,g)be a connected spherically symmetric photon surface. We say thatPnis maximal ifPndoes not lie inside a strictly larger connected spherically symmetric photon surface.

If ζ:JR×Mn is a null geodesic in (R×Mn,g)S defined on some interval JR, then the energyE:=gζ(t|ζ,ζ̇) is constant along ζ. Exploiting the spherical symmetry of the spacetime, one can find a set {Xi}i=1(n1)(n2)2 of Killing vector fields tangential to the orbits of the spherical symmetry, which span (over R) the Killing subalgebra of the spacetime corresponding to spherical symmetry. Now, locally select a linearly independent system {XiI}I=1n1 among these and observe that this gives us

[XiI,XiJ]=:αIJKXiK
(3.61)

for some smooth coefficients αIJK satisfying αIJK=αJIK, where I, J, K = 1, …, n − 1, and where we have used that the spheres of symmetry are n − 1-dimensional. Next, we set I:=g(XiI,ζ̇), and note that I is constant along ζ. We then define the (total) angular momentumofζ as :=(GIJIJ)120, where (GIJ) is the inverse of (GIJ):=(gζ(XiI|ζ,XiJ|ζ)). Then, is constant along ζ because, locally, we have

dds2=GIAGJBddsGABIJ=GIAGJBgζ(ζ̇gXiA,XiB|ζ)+gζ(ζ̇gXiB,XiA|ζ)IJ=GIAGJBgζ(XiBgXiAζ,ζ̇)+gζ(XiAgXiBζ,ζ̇)IJ=GIAGJBαBAKgζ(XiK|ζ,ζ̇)IJ=GIAGJBαBAKIJK,

where s denotes the parameter along ζ and where we have used the Killing property of XiA and XiB to get from the second to the third line, (3.61) to get from the third to the fourth, and the definition of K to get from the fourth to the last one. As α is antisymmetric in its lower indices, must be constant along ζ. Moreover, is independent of the local choice of the linearly independent system {XiI}I=1n1, which can be seen as follows: Suppose that {YI}I=1n1 is another local linearly independent system of spherical Killing vector fields sharing an open domain with {XiI}I=1n1, set ĜIJ:=(gζ(YI|ζ,YJ|ζ)), let (ĜIJ) be the inverse of (ĜIJ), and set ̂I:=g(YI,ζ̇) for I, J = 1, …, n − 1. Observe that, by linear independence, YI=:TIKXiK for a smooth family of invertible matrices (TIK) with inverses ((T1)KI). Then,

ĜIJ=GKLTIKTJL,̂I=TIKK, and hence,ĜIĴÎJ=GAB(T1)AI(T1)BJTIKKTJLL=GABAB=2.

We will use the following definition of null geodesics generating a hypersurface.

Proposition 3.13.
Let(R×Mn,g)Sand letζ:JR×Mnandζ(s) = (t(s), r(s), ξ(s)) be a (not necessarily maximal) null geodesic defined on some open intervalJRwith angular momentum, whereξ(s)Sn1. Then,ζgenerates the hypersurfaceHζndefined as
Hζn:={(t,p)R×Mn|s*J,ξ*Sn1:t=t(s*),p=(r(s*),ξ*)},
which is a smooth, connected, and spherically symmetric hypersurface inR×Mn, which is timelike if > 0 and null if = 0.

Proof.

The claim that Hζn is a smooth spherically symmetric hypersurface is verified by recalling that r > 0 for all (t,p)R×Mn so that ζ*Sn1 is unique and by realizing that t = t(s) is injective along the null geodesic ζ, which shows that s* is unique. It is connected because the geodesic ζ is defined on an interval.

To show the claims about its causal character, set e0=ζ̇ and extend it to all of Hζn by making it invariant under the spherical symmetries. Thus, e0 is a null tangent vector field to Hζn, orthogonal to each time slice {t(s) = const.} of Hζn. Now, if = 0, it is easy to see that ξ̇=0 along ζ and, hence, e0gSn1 or, in other words, g(e0,X)=0 for any vector field X tangent to Hζn whence Hζn is null. Otherwise, if > 0, we will have ξ̇0 everywhere along ζ, and hence, e0 will not be g-orthogonal to Sn1 and, hence, g will induce a Lorentzian metric on Hζn.□

With these concepts at hand, we can write down a characterization of spherically symmetric photon surfaces via generating null geodesics.

