Butterfly Velocity Bound and Reverse Isoperimetric Inequality

0.23 MB·

by  Xing-Hui Feng

Butterfly Velocity Bound and Reverse Isoperimetric
Inequality
Xing-Hui Feng and H. Lu¨
∗
Center for Advanced Quantum Studies, Department of Physics,
7 Beijing Normal University, Beijing 100875, China
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n
a
J ABSTRACT

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WestudythebutterflyeffectoftheAdSplanarblackholesintheframeworkofEinstein’s

]
h general relativity. We find that the butterfly velocities can be expressed by a universal
t
- formula v2 = TS/(2V P). In doing so, we come upon a near-horizon geometrical formula
p B th
e
for the thermodynamical volume V . We verify the volume formula by examining a variety
h th
[
of AdS black holes. We also show that the volume formula implies that the conjectured

2
reverseisoperimetricinequality follows straightforwardly fromthenull-energycondition, for
v
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static AdS black holes. The inequality is thus related to an upper bound of the butterfly
0
2
velocities.
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0
.
1
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7
1
:
v
i
X
r
a
∗[email protected]
Contents
1 Introduction 2
2 A universal formula for butterfly velocities 5
2.1 Shockwave and butterfly effect . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Butterfly velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Subtleties involving scalar hair . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 A geometrical formula for thermodynamical volumes 9
4 Testing the identity: Kaluza-Klein dyonic AdS black holes 11
4.1 Spherically-symmetric black hole . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Toroidally-symmetric black hole . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Testing the identity: further AdS black holes 14
5.1 R-charged black holes in gauged supergravities . . . . . . . . . . . . . . . . 15
5.2 Charged dilatonic AdS black holes . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 Scalar hairy AdS black holes . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Reverse isoperimetric inequality and the v bound 19
B
7 Conclusions 20
1 Introduction
The butterfly effect is associated with the exponential growth of a small perturbation to a
quantumsystem. Inthecontext of holography, thiseffect hasabeautifulrealization [1–5]in
terms of a gravitational shock wave near thehorizon of an AdS black hole [6]. Thebutterfly
velocity, i.e. the velocity for the shockwave, for the Schwarzschild-AdS planar black hole in
general D dimensions, turns out to be constant, given by [1]
D 1
v2 = − . (1.1)
B 2(D 2)
−
The butterfly velocities for a variety of AdS planar black holes of various matter energy-
momentum tensor were obtained [7–14]. The study has been further generalized to include
higher-order gravities [3,15]. The expression of v can be simple or complicated depending
B
on the detail structures of the black holes.
2
On the other hand, based on the no-hair theorem, black holes are supposed to be the
purest and hence simplest thermodynamical systems. The physics are specified only by
their thermodynamical quantities including mass and charges. It is thus not unreasonable
to expect that the butterfly velocity be a dimensionless ratio of some (dimensionful) ther-
modynamical quantities of the black holes. In fact, based on this principle, it was shown
that the holographic sheer viscosity-entropy ratio [16,17] is a simple consequence of the
generalized Smarr relation of the thermodynamical variables [18].
In this paper, we study the butterfly effect of planar black holes in the framework of
Einstein’sgeneralrelativity, withonlyminimallycoupledmatter. Welimitthediscussionto
isotropic solutions where the toroidal (or Euclidean) plane is uniform. We find a universal
formula for the butterfly velocities
1 TS
v2 = . (1.2)
B 2V P
th
In this formula, (T,S) are the temperature and entropy of the black holes. For Einstein’s
gravity, the entropy is simply one quarter of the area of the event horizon. The quantity P
is the pressure associated with the cosmological constant Λ (as in = √ g(R 2Λ)). It is
L − −
given by
Λ
P = . (1.3)
−8π
Thequantity V is thethermodynamicalvolume conjugate toP. Treatingthecosmological
th
constant as a thermodynamical pressure was proposed in [19,20]. The first law of black
hole thermodynamics expands to
dM = TdS +Φ dQ +V dP + , (1.4)
α α th
···
where the dots denote contributions from further black hole hair, and the repeated α index
implies summation. In this picture, the mass of the black hole should be better interpreted
as theenthalpy [19]. Thuswesee thatthe butterflyvelocity is related to thevolume density
of TS, which is precisely the Gibbons-Hawking surface-term [21] contribution to the action
growth evaluated on the horizon [22].
