Entanglement entropy of two black holes and entanglement entropic force

Journal Article · Tue Mar 15 00:00:00 EDT 2011 · Physical Review. D, Particles Fields

DOI:https://doi.org/10.1103/PHYSREVD.83.065002· OSTI ID:21537564

Shiba, Noburo ^[1]

---

1. Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043 (Japan)

We study the entanglement entropy S{sub C} of the massless free scalar field on the outside region C of two black holes A and B whose radii are R{sub 1} and R{sub 2} and how it depends on the distance r(>>R{sub 1},R{sub 2}) between two black holes. If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force acting on the two black holes from the r dependence of S{sub C}. We develop the computational method based on that of Bombelli et al. to obtain the r dependence of S{sub C} of scalar fields whose Lagrangian is quadratic with respect to the scalar fields. First, we study S{sub C} in (d+1)-dimensional Minkowski spacetime. In this case the state of the massless free scalar field is the Minkowski vacuum state, and we replace two black holes by two imaginary spheres and take the trace over the degrees of freedom residing in the imaginary spheres. We obtain the leading term of S{sub C} with respect to 1/r. The result is S{sub C}=S{sub A}+S{sub B}+(1/r{sup 2d-2})G(R{sub 1},R{sub 2}), where S{sub A} and S{sub B} are the entanglement entropy on the inside region of A and B, respectively, and G(R{sub 1},R{sub 2}){<=}0. We do not calculate G(R{sub 1},R{sub 2}) in detail, but we show how to calculate it. In the black hole case we use the method used in the Minkowski spacetime case with some modifications. We show that S{sub C} can be expected to be the same form as that in the Minkowski spacetime case. But in the black hole case, S{sub A} and S{sub B} depend on r, so we do not fully obtain the r dependence of S{sub C}. Finally, we assume that the entanglement entropy can be regarded as thermodynamic entropy and consider the entropic force acting on two black holes. We argue how to separate the entanglement entropic force from other forces and how to cancel S{sub A} and S{sub B} whose r dependences are not obtained. Then we obtain the physical prediction, which can be tested experimentally in principle.

• OSTI ID: 21537564
• Journal Information: Physical Review. D, Particles Fields, Vol. 83, Issue 6; Other Information: DOI: 10.1103/PhysRevD.83.065002; (c) 2011 American Institute of Physics; ISSN 0556-2821
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
79 ASTROPHYSICS
COSMOLOGY AND ASTRONOMY
BLACK HOLES
DEGREES OF FREEDOM
ENTROPY
FORECASTING
LAGRANGIAN FUNCTION
MINKOWSKI SPACE
MODIFICATIONS
QUANTUM ENTANGLEMENT
SCALAR FIELDS
SPACE-TIME
FUNCTIONS
MATHEMATICAL SPACE
PHYSICAL PROPERTIES
SPACE
THERMODYNAMIC PROPERTIES