Supplemental Materials: Constraints on low-energy effective theories from weak
cosmic censorship
Baoyi Chen,
1
Feng-Li Lin,
2
Bo Ning,
3
and Yanbei Chen
1
1
Burke Institute of Theoretical Physics and Theoretical Astrophysics 350-17,
California Institute of Technology, Pasadena, California 91125
2
Department of Physics, National Taiwan Normal University,
No. 88, Sec. 4, Ting-Chou Road, Taipei 11677, Taiwan
3
College of Physics, Sichuan University, Chengdu, Sichuan 610064, China
I. CORRECTIONS TO THE MAXWELL SOURCE AND STRESS TENSOR
We consider the most general fourth-derivative higher order corrections to Einstein-Maxwell theory, namely,
I
=
∫
d
4
x
√
−
g
(
1
2
κ
R
−
1
4
F
μν
F
μν
+ ∆
L
)
(1)
where
∆
L
=
c
1
R
2
+
c
2
R
μν
R
μν
+
c
3
R
μνρσ
R
μνρσ
(2)
+
c
4
RF
μν
F
μν
+
c
5
R
μν
F
μρ
F
ν
ρ
+
c
6
R
μνρσ
F
μν
F
ρσ
+
c
7
F
μν
F
μν
F
ρσ
F
ρσ
+
c
8
F
μν
F
νρ
F
ρσ
F
σμ
.
The field equations obtained by the variation of the action (1) with respect to
A
μ
and
g
μν
are given respectively by
∇
ν
(
F
μν
−
S
μν
) = 0
,
(3)
and
R
μν
−
1
2
g
μν
R
=
κT
μν
=
κ
(
̃
T
μν
+ ∆
T
μν
)
.
(4)
In the above
̃
T
μν
=
F
μ
ρ
F
νρ
−
1
4
g
μν
F
ρσ
F
ρσ
is the stress tensor of the Maxwell theory, and ∆
T
μν
and
S
μν
are the
corrections respectively to the stress tensor and Maxwell source field from the higher-dimension operators.
Here we list the details of the corrections to the Maxwell source field and stress tensor and , i.e.,
S
μν
in Eq. (9) and
∆
T
μν
in Eq. (10) of the main text:
S
μν
= 4
c
4
RF
μν
+ 2
c
5
(
R
μρ
F
ρ
ν
−
R
νρ
F
ρ
μ
) + 4
c
6
R
μνρσ
F
ρσ
+
+ 8
c
7
F
ρσ
F
ρσ
F
μν
+ 8
c
8
F
ρσ
F
ρν
F
μσ
,
(5)
and
∆
T
μν
=
c
1
(
g
μν
R
2
−
4
RR
μν
+ 4
∇
ν
∇
μ
R
−
4
g
μν
R
)
+
+
c
2
(
g
μν
R
ρσ
R
ρσ
+ 4
∇
α
∇
ν
R
α
μ
−
2
R
μν
−
g
μν
R
−
4
R
α
μ
R
αν
)
+
+
c
3
(
g
μν
R
αβγδ
R
αβγδ
−
4
R
μαβγ
R
ν
αβγ
−
8
R
μν
+4
∇
ν
∇
μ
R
+ 8
R
α
μ
R
αν
−
8
R
αβ
R
μανβ
)
+
+
c
4
(
g
μν
RF
2
−
4
RF
μ
σ
F
νσ
−
2
F
2
R
μν
+ 2
∇
μ
∇
ν
F
2
−
2
g
μν
F
2
)
+
+
c
5
(
g
μν
R
κλ
F
κρ
F
λ
ρ
−
4
R
νσ
F
μρ
F
σρ
−
2
R
αβ
F
αμ
F
βν
)
−
g
μν
∇
α
∇
β
(
F
α
ρ
F
βρ
+ 2
∇
α
∇
ν
(
F
μβ
F
αβ
)
−
(
F
μρ
F
ν
ρ
)
)
+
+
c
6
(
g
μν
R
κλρσ
F
κλ
F
ρσ
−
6
F
αν
F
βγ
R
α
μβγ
−
4
∇
β
∇
α
(
F
α
μ
F
β
ν
)
)
+
+
c
7
(
g
μν
(
F
2
)
2
−
8
F
2
F
μ
σ
F
νσ
)
+
+
c
8
(
g
μν
F
ρκ
F
ρσ
F
σλ
F
κλ
−
8
F
μ
ρ
F
ν
σ
F
ρ
κ
F
σκ
)
.
