Entropy for curvature squared gravity
using surface term and auxiliary ﬁeld
arXiv:1304.7085 [hep-th]

Kyosuke hirochi (D3)
QG lab. (Gravity and Particle Cosmology Group)
Department of Physics, Nagoya University

Introductionaim

Calculate the black hole entropy for the curvature squared gravity
methodUsing the surface term of the action

Introductionaim

Calculate the black hole entropy for the curvature squared gravity
methodUsing the surface term of the action
surface term

Noether charge

Introductionaim

Calculate the black hole entropy for the curvature squared gravity
methodUsing the surface term of the action
Noether charge

surface term

Virasoro algebra

central charge
zero mode eigenvaluesof the Noether charge

Introductionaim

Calculate the black hole entropy for the curvature squared gravity
methodUsing the surface term of the action
Noether charge

surface term

Virasoro algebra

central charge
zero mode eigenvaluesof the Noether charge

Cardy formula

Entropy

Gravitational action (with curvature-squared term)
E-H term

curvature-squared termGravitational action (with curvature-squared term)
E-H termthe surface term
for the Einstein-Hilbert term

“Gibbons-Hawking term”

curvature-squared termGravitational action (with curvature-squared term)
E-H term

curvature-squared term

the surface term
for the Einstein-Hilbert term

for the curvature-squared term??
“Gibbons-Hawking term”

cannot be obtained directly !
reasonFor theories including higher curvature term,
there is no corresponding surface termGravitational action (with curvature-squared term)
E-H term

curvature-squared term

the surface term
for the Einstein-Hilbert term

for the curvature-squared term??
“Gibbons-Hawking term”

cannot be obtained directly !
reasonFor theories including higher curvature term,
there is no corresponding surface termalternative method

construct the action including up to second-derivative terms by introducing an
auxiliary field which is equivalent with the curvature squared action (on-shell)

Action (including second-order derivative terms)… auxiliary field (same symmetry properties with the Riemann tensor)
Equation of Motion forAction (including second-order derivative terms)… auxiliary field (same symmetry properties with the Riemann tensor)
Equation of Motion forparameter choiceequivalencesurface term forsurface term forsurface term for the curvature squared action

Generalized Gibbons Hawking term

(A)dS backgrounds solutions

Noether charge
General form of the surface term

diﬀeomorphism(leaves the horizon structure invariant)

Noether charge

Schwarzschild type metricconserved charge

Fourier expansion

Virasoro algebra... surface area of black hole

Entropy (Using the Cardy formula)
Virasoro algebra

central charge

zero mode eigenvalues

Cardy formula

entropy for the curvature squared gravity

several specific examples
Einstein+Gauss-Bonnet

New Massive Gravity (3-dim)

Critical Gravity (4-dim)consistent with the Wald entropy!

Conclusion
we have calculated the entropy for D-dimensional gravity
with curvature squared term
introducing an auxiliary field, we have obtained the secondderivative formed action which is equivalent with the curvaturesquared action (on-shell)

we have calculated the surface action and the Black Holeentropy for the Schwarzschild type metric.

special case of the parameters