A Second Law for Higher Curvature Gravity

Aron C. Wall∗
School of Natural Sciences, Institute for Advanced Study
1 Einstein Dr, Princeton NJ, 08540 USA
arXiv:1504.08040v2 [gr-qc] 1 Oct 2015

October 5, 2015

Abstract
The Second Law of black hole thermodynamics is shown to hold for arbi-
trarily complicated theories of higher curvature gravity, so long as we allow
only linearized perturbations to stationary black holes. Some ambiguities in
Wald’s Noether charge method are resolved. The increasing quantity turns out
to be the same as the holographic entanglement entropy calculated by Dong.
It is suggested that only the linearization of the higher-curvature Second Law
is important, when consistently truncating a UV-complete quantum gravity
theory.

You’ve just invented a new theory of gravity. Like Einstein’s, your theory is
generally covariant and formulated in terms of a metric gab . But the action is more
complicated; it is an arbitrary function of the Riemann curvature tensor, perhaps
some scalars φ, and their derivatives:
√
Z
I = dD x −g (Lg (gab , Rabcd , ∇a Rbcde , ∇a ∇b Rcdef . . . φ, ∇a φ, ∇a ∇b φ . . .) + Lm )
(1)
where Lg is the exciting gravitational piece, while Lm is a boring minimally coupled
matter sector obeying the null energy condition Tab k a k b ≥ 0 (NEC) (k a being null).
Such “higher curvature” corrections are known to occur with small coefficients due
to quantum and/or stringy corrections.
The equation of motion from varying the metric is
2 δLg −2 δLm
Hab ≡ √ ab
=√ ≡ Tab . (2)
−g g −g g ab
∗
aroncwall@gmail.com

1
To make up for all the excitement in the gravitational sector, you decide the matter
sector should be an ordinary field theory minimally coupled to gab , obeying the null
energy condition Tab k a k b ≥ 0 (NEC) for k a null.
A lesser mind would ask whether this theory is in agreement with observation, or
perhaps whether the vacuum is even stable. But not you! You are concerned with
a far deeper question: do black holes in your theory still obey the Second Law of
horizon thermodynamics?
In GR, the NEC (plus a version of cosmic censorship) implies that the area of any
future event horizon H is always increasing [1]. So Bekenstein [2] postulated that
black holes have entropy proportional to their area; Hawking radiation [3] showed that
it was more than just an analogy, and that in turn had all kinds of ramifications [4]!
Does this only make sense for the Einstein-Hilbert action, or is it true more broadly?
We shall see that there is indeed a Second Law in all such theories of higher curvature
gravity theories, provided that you only consider linearized perturbations δgab , δφ
of the gravitational fields (possibly sourced by a first order perturbation to δTab )
evaluated on a stationary black hole background (or more precisely, a bifurcate Killing
horizon1 ). This has previously been done for f (Lovelock) gravity [5], and quadratic
curvature gravity [6].
Pick a gauge so that u and v are null coordinates which increase as one moves
spacelike away from the horizon, so that u = 0 is the future horizon H, v is an affine
parameter along the null generators of the horizon, v = 0 is the past horizon, i, j
indices point in the D − 2 transverse directions, the metric obeys

gvv = gvi = gvv,u = 0, guv = 1 (u = 0); (3)

guu = gui = 0 (everywhere); (4)

and the Killing symmetry acts like a standard Lorentz boost on the null coordinates:
v → av, u → u/a.2
The Killing weight n of a tensor (with all indices lowered) is given by the number
of v-indices minus the number of u-indices, and will sometimes be indicated by an
(n)
superscript. A key feature of this gauge choice is that on the horizon, any tensor
with positive weight n always has at least n v-derivatives acting on it.3
1
At least through Eq. (11), the arguments below can also be adapted to linearized perturbations
of stationary null surfaces in pp-wave spacetimes, using the null translation symmetry instead of
the Killing boost.
2
On a slice of the horizon, this gauge implies that the only nonvanishing Christoffel symbols are
Γkij (the intrinsic geometry), Γvij , Γjiu , Γuij , Γjiv (which can be calculated from the extrinsic curvatures
Kij(u) and Kij(v) ), and Γviv , Γuiu , Γiuv (the twist).
3
If your theory had any gravitational fields with spin besides the metric, it would probably be
necessary to provide them with a gauge symmetry too, in order to permit an analogous gauge-fixing.
For example, in the case of a vector potential you would need to impose the null gauge Av = 0.
But this would raise additional questions e.g. is the entropy always gauge-invariant even when Aa
is nonminimally coupled? So for now, you reluctantly stick to a metric-scalar theory.