Assuming that the theory does not involve further dimensionful coupling constants, the
first-law implies the Smarr relation
D 2 2
M = − TS +Φ Q PV . (1.5)
α α th
D 3 − D 3
− −
(See [23] for the original discussion of the Smarr formula in four dimensions.) For AdS
planar black holes, there exists a further generalized Smarr relation, associated with the
3
new scaling symmetry of the AdS planar black hole. It is given by [18,24]
D 2
M = − (TS +Φ Q ). (1.6)
α α
D 1
−
These two relations enable us to express the butterfly velocity in terms of mass, charges
and their chemical potentials, namely
D 1 (D 3)Φ Q
v2 = − − α α
B 2(D 2) − 2(D 2)(2M Φ Q )
α α
− − −
(D 3)M
= 1 − . (1.7)
− (D 2)(2M Φ Q )
α α
− −
(Note that when the black holes involve more hair, the Smarr relations and the above
formula may involve further terms.)
In establishing (1.2), we find an identity that relates the thermodynamical volume to
the Euclidean boundedvolume. Thisrelation is purely geometrical and it dependson solely
the metric functions of the near-horizon geometry. We test this identity against a variety of
static AdS black holes with both planar or spherical topologies, and find no exception. It
thus provides a simple method of calculating the thermodynamical volume even for black
holes with no analytical expressions. Interestingly, if we assume that this formula is valid,
the conjectured reverse isoperimetric inequality [20] follows directly from the null-energy
condition, for static AdS black holes.
The paper is organized as follows: In section 2, we first review the butterfly effects
in static and isotropic AdS planar black holes. We then obtain the universal formula
(1.2) for the butterfly velocities, by making use of the geometrical identity that relates the
thermodynamical and Euclidean bounded volumes. We also address the subtleties that can
arise when black holes involve non-trivial scalar hair. In section 3, we further elaborate the
formula of thermodynamicalvolume. We also pointout that it is natural to introduceblack
hole volume, in addition to (T,S) in order to fully specify the near-horizon geometry. In
sections 4and5, welistalargenumberofAdSblackholesandtesttheidentityagainsttheir
thermodynamical volumes that are derived from the first law. In section 6, we demonstrate
that the identity, together with the null-energy condition, imply the reverse isoperimetric
inequality for static AdS black holes. We then obtain a universal bound for the butterfly
velocities in terms of temperature and entropy. We conclude the paper in section 7.
4
2 A universal formula for butterfly velocities
2.1 Shockwave and butterfly effect
In this section, we consider a generic static AdS planar black hole in general D dimensions.
We focus on the isotropic configuration in the planar directions. The metric takes the form
dρ2
ds2 = h(ρ)dt2+ +ρ2dx˜idx˜i. (2.1)
D − f(ρ)
In this Schwarzschild-type metric, ρ and t have the dimension of length, whilst (h,f) and
x˜i are dimensionless. Asymptotically at large ρ, the functions (h,f) behave as
α α˜
h= g2ρ2+ + , f = g2ρ2+ + , (2.2)
···− ρD 3 ··· ···− ρD 3 ···
− −
where ℓ = 1/g is the AdS radius. The mass of the black hole is proportional to the constant
α. This definition of mass is in most cases consistent with the first law. The situation
becomes more subtle where there are non-trivial scalar charges, which we shall clarify in
section 2.3.
