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Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Energy in Higher-Derivative Gravity via
Topological Regularization
D. Rivera-Betancour
Universidad Andrés Bello
based on Anastasiou, Olea and D.R.B (UNAB), PLB 788, 021 (2018),
Giribet (NYU), Miskovic (PUCV), Olea and D.R.B (UNAB), PRD 98,044046
(2018) and arXiv:1904.XXXX.
April 4, 2019
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Overview
1
Introduction
2
Quadratic-Curvature Gravity in 4D
3
ADT energy
4
Iyer-Wald Charges and Topological Regularization
5
Conclusions and future directions
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topology and its applications
Topology plays a fundamental role in Mathematics and Physics.
Some examples are
Four color theorem
Topological Networks
Quantum Hall effect
Topological Insulators
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topology and gravity
How to use top. terms in gravity?
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topology and gravity
Problems with the application of Noether theorem to gravity.
2M G
ds = − 1 −
r
2
2M G −1 2
dt + 1 −
dr + rdΩ2
r
2
The Komar charge gives
Q[∂t ] =
Incorrect overall factor.
D. Rivera-Betancour
M
.
2
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topology and gravity
In the presence of a negative cosmological constant
−1
2M G r2
2M G r2
2
ds = − 1 −
+ 2 dt + 1 −
+ 2
dr2 +rdΩ2 .
r
`
r
`
2
We have
Q[∂t ] =
M
+ ∞.
2
The energy is divergent.
Background subtraction
Holographic renormalization [S. de Haro, K. Skenderis and S.
Solodukhin, CMP 788, 217 (2000)]
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topology and gravity
The inclusion of topological terms to the Einstein-Hilbert action
with a negative cosmological constant render the variational
problem well-posed and regularizes the Noether charges.
Z
Z
1
4 √
d x −g (R − 2Λ) + γ E4 ,
(1)
I=
16πG
M
where γ = `2 /(64πG).
[ R. Aros et al., PRL 84, 1647 (2000)]
D. Rivera-Betancour
M
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Quadratic-Curvature Gravity
QCG action is defined by a combination of quadratic couplings in
the curvature added on top to the Einstein-Hilbert action.
Z
√
1
IQCG =
d4 x −g R − 2Λ + αRµν Rµν + βR2
16πG
where α y β are coupling constants.
The inclusion of the curvature squared terms improve the UV
behavior. [ K.S. Stelle, PRD 16, 953 (1977)]
This theory has a massive scalar field and massive and
massless spin-2 fields.
Higher-Derivative gravity introduces new couplings and
sources at the boundary.
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Field equations and surface terms
Taking arbitrary variations of the action one gets
Z
√
1
δI =
d4 x −g (E µν δgµν + ∇α Θα ) ,
16πG
where
E µν = Gµν + α (g µν − ∇µ ∇ν ) R + αGµν +
1 µν σρ
1 µν
µσνρ
µν
+2α R
− g R
Rσρ + 2βR R − g R +
4
4
+2β (g µν − ∇µ ∇ν ) R.
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Field equations
For the vacuum state of the theory that satisfies
µν
Rαβ
=−
1 [µν]
δ
,
2
`ef f [αβ]
the relation between the cosmological constant and the AdS radius
is not affected by the quadratic curvature contribution.
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Abbott-Deser-Tekin mass
Consider the linearization the EOMs through gµν = ḡµν + hµν
δ(Gµν + E µν ) = T µν
J µ = T µν ξ¯ν
¯ ν q µν
Jµ = ∇
ADT
The charge is written as a surface integral of the 2-form as
Z
1
α
αβ
dSβ qADT
QADT [ξ] =
8πG
Σ
[ S. Deser and B. Tekin, PRD 67, 084009 (2003) ]
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Abbott-Deser-Tekin mass
Expressing the full prepotential in terms of α y β for QCG action
we obtain
αβ
¯ [α hβ]λ + ξ¯[α ∇
¯ β] h + hλ[α ∇
¯ β] ξ¯λ −
qADT
= (1 + 2Λ (4β + α)) ξ¯λ ∇
!
1 ¯ α ¯β
[α ¯
β]λ
¯
¯ β] RL + RL ∇
¯ α ξ¯β −
+ (α + 2β) 2ξ¯[α ∇
− ξ ∇λ h + h∇ ξ
2
¯ [α Gβ]λ + 2Gλ[α ∇
¯ β] ξ¯λ .
−α 2ξ¯λ ∇
L
L
For the Schwarzschild-AdS black hole the mass is
M = m [1 + 2Λ(α + 4β)] .
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Critical Gravity
QCG suffers from having ghosts (Massive spin-2 field). This can be
eliminate at the points α = −3β and β = −1/2Λ (Critical
Gravity).
The massive spin-2 field becomes massless and the scalar
mode is decoupled.
The mass of the black hole and the energy of the propagating
modes vanish.
The linearized EOMs degenerates and,
Additional propagating modes appear (Logarithmic modes).