(6)
Note that
F
2
=
F
ρσ
F
ρσ
and
=
∇
a
∇
a
.
2
II. CORRECTIONS TO THE REISSNER-NORDSTR
̈
OM BLACK HOLE
The functions
λ
(
r
) and
ν
(
r
) are related to the components of Ricci curvature tensor
R
μν
via
1
2
(
R
t
t
−
R
r
r
)
−
R
θ
θ
=
1
r
2
d
dr
[
r
(
e
−
λ
(
r
)
−
1)
]
,
(7)
R
t
t
−
R
r
r
=
−
e
−
λ
(
r
)
r
[
ν
′
(
r
) +
λ
′
(
r
)]
.
To solve for
λ
and
ν
explicitly, we need an additional boundary condition. Assuming that at
r
→ ∞
the metric
approaches the Schwarzschld solution, the results are then given by
e
−
λ
(
r
)
= 1
−
κM
4
πr
−
1
r
∫
∞
r
dr r
2
[
1
2
(
R
t
t
−
R
r
r
)
−
R
θ
θ
]
,
(8)
ν
(
r
) =
−
λ
(
r
) +
∫
∞
r
dr r
(
R
t
t
−
R
r
r
)
e
λ
(
r
)
.
We further take the trace-reverse of Eq. (10) from the main text and obtain that
R
μν
=
κ
(
T
μν
−
1
2
Tg
μν
)
,
(9)
where
T
is the trace of the total energy-momentum tensor
T
μν
, and is given by
T
=
T
t
t
+
T
r
r
+ 2
T
θ
θ
. Plugging the
trace-reversed Einstein field equation into the integral expression (8), we get
e
−
λ
(
r
)
= 1
−
κM
4
πr
−
κ
r
∫
∞
r
dr r
2
T
t
t
,
(10)
ν
(
r
) =
−
λ
(
r
) +
κ
∫
∞
r
dr r
(
T
t
t
−
T
r
r
)
e
λ
(
r
)
.
Once we know the diagonal components of the energy-momentum tensor, it will be straightforward to compute the
corrections to the spherically symmetric static spacetime as induced by
T
μν
.
We now take our background spacetime to be Reissner-Nordstr ̈om black hole in four-dimension. That is,
e
ν
(0)
=
e
−
λ
(0)
= 1
−
κM
4
πr
+
κQ
2
32
π
2
r
2
,
(11)
F
(0)
μν
dx
μ
∧
dx
ν
=
Q
4
πr
2
dt
∧
dr.
Here
ν
(0)
(
r
) and
λ
(0)
(
r
) refer to the metric components in the unperturbed black hole spacetime, and
F
(0)
μν
is the
background electromagnetic energy-momentum tensor. Considering the action in Eq. (2) of the main text, we treat
the corrections from higher-dimension operators as perturbations. For convenience, we also introduce a power counting
parameter
ε
, and consider a one-parameter family of actions
I
ε
, which is given by
I
ε
=
∫
d
4
x
√
−
g
(
L
0
+
ε
∆
L
)
.
(12)
The original action will be recovered after setting
ε
= 1. We then expand everthing into powers series in
ε
. For
instance,
g
μν
=
g
(0)
μν
+
εh
(1)
μν
+
O
(
ε
2
)
,
F
μν
=
F
(0)
μν
+
εf
(1)
μν
+
O
(
ε
2
)
.
(13)
At order
ε
1
, the stress energy tensor is given by
T
(1)
μν
=
̃
T
μν
[
g
(0)
,f
(1)
,F
(0)
] +
̃
T
μν
[
h
(1)
,F
(0)
,F
(0)
] + ∆
T
μν
[
g
(0)
,F
(0)
]
.