Proposition 3.14.

Let(R×Mn,g)Sand letHn(R×Mn,g)be a connected spherically symmetric timelike hypersurface. Then,Hnis generated by a null geodesicζ:JR×Mnif and only ifHnis a photon surface. Moreover,Hnis a maximal photon surface if and only if any null geodesicζ:JR×MngeneratingHnis maximal.

The umbilicity factorλof a photon surfacePnis related to the energyEand angular momentumof its generating null geodesics byλ=E.

Proof.
First, assume that Hn is generated by a null geodesic ζ:JR×Mn so that Hn=Hζn and observe that Hn must then actually be ruled by the null geodesics arising by rotating ζ around Sn1. As Hn is timelike by assumption, we know from Definition and Proposition 3.13 that the angular momentum of ζ satisfies > 0. Proceeding as above, let {eI}I=1n1 be a local orthonormal system tangent to Sn1 along ζ, and set e0:=ζ̇ and extend it to all of Hζn by making it invariant under the spherical symmetries so that {e0,eI}I=1n1 is a local frame along ζ. Writing ζ(s) = (t(s), r(s), ξ(s)) for sJ as before, we can write
ζ̇(s)=ṫ(s)t+ṙ(s)r+ξ̇(s)
(3.62)
for sJ so that ζ being a null curve is equivalent to
f(r)ṫ2+ṙ2f(r)+r2|ξ̇|Ω2=0
(3.63)
along ζ. From this and the definition of energy E and angular momentum , we obtain
ṫ=Ef(r),
(3.64)
ṙ2=E22f(r)r2,
(3.65)
|ξ̇|Ω2=2r4
(3.66)
along ζ. To compute the second fundamental form h of Hn(R×Mn,g), let us first compute the outward (growing r) spacelike unit normal η to Hn. By spherical symmetry, η must be a linear combination of t and r with no angular contribution. It must also be orthogonal to e0=ζ̇. From this, we find
η=1rṙf(r)t+Err,
(3.67)
where we have used (3.64) and (3.65). For I = 1, …, n − 1, we find by a direct computation (exploiting spherical symmetry) that
eIgη=EeI,
(3.68)
and hence,
h(eI,eβ)=g(eIgη,eβ)=Eg(eI,eβ)
(3.69)
for I = 1, …, n − 1 and β = 0, …, n − 1. On the other hand, smoothly extending e0 to a neighborhood of Hn, one finds that
h(e0,e0)=g(e0gη,e0)=g(e0ge0,η)=0
(3.70)
holds as e0=ζ̇ and ζ is a geodesic. Hence, by symmetry of h and g,
h(eα,eβ)=Eg(eα,eβ)
(3.71)
for α, β = 0, …, n − 1 so that Hn is totally umbilic with umbilicity factor λ=E.

Conversely, assume that Hn is a spherically symmetric photon surface. Let qHn and let XTqHn be any null tangent vector. Let ζ:Jmax0R×Mn be the maximal null geodesic with ζ(0) = q, ζ̇(0)=X. As Hn is totally umbilic, ζ must remain tangent to Hn on some maximal open interval 0 ∈ JJmax by Proposition 2.2. By spherical symmetry, Hn is, in fact, generated by ζ|J.

It remains to discuss the claim about maximality, which is a straightforward consequence of the above argument about the choice of the interval JJmax.□

Remark 3.15.
In view of effective one-body dynamics (see Ref.37 ), it may be of interest to point out, as a computation shows, that a spherically symmetric photon surfacePnin a static spherically symmetric spacetime with metric of the form
g=Gdt2+1fdr2+r2Ω,
whereG = G(r) andf = f(r) are smooth, positive functions on an open intervalI ⊆ (0, ), will have a constant umbilicity factorλif and only ifG = κfalongPnfor someκ > 0, withλ=Ef(r)G(r)characterizing the umbilicity factor along any generating null geodesic with energyEand angular momentum.

