Slowly Rotating Black Holes in 4D Einstein Gauss-Bonnet Gravity Michael Gammonand Robert B. Mann

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Slowly Rotating Black Holes

in 4D Einstein Gauss-Bonnet Gravity

Michael Gammon∗and Robert B. Mann†

University of Waterloo

Department of Physics

(Dated: April 2, 2024)

Since the recent derivation of a well-deﬁned D→4 limit for regularized 4D Einstein Gauss-Bonnet

(4DEGB) gravity, there has been considerable interest in testing it as an alternative to Einstein’s

general theory of relativity. In this paper we construct slowly rotating black hole solutions for

4DEGB gravity in asymptotically ﬂat, de Sitter, and anti-de Sitter spacetimes. At leading order

in the rotation parameter, exact solutions of the metric functions are derived and studied for all

three of these cases. We compare how physical properties (innermost stable circular orbits, photon

rings, black hole shadow, etc.) of the solutions are modiﬁed by varying coupling strengths of the

4DEGB theory relative to standard Einstein gravity results. We ﬁnd that a vanishing or negative

cosmological constant in 4DEGB gravity enforces a minimum mass on the black hole solutions,

whereas a positive cosmological constant enforces both a minimum and maximum mass with a

horizon root structure directly analogous to the Reissner-Nordstr¨om de Sitter spacetime. Besides

this, many of the physical properties are qualitatively similar to general relativity, with the greatest

deviations typically being found in the low (near-minimal) mass regime.

I. INTRODUCTION

Despite the empirical success and predictive power of

Einstein’s general theory of relativity (GR), its modiﬁ-

cations continue attract attention. Early attempts can

be seen in the writings of Weyl [1] or Eddington [2], and

continue through to present day, motivated by the sin-

gularity problem (ie. geodesic incompleteness), the need

for reconciling quantum physics and gravity, and the need

for phenomenological competitors so as to test GR in the

most stringent manner possible.

A common class of modiﬁcations to GR are higher cur-

vature theories (or HCTs), in which it is assumed that

a sum of powers of the curvature tensor is proportional

to stress-energy, extending the assumed linear relation-

ship between spacetime curvature (the Einstein tensor)

and stress-energy in GR. Logically this linear relation-

ship is not required, and it is conceivable that the empir-

ical success of Einstein’s theory could be improved upon

by modifying the left-hand-side of the Einstein equations

with a sum of powers of the curvature tensor.

These HCTs play important roles in many diﬀerent ar-

eas of physics - they appear in a number of proposals for

quantum gravity [3], and may be necessary to account

for observational evidence of dark matter, early-time in-

ﬂation, or late-time acceleration [4]. Lovelock theories

[5] are the best-known examples of HCTs, and have the

distinct feature that their diﬀerential equations are of

2nd order. However, until recently, their equations only

had non-trivial solutions in spacetime dimensions larger

than four (D > 4) [6], so their physical signiﬁcance had

been unclear. Recently, a new class of HCTs have been

∗mgammon@uwaterloo.ca

†rbmann@uwaterloo.ca

proposed that do allow for higher-order gravity in four

dimensions and satisfy reasonable physical requirements

such as positive energy excitations on constant curva-

ture backgrounds. Such higher curvature theories are

referred to as “generalized quasi-topological gravities”,

or GQTGs [6]. The original HCT was cubic in curvature

[5, 7]; it was soon followed by a class quartic in curvature

[3]. Most recently a procedure was found for construct-

ing physically reasonable HCTs to any desired power of

curvature [6].

Amidst the plethora of higher curvature gravity theo-

ries, the quadratic Lovelock theory, or so-called “Einstein

Gauss-Bonnet” gravity has been of special interest. In

addition to having 2nd order equations of motion, it is

the simplest HCT. For a long time it was thought that

the Gauss-Bonnet (GB) action term

SGB

D=αZdDx√−gRµνρτ Rµνρτ −4Rµν Rµν +R2

≡αZdDx√−gG

(1)

could not contribute to a system’s gravitational dynam-

ics in D≤4 (since it becomes a total derivative in such

cases), and hence the GB contribution (1) is often re-

ferred to as a topological term of no relevance. Further-

more, a new 3+1-dimensional gravity theory which pos-

sesses diﬀeomorphism invariance, metricity, and second

order equations of motion would be a violation of the

Lovelock theorem [5] and thus should only be possible

by introducing an additional ﬁeld into the theory besides

the metric tensor.