(14)
Noting that in order to compute the corrections to the metric, we need to calculate
T
μ
ν
instead of
T
μν
. At order
ε
1
,
T
μ
ν
(1)
is given by
T
μ
ν
(1)
=
̃
T
μ
ν
[
g
(0)
,F
(1)
] + ∆
T
μ
ν
[
g
(0)
,F
(0)
]
.
(15)
3
We solve for the corrections to Maxwell equations, and obtain that the nonzero components of
f
(1)
μν
are
f
(1)
tr
=
−
f
(1)
rt
=
1
32
π
3
r
6
(
c
5
κQ
3
−
16
πc
6
κMQr
+ 6
c
6
κQ
3
+ 8
c
7
Q
3
+ 4
c
8
Q
3
)
.
(16)
This corresponds to the gauge field
A
a
given by
A
t
=
−
q
r
+
2
q
3
5
r
5
(
c
5
κ
+ 6
c
6
κ
−
5
c
6
κmr
q
2
+ 8
c
7
+ 4
c
8
)
,
A
r
=
A
θ
=
A
φ
= 0
.
(17)
With the corrections to
F
μν
, we can solve for the corrected energy-momentum tensor
T
μ
ν
(1)
. We then find the
corrected metric tensor component to be
e
−
λ
=1
−
κm
r
+
κq
2
2
r
2
+
c
2
(
3
κ
3
mq
2
r
5
−
6
κ
3
q
4
5
r
6
−
4
κ
2
q
2
r
4
)
+
c
3
(
12
κ
3
mq
2
r
5
−
24
κ
3
q
4
5
r
6
−
16
κ
2
q
2
r
4
)
+
c
4
(
14
κ
2
mq
2
r
5
−
6
κ
2
q
4
r
6
−
16
κq
2
r
4
)
+
c
5
(
5
κ
2
mq
2
r
5
−
11
κ
2
q
4
5
r
6
−
6
κq
2
r
4
)
+
c
6
(
7
κ
2
mq
2
r
5
−
16
κ
2
q
4
5
r
6
−
8
κq
2
r
4
)
+
c
7
(
−
4
κq
4
5
r
6
)
+
c
8
(
−
2
κq
4
5
r
6
)
,
e
+
ν
=1
−
κm
r
+
κq
2
2
r
2
+
c
2
(
κ
3
mq
2
r
5
−
κ
3
q
4
5
r
6
−
2
κ
2
q
2
r
4
)
(18)
+
c
3
(
4
κ
3
mq
2
r
5
−
4
κ
3
q
4
5
r
6
−
8
κ
2
q
2
r
4
)
+
c
4
(
−
6
κ
2
mq
2
r
5
+
4
κ
2
q
4
r
6
+
4
κq
2
r
4
)
+
c
5
(
4
κ
2
q
4
5
r
6
−
κ
2
mq
2
r
5
)
+
c
6
(
κ
2
mq
2
r
5
−
κ
2
q
4
5
r
6
−
2
κq
2
r
4
)
+
c
7
(
−
4
κq
4
5
r
6
)
+
c
8
(
−
2
κq
4
5
r
6
)
.
(19)
In the above we have defined the reduced quantities
m
=
M/
4
π
and
q
=
Q/
4
π
. Note that the
R
2
-term in the action
has no contributions to the equation of motion at leading order in
ε
. The contributions from
R
μν
R
μν
and
R
μνρθ
R
μνρθ
can be canceled out by choosing
c
2
=
−
4
c
3
. This directly confirms that the Gauss-Bonnet term is a topological
invariant and does not influence the equation of motion. Due to the fact that only the
tr
- and
rt
−
component of
F
μν
are nonzero, the term
F
μν
F
μν
F
ρσ
F
ρσ
always have twice the contributions from
F
μν
F
νρ
F
ρσ
F
σμ
towards the equation
of motion.
III. EXPLICIT FORMS OF Q
ξ
AND C
a
FOR THE HIGHER THEORY
The Lagrangian 4-form
L
for the higher theory can be written as
L
=
L
0
+
∑
i
c
i
L
i
. In this appendix, by following
the canonical method developed by Iyer and Wald, we derive and present the Noether charge and constraint associated
with each term in
L
.