2
 Now the first question is which entropy should you use, which might always in-
crease? It should be some local geometrical expression which can be integrated along
a given horizon slice, but which one?
Unfortunately, the Noether charge method [7, 8] used to derive the black hole en-
tropy is subject to a number of ambiguities, identified by Jacobson, Kang, and Myers
(JKM) [8,9]. None of these ambiguities matter for compact, stationary horizons. For
compact but nonstationary horizons there are ambiguities of the form [5, 9]
Z
(JKM) √ X (n)
S = dD−2 x g X · Y (−n) , (5)

where the expression is a boost invariant (i.e. Killing weight 0) product of terms
which are not separately boost invariant. Such ambiguity terms vanish on Killing
horizons. At first order they vanish on the bifurcation surface v = 0, but they may
be important at v 6= 0. Possible Noether charges include:
Wald Differentiate with respect to the Riemann tensor (integrating by parts where
necessary) [7, 8]:
δLg
Z
√
SWald = −2π dD−2 x g 4 . (6)
δRuvuv
Iyer-Wald Expand the Wald entropy in fields and their derivatives, and keeping the
terms which depend only on boost-invariant (weight 0) fields [8].
Dong The holographic entanglement entropy in f (Riemann) theories, derived by
analytically continuing certain gravitational instantons [10].4
Every entropy in the class SWald + S (JKM) obeys the physical process version of the
First Law,5 which says that if the horizon begins and ends in a stationary configura-
tion, the increase in entropy from v = 0 to v = +∞ due to a first order perturbation
δgab , δφ is given by the Killing energy flux across the horizon.
Z +∞
√
δS(+∞) − δS(0) = δTvv v dv g dD−2 x. (7)
v=0

Given the NEC, it follows that ∆S ≥ 0. The JKM ambiguity doesn’t matter because
factors with negative weight vanish at v = 0 while factors with positive weight vanish
as v → +∞.
So does this prove the Second Law? Not yet, because the Second Law requires the
entropy to be increasing at every instant of time: dS/dv ≥ 0. At intermediate times,
you must fix the JKM ambiguity (up to higher order terms like K 4 which vanish at
linear order).
4
Some special cases are given in [11–16], while progress for actions with derivatives of Riemann
is in [17–19].
5
For higher curvature gravity theories, this was shown explicitly in [20], and is implied by the
Appendix of [8] or section 2 of [21].

3
 The key is to rewrite Hvv = Tvv ≥ 0 on the horizon as the second derivative of
some quantity ς(gab , δgab , φ, δφ):
Z
√
δ dD−2 x gHvv = −2π∂v ∂v ς ≥ 0. (8)
√
You can show this by expanding δ( gHvv ) (which has weight 2) as a sum of product
of the gravitational fields and their variations:
√ X
δ( gHvv ) = X (−n) δY (2+n) , (9)
n≥0

where the positivity of n arises due to the fact that all fields with positive weight
vanish on a future Killing horizon (since otherwise the Killing symmetry would require
it to diverge at the bifurcation surface), and Y takes the form gab,... or φ,ab... . By virtue
of the gauge fixing, the total number of v-derivatives must always be at least 2 + n.
By repeatedly differentiating by parts, you can move each of these derivatives either
(a) to the X term or (b) outside the expression entirely. But (a) can happen at most
n times since the X term cannot have positive weight. So at least two ∂v ’s end up
outside the expression, proving (8).
If the black hole becomes stationary at late times, you can impose the final bound-
ary condition ∂v ς(+∞) = 0. (Since the Killing time is t = ev , this condition requires
only that the perturbation to the black hole grows slower than exponentially with re-
spect to Killing time.) By integrating backwards in time using (8), you can therefore
conclude that ∂v ς ≥ 0, which looks very much like a Second Law.
However, you probably would also like to know that ς is exact, i.e. a variation of
some local geometric entropy: ς = δS. To see this, expand ς as sum of a product of
terms of the form X X
ς= X (0) δY (0) + X (−n) δY (n) . (10)
n≥1

where the X terms contain no factors with positive weight. In the terms with n ≥ 1,
since on the background spacetime Y = 0 on H, (δX)Y = 0 and thus you can simply
move the δ to the outside, obtaining a term of the JKM form:
X
δ X (−n) Y (n) = δS (JKM) . (11)
n≥1

The term made of boost-invariant factors (call it ς0 ) is trickier, but using the
physical process version of the First Law, you can see that it is the variation of the
Iyer-Wald entropy SIW . Let B be the space of all possible boost-invariant metric
data for first order variations on a given slice with v = const, with arbitrary matter
sources. Considering only boost-invariant data you can’t tell whether or not you are
on the bifurcation surface v = 0, so without loss of generality let’s suppose you are
(for purposes of evaluating ς0 ).