In 2020, Glavan and Lin [8] claimed to have bypassed

the Lovelock theorem via the following rescaling of the

Gauss-Bonnet coupling constant:

(D−4)α→α, (2)

arXiv:2210.01909v3 [gr-qc] 30 Mar 2024

and then taking the D→4 limit of exact solutions

to Einstein-Gauss-Bonnet gravity. This interesting ap-

proach allowed for non-vanishing contributions from the

GB action term in D= 4. In doing so, a number of 4-

dimensional metrics can be obtained (for spherical black

holes [8–12], cosmological solutions [8, 13, 14], star-like

solutions [15, 16], radiating solutions [17], collapsing so-

lutions [18], etc.) carrying imprints of higher curvature

corrections inherited from their D > 4 counterparts.

Unfortunately the existence of limiting solutions does

not actually imply the existence of a 4D theory, and a

number of objections in this vein quickly appeared [19–

21]. However, the conclusion that there ultimately was no

four-dimensional Gauss-Bonnet theory of gravity proved

to be premature when it was shown that a D→4 limit

of the action (2) could be taken [22, 23], generalizing

a previous procedure for taking the D→2 limit of GR

[24]. It is also possible to employ a Kaluza-Klein-like pro-

cedure [25], compactifying D-dimensional Gauss-Bonnet

gravity on a (D−4)-dimensional maximally symmetric

space and then re-scaling the coupling constant according

to equation (2). The resultant 4D scalar-tensor theory is

a special case of Horndeski theory [26], which surprisingly

has spherical black hole solutions whose metric functions

match those from the na¨ıve D→4 limiting solutions de-

rived by Glavan & Lin [8]. Such solutions can be obtained

without ever referencing a higher dimensional spacetime

[22].

This theory, referred to as 4D Einstein Gauss-Bonnet

gravity, provides an interesting phenomenological com-

petitor to GR [27]. Conformally transforming the metric

gµν →e−2ϕgµν in (1) and subtracting it from the original

GB action (1) yields the 4DEGB action after trivial ﬁeld

redeﬁnitions: [22]

4=αZd4x√−ghϕG+ 4Gµν ∇µϕ∇νϕ−4(∇ϕ)2□ϕ

+ 2(∇ϕ)4i(3)

using (2), where ϕis an additional scalar ﬁeld. No further

assumptions about particular solutions to higher dimen-

sional theories or background spacetimes are required.

Adding to this the Einstein-Hilbert action with a cos-

mological term,

S=Zd4x√−g[R−2Λ] + SG

4(4)

the equations of motion follow from the standard varia-

tional principle, with that for the scalar being given by

Eϕ=−G + 8Gµν ∇ν∇µϕ+ 8Rµν ∇µϕ∇νϕ−8(□ϕ)2

+ 8(∇ϕ)2□ϕ+ 16∇aϕ∇νϕ∇ν∇µϕ+ 8∇ν∇µϕ∇ν∇µϕ

= 0

(5)

and the variation with respect to the metric yields

Eµν = Λgµν +Gµν +α"ϕHµν −2R[(∇µϕ) (∇νϕ) + ∇ν∇µϕ]

+ 8Rσ

(µ∇ν)∇σϕ+ 8Rσ

(µ∇ν)ϕ(∇σϕ)−2Gµν (∇ϕ)2+ 2□ϕ

−4 [(∇µϕ) (∇νϕ) + ∇ν∇µϕ]□ϕ+ 8 ∇(µϕ∇ν)∇σϕ∇σϕ

−gµν (∇ϕ)2−4 (∇µϕ) (∇νϕ)(∇ϕ)2+ 4 (∇σ∇νϕ) (∇σ∇µϕ)

−4gµν Rσρh∇σ∇ρϕ+ (∇σϕ) (∇ρϕ)i−2gµν (∇σ∇ρϕ) (∇σ∇ρϕ)

−4gµν (∇σϕ) (∇ρϕ) (∇σ∇ρϕ)+4Rµνσρ [(∇σϕ) (∇ρϕ) + ∇ρ∇σϕ]

+ 2gµν (□ϕ)2#= 0

(6)

where His the Gauss-Bonnet tensor:

Hµν = 2hRRµν −2Rµανβ Rαβ +RµαβσRαβσ

ν−2RµαRα

−1

4gµν RαβρσRαβρσ −4Rαβ Rαβ +R2i.

(7)

These ﬁeld equations satisfy the following relationship

0 = gµν Eµν +α

2Eϕ= 4Λ −R−α

2G(8)

which can act as a useful consistency check to see whether

prior solutions generated via the Glavin/Lin method are

even possible solutions to the theory. For example, using

(8) it is easy to verify that the rotating metrics generated

from a Newman-Janis algorithm [11, 28] are not solutions

to the ﬁeld equations of the 4DEGB theory.