Variation of the Lagrangian 4-form
L
0
yields
δ
L
0
=
δg
ab
(
−
1
2
κ
G
ab
+
1
2
T
EM
ab
)
+
δA
a
(
∇
b
F
ba
)
+
dΘ
0
,
(20)
where
G
ab
=
R
ab
−
1
2
g
ab
R
is the Einstein tensor, and
T
EM
ab
is the electro-magnetic stress-energy tensor, which is
defined by
T
EM
ab
=
F
ac
F
b
c
−
1
4
g
ab
F
de
F
de
.
(21)
The symplectic potential can be written as
Θ
0
=
Θ
GR
+
Θ
EM
,
(22)
4
where
Θ
GR
abc
(
φ,δφ
) =
1
2
κ
dabc
g
de
g
fg
(
∇
g
δg
ef
−∇
e
δg
fg
)
,
(23)
Θ
EM
abc
(
φ,δφ
) =
−
dabc
F
de
δA
e
.
(24)
Let
ξ
a
be any smooth vector field on the spacetime. We find that the Noether charges associated with the vector field
are respectively,
(
Q
GR
ξ
)
ab
=
−
1
2
κ
abcd
∇
c
ξ
d
,
(25)
(
Q
EM
ξ
)
ab
=
−
1
2
abcd
F
cd
A
e
ξ
e
.
(26)
The equations of motion and constraints are given by
E
0
δφ
=
−
(
1
2
T
ab
δg
ab
+
j
a
δA
a
)
,
(27)
C
bcda
=
ebcd
(
T
e
a
+
j
e
A
a
)
,
(28)
where we have defined
T
ab
=
1
κ
(
G
ab
−
κT
EM
ab
)
as the non-electromagnetic stress energy tensor, and
j
a
=
∇
b
F
ab
is the
charge-current of the Maxwell sources.
We similarly obtain the Noether charges and constraints for all higher-derivative terms. The results are presented
below.
a.
L
1
Variation of
L
1
yields
δ
L
1
=
δg
ab
(
E
1
)
ab
+
dΘ
1
,
(29)
where we have defined
(
E
1
)
ab
=
1
2
g
ab
R
2
−
2
RR
ab
+ 2
∇
b
∇
a
R
−
2
g
ab
∇
c
∇
c
R.
(30)
The Noether charge associated with the vector field
ξ
a
is
(
Q
1
ξ
)
ab
=
abcd
(
−
4
ξ
c
∇
d
R
+ 2
R
∇
d
ξ
c
)
.
(31)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
1
)
e
a
.
(32)
b.
L
2
Variation of
L
2
yields
δ
L
2
=
δg
ab
(
E
2
)
ab
+
dΘ
2
,
(33)
where we have defined
(
E
2
)
ab
=
1
2
g
ab
R
cd
R
cd
+
∇
c
∇
b
R
ac
+
∇
c
∇
a
R
bc
−
g
ab
∇
d
∇
c
R
cd
−∇
c
∇
c
R
ab
−
2
R
ac
R
b
c
.
(34)
The Noether charge associated with the vector field
ξ
a
is
(
Q
2
ξ
)
ab
=
abcd
(
4
ξ
[
f
∇
c
]
R
f
d
+
R
f
d
∇
f
ξ
c
+
R
f
c
∇
d
ξ
f
)
.
(35)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
2
)
e
a
.
(36)
5
c.
L
3
Variation of
L
3
yields
δ
L
3
=
δg
ab
c
3
(
E
3
)
ab
+
dΘ
3
,
(37)
where we have defined
(
E
3
)
ab
=
1
2
g
ab
R
2
+ 2
g
ab
R
cd
R
cd
+ 2
R
ab
R
−
8
R
cd
R
acbd
+ 2
∇
b
∇
a
R
−
4
R
ab
.
(38)
The Noether charge associated with the vector field
ξ
a
is
(
Q
3
ξ
)
ab
=
abcd
(
−
4
ξ
e
∇
f
R
e
fcd
+ 2
R
ef
cd
∇
f
ξ
e
)
.
(39)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
3
)
e
a
.
(40)
d.
