Thermodynamics of entanglement in Schwarzschild spacetime

Extending the analysis in our previous paper, we construct the entanglement thermodynamics for a massless scalar field on Schwarzschild spacetime. Contrary to the flat case, the entanglement energy $E_{\mathrm{ent}}$ turns out to be proportional to the area radius of the boundary if it is near the horizon. This peculiar behavior of $E_{\mathrm{ent}}$ can be understood by the redshift effect caused by the curved background. Combined with the behavior of the entanglement entropy, this result yields, quite surprisingly, the entanglement thermodynamics of the same structure as black hole thermodynamics. On the basis of these results, we discuss the relevance of the concept of entanglement as the microscopic origin of black hole thermodynamics.