Despite much exploration of the theory [29], there has

been relatively little work investigating rotating black

hole solutions. Attempts have been made using a na¨ıve

rescaling of the 4DEGB coupling constant [30], or by

implementing the Newman-Janis approach [11, 28, 31],

neither of which produce valid solutions to this theory in

general (the latter case not satisfying the positive energy

condition). A notable exception is a recently obtained

class of asymptotically ﬂat slowly rotating black hole so-

lutions [16] that were obtained for the 4DEGB theory.

However, a detailed study of the geodesics of particles

surrounding such black holes (particularly in de Sitter

and anti-de Sitter spacetimes) was not done.

In this paper we address this issue, obtaining slowly

rotating black hole solutions to the ﬁeld equations of the

4DEGB theory in asymptotically ﬂat/(A)dS space. We

then analyze their physical properties (innermost stable

circular orbits, photon rings, black hole shadow, etc.)

to see how they diﬀer from standard results in Einstein

gravity. We ﬁnd for Λ ≤0 that 4DEGB gravity enforces

a minimum mass on the black hole solutions, whereas

for Λ >0 both a minimum and maximum mass occur,

whose horizon structure is directly analogous to that of

Reissner-Nordstr¨om de Sitter spacetime. Besides this,

the results are similar in form to general relativity. The

incredible empirical success of GR necessitates this simi-

larity of solutions for a correct theory, but makes diﬀeren-

tiation via measurement diﬃcult. We ﬁnd that the great-

est deviations from GR are typically in the low (near-

minimal) mass regime, motivating a search for the small-

est observable astrophysical black holes.

II. SOLUTIONS

A. Metric Functions

To construct slowly rotating solutions for the new

4DEGB theory, we begin with the following metric

ansatz:

ds2=−f(r)dt2+dr2

h(r)+ 2ar2p(r) sin2θdtdϕ

+r2dθ2+ sin2θdϕ2

(9)

where ais a small parameter governing the rate of rota-

tion. Of particular interest are Schwarzschild-like solu-

tions where h(r) = f(r). With this, and the substitution

x= cos θ, our line element can be written

ds2=−f(r)dt2+dr2

f(r)+ 2ar2p(r)(1 −x2)dtdϕ

+r2hdx2

1−x2+ (1 −x2)dϕ2i.

(10)

Inserting this into the equations of motion eqs. (5)

and (6) and considering the combination E0

0− E1

1, we

derive the following equation for the scalar ﬁeld

(ϕ′2+ϕ′′)(1 −(rϕ′−1)2f) = 0 (11)

which admits either the solution ϕ= ln( r−r0

l) (with r0, l

integration constants), or the solutions

ϕ±=Z√f±1

√fr dr, (12)

where the latter solution with a minus sign reproduces

previous results [8] in the spherically symmetric case, and

falls oﬀ as 1/r when Λ = 0. We shall choose this solution

henceforth.

Using (11) we can solve for the metric function f(r)

from the geometric expression (8). It can easily be shown

that

r2(1 −f(r)) −Λr4

3+αf(r)2−2αf(r)−αC2r+C1= 0.

(13)

As we wish to recover the Schwarzschild-AdS solution

when α= 0, we set C1=αand C2=2M

αyielding

f±= 1 + r2

2α1±r1 + 8αM

r3+4

3αΛ(14)

where the f−(or Einstein) branch is the one yielding the

Schwarzschild AdS solution in the limit α→0. From

here it is straightforward to show that ϕ−falls oﬀ as 1/r

when Λ = 0, whereas all other solutions for ϕdiverge

logarithmically at large r.

Remarkably the solution (14) is still valid to leading

order in a. The only remaining independent equation

from (6) to this order is given by E03:

rp′′ 24αM +r3(4αΛ + 3)+4p′15αM +r3(4αΛ + 3)= 0

(15)

and admits the exact solution

p=C2−C1q1 + 8αM

r3+4

3αΛ

12αM .(16)

We require that the metric match the slowly rotating

Kerr-(A)dS metric function pin the large rlimit. An

expansion of the function f−(r) in this limit indicates

that

Λeﬀ =3

2α"r1 + 4αΛ

3−1#=2Λ

1 + q1 + 4αΛ

(17)

is the eﬀective cosmological constant, yielding

C1= 2M√12αΛ+9 C2=3+4αΛ−2αΛeﬀ

6α(18)

This leaves us with the ﬁnal expression

p(r) = 2Λ

3

q1 + 4αΛ

1 + q1 + 4αΛ



(19)

2α"1−r1 + 4αΛ

3s1 + 4α

3Λ + 6M

r3#

which, in the Λ = 0 limit, matches1the result [16] for

asymptotically ﬂat spacetime. Additionally we note that

we can rewrite the metric functions in terms of Λeﬀ as

f±= 1 + r2

2α

1±s2

3αΛeﬀ + 12

+8αM

r3

(20)

p(r) = Λeﬀ

32

3αΛeﬀ + 1(21)

2α

1−2

3αΛeﬀ + 1s2

3αΛeﬀ + 12

+8αM

r3

.