Remark 3.16.

Proposition 3.14 allows us to conclude the existence of (maximal) spherically symmetric photon surfaces in spacetimes of classSfrom the existence of (maximal) null geodesics.

A different view on spherically symmetric photon surfaces in spacetimes of class S can be gained by lifting them to the phase space (i.e., to the cotangent bundle). We will end this section by proving a partitioning property of the null section of phase space by maximal photon surfaces and by the so-called maximal principal null hypersurfaces.

Definition 3.17

[(Maximal) Principal null hypersurfaces]. Let(R×Mn,g)S,ζ:JR×Mnbe a null geodesic, andbe its angular momentum. Then,ζis called a principal null geodesic if = 0. A hypersurfaceHζngenerated by a (maximal) principal null geodesicζwill be called a (maximal) principal null hypersurface.

Recall that from the Definition and Proposition 3.13, we know that principal null hypersurfaces are spherically symmetric and indeed null. Arguing as in the Proof of Proposition 3.14, one sees that a maximal principal null hypersurface will be maximal in the sense that it is not contained in any strictly larger principal null hypersurface. In particular, if one generating null geodesic is maximal, then all of them are. Finally, principal null hypersurfaces are connected by definition.

With these considerations at hand, let us prove the following partitioning property of the null section of the phase space of any spacetime of class S. To express this, we will canonically lift the null bundles over the involved spherically symmetric photon surfaces and principal null hypersurfaces to the null section of the phase space. Recall that maximal spherically symmetric photon surfaces are connected by definition.

Proposition 3.18.

Let(R×Mn,g)S. Then, the null section of the phase space of(R×Mn,g),N:={ωT*(R×Mn)|g(ω#,ω#)=0}, is partitioned by the canonical liftsN(Pn)Nof the null bundles over all maximal spherically symmetric photon surfacesPnand the canonical liftsN(Hn)Nof the null bundles over all maximal principal null hypersurfacesHn.

Proof.

Consider ωN and let ζ:J0R×Mn be the unique maximal null geodesic satisfying ζ̇(0)=ω#. Let Hζn be the hypersurface of R×Mn generated by ζ, and note that ωN(Hζn) holds by construction. By Definition and Propositions 3.13 and 3.14 and Definition 3.17, we know that Hζn is a maximal photon surface if the angular momentum of ζ is positive and a maximal principal null hypersurface if = 0 vanishes. As ≥ 0, this shows that ω is either contained in at least one canonical lift of a maximal photon surface or contained in at least one canonical lift of a maximal principal null hypersurface. Furthermore, ω cannot lie in the canonical lifts of two different maximal spherically symmetric photon surfaces because these would both be generated by ζ and hence coincide by Proposition 3.14. Finally, ω cannot lie in the canonical lifts of two different maximal principal null hypersurfaces, which can be seen by repeating the arguments in the Proof of Proposition 3.14.□

As discussed in Sec. III (see also Fig. 1), the Schwarzschild spacetime of mass m > 0 in n + 1 dimensions possesses not only the well-known photon sphere at r=(nm)1n2 but also many other photon surfaces. Except for the planar ones, all these Schwarzschild photon surfaces are spherically symmetric and thus particularly equipotential, as defined in Sec. II. In this section, we will prove the following theorem that can be considered complementary to Corollary 3.9 in the context of static, vacuum, and asymptotically flat spacetimes.

Theorem 4.1.

Let(Ln+1,g)be a static, vacuum, and asymptotically isotropic spacetime of massm. Assume that(Ln+1,g)is geodesically complete up to its inner boundaryL, which is assumed to be a (possibly disconnected) photon surface,L=:Pn. Assume, in addition, thatPnis equipotential, outward directed, and has compact time slices Σn−1(t) = PnMn(t). Then,(Ln+1,g)is isometric to a suitable piece of the Schwarzschild spacetime of massm, and in fact,m > 0. In particular,Pnis connected and is (necessarily) a spherically symmetric photon surface in Schwarzschild spacetime.

