arXiv:1005.1132v3 [gr-qc] 29 Jun 2010 Tolman mass, generalized surface gravity, and entropy bounds Gabriel Abreu ∗ and Matt Visser † School of Mathematics, Statistics, and Operations Research; Victoria University of Wellington, Wellington, New Zealand. (Dated: 7 May 2010; 25 May 2010; 28 June 2010; L A T E X-ed October 28, 2018) In any static spacetime the quasi-local Tolman mass contained within a volume can be reduced to a Gauss-like surface integral involving the flux of a suitably defined generalized surface gravity. By introducing some basic thermodynamics and invoking the Unruh effect one can then develop elementary bounds on the quasi-local entropy that are very similar in spirit to the holographic bound, and closely related to entanglement entropy. Tolman mass is one of the standard notions of quasi- local mass in common use in general relativity [1]. Using classical general relativity, this quasi-local Tolman mass can, in any static spacetime (either with or without a black hole region), be reduced to a flux integral of (gener- alized) surface gravity across the boundary of the region of interest. (This is closely related to the classical laws of black hole mechanics [2].) General relativistic thermo- dynamics, together with a minimal appeal to quantum physics as embodied in the Unruh effect [3], is then suf- ficient to develop elementary but powerful bounds on a suitably defined notion of quasi-local entropy — bounds very similar in spirit to the holographic bound [4–8], and closely related to entanglement entropy [9]. In a static spacetime where the metric is taken to be of the form ds 2 = −e −2Ψ dt 2 + g ij dx i dx j , (1) the Tolman mass contained in a region Ω is defined in terms of the orthonormal components of stress-energy by first taking ρ = T ˆ 0 ˆ 0 and p = 1 3 tr{T ˆ ıˆ  }; and then setting m T (Ω) =  Ω √ −g 4 {ρ +3p} d 3 x. (2) The Einstein equations then imply the purely geometrical statement m T (Ω) = 1 4π  Ω √ −g 4 R ˆ 0 ˆ 0 d 3 x. (3) The Tolman mass is intimately related to the Komar mass [10], though we will not be phrasing any of the dis- cussion below in terms of Killing vectors. It is a very old result, going back at least as far as Landau–Lifshitz [11] that in any stationary metric R 0 0 = 1 √ −g 4 ∂ i (√ −g 4 g 0a Γ i a0 ) . (4) (Here a ∈{0, 1, 2, 3}; i ∈{1, 2, 3}.) Adopting the mani- fest static coordinates of equation (1), and then going to an orthonormal basis, this is more simply phrased as R ˆ 0 ˆ 0 = 1 √ −g 4 ∂ i (√ −g 4 g 00 Γ i 00 ) . (5) * gabriel.abreu@msor.vuw.ac.nz † matt.visser@msor.vuw.ac.nz To get a clean physical interpretation of this formula, consider a fiducial observer (FIDO) with 4-velocity V a =   |g 00 |;0, 0, 0  . (6) By definition the 4-acceleration of these FIDOs is A a =(∇ V V ) a = V b ∇ b V a = V 0 (∂ 0 V a +Γ a c0 V c ) =  |g 00 | Γ a 00  |g 00 | = |g 00 | Γ a 00 . (7) But then, since V is 4-orthogonal to A, we have A 0 = 0; A i = |g 00 | Γ i 00 ; (8) where A i are the 3 spatial components of 4-acceleration. Therefore in any static spacetime, in the region outside the horizon, the Landau–Lifshitz result is R ˆ 0 ˆ 0 = 1 √ −g 4 ∂ i (√ −g 4 A i ) . (9) Then for any 3-volume Ω (if a horizon is present then for convenience we confine ourselves to a region that lies out- side the horizon) we can use ordinary partial derivative integration by parts to deduce  Ω √ −g 4 R ˆ 0 ˆ 0 d 3 x =  Ω ∂ i (√ −g 4 A i ) d 3 x =  Ω ∂ i ( √ g 3 {e −Ψ A i } ) d 3 x =  ∂Ω {e −Ψ A i } ˆ n i √ g 2 d 2 x, (10) where ˆ n is the unit normal, (defined in terms of the 3- metric g ij ), and √ g 2 is the induced area measure on ∂ Ω. Define a (generalized) surface gravity (3-vector) and its norm by κ i = e −Ψ A i ; κ =  g ij κ i κ j = e −Ψ  g ij A i A j . (11) This is just the “redshifted” 4-acceleration of the FIDOs, and is a natural generalization of surface gravity, not just for any event horizon that might be present, but also applying to FIDOs skimming along the boundary ∂ Ω. In terms of this generalized surface gravity we now have m T (Ω) = 1 4π  ∂Ω κ i ˆ n i √ g 2 d 2 x = 1 4π  ∂Ω κ· ˆ n dA . (12)