We assume that the metric describes a black hole with the event horizon located at
ρ = ρ > 0. We may perform Taylor expansions near the horizon
0
h = h (ρ ρ )+h (ρ ρ )2+ , f = f (ρ ρ )+f (ρ ρ )2+ . (2.3)
1 0 2 0 1 0 2 0
− − ··· − − ···
Inotherwords,h = h(ρ )andf =f (ρ ). Thetemperatureandentropycanbecalculated
1 ′ 0 1 ′ 0
using the standard technique, given by
√h f
T = 1 1 , S = 1ρD 2 , (2.4)
4π 4 0− AD−2
where is the volume of the metric dx˜idx˜i. If this metric is compact describing a
D 2
A −
(D 2)-torus, we can scale the coordinate r to set to be any fixed value. For
D 2
− A −
example, we can choose it to be = 1 or be the same as the volume of the unit round
D 2
A −
SD 2, namely
−
2π12(D−1)
= . (2.5)
AD−2 Γ[1(D 1)]
2 −
Ifdx˜idx˜i isnoncompactlike Euclidean,wecanalsochoose tobetheabovevalues, but
D 2
A −
now with the understanding that the extensive quantities such as mass, charge or entropy
describe the corresponding densities.
Before proceeding, we would like to point out the fact that the near-horizon geometry
is in general specified by three parameters (h ,f ,ρ ). However, the well-established tem-
1 1 0
perature and entropy only give two parameters, leaving the combination h /f unspecified.
1 1
5
This demonstrates that the temperature and entropy are not enough to characterize the
black hole horizon and a new quantity is called for. We shall come back to this point in
section 3.
Tostudythebutterflyeffects, itisconvenienttointroducetheKruskalcoordinates(u,v)
dρ
u= eκ(ρ∗ t), v = eκ(ρ∗+t), with dρ = . (2.6)
−
− ∗ √hf
Here κ = 2πT = 1√h f is the surface gravity on the horizon ρ = ρ , which corresponds
2 1 1 0
to uv = 0. Near the horizon, we have
f h
uv = (ρ ρ ) 1 2 + 2 (ρ ρ )2+ ,
− 0 − 2 f h − 0 ···
1 1
f (cid:0)h (cid:1)
ρ ρ = uv+ 1 2 + 2 (uv)2 + . (2.7)
− 0 2 f h ···
1 1
(cid:0) (cid:1)
The metric (2.1) can now be expressed as
ds2 = A(uv)dudv +B(uv)dxidxi, (2.8)
D
where the coordinates xi x˜i/g = ℓx˜i have dimension of length, and
≡
1 h
A(uv) = , B(uv) = g2ρ2. (2.9)
κ2uv
We can expand the functions A and B on the horizon uv = 0,
A= A +A (uv)+A (uv)2+ , B = B +B (uv)+B (uv)2 + . (2.10)
0 1 2 0 1 2
··· ···
It is clear that the coefficients of the Taylor expansions (2.3) and (2.10) are related. We
find
4 2f 6h 1 3f2 f 13f h 19h2 5h 1
A = , A = 2 + 2 , A = 2 + 3 + 2 2 + 2 + 3 ,
0 f 1 f h f 2 4f2 f 2f h 4h2 h f
1 1 1 1 1 1 1 1 1 1 1
B = g2ρ2, B =(cid:0) 2g2ρ , B(cid:1) = g2 1+ f(cid:0)2ρ + h2ρ . (cid:1) (2.11)
0 0 1 0 2 f 0 h 0
1 1
(cid:16) (cid:17)
This allows us to translate the butterfly velocity, typically calculated using the metric of
the Kruskal (u,v) coordinates, in terms of variables in the more standard black hole metric
of the (t,r) coordinates.
The butterfly effect emerges if one releases a particle from x = 0 on the boundary of
asymptotical AdS black hole at a time t in the past. As was described in [7], for late times
w
(i.e. t > β = 1 = 2π/κ), the energy density of this particle in Kruskal coordinates is
w T
localised on the u= 0 horizon and it is exponentially boosted:
δTuu Ee2βπtwδ(u)δ(~x), (2.12)
∼
6
whereE is theinitial asymptotic energy of theparticle. Consequently, even theeffects of an
initially-small perturbationcannotbeneglected andafter thescramblingtimet βlogN2
∗ ∼
the back-reaction of the stress tensor on the metric becomes significant.