L
4
Variation of
L
4
yields
δ
L
4
=
δg
ab
(
E
g
4
)
ab
+
δA
a
(
E
A
4
)
a
+
dΘ
4
,
(41)
where we have defined the equation of motions for
g
ab
and
A
a
respectively as
(
E
g
4
)
ab
=
[
−
R
ab
+
1
2
g
ab
R
−
g
ab
∇
2
+
∇
(
a
∇
b
)
]
F
2
−
2
RF
ac
F
b
c
,
(42)
(
E
A
4
)
a
= 4
∇
b
(
RF
ab
)
.
(43)
The Noether charge associated with the vector field
ξ
a
is
(
Q
4
ξ
)
ab
=
abcd
(
F
2
∇
d
ξ
c
−
2
ξ
c
∇
d
F
2
+ 2
RF
cd
A
e
ξ
e
)
.
(44)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
g
4
)
e
a
−
ebcd
(
E
A
4
)
e
A
a
.
(45)
e.
L
5
Variation of
L
5
yields
δ
L
5
=
δg
ab
(
E
g
5
)
ab
+
δA
a
(
E
A
5
)
a
+
dΘ
5
,
(46)
where we have defined the equation of motions for
g
ab
and
A
a
respectively as
(
E
g
5
)
ab
= 2
F
(
bc
F
c
d
R
a
)
d
−
F
ac
F
bd
R
cd
+
1
2
F
c
e
F
cd
g
ab
R
de
(47)
−∇
(
a
F
b
)
c
∇
d
F
c
d
−
F
cd
∇
d
∇
(
a
F
b
)
c
−
F
(
bc
∇
d
∇
a
)
F
c
d
−
F
(
bc
F
a
)
c
−∇
(
b
F
cd
∇
d
F
a
)
c
−
F
cd
g
ab
∇
(
d
∇
e
)
F
c
e
−∇
d
F
b
c
∇
d
F
ac
+
1
2
g
ab
∇
c
F
cd
∇
e
F
d
e
−
1
2
g
ab
∇
d
F
ce
∇
e
F
cd
,
(
E
A
5
)
a
= 2
∇
c
(
R
bc
F
a
b
+
F
bc
R
a
b
)
.
(48)
The Noether charge associated with the vector field
ξ
a
is
(
Q
5
ξ
)
ab
=
abcd
[
−
2
ξ
e
A
e
F
fc
R
f
d
−
2
ξ
c
F
f
(
e
∇
e
F
f
d
)
+
ξ
e
∇
d
(
F
fc
F
ef
)
+
F
f
d
F
e
f
∇
[
c
ξ
e
]
]
.
(49)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
g
5
)
e
a
−
ebcd
(
E
A
5
)
e
A
a
.
(50)
6
f.
L
6
Variation of
L
6
yields
δ
L
6
=
δg
ab
(
E
g
6
)
ab
+
δA
a
(
E
A
6
)
a
+
dΘ
6
,
(51)
where we have defined the equation of motions for
g
ab
and
A
a
respectively as
(
E
g
6
)
ab
=
1
2
F
cd
F
ef
g
ab
R
cdef
−
3
F
(
ac
F
de
R
b
)
cde
(52)
−
2
F
(
ac
∇
c
∇
d
F
b
)
d
−
2
F
(
ac
∇
d
∇
c
F
b
)
d
−
4
∇
c
F
(
ac
∇
d
F
b
)
d
,
(
E
A
6
)
a
= 4
∇
d
(
F
bc
R
ad
bc
)
.
(53)
The Noether charge associated with the vector field
ξ
a
is
(
Q
6
ξ
)
ab
=
abcd
[
2
ξ
e
A
e
F
fg
R
fg
cd
−
2
ξ
e
∇
f
(
F
cd
F
e
f
)
+
F
cd
F
ef
∇
f
ξ
e
]
.
(54)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
g
6
)
e
a
−
ebcd
(
E
A
6
)
e
A
a
.
(55)
g.
L
7
Variation of
L
7
yields
δ
L
7
=
δg
ab
(
E
g
7
)
ab
+
δA
a
(
E
A
7
)
a
+
dΘ
7
,
(56)
where we have defined the equation of motions for
g
ab
and
A
a
respectively as
(
E
g
7
)
ab
=
1
2
g
ab
F
2
F
2
−
4
F
ac
F
b
c
F
2
,
(57)
(
E
A
7
)
a
= 8
∇
b
(
F
ab
F
2
)
.