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 Now if the horizon becomes stationary at late times, you can always choose δgab ,
δφ to vanish at late times on the horizon, so that for any δgab , δφ ∈ B,
Z +∞
√
ς0 = − δTvv v dv g dD−2 x = δSIW , (12)
v=0

where the first equality comes from integrating (8), and the second from (7). You
can thus conclude that there is an increasing Noether charge entropy, of the form

S = SIW + S (JKM) , (13)

where the JKM ambiguity term can be constructed explicitly by the procedure given
above, using only those terms in Hvv which have at least three ∂v ’s acting on the same
field.6
In the special case of f (Riemann) gravity, with some effort (see Appendix A if you
get stuck) you can show that (up to a total derivative, which vanishes if the horizon
is compact):

∂ 2 Lg
 
∂Lg
Z
D−2 √
S = −2π d x g 4 + 16 Kij(u) Kkl(v) , (14)
∂Ruvuv ∂Ruiuj ∂Rvkvl
where Kij(a) is the extrinsic curvature in the a direction. This matches SDong exactly
at linear order in the metric perturbation!7 This is a rather remarkable coincidence,
considering that the two entropies were calculated using completely different tech-
niques.
If your gravitational theory is coupled semiclassically to a a free quantum field the-
ory, a Generalized Second Law (which accounts for the entropy in Hawking radiation
and matter fields) also follows, as in [5, 22].
At second order in the metric perturbation, one could consider the effects of
gravity waves on the Second Law. But some theories (e.g. GR with a negative GN ,
in which gravity waves carry negative energy) will fail this test. But at very nonlinear
order (e.g. black hole mergers), even Lovelock gravity violates the Second Law [23–
25]! Perhaps this indicates that it is only consistent to treat the non-GR couplings
perturbatively, in a consistent truncation of a UV-complete theory of quantum gravity
6
By replacing v with a general affine coordinate λ = a + bv, b > 0, where a, b can depend on
the generator of the horizon, and by performing the same series of steps, it follows that S is also
increasing from any initial slice on the horizon to any final slice. (Note that, in the step which uses
the physical process First Law, the bifurcation surface is not generally at a constant value of λ,
but this does not invalidate the proof.) It is perhaps no longer totally obvious that the increasing
entropy formula is a covariant functional of the horizon slice (i.e. that it does not depend on b),
but a direct calculation for f (Riemann) shows that the entropy is covariant at least in this case.
7
Note that O(K 4 ) = K(u)
2 2
K(v) and higher order terms cannot be determined by the linearized
second law [5]. These terms are given in [10] by a more complicated expression involving a symmetry
factor. According to [17,18], there is a “splitting problem” which renders Dong’s method ambiguous
for terms of order K 4 and higher, but fortunately this problem does not matter at O(K 2 ).

5
[26, 27]. But in adiabatic (i.e. thermodynamically reversible) quantum processes
for which δ(S + Smatter ) = 0, the linearized Second Law could still be a necessary
consistency condition, even in a perturbative treatment [6].

Acknowledgments
I am grateful for interactions with Sudipta Sarkar, Srijit Bhattacharjee, Ted Jacobson, Zach
Fisher, and Will Kelly, and for support from NSF grants PHY-1314311, PHY11-25915, the
Institute for Advanced Study, UC Berkeley and the KITP.

A The entropy formula for f(Riemann)

This appendix details the calculation of the increasing entropy S = SIW + S (JKM) for
f (Riemann) gravity. In a moment we will catalogue the terms that arise, but first
we will comment on their structure.
Each term of Hvv arises either from varying either an inverse metric g ab or a
Riemann tensor Rabcd with respect to g vv . Only the later process produces terms
with derivatives of the Riemann tensor, which is needed to get a S (JKM) term.
Thus we look for a Riemann term with the index structure Ruaub , and eliminate
it while integrating by parts to move the ∇a and ∇b derivatives so that they act on
the remaining parts of the expression, using the relation
√ 1 √
Z Z
D abcd
d x −g X δRcadb = − dD x −g ∇a ∇b X abcd δgcd (15)
2
+(a ↔ −c, b ↔ −d) permutations + curvature terms,