It is straightforward to show that the metric (9) is

asymptotically of constant curvature, with

Rµν →Λeﬀ gµν (22)

1up to an errant factor of the rotation parameter included in the

metric function in [16]

as r→ ∞.

We pause to comment on the range of validity of the

metric (9) (with f−(r) and p(r) given by respectively by

(14) and (19)), which is within the class of generalized

Lense-Thirring metrics recently discussed in [32]. Unlike

Einstein gravity, the solution (9) is characterized by two

metric functions, as was shown for this class [32]. It gen-

eralizes the asymptotically ﬂat case previously obtained

[16] and reduces to the slowly rotating Kerr-(A)dS met-

ric if α= 0. We ﬁrst note that if M= 0 the metric is

that of a spacetime of constant curvature Λeﬀ in rotating

coordinates, and that if a= 0 the metric is an exact so-

lution to the ﬁeld equations (5), (6), with event horizons

given by the real solutions of

Λr4−3r2+ 6Mr −3α= 0 (23)

where f−(r) = 0. If Λ >0 there are, in decreasing order

of magnitude, three solutions rc,r+, and r−to (23); if

Λ<0 only the latter two are present. In both cases r+is

the outer horizon of the black hole. Note that the values

of these solutions to (23) are independent of a, as is the

case for the Kerr solution to leading order in a, though

the latter has only a single horizon in this approximation.

Using analytic continuation via a Kruskal-type extension

to continue to values of r < r+, our solution will be

valid for all values of r > r−provided a << r−. In

analyzing the structure of the metric, we shall make this

assumption2. However phenomenologically we need only

consider physics in regions where r > r+, in which case

it is suﬃcient to require a << r+, ensuring that the gtϕ

component of the metric remains small in this region.

For α= 0 = Λ this criterion becomes a < 2m.

B. Analytic Properties

The slowly rotating 4DEGB metric (10), with pgiven

by (19) and f=hgiven by (14), is singular near r= 0,

which can be seen by computing the Ricci scalar to lowest

order in r. In doing so we ﬁnd that

R(r)r→0∼15

4r2M

αr3(24)

regardless of the value of Λ. We consider here only solu-

tions where this singularity is behind the horizon. Com-

puted explicitly to leading order in the rotation parame-

ter, the Kretschmann scalar is

K=f′′(r)2+4f′(r)2

r2+4(f(r)−1)2

r4,(25)

2It is possible to consider values of the scalar ﬁeld for r < r+

if the solution (12) is extended to the time-dependent solution

ϕ=qt +Rf−√f+q2r2

f r dr where qis a constant of integration.

Remarkably this does not aﬀect the solution for the metric [16].

and the only other condition leading to a curvature sin-

gularity is

4αΛ + 24αM

r3+ 3 = 0 (26)

which always occurs inside the horizon and is never true

since the quantity 4αΛ+3≥0 in the allowed region of

(α, Λ) parameter space (see section III). Therefore the

4DEGB metric is regular everywhere but the black hole

singularity.

The metric (9) has two Killing vectors ξ(t)=∂/∂t and

ξ(ϕ)=∂/∂ϕ. It can be shown easily that parameter

M=MK≡ Q(ξ(t)) is the Komar mass, where

Q(ξ) = 1

8πZ∂Σ

⋆K =c

8πZ∂Σ

⋆dξ +3∇2dξ

2Λeﬀ (27)

is the Komar charge [33] associated with the Killing form

ξ=ξµdxµ(with ⋆the Hodge dual); the normalization

constant c=1

2q1 + 4

3αΛ, and ∂Σ is the 2-surface of

constant tat spatial inﬁnity. Likewise, it is straightfor-

ward to show that J= 2Mac is the angular momentum

associated with the Komar charge Q(ξ(ϕ)).

For black hole solutions, the scalar ﬁeld (12) is ill-

deﬁned inside the horizon where f(r)<0. Such solutions

can be made regular across the horizon [16]. As we are

primarily interested in the phenomenological aspects of

the slowly rotating solutions, we shall focus on exterior

solutions in the sequel.