The proof relies on the following theorem by the first author.

Theorem 4.2
(Ref. 23). Assumen ≥ 3 and letMnbe a smooth connectedn-dimensional manifold with non-empty, possibly disconnected, smooth compact inner boundaryM=.i=1IΣin1. Letgbe a smooth Riemannian metric onMn. Assume that (Mn, g) has a non-negative scalar curvature,
R0,
and that it is geodesically complete up to its inner boundary∂M. Assume, in addition, that (Mn, g) is asymptotically isotropic with one end of massmR. Assume that the inner boundary∂Mis umbilic in (Mn, g) and that each componentΣin1has constant mean curvatureHiwith respect to the outward pointing unit normalνi. Furthermore, assume that there exists a functionu:MnRwithu > 0 away from∂M, which is smooth and harmonic on (Mn, g),
gu=0.
We ask thatuis asymptotically isotropic of the same massmsuch thatu|Σin1:uiand the normal derivative ofuacrossΣin1,νi(u)|Σin1:ν(u)i, are constant on eachΣin1. Finally, we assume that for eachi = 1, …, I, we are either in the semi-static horizon case,
Hi=0,ui=0,ν(u)i0,
(4.1)
or in the true CMC case Hi > 0,ui > 0, and there existsci>n2n1such that
Rσi=ciHi2,
(4.2)
2ν(u)i=cin2n1Hiui,
(4.3)
whereRσidenotes the scalar curvature ofΣin1with respect to its induced metricσi.

Then,m > 0 and (Mn, g) is isometric to a suitable piece(M̃mn\BS(0),g̃m)of the (isotropic) Schwarzschild manifold of massmwithSsm. Moreover,ucoincides with the restriction ofũm(up to the isometry), and the isometry is smooth.

Remark 4.3

(Generalization). Our Proof of Theorem 4.1 makes use of the static vacuum Einstein equations(2.4),(2.5), and(2.6). In fact, as we will see in the proof in the following, it is sufficient to ask that the vacuum Einstein equations hold in a neighborhood ofPn; outside this neighborhood, it suffices thatgN = 0 and that the dominant energy conditionR ≥ 0 holds.

Remark 4.4

(Multiple ends). As Theorem 4.2 generalizes to multiple ends (see Ref.23 ), Theorem 4.1 also readily applies in the case of multiple ends satisfying the decay conditions(2.14)and(2.17)with potentially different massesmiin each endEi. Note that, in each endEi, it is necessary that bothgijandNhave the same massmiin their expansions.

Remark 4.5
[Discussion of η(N) > 0]. The assumption thatPnis outward directed,η(N) > 0 (hencedN ≠ 0) alongPn, can be removed if instead, one assumes thatm > 0 a priori and thatPnis connected. Using the Laplace equationgN = 0 and the divergence theorem as well as the asymptotics(2.14)and(2.17), one computes
1ωn1Σn1ν(N)dA=m>0,
where Σn−1 := Pn ∩ {t = const} andωn−1is the volume of(Sn1,Ω)(see Definition 4.2.1 of Ref.7 for then = 3 case); the argument is identical in higher dimensions. From this and connectedness of Σn−1, we can deduce thatν(N) > 0 and thus, particularly,η(N) > 0 at least in an open subset of Σn−1. However, we will see in the Proof of Theorem 4.1 that this necessarily implies thatν(N) ≡ const > 0 in this open neighborhood (noting that all computations performed there are purely local). As Σn−1is connected, we obtainν(N) ≡ const > 0, and thus, in particular,η(N) > 0 everywhere on Σn−1[seeEq. (4.6)in the following].

Proof.

We write Mn(t) for the time slice {t} × M (cf. Remark 2.4) and consider each connected component Pin, i = 1, …, I of Pn separately. For the component of Pin under consideration, let Σin1(t):=PinMn(t). We will drop the explicit reference to i in the following and only start reusing it toward the end of the proof, where we bring in global arguments.

