aa r X i v : . [ g r- q c ] S e p Rindler horizon entropy from nonstationarity
Hristu Culetu,Ovidius University, Dept.of Physics,B-dul Mamaia 124, 900527 Constanta, Romania,e-mail : [email protected] 3, 2018
Abstract
Finite entropy and energy are obtained for the horizon of a Rindlerobserver on the grounds of the nonstatic character of the geometry be-yond the horizon. Edery - Constantineau prescription is used to find thedynamical phase space of this particular spacetime. The number of mi-crostates rooted from the ignorance of a Rindler observer of the parameter t from the nonstationary region are calculated. The entropy expression isalso obtained from the electric field on the Rindler horizon generated inthe comoving system of a uniformly accelerated charge.We suggest that the gravitational energy density constructed by meansof the horizon energy and using the Holographic Principle is proportionalto g , similar with a result recently obtained by Padmanabhan. Keywords : nonstationary metric, field configurations, horizon degreesof freedom .
The black hole (BH) physics after Bekenstein and Hawking has implied thatthere is a deep connection among gravitation, thermodynamics and quantuminformation theory. The Hawking formula for the BH horizon temperature isan evidence (it includes all the fundamental constants of physics).Since horizons block informations to certain observers, it seems reasonableto associate an entropy with any event horizon [1]. If a family of observers haveno access to a part of spacetime, then they will attribute an entropy to thegravitational field because of the degrees of freedom (DOF) which are hiddenbehind the horizon. Padmanabhan goes further and shows [2] that the space-time has microscopic DOF and the Einstein field equations in the continuumlimit are to be obtained as the coarse-grained, thermodynamic limit of the (un-known) microscopic laws. Therefore, there should exist a relation similar to theequipartition law E = (1 / nk B T connecting the spacetime energy, temperatureand the number of the microscopic DOF within that spacetime.1t is generally accepted that the BH horizon has entropy but there is noconsensus whether this is valid for Rindler’s or deSitter’s horizon, too (the factthat the spacetime has a microstructure allows one to obtain the dynamicsextremizing a suitable thermodynamic potential, for example entropy [2]).Edery and Constantineau [3] showed that non-extremal BHs contain a non-stationary region hidden behind the event horizon where the Killing vector be-comes spacelike. In their view, the Schwarzschild BH stems from the nonstaticinterior region: it is a measure of an outside observer’s ignorance of the value ofthe time t ∈ (0 , m ) which labels a continuous set of classical microstates. Theauthors of [3] applied the above idea to Schwarzschild, Reissner - Nordstromand Kerr BHs, stressing that the extremal BHs have zero entropy because theydo not contain nonstationary regions (that corresponds to a single metric con-figuration).We apply Edery-Constantineau ideas to the spacetime felt by a uniformlyaccelerated observer who possesses a horizon and, beyond it, the geometry isnonstationary. We further show that the number of DOF on the Rindler horizonis proportional to 1 /g where g is the observer constant acceleration. Moreover,the horizon has energy and entropy thanks to the nonstatic character of themetric beyond it. A timelike congruence of geodesic observers is endowed withexpansion and shear, depending on the acceleration g . Throughout the paperwe take G = c = ~ = k B = 1. In Ref. [4] we have shown how the well known Rindler geometry ds = − (1 − gX ) dT + dX + dY + dZ (2.1)is obtained from the usual Minkowski metric by the coordinate transformation x M = ( 1 g − X ) cosh gT, t M = ( 1 g − X ) sinh gT, | x M | > | t M | . (2.2)(the X = const. observers move along the hyperbola x M − t M = (1 /g − X ) where ( x M , t M ) are Minkowski coordinates. The transformation1 − gX = p − g ¯ x (2.3)brings the geometry (2.1) under the form ds = − (1 − g ¯ x ) d ¯ t + d ¯ x − g ¯ x + d ¯ y + d ¯ z (2.4)where X ≺ /g, ¯ x ≺ / g and ¯ t ≡ T, ¯ y = Y, ¯ z = Z . In the region ¯ x > / g ,1 − g ¯ x becomes negative, ¯ x - timelike and ¯ t - spacelike. Therefore, we replace¯ x with t and ¯ t with x . One obtains ds = − dt gt − gt − dx + dy + dz (2.5)2here t ≻ / g, y = ¯ y, z = ¯ z . The geometry (2.5) is flat, nonstationary and isvalid beyond the Rindler horizon X = 1 /g (or ¯ x = 1 / g ).It is worth noting that, for ¯ x > / g , 1 − gX from (2.1) becomes imaginaryand, therefore, (1 − gX ) <