The structure of the ﬁeld equations is such that all

quantities can be rescaled into a unitless form, relative

to some length scale. Writing

Λ = ±3

L2(28)

where the positive and negative branches correspond to

de Sitter and anti-de Sitter space respectively, and Lis

the Hubble length, we shall perform the rescalings:

α→¯αL2M→¯

ML r →¯rL a →¯aL (29)

in units of L. The 4DEGB metric functions then become

f(¯r) = 1 + ¯r2

2¯α 1−r1±4¯α+8¯α¯

¯r3!(30)

and r2p(r)→¯r2¯p(¯r), where

¯p(¯r) = ±2√1±4¯α

1 + √1±4¯α+1

2¯α 1−√1±4¯αr1±4¯α+8¯α¯

¯r3!.

(31)

If instead we consider an asymptotically ﬂat spacetime,

we can directly set Λ = 0 in eqs. (14) and (19) and can

rescale all quantities in units of some ﬁducial mass. In

what follows we shall take this to be a solar mass.

III. PROPERTIES OF THE SOLUTION

In this section we study physical properties of the so-

lutions derived above. We discuss the location and an-

gular velocity of the black hole horizons, the equatorial

geodesics – including the innermost stable circular orbit,

photon rings (and associated Lyapunov exponents) – as

well as the black hole shadow. In each case we compare

the properties of the 4DEGB solution for multiple val-

ues of the coupling constant to the analogous GR result,

and discuss how the rotational corrections aﬀect these

properties.

A. Location and Angular Velocity of the Black

Hole Horizons

The angular velocity of the black hole horizon is de-

ﬁned as

Ωh=−gtϕ

gϕϕ |r=rh=−ap(rh) (32)

where rhis the radius at which grr diverges (ie. f(rh) =

0).

To determine the locations of the horizons, it is conve-

nient to rewrite the metric function f−(r) as

f−(r) = r2

F(r)

f+(r)(33)

where

F(r)=1−2M

r+α

r2−1

3Λr2.

The denominator f+does not vanish, and the horizons

are given by the roots of the numerator, which obey the

equation

Λr4

h−3r2

h+ 6Mrh−3α= 0 (34)

which has exact solutions for all values of Λ.

1. Λ=0

In asymptotically ﬂat space, (34) admits the following

simple solutions:

rh=M±pM2−α(35)

with r+(the outer horizon) recovering the Schwarzschild

value as α→0. Assuming α > 0, this equation sets a

minimum value for black hole mass in the theory, namely

Mmin =√α(36)

when Λ = 0. For smaller masses, the metric function

f(r) does not vanish anywhere and thus no horizon exists.

These solutions have naked singularities.

Since we have an exact solution for p(r) from (19) and

a simple analytic form for rh, the angular velocity of the

horizon

Ωh=a

1−r1 + 8αM

(M+√M2−α)3

2α(37)

is straightforward to compute. In ﬁgure 1 we plot Ω

χ(in

units of inverse seconds) as a function of M(in units of

solar mass) for a variety of values of αalongside the GR

solution for comparison (where χ=a/M). The main

new feature introduced by the 4DEGB theory is the ex-

istence of a maximal angular velocity, indicated by the

termination points of the blue curves at any given α, due

to the presence of a minimum mass. As αincreases, this

maximal value decreases.

For a ﬁxed αwe observe that the 4DEGB theory pre-

dicts a signiﬁcantly larger angular velocity at a given

(small) mass than does GR, but quickly converges to the

GR result when the mass is large.

0.0 0.5 1.0 1.5 2.0

M(M

⊙

2×108

4×108

6×108

8×108

Ω/χ(s-1)

FIG. 1: Angular velocity of the black hole horizon as a

function of mass when Λ = 0 for α/M 2

⊙= 0.01, 0.1, 0.2,

0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1 (in blue from left to right)

plotted against the Einstein (α= 0) solution Ω = χ/4M

(in red).

2. AdS (Λ<0)

Solutions to (34) yield rather cumbersome expressions

for the two diﬀerent positive values of rh. For this reason,

we solve numerically for the horizon as a function of black

hole mass, illustrating the results in ﬁgure 2a. As in

the asymptotically ﬂat case, a minimum mass black hole

exists at which the inner and outer horizons merge, given

Mmin =q1 + 12αΛ−p(1 −4αΛ)3

3√2Λ (38)

which yields (36) in the limit Λ →0.

There are also upper and lower bounds on the Gauss-

Bonnet coupling constant αfor any ﬁxed Λ. These are

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Slowly Rotating Black Holes in 4D Einstein Gauss-Bonnet Gravity Michael Gammonand Robert B. Mann

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