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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
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 2 / 15
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
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 3 / 15
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)
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 4 / 15
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′
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 5 / 15
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
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 6 / 15
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
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 7 / 15
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+
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 8 / 15
Horizon area
AH = 8πM2 +π`4
M2
[α21∆A(1) + α2
2∆A(2) + λev ∆A(ev)]
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 9 / 15
Surface gravity
κ =
√1− χ2
2M(
1 +√
1− χ2) +
`4
M5
[α21∆κ(1) + α2
2∆κ(2) + λev ∆κ(ev)]
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 10 / 15
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
]
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 11 / 15
Isometric embedding of the horizon in E3: parity even case
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 12 / 15
Isometric embedding of the horizon in E3: parity odd case
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 13 / 15
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?
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 14 / 15
Thanks for your attention
Alejandro Ruiperez (IFT UAM/CSIC) 17/01/2020 15 / 15