0. Consequently, x M − t M = (1 /g − X ) <
0. Inaddition, dX = − d ¯ x / (2 g ¯ x − < x istimelike) beyond X = 1 /g or x M = t M . In other words, the spacetime (2.5)is valid beyond the Minkowski causal horizon x M = t M . We indeed obtain, bymeans of the transformation [4, 5] p gt − gτ, x = τ, (2.6)the metric ds = − dτ + g τ dη + dy + dz (2.7)which is nothing else but the nonstationary Milne geometry [5, 6] (or the de-generate Kasner geometry), well known from Cosmology and also from the Rel-ativistic heavy Ions Colissions (RHIC), where η represents the rapidity. Wecould, of course, have obtained it directly from (2.1).A similar situation is encountered when the Schwarzschild horizon r = 2 m iscrossed - the timelike Killing vector becomes spacelike and the geometry insidethe BH is nonstationary [7].According to Edery-Constantineau prescription, the phase space beyond theRindler horizon where the metric (2.5) is valid does not correspond to a singlemicrostate but to a continuous set of states parameterized by the time t , with t ∈ (1 / g, ∞ ). Therefore, Rindler’s horizon must have an entropy due to theinaccessibility to have informations about internal configurations beyond theevent horizon [8]. Let us take a family of spacelike hypersurfaces Σ of constant t . The inducedmetric on Σ is given by h ab = g ab + u a u b , (3.1)where u a = ( √ gt − , , ,
0) is the velocity field of the congruence ( h ab u b = 0)which is orthogonal to Σ. The indices a, b run from 0 to 3.The dynamical phasespace ( h ab , P ab ) is defined by [3] P ab = √ h π ( K ab − Kh ab ) , (3.2)where P ab is the momentum conjugate to h ab , K ab = ˙ h ab / N is the extrin-sic curvature of Σ, h = det ( h ab and N ( t ) is the lapse function, that is N =1 / √ gt −
1. We have h ab = (0 , gt − , , , K xx = g p gt − , K yy = K zz = 0 , K = g √ gt − . (3.3)3q. (3.2) yields the only nonzero components P yy = P zz = − g π . (3.4)The Hamiltonian constraint R + K ab K ab − K = 0 is obeyed nontrivially inthe spacetime (2.5) because K ab K ab = K = g / (2 gt − = 0 ( R , constructedwith h ab , is vanishing). We also note that P ab ’s are constants and depend onlyon the acceleration g .For the other kinematical parameters of the congruence one obtains (bymeans of the software package Maple - GRTensor)- the scalar expansion Θ ≡ ∇ a u a = g √ gt − , (3.5)namely Θ = K ≡ K aa .- the shear tensor σ ab = 12 ( h cb ∇ c u a + h ca ∇ c u b ) −
13 Θ h ab (3.6)has the nonzero components σ xx = 2 g √ gt − , σ yy = σ zz = − g √ gt − , (3.7)with σ ≡ σ ab σ ab = 2 g / gt − a b = u a ∇ a u b = 0, showingthat the congruence is geodesic, that is the ”static” observer with u i = 0 ( i =1 , ,
3) move along a geodesic (the situation resembles the BH interior case,where r = const. observers are geodesic [9]Let us noting that all the kinematical parameters vanish when g = 0 or when t → ∞ . In addition, the time variation of the scalar expansion˙Θ ≡ u a ∇ a Θ = − g gt − t ) decreases (otherwise the Raychaudhuri equation˙Θ − ∇ b a b + σ − ω + 13 Θ = − R ab u a u b (3.9)will not be satisfied (we have here a b = 0, the Ricci tensor R ab = 0 and thevorticity tensor ω ab = 0, with ω ≡ ω ab ω ab )). Using the Edery - Constantineau paradigm, the entropy is a measure of theRindler observer ignorance on the value of the parameter t which labels the4onstationarity of the metric fields beyond the horizon. However, their modeldoes not give us a method to calculate the entropy or the gravitational energy,in general.Having known that the entropy of the Rindler spacetime should be nonzero,we have nothing else to do than to take its expression from [10] S = π g (4.1)Keeping in mind that the Rindler horizon temperature is T = g/ π , we imme-diately obtain E = 2 T S = 14 g (4.2)for the energy of the Rindler spacetime (see also [11]). We may of courseget the expression of the entropy directly from the nonstatic geometry (2.5).The boundary of the spacetime corresponds to t = 1 / g (that is, the horizon¯ x max = 1 / g ). As Edery and Constantineau have shown, most of the energycontribution comes from a thin slice in the interior region near the event horizon(for the Schwarzschild BH), which corresponds in our case to the