Let ν denote the outward unit normal to Σn−1(t) ↪ (Mn(t), g), pointing to the asymptotically isotropic end. Let η denote the outward unit normal of Pn(R×Mn,g). As N =: u(t) on Σn−1(t) and because we assumed η(N) > 0 and hence ν(N) > 0 [see (4.6) below] on Pn, we have
ν=gradN|gradN|=gradN|dN|=gradNν(N).
(4.4)
Now, let μ(s) = (s, x(s)) be a curve in Pn, i.e., Nμ(s) = u(s). This implies by chain rule that dN(μ̇)=u̇. If μ̇(t)Σn1(t), the tangent vector of μ can be computed explicitly as
Z:=μ̇=t+ẋ=t+u̇ν(N)νΓ(TPn).
(4.5)
Expressed in words, Z is the vector field going “straight up” along Pn.
The explicit formula (4.5) for Z allows us to explicitly compute the spacetime unit normal η to Pn too: It has to be perpendicular to Z and its projection onto Mn has to be proportional to ν. From this, we find
η=ν+u̇u2ν(N)2t1u̇2u2ν(N)2.
(4.6)
From umbilicity of Pn(R×Mn,g), it follows that the corresponding second fundamental form h of Pn(R×Mn,g) satisfies
h=1nHp,
(4.7)
where p is the induced metric on Pn and H:=trph. From Proposition 3.3 in Ref. 9, we know that Hconst. Equations (4.7) implies particularly that, for any tangent vector fields X, Y ∈ Γ(TΣn−1(t)), we have
h(X,Y)=1nHσ(X,Y),
(4.8)
where σ denotes the induced metric on Σn−1(t). Now, extend X, Y arbitrarily smoothly along Pn such that they remain tangent to Σtn1. We compute
h(X,Y)=gXgY,η=11u̇2u2ν(N)2gXgY,ν+u̇u2ν(N)2t=11u̇2u2ν(N)2gXgY,ν+u̇uν(N)2gXgY,1ut=11u̇2u2ν(N)2gXgY,νu̇uν(N)2KX,Y=K=011u̇2u2ν(N)2gXgY,ν=g static11u̇2u2ν(N)2gXgY,ν=11u̇2u2ν(N)2h(X,Y),
(4.9)
where K = 0 denotes the second fundamental form of Mn(t)(R×Mn,g) and h denotes the second fundamental form of Σn−1(t) ↪ ({tMn, g). In particular, Σn−1(t) ↪ ({tMn, g) is umbilic and its mean curvature H inside Mn can be computed as
H=n1nH1u̇2u2ν(N)2
(4.10)
when combining (4.8) with (4.9). Of course, then h=1n1Hσ. We will now proceed to show that H and ν(N) are constant for each fixed t. Consider first the contracted Codazzi equation for Σn−1(t) ↪ ({tMn, g). It gives
Ric(X,ν)=n2n1X(H).
(4.11)
On the other hand, using the static equation gives
X(ν(N))=Nu(t)X(ν(N))Xgν(N)=2gN(X,ν)=NRic(X,ν)=(4.11)n2n1uX(H).
(4.12)
Furthermore, (4.10) allows us to compute
X(H)=(n1)Hn1u̇2u2ν(N)2u̇2u2ν(N)3X(ν(N))=(4.12)(n2)u̇2Hnuν(N)31u̇2u2ν(N)2X(H).
(4.13)
Assume that X(H) ≠ 0 in some open subset U ⊂ Σn−1(t). Then, in U, we have
nuν(N)31u̇2u2ν(N)2=(n2)u̇2Hν(N)61u̇2u2ν(N)2=(n2)2u̇4H2n2u2ν(N)6u̇2u2ν(N)4(n2)2u̇4h2n2u2=0.
This is a polynomial equation for ν(N) with coefficients that only depend on t. As a consequence, ν(N) has to be constant in U. However, from (4.12), we know that then, also H has to be constant in U, a contradiction to X(H) ≠ 0 in U. Thus, H and, by (4.12), also ν(N) are constants along Σn−1(t) and only depend on t. From now on, we will drop the explicit reference to t and also go back to using N instead of u(t) as the remaining part of the proof applies to each t separately.
Thus, each Σn−1 = Σn−1(t) is an umbilic, CMC, equipotential surface in Mn with ν(N) being constant too. From the usual decomposition of the Laplacian on functions and the static vacuum Eq. (2.6), we find
0=gN=σN+NRic(ν,ν)+Hν(N)=NRic(ν,ν)+Hν(N)
so that Ric(ν,ν)=Hν(N)N must also be constant along Σn−1.
Plugging this into the contracted Gauß equation and using R = 0, we obtain
2Ric(ν,ν)=Rσn2n1H2Rσ=2Hν(N)N+n2n1H2.
(4.14)
This shows that (Σn−1, σ) also has constant scalar curvature. Now, define the constant c>n2n1 by
c:=n2n1+2ν(N)NH.
(4.15)
Together with (4.14), this definition of c ensures that
Rσ=cH2,
(4.16)
2ν(N)=cn2n1HN.
(4.17)