(58)
The Noether charge associated with the vector field
ξ
a
is
(
Q
7
ξ
)
ab
=
abcd
(
4
ξ
e
A
e
F
cd
F
2
)
.
(59)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
g
7
)
e
a
−
ebcd
(
E
A
7
)
e
A
a
.
(60)
h.
L
8
Variation of
L
8
yields
δ
L
8
=
δg
ab
(
E
g
8
)
ab
+
δA
a
(
E
A
8
)
a
+
dΘ
8
,
(61)
where we have defined the equation of motions for
g
ab
and
A
a
respectively as
(
E
g
8
)
ab
=
1
2
g
ab
F
c
d
F
d
e
F
e
f
F
f
c
−
4
F
ac
F
bd
F
c
e
F
de
,
(62)
(
E
A
8
)
a
=
−
8
∇
d
(
F
a
b
F
b
c
F
cd
)
.
(63)
The Noether charge associated with the vector field
ξ
a
is
(
Q
8
ξ
)
ab
=
abcd
(
4
ξ
e
A
e
F
f
d
F
g
c
F
gf
)
.
(64)
The constraints are given by
C
bcda
=
−
2
ebcd
(
E
g
8
)
e
a
−
ebcd
(
E
A
8
)
e
A
a
.
(65)
Finally, the above results can be summarized in the following compact form:
(
Q
ξ
)
c
3
c
4
=
abc
3
c
4
(
M
abc
ξ
c
−
E
abcd
∇
[
c
ξ
d
]
)
,
(66)
where
M
abc
≡−
2
∇
d
E
abcd
+
E
ab
F
A
c
,
(67)
and
(
C
d
)
abc
=
eabc
(2
E
pqre
R
d
pqr
+ 4
∇
f
∇
h
E
efdh
+ 2
E
eh
F
F
d
h
−
2
A
d
∇
h
E
eh
F
−
g
ed
L
)
(68)
with
E
abcd
≡
δ
L
δR
abcd
, E
ab
F
≡
δ
L
δF
ab
.
(69)
7
Naked singularity region
area
decrease
area
increase
constant
area
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
FIG. 1: Extremality contour and constant area contours. Extremal black holes live on the red solid line which divides the
whole parameter space into the naked singularity region and the non-extremal black hole region. The constant area contours
are always tangent to the extremal line. A small perturbation around an extremal point then shifts the spacetime to one of
the following: (i) a naked singularity when the horizon area is decreased; (ii) another extremal solution when the area is
unchanged; and (iii) a nonextremal black hole when the area is increased.
IV. PROOF THAT CONSTANT AREA DIRECTION IS ALONG THE EXTREMALITY CURVE
Suppose the radius, hence area
A
of the horizon is determined implicitly by the following equation
F
(
M,Q,A
) = 0
.
(70)
Extremality condition requires, in addition, that
∂
A
F
(
M,Q,A
) = 0
.
(71)
This is because the two roots of 1
/g
rr
coincide at this location.
Extremal black holes is a one-parameter family, with
Q
ext
(
M
),
A
ext
(
M
) determined jointly by Eqs. (70) and (71).
In practice, when
Q < Q
ext
(
M
), we will have contours of constant
A
(as shown in Fig. 1), determined by
∂
M
FdM
+
∂
Q
FdQ
= 0
,
(72)
or
(
dQ/dM
)
A
=
−
∂
M
F/∂
Q
F .
(73)
On the other hand, we can find out the direction of the extremality curve in the (
M,Q,A
) space. The tangent
vector satisfies
∂
M
F
∆
M
+
∂
M
F
∆
Q
+
∂
A
F
∆
A
= 0
.
(74)
However, because we have
∂
A
F
on that curve, we have
∂
A
F
= 0 and also
(
dQ/dM
)
ext
=
−
∂
M
F/∂
Q
F .
(75)
This means, on the extremality contour, the direction at which area remains constant is the same as the contour
itself. This does not mean that the contour all has the same area — instead, constant area contours reach the
extremality contour in a tangential way, as shown in the figure.