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.

1.
W.
Israel
, “
Event horizons in static vacuum space-times
,”
Phys. Rev.
164
(
5
),
1776
1779
(
1967
).
2.
G. L.
Bunting
and
A. K. M.
Masood-ul Alam
, “
Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time
,”
Gen. Relativ. Gravitation
19
(
2
),
147
154
(
1987
).
3.
M.
Heusler
,
Black Hole Uniqueness Theorems
, Cambridge Lecture Notes in Physics (
Cambridge University Press
,
1996
).
4.
D.
Robinson
, “
Four decades of black hole uniqueness theorems
,” in
The Kerr Spacetime: Rotating Black Holes in General Relativity
, edited by
S. M.
Scott
,
D. L.
Wiltshire
, and
M.
Visser
(
Cambridge University Press
,
2009
), pp.
115
143
.
5.
S.
Raulot
, “
A spinorial proof of the rigidity of the Riemannian Schwarzschild manifold
,” arXiv:2009.09652v1 (
2020
).
6.
V.
Agostiniani
and
L.
Mazzieri
, “
On the Geometry of the level sets of static potentials
,”
Commun. Math. Phys.
355
(
1
),
261
301
(
2017
).
7.
C.
Cederbaum
, “
The Newtonian limit of geometrostatics
,” Ph.D. thesis,
FU Berlin
,
2012
; arXiv:1201.5433v1.
8.
C.-M.
Claudel
,
K. S.
Virbhadra
, and
G. F. R.
Ellis
, “
The Geometry of photon surfaces
,”
J. Math. Phys.
42
(
2
),
818
839
(
2001
).
9.
C.
Cederbaum
, “
Uniqueness of photon spheres in static vacuum asymptotically flat spacetimes
,” in
Complex Analysis and Dynamical Systems VI
, Contemporary Mathematics Vol. 667 (
AMS
,
2015
), pp.
86
99
.
10.
C.
Cederbaum
and
G. J.
Galloway
, “
Uniqueness of photon spheres via positive mass rigidity
,”
Commun. Anal. Geom.
25
(
2
),
303
320
(
2017
).
11.
C.
Cederbaum
and
G. J.
Galloway
, “
Uniqueness of photon spheres in electro-vacuum spacetimes
,”
Classical Quantum Gravity
33
(
7
),
075006
(
2016
).
12.
G. W.
Gibbons
and
C. M.
Warnick
, “
Aspherical photon and anti-photon surfaces
,”
Phys. Lett. B
763
,
169
173
(
2016
).
13.
S. S.
Yazadjiev
, “
Uniqueness of the static spacetimes with a photon sphere in Einstein-scalar field theory
,”
Phys. Rev. D
91
(
12
),
123013
(
2015
).
14.
S.
Yazadjiev
and
B.
Lazov
, “
Uniqueness of the static Einstein–Maxwell spacetimes with a photon sphere
,”
Classical Quantum Gravity
32
(
16
),
165021
(
2015
).
15.
S. S.
Yazadjiev
and
B.
Lazov
, “
Classification of the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes with a photon sphere
,”
Phys. Rev. D
93
(
8
),
083002
(
2016
).
16.
S.
Jahns
, “
Photon sphere uniqueness in higher-dimensional electrovacuum spacetimes
,”
Classical Quantum Gravity
36
(
23
),
235019
(
2019
).
17.
A. A.
Shoom
, “
Metamorphoses of a photon sphere
,”
Phys. Rev. D
96
(
8
),
084056
(
2017
).
18.
Y.
Tomikawa
,
T.
Shiromizu
, and
K.
Izumi
, “
On uniqueness of static spacetimes with non-trivial conformal scalar field
,”
Classical Quantum Gravity
34
(
15
),
155004
(
2017
).
19.
Y.
Tomikawa
,
T.
Shiromizu
, and
K.
Izumi
, “
On the uniqueness of the static black hole with conformal scalar hair
,”
Prog. Theor. Exp. Phys.
2017
(
3
),
033E03
.
20.
H.
Yoshino
, “
Uniqueness of static photon surfaces: Perturbative approach
,”
Phys. Rev. D
95
,
044047
(
2017
).
21.
V.
Perlick
, “
On totally umbilic submanifolds of semi-Riemannian manifolds
,”
Nonlinear Anal.
63
(
5–7
),
e511
e518
(
2005
).
22.
C.
Cederbaum
,
S.
Jahns
, and
O. V.
Martínez
, “
On equipotential photon surfaces in electrostatic spacetimes of arbitrary dimension
” (unpublished) (
2020
).
23.
C.
Cederbaum
, “
Rigidity properties of the Schwarzschild manifold in all dimensions
” (unpublished) (
2020
).
24.
G. W.
Gibbons
,
D.
Ida
, and
T.
Shiromizu
, “
Uniqueness and non-uniqueness of static vacuum black holes in higher dimensions
,”
Prog. Theor. Phys. Suppl.
148
,
284
290
(
2002
).
25.
R.
Schoen
and
S.-T.
Yau
, “
On the Proof of the positive mass conjecture in general relativity
,”
Commun. Math. Phys.
65
(
1
),
45
76
(
1979
).
26.
E.
Witten
, “
A new proof of the positive energy theorem
,”
Commun. Math. Phys.
80
(
3
),
381
402
(
1981
).
27.
R. M.
Schoen
and
S.-T.
Yau
, “
Positive scalar curvature and minimal hypersurface singularities
,” arXiv:1704.05490v1 (
2017
).
28.
P.
Miao
, “
Positive mass theorem on manifolds admitting corners along a hypersurface
,”
Adv. Theor. Math. Phys.
6
,
1163
1182
(
2002
).
29.
D.
McFeron
and
G.
Székelyhidi
, “
On the positive mass theorem for manifolds with corners
,”
Commun. Math. Phys.
313
(
2
),
425
443
(
2012
).
30.
D. A.
Lee
and
P. G.
LeFloch
, “
The positive mass theorem for manifolds with distributional curvature
,”
Commun. Math. Phys.
339
,
99
120
(
2015
).
31.
F. R.
Tangherlini
, “
Schwarzschild field in n dimensions and the dimensionality of space problem
,”
Nuovo Cimento
27
,
636
651
(
1963
).
32.
D.
Kennefick
and
N. Ó.
Murchadha
, “
Weakly decaying asymptotically flat static and stationary solutions to the Einstein equations
,”
Classical Quantum Gravity
12
(
1
),
149
(
1995
).
33.
R.
Arnowitt
,
S.
Deser
, and
C. W.
Misner
, “
Coordinate invariance and energy expressions in general relativity
,”
Phys. Rev.
122
(
3
),
997
1006
(
1961
).
34.
R.
Bartnik
, “
The mass of an asymptotically flat manifold
,”
Commun. Pure Appl. Math.
39
,
661
693
(
1986
).
35.
C.
Cederbaum
and
M.
Wolff
, “
Some new perspectives on the Kruskal–Szekeres extension with applications to photon surfaces
” (unpublished) (
2020
).
36.
T.
Foertsch
,
W.
Hasse
, and
V.
Perlick
, “
Inertial forces and photon surfaces in arbitrary spacetimes
,”
Classical Quantum Gravity
20
(
21
),
4635
4651
(
2003
).
37.
A.
Buonanno
and
T.
Damour
, “
Effective one-body approach to general relativistic two-body dynamics
,”
Phys. Rev. D
59
,
084006
(
1999
).
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