Let us summarize as follows: Fix t and keep dropping the explicit reference to it. Then, each component of (Σn−1, σ)↪(Mn, g) is umbilic, CMC, equipotential and has constant scalar curvature and constant ν(N), and all these constants together satisfy Eqs. (4.16) and (4.17) with constant c given by (4.15), which is potentially different for each component of Σn−1. Recall the assumption that (Mn, g) is geodesically complete up to its inner boundary, so in particular, (Mn \ K, g) is geodesically complete up to Σn−1, where K is the compact set such that Σn1=Mn\K. Moreover, (Mn \ K, g) satisfies the static vacuum equations. Altogether, these facts ensure that Theorem 4.2 applies. Thus, (Mn \ K, g) is isometric to a spherically symmetric piece of the spatial Schwarzschild manifold of mass m given by the asymptotics (2.14) and (2.17), and N corresponds to the Schwarzschild lapse function of the same mass m under this isometry. The area radius r of the inner boundary Σn−1 in the spatial Schwarzschild manifold is determined by Rσ=(n1)(n2)r2.

Thus, recalling the dependence on t, the manifold [{t}× (Mn \ K(t)), g] outside the photon surface time slice Σn−1(t) is isometric to a piece of the spatial Schwarzschild manifold of mass m with inner boundary area radius r(t). In particular, Σn−1(t) is a connected sphere and we find m > 0 (as per Theorem 1.1 in Ref. 23). Recombining the time slices Σn−1(t) to the photon surface Pn, this shows that the part of the spacetime (R×Mn,g) lying outside the photon surface Pn is isometric to a piece of the Schwarzschild spacetime of mass m and m > 0 necessarily. Moreover, Pn is connected and its isometric image in the Schwarzschild spacetime is spherically symmetric with the radius profile r(t).□

We point out in connection with Remarks 4.4 and 4.3 that the assumptions of Theorem 4.2 keep being met if we start with several ends and the static vacuum equations only holding near Σn−1, with △gN = 0 everywhere in Mn as all the above computations and arguments were purely local near Pn.

The authors thank Gary Gibbons, Sophia Jahns, Volker Perlick, Volker Schlue, Olivia Vičánek Martínez, and Bernard Whiting for helpful comments and questions.

The authors thank the Mathematisches Forschungsinstitut Oberwolfach, the University of Vienna, and the Tsinghua Sanya International Mathematics Forum for allowing us to collaborate in stimulating environments.

C.C. is indebted to the Baden-Württemberg Stiftung for the financial support of this research project by the Elite Programme for Postdocs. The work of C.C. was supported by the Institutional Strategy of the University of Tübingen (Deutsche Forschungsgemeinschaft, Grant No. ZUK 63) and the focus program on Geometry at Infinity (Deutsche Forschungsgemeinschaft, Grant No. SPP 2026). The work of G.J.G. was partially supported by the NSF (Grant Nos. DMS-1313724 and DMS-1710808).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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