where the curvature terms involve Riemann instead of derivatives of X abcd . This
relation can be derived most easily in local inertial coordinates.
To obtain the JKM terms, we then look for terms with a factor of weight 3 or
greater; these either involve 2 derivatives of Riemann (which is straightforward) or
else expressions like like Γijv,vv Γklu ; after removing the two derivatives from this term,
we obtain Γijv = −Γvij = −Kij(v) and Γklu = −Γukl = −Kkl(u) in our gauge choice
(where Γabc = gcd Γdab ). We can ignore terms with derivatives acting on the area
element because they cannot have weight more than 2.
The end result is an expression which involves at most two differentiations with
respect to the Riemann tensor: one to get Hvv and a second to identify the term with
weight 3 or higher which gets differentiated twice with respect to v. Recall that we
can drop any term with two or more factors with positive Killing weight.
We use the standard convention where ∂R∂abcd has the same symmetries of the
Riemann tensor, and is normalized so that δX = δRabcd ∂R∂X abcd
. This convention gives

6
us the following symmetry factors when differentiating:
∂ ∂ ∂ ∂
, , , :4 (16)
∂Ruvuv ∂Ruiuj ∂Rvivj ∂Rvijk
∂ ∂
, :8 (17)
∂Ruvui ∂Rvuvi
from counting the number of equivalent ways to order the indices. In addition, all
terms have a factor of 1/2 coming from differentiating Riemann with respect to the
inverse metric using 15, a factor of 2 coming from the coefficient in Eq. (2), a factor
of −2π coming from the coefficient in Eq. (8), and some will have signs flipped due
to manipulating Γ’s, either from taking the covariant derivative of a covector, or
rearranging the order of indices of a Γ.
The first term in the entropy density s comes from differentiating the Lagrangian
with respect to Ruvuv . This places a ∇v ∇v outside the rest of the expression. Remov-
ing these derivatives does not give us sWald due to the fact that ∇v 6= ∂v . However,
for purposes of calculating the JKM piece of the entropy for f (Riemann), only the
∂v ∂v piece contributes. Thus we remove the factors of ∂v ∂v , and keep only those terms
which are of the JKM form. This gives us all terms coming from differentiating with
respect to Ruvuv (i.e. the Wald entropy density) except for those terms which are fully
boost-invariant (i.e. the Iyer-Wald entropy density):
(JKM)
s1 = sWald − sIW . (18)

It is convenient to add the sIW term back in at this stage to obtain sWald .
The remaining JKM terms are:

(JKM) ∂ 2 Lg
s2 = 32π Γiju Γklv , (19)
∂Rvkvl ∂Ruiuj

which comes from the Γvij ∂v term of ∇i ∇j ,

(JKM) ∂ 2 Lg
s3 = 128π Γiju Γklv , (20)
∂Rvjkl ∂Ruvui

which comes from the terms in ∇v ∇i Rvjkl containing Γvik or Γvil ,

(JKM) ∂ 2 Lg
s4 = −128π Γkiu Γjkv , (21)
∂Rvuvj ∂Ruvui

where the Γkiu comes from ∇i acting on the u index of Rvuvj , and finally there is a
derivative term
∂ 2 Lg
 
64π∇i Γjkv , (22)
∂Rvjvk ∂Ruvui

7
which arises if the ∇i acts on a different Riemann curvature term than the ∇v acts.
We can break this term up further using the formal expression ∇i = Di + Ki , where
Di is the derivative associated with parallel translating i-, u-, or v- indices along the
D − 2 dimensional horizon slice, while Ki is the connection corresponding to the
extrinsic curvature, i.e. that part of Γbia which contains exactly one v- or u- index.
The Di term is a total derivative:
∂ 2 Lg
 
(JKM)
s5 = 64πDi Γjkv , (23)
∂Rvjvk ∂Ruvui
which may be dropped when integrated along a compact horizon (or when evolving
between slices of the horizon that differ only in a region with compact support).
The remaining pieces come from acting with Γbia on the expression in parentheses
(bearing in mind, when acting with the Γ’s that in the denominator reverses the role
of downstairs and upstairs indices):

(JKM) ∂ 2 Lg
s6 = −64π Γilu Γjkv (24)
∂Rvjvk ∂Rului
∂ 2 Lg ∂ 2 Lg
+128π Γilu Γjkv + 128π Γk Γjkv.
∂Rvjlk ∂Ruvui ∂Rvuvj ∂Ruvui iu
Adding up sWald + s2 + s3 + s4 + s5 + s6 , there are some cancellations and we obtain
the total entropy Z
√
S = dD−2 x g (sWald − s2 + s5 ) (25)

which matches Eq. (14) after removing the total derivative term s5 .
Calculating the increasing entropy for actions which include derivatives of the
Riemann tensor should be straightforward, if tedious.

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