Thisresultsintheformationofashockwave geometry, whosemetriccanthenbewritten
as [6]
ds2 = A(uv)dudv +B(uv)dxidxi A(uv)δ(u)h(~x)du2. (2.13)
−
For Einstein gravity with minimally-coupled matter, one finds that the shockwave satisfies
the wave equation [6–8]
((cid:3) m2)h(~x) Ee2βπtwδ(~x). (2.14)
− ∼
Here (cid:3)is theLaplacian on the(D 2)-dimensional Euclidean metric dxidxi andthescreen-
−
ing length m is given by
B
m2 = (D 2) 1 , (2.15)
− A
0
where B and A are defined by (2.10). Thus the properties of the shockwave is encoded in
1 0
the equation (2.14). At long distances x m 1 the metric is simply given by
−
≫
Ee2βπ(tw−t∗)−m|~x|
h(~x) (2.16)
∼ ~x D2−3
| |
One can immediately read off the Lyapunov exponent λ and velocity v of these holo-
L B
graphic theories as [3]
2π 2π
λ = , v = . (2.17)
L β B βm
2.2 Butterfly velocity
As we have discussed in the introduction, we expect that the dimensionless v may be
B
a ratio of some thermodynamical quantities. It follows from (2.17) that the numerator
has a factor of T, suggestive of a dimensionless ratio of TS and the product of another
thermodynamical conjugate pair. In fact, it follows from (2.17), (2.15) and (2.11) that the
butterfly velocity satisfies
h 2πT 1 h TS
v2 = 1 = 1 , (2.18)
B sf1 (D 2)g2ρ0 2sf1 VEP
−
where
(D 1)(D 2)g2 ρD 1
P = − 16π− , S = 14ρD0−2AD−2, VE = D0−1AD−2. (2.19)
−
Here V denotes the Euclidean bounded volume and its terminology will be explained in
E
section 3. For non-dilatonic black holes such as Schwarzschild or Ressner-Nordstrøm (RN)
7
metrics, one has h= f, hence h /f = 1, and V is precisely the thermodynamical volume.
1 1 E
In general, we claim that the thermodynamical volume is given by
f
1
V = V . (2.20)
th E
h
r 1
This then leads straightforwardly to the general formula (1.2) for butterfly velocities. We
shall elaborate this formulafurtherin section 3, andthen insections 4 and5, we shallverify
this formula using a variety of AdS black holes.
Intheintroduction,wethenmadeuseofthetheSmarrrelation(1.5)andthegeneralized
Smarr relation (1.6) to express the v2 as (1.7). However, there is a subtlety when the black
B
hole contains non-trivial scalar hair.
2.3 Subtleties involving scalar hair
A rather general class of AdS black holes come from Lagrangians with a scalar potential.
For simplicity, we shall consider only one scalar φ, and its scalar potential is (φ). The
V
existence of the AdS vacuum of radius ℓ = 1/g requires that has a stationary point, say
V
φ = 0, with
(0) = 0, (0) = (D 1)(D 2)g2. (2.21)
′
V V − −
Depending on the mass parameter of the scalar m˜2 = (0), at large ρ, the scalar behaves
′′
V
as
φ φ
1 2
φ(r) + + , (2.22)
∼ r21(D−1−σ) r12(D−1+σ) ···
where σ is related to the mass parameter as σ = 4m˜2g 2+(D 1)2. It was shown that
−
−
the first law of the black hole thermodynamics tapkes the form [25,26]
dM = TdS +Z + , (2.23)
···
with
σg2
Z = g2 (D 1+σ)φ dφ (D 1 σ)φ dφ . (2.24)
2 1 1 2
32π(D 1) − − − −
− (cid:16) (cid:17)
Here the mass is given by
(D 2)
D 2
M = − A − α. (2.25)
16π
The above result is controversial owing to the fact that the variation of the on-shell Hamil-
tonian
(d ) = dM Z, (2.26)
ρ
H →∞ −
is notintegrable, andhence theenergy is notwell defined. In this paper, we adoptthe same
strategy of [26], and introduce a “gravitational mass” M that describes the condensate
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of the spin-2 massless graviton. It is then clear from the dimensional analysis that the
integration constants (φ ,φ ) do not involve in the Smarr relation, and hence relation (1.5)
1 2
holds even for black holes with such non-trivial scalar hair, provided that M is defined as
the gravitational mass, satisfying (2.23).