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Critical Gravity
Can we trust linearized charges?
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Iyer-Wald Charges
For a Lagrangian of the type, L (gµν , Rµναβ ), the conserved
current reads
αβ µ ν
αβ ν
J α = 2∇β Eµν
∇ ξ + 2∇µ Eµν
ξ + (EOM )αβ ξβ ,
αβ
where Eµν
correspond to derivatives of the Lagrangian with
respect to the Riemann tensor
αβ
Eµν
=
D. Rivera-Betancour
∂L
µν .
∂Rαβ
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Iyer-Wald Charges
For QCG the Iyer charge reads
1
QIW [ξ] =
8πG
α
Z
αβ
dSβ qIW
,
Σ
for the Iyer-Wald prepotential
αβ
αβ µ ν
αβ ν
= Eµν
∇ ξ + 2∇µ Eµν
ξ .
qIW
The mass of Schwarzchild-AdS black hole is divergent and has the
incorrect overall factor
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topological invariants and conserved quantities
In Gauss-Bonnet we trust!
Consider now the addition of the Euler density in four dimensions
by an arbitrary coupling.
√ µν αβ
E4 = −g Rαβ
Rµν − 4Rµν Rµν + R2 .
This does not affect the EOMs, but
It does modify the surface terms.
Z
√ [αβγδ] σλ
δItot = d4 x −g · · · + γ g νλ δ[µνσλ] Rγδ
δΓµβλ ,
M
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topological invariants and conserved quantities
The minimum requirement for finite conserved charges is a finite
energy for the vacumm.
Demanding that δItot = 0 for the vacuum solution (well-posed
variational principle) one can fix the coupling γ
γ=
`2
[1 + 2Λ (4β + α)] .
4
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topological invariants and conserved quantities
The prepotential now is
αβ
qtop
=
i
1 µ ν h [αβ]
[α β]
[αβ]
[αβγδ] σλ
∇ ξ δ[µν] + αR[µ δν] + 2βRδ[µν] + γδ[µνσλ] Rγδ
+
2
[α β]
[αβ]
+ ∇µ αR[µ δν] + 2βRδ[µν] ξ ν ,
and the conserved charges read
Qα
top [ξ]
1
=
8πG
Z
Σ
D. Rivera-Betancour
αβ
dSβ qtop
.
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Topological invariants and conserved quantities
For the Schwarzchild-AdS black hole the mass reads
Qt [∂t ] = m [1 + 2Λ (4β + α)] .
Coming back to Critical Gravity, Noether charges are written as
Z
`2
1 [γδ]
[µνσλ] α β
γδ
γδ
µ
Q [ξ] =
dSν δ[αβγδ] ∇ ξ
Rσλ + 2 δ[σλ] − Wσλ .
64πG
`
Σ
All Einstein spaces give a vanishing Noether charge. [G.
Anastasiou, R. Olea and D.R.B, PLB 788 (2018)].
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Non-Einstein solutions
Considering the non-Einstein space that solves the EOM and
represent the propagation of gravitational waves
ds2 =
`2 −(1 + F (z))dt2 + 2dtdu + dz 2 + dx2 ,
2
z
where F is a function that does not depend on the lightlike
coordinate u.
The only non-vanishing part of the EOM is
( − m̃2 )F = 0 ,
with the effective mass parameter
m̃2 =
6(α + 4β) − `2
.
α
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Non-Einstein solutions
Two sectors appear from the above equation :
F (z) = cte and F (z) = z 3 (Einstein).
F (z) = z k (non-Einstein).
q
2
Here k± = 32 ± 94 + 6(α+4β)−`
. [E. A. Beato, G. Giribet and M.
α
Hassaine, arXiv:1207.0475 (2012)]
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Non-Einstein solutions
For the non-Einstein sector of the solution one finds the
prepotential
[αβ]
αβ
= z k+1 `−4 k(αk 2 − 3αk + `2 − 6α − 24β) δ[uz] .
qtop
It vanishes for k = k± . This is in total agreement with the ADT
method.
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Conclusions
It was provided a new energy definition for Higher-Derivative
Gravity that does not come from linearization.
Topological regularization contributes for gravity theories with
non-linear contribution on the curvature.
Our method works for Einstein and non-Einstein spaces.
D. Rivera-Betancour
Introduction
Quadratic-Curvature Gravity in 4D
ADT energy
Iyer-Wald Charges and Topological Regularization
Conclusions and future directions
Future directions
Check this energy definition for other massive non-Einstein
solutions in QCG, for example non-Einstein black holes in
Critical Gravity [R. Svarc, J. Podolsky, V. Pravda and A.
Pravdova, PRL 121, 231104 (2018)] .
Extend this definition to higher dimensions.
[arXiv:1904.XXXX].
Prove that for certain cases that the charge is proportional to
the Weyl tensor (Ashtekar-Magnon-Das mass) in QCG [ Y.
Pang, PRD 81, 087501 (2011)].
D. Rivera-Betancour
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