Leading higher-derivative corrections to the Kerr geometry

Leading higher-derivative corrections tothe Kerr geometry

Alejandro Ruiperez

IFT UAM/CSIC

17/01/2020

Iberian Strings (Santiago de Compostela)

Based on JHEP 1905 (2019) 189 in collaboration with Pablo A. Cano

Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 1 / 15

Outline

1 Introduction

2 Effective field theory approach

3 Corrections to the Kerr solution

4 Conclusions


Introduction

There are very good theoretical reasons to expect that Einstein’sGeneral Relativity (GR) will be modified at high-energies and atstrong curvature regimes

E.g. String Theory predicts the appearance of higher-curvature terms

In the coming decades, the increasing sensibility of our gravitationalwave detectors will allow us to test gravity in strong-field regimes

There exists a possibility that we can observe deviations with respectto the predictions of GR

It is thus an important task to obtain a general parametrization of thecorrections to GR that we can test using GW data

A preliminar exercise that one can do is to compute the corrections tothe Kerr solution


Effective field theory approach

This approach relies on the assumption that the corrections can betreated perturbatively: GM >> `, where ` is the length scaleassociated to the corrections (e.g.: ` =

√α′ in string theory).

Then, it is natural to consider an effective action of the form

Seff =

∫d4x√−g

R +∑n≥2

`2n−2L(2n)hd

where L(2n)

hd contains 2n derivatives

If we assume diff-invariance and no extra matter dofs, L(2n)hd will be

constructed out of contractions of the Riemann tensor and itscovariant derivatives

These assumptions can be relaxed by allowing terms involvingRµνρσ = 1

2εµναβRµναβ and dynamical couplings (scalars)


Effective action

Under these assumptions, the leading higher-derivative corrections to theKerr geometry are captured by the following action

S =

∫d4x

√|g |R + α1φ1`

2X4 + α2 (φ2 cos θm + φ1 sin θm) `2RµνρσRµνρσ

+ λev`4Rµν

ρσRρσδγRδγ

µν + λodd`4Rµν

ρσRρσδγRδγ

µν − 1

2

2∑i=1

(∂φi )2

where X4 = RµνρσRµνρσ − 4RµνR

µν + R2

It depends on five parameters: α1, α2, θm, λev and λodd

λodd and θm parametrize the parity-breaking corrections

Heterotic string theory on a 6-torus: α1 = −α2 = −18 , θm = 0,

λev = λodd = 0 and ` =√α′


Corrected solution

Since we work perturbatively, the corrected metric will have the form

gµν = g(0)µν + `4 g

(4)µν + . . . where g

(0)µν is the Kerr metric:

g (0)µν dx

µdxν = −(

1− 2Mρ

Σ

)dt2 − 4Maρ(1− x2)

Σdtdφ

+Σ

(dρ2

∆+

dx2

1− x2

)+

(ρ2 + a2 +

2Mρa2(1− x2)

Σ

)(1− x2)dφ2

with

Σ = ρ2 + a2x2 , ∆ = ρ2 − 2Mρ+ a2


Corrected solutionand

`4g (4)µν =H1dt

2 − H24Maρ(1− x2)

Σdtdφ+ H3Σ

(dρ2

∆+

dx2

1− x2

)+ H4

(ρ2 + a2 +

2Mρa2(1− x2)

Σ

)(1− x2)dφ2

To find the solution, we first assume it admits a slowly-spinning

expansion, i.e.: Hi =∑∞

n=0H(n)i χn and φ1,2 =

∑∞n=0 φ

(n)1,2χ

n

Then we plug these expansions into the corrected eqs. of motion tofind that the form of the corrected solution is:

H(n)i =

n∑p=0

kmax∑k=0

H(n,p,k)i xpρ−k , φ

(n)1,2 =

n∑p=0

kmax∑k=0

φ(n,p,k)1,2 xpρ−k


Horizon

The main advantage of our choice of coordinates is that the horizon

is still placed at ρ = ρ+ = M(

1±√

1− χ2)

Induced metric at the horizon:

ds2H = (1 + H3)∣∣∣ρ+

ρ2+ + a2x2

1− x2dx2 + (1 + H4)

∣∣∣ρ+

4M2ρ2+(1− x2

)ρ2+ + a2x2

dφ2

Area:

AH = 4πMρ+

∫ 1

−1dx (1 + H3/2 + H4/2)

∣∣∣ρ+


Horizon area

AH = 8πM2 +π`4

M2

[α21∆A(1) + α2

2∆A(2) + λev ∆A(ev)]


Surface gravity

κ =

√1− χ2

2M(

1 +√

1− χ2) +

`4

M5

[α21∆κ(1) + α2

2∆κ(2) + λev ∆κ(ev)]


Angular velocity: ΩH =|gtφ|gφφ

∣∣∣ρ+

ΩH =χ

2M(

1 +√

1− χ2) +

`4

M5

[α21 ∆Ω

(1)H + α2

2 ∆Ω(2)H + λev ∆Ω

(ev)H

]


Isometric embedding of the horizon in E3: parity even case


Isometric embedding of the horizon in E3: parity odd case


Main results and future directions

The main results of our paper are the following:

1 We have constructed an EFT that captures the leadinghigher-derivative corrections to vacuum solutions of GR

2 We computed analytically the corrections to the Kerr solution in aslowly-spinning expansion

3 We have studied some properties of the corrected solution: horizon,ergoregion, photon rings, etc . . .

Future directions:

1 Detailed analysis of the geodesics (e.g.: obtain the black hole shadow)

2 Study quasinormal modes

3 Extremal limit?


Thanks for your attention


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Leading higher-derivative corrections to the Kerr geometry