initial time t min . Taking therefore 4 πt min as the area of the initial surface, one obtains S = 14 4 πt min = π g , (4.3)as in Eq. (4.1).We are now in a position to find the number n of the DOF (or the numberof internal configurations) of (2.4) due to the time dependence of the metric (aset of classical microstates which express the ignorance of an observer locatedin the static region).From the equipartition rule we have E = (1 / nT . Using (4.2) and therelation for the horizon temperature T , we obtain n = πg (4.4)When all fundamental constants are recovered, (4.4) becomes n = ( c /G ~ )( π/g ).For example, the value g = 10 m/s leads to n ≈ bits. In other words, thenonstationarity of Rindler spacetimes (2.1) or (2.3) beyond the horizon ¯ x = 1 / g leads to that enormous value of the number of microstates.Incidentally, when we try to write down the energy density ¯ ρ rooted from E , the following expression is reached¯ ρ = EV = 14 g π g = 3 g π , (4.5)where we have used the Holographic Principle taking E to be located uniformlyon a sphere of radius 1 /g (we stress here the special role played by the distance1 /g [12]), instead of being concentrated on the horizon. But the relation (4.5)5or the energy density of the gravitational field resembles that one obtained byPadmanabhan [2] for a nongeodesic observer at rest in a static spacetime. Thatis not surprising because a static observer in a gravitational field is equivalentto an accelerated one in flat spacetime.Moreover, even the location of the radiation emitted by a uniformly acceleratedcharge may be explained using the nonstationarity of the Rindler metric beyondthe horizon X = 1 /g (or x M = t M ). It is a well established fact that an observerwho is comoving with an accelerating charge will not detect any electromagneticradiation since the radiation field is confined to a region inaccessible to him,namely | x M | < t M , where the geometry is time dependent (see [5, 6]). On theline of Edery - Constantineau view, we could assert that the classical radiationis localized in the nonstationary Milne spacetime, due to the continuous set ofmicrostates parameterized by the time t . The radiation field (containing theradiating part extracted by separating the components dropping of as 1 /r fromthe usual 1 /r fields) have been obtained in many papers [13, 14, 6].To simplify the analysis, let us put 1 − gX = gξ in the metric (2.1). Oneobtains ds = − g ξ dT + dξ + dY + dZ (4.6)We give here only the component E ξ of the static electric field in the (comovingwith the charge) coordinates ( ξ is in the direction of acceleration). It is givenby [13, 14] E ξ = 4 eg ( ξ − ρ − g − )[ g ( ξ + ρ + g − ) − ξ ] / (4.7)where e is the charge of the electric particle and ρ = Y + Z . The sameexpression is obtained for the x M - component of the electric field in Minkowskicoordinates, with only one difference: ξ has to be replaced with x M − t M . Wemention also that the lack of radiation in the comoving system is due to thenull values of the components of the magnetic field which however are nonzeroin the nonstatic region ¯ x > / g . On the surface ξ = 0 (the event horizon H )the electric field acquires the value E ξH = − eg (1 + g ρ ) (4.8)with the surface charge density σ H = E ξH π = − eg π (1 + g ρ ) (4.9)For small values of Y, Z (or for ρ << /g ), (4.9) yields σ H ≈ − eg π (4.10)We observe from (4.10) that π/g plays the role of a surface (it is a portion ofthe horizon, where | E ξH | acquires the highest values. Hence, taking π/g as the6rea of the horizon, its entropy will be given by S = area/ π/ g , i.e. thevalue given Eq. (4.3), which has been obtained using different arguments. The prescription of Edery and Constantineau is applied in this paper to provethat the Rindler horizon possesses microscopic DOF and, from here, an entropyproportional to 1 /g . It is rooted from the nonstationary character of the ge-ometry beyond the event horizon which leads to a continuous set of classicalmicrostates. The entropy measures the ignorance of a Rindler observer to knowthe value of the label t ∈ (1 / g, ∞ ).Using a congruence of ”static” geodesic observers labeled by the velocity field u a , we have written the dynamic phase space ( h ab , P ab ) and the extrinsic curva-ture tensor of the hypersurface Σ of constant time. The kinematical parametersof the congruence have a nontrivial form and ˙Θ <