The first law (2.23) and the Smarr relation (1.5) are independent of whether the AdS
black hole is spherically symmetric or is of the planar type. For the planar AdS black holes
with non-trivial scalar hair, it was shown in [18] that the generalized Smarr relation (1.6)
also works. In other words, the scalar hair variables (φ ,φ ) do not enter the Smarr nor
1 2
the generalized Smarr relation. Thus the formula (1.7) is valid even for black holes with
non-trivial scalar hair provided that M is the gravitational mass defined by (2.25).
3 A geometrical formula for thermodynamical volumes
In the previous section, we obtained the butterfly velocity (2.18). We find that it can be
expressed universally as a simple ratio (1.2) of thermodynamical variables, provided that
the identity (2.20) for calculating the thermodynamical volume is valid. In this section, we
discuss this identity in more detail. A static black hole metric may not always be expressed
analytically in terms of the Schwarzschild-type coordinate as in (2.1). Instead it takes a
more general form
dr2
ds2 = h(r)dt2+ +ρ(r)2dΩ2 . (3.1)
− f(r) D−2,k
The statement of the identity (2.20) in this more general coordinate system becomes
V f dρ
th
= , (3.2)
VE rh dr r r0
(cid:12) →
(cid:12)
where r = r is the event horizon. It is perhaps(cid:12)better to express the above identity in a
0
more abstract notation
V 1 dρ
th
= . (3.3)
VE √−gttgrr dr(cid:12)r→r0
As explained in the introduction, the quantity V is(cid:12)the thermodynamical volume derived
th (cid:12)
from the first law of black hole thermodynamics
dM = TdS +Φ dQ +V dP + , (3.4)
α α th
···
where P is the thermodynamical pressure
Λ (D 1)(D 2)g2
P = = − − . (3.5)
−8π 16π
9
The quantity V is defined as
E
ρD 1
V = 0− . (3.6)
E D 2
D 1A −
−
where ρ = ρ(r ) and is the volume of the metric dΩ2 which is a unit Einstein
0 0 AD−2 D−2,k
metric with
R = (D 3)kg˜ , k = 1,0,1. (3.7)
ij ij
− −
When k = 1, and the metreic is a unit round SD−2, VE is precisely the volume of a ball
of radius ρ in the (D 1)-dimensional Euclidean space. For this reason, V was called
0 E
−
Euclidean bounded volume in [20]. Here we generalize the concept to include the k = 1,0
−
topologies as well. Thus the identity (3.2) or (3.3) provides a purely horizon-geometric
formula for calculating the thermodynamical volume. An important advantage is that this
method does not require an exact analytical expression of a black hole in order to calculate
its thermodynamical volume. By contrast, determining V through the first law would
th
necessarily require that the exact solution of the black hole be known, since the first law
involves both horizon quantities such as (T,S) and the asymptotic quantities such as mass
and charges.
For Schwarzschild-AdS or RN-AdS black holes, one has h = f in the r = ρ coordinate
gauge choice. The thermodynamical volume is precisely the Euclidean bounded volume.
Such black holes are typically non-dilatonic and include, for examples, [27–31]. For these
examples, the formula (3.2) is thus automatically valid.
The situations becomes more complicated when the theory involve radially-dependent
scalar fields. For example, let us consider
= √ g R 1(∂φ)2 V(φ)+ . (3.8)
L − − 2 − ···
(cid:16) (cid:17)
The Einstein equation of motion implies that (in the r = ρ gauge)
(D 2) h f
φ′2 = − ′. (3.9)
− ρ f h
(cid:16) (cid:17)
Thus for these black holes, V = V , but rather their ratio satisfies the identity (3.2). In
th E
6
thenexttwosections, weshalltestthisidentity withavariety ofblack holesinvolvingscalar
fields, by computing their thermodynamicalvolumes usingthefirstlaw (3.4). Aformulafor
calculating the thermodynamical volume based on the Wald formalism was also obtained
in [32,33]. As in the case of using the first law to computing the volume, this formula
of [32,33] is useful only when one has an exact solution, since it requires an integration
from the horizon to asymptotic infinity, where potential divergent terms cancel out in a
10