Unified first law and some general prescription: a redefinition of surface gravity

The paper contains an extensive study of the unified first law (UFL) in the Friedmann-Robertson-Walker spacetime model. By projecting the UFL along the Kodama vector the second Friedmann equation can be obtained. Also studying the UFL on the event horizon it is found that Clausius relation cannot be obtained from the UFL by projecting it along the tangent to the event horizon as it can be for the trapping horizon. However, it is shown in the present work that Clausius relation can be obtained by projecting the UFL along the Kodama vector on the horizon and the result is found to be true for any horizon. Finally motivated by the Unruh temperature for the Rindler observer, surface gravity is redefined and a Clausius relation is obtained from the UFL by projecting it along a vector analogous to the Kodama vector.

In 1970's Hawking [1] showed that BH is not totally black, rather emits thermal radiations by a combine application of quantum mechanics and general relativity at semi classical level. Interestingly, the temperature of the radiation (known as Hawking temperature) and the entropy of the horizon (known as Bekenstein entropy) have a certain universality in the sense that surface gravity (proportional to Hawking temperature) and horizon area (proportional to Bekenstein entropy) [1,2] are purely geometric entity characterized by the space-time geometry. Also this entropy and temperature are related to the BH mass through the first law of BH thermodynamics: dM = T dS [3]. Moreover this fantastic discovery gave rise to (i) a speculation for a deep interrelationship between gravity theories and thermodynamics and (ii) a clue to the nature of quantum gravity.
However, the first possibility came true in 1995 when Jacobson [4] derived Einstein equations from clausius relations: δQ = T dS for all the local Rindler causal horizon through spacetime point (δQ → the energy flux, T → Unruh temperature seen by the accelerated observer just inside the horizon). Subsequently, Padmanabhan [5,6] was able to show the first law of thermodynamics on the horizon, starting from Einstein equations, for a general static spherically symmetric space-time.
Assuming the Universe as a thermodynamical system, this nice interrelation between Einstein equations and thermodynamic laws has been extended in the context of cosmology. For homogeneous and isotropic FRW model it was found [7] that the Friedmann equations are equivalent to the first law of thermodynamics on the apparent horizon having Hawking temperature A G (R A = geometric radius of the apparent horizon). Then in higher dimensional space-time, this equivalence was established for gravity with the Gauss-Bonnet term and for the Lovelock gravity [7][8][9][10].
On the contrary, the situation is totally different for universe bounded by the event horizon (which exists only in accelerating phase of the expansion). Wang et al [11] showed that universe bounded by apparent horizon is a perfect thermodynamical system as both 1st and 2nd law of thermodynamics hold for perfect fluid with constant equation of state and holographic dark energy models. However, according to them both the thermodynamical laws failed to satisfy on the event horizon. Then assuming first law, Mazumdar et al [12][13][14] were able to satisfy second law of thermodynamics on the event horizon with some realistic restrictions. In analogy with apparent horizon, the entropy and temperature at the event horizon were chosen as S E = πR 2 E G and T E = 1 2πR E . Later, it was found [15,16] that the temperature taken on the event horizon (i.e. T E = 1 2πR E ) is not correct and taking the corrected form (i.e. T ) the thermodynamics on the event horizon has been studied. It has been shown [17] that for the following two choices both the thermodynamical laws are satisfied on the event horizon. Also for infinitesimal thermal fluctuation, there is a logarithmic correction to the Bekenstein entropy in the 2nd choice [17].
On the other hand, in the context of dynamical BH, Hayward [18][19][20][21] introduced the notion of trapping horizon and proposed a method to deal with thermodynamics associated with a trapping horizon. According to him, for spherically symmetric space-times, Einstein equations can be rewritten in a form termed as "Unified first law". Then projecting this Unified first law (UFL) along a trapping horizon, the first law of thermodynamics was derived. Further, from the point of view of universal thermodynamics we consider our universe as a non-stationary gravitational system and FRW model may be considered as dynamical spherically symmetric space-time. Moreover, in FRW model we have only inner trapping horizon which coincides with the apparent horizon [18][19][20][21][22][23] and Friedmann equations are equivalent to the UFL on the apparent horizon [22,26]. Also projection of UFL along the tangent to the apparent horizon gives the clausius relation [22].
Further, there is no preferred time coordinate in an evolving time dependent space-time as there is no longer any (asymptotically time-like) Killing vector field. To resolve this problem, Kodama [27] came forward with a geometrically natural divergence free vector field which exists in any time-dependent spherically symmetric space-time. This vector in the literature is popularly known as Kodama vector, and it identifies a natural time like direction outside a dynamic BH. Also there is a conserved current associated with this vector field [27,28].
In the present work, we shall study the Unified first law (UFL) on the event horizon for FRW model of the universe. The line element for FRW space-time can be written as [24] where R = ar is the area radius, h ab = diag(−1, a 2 1−kr 2 ) is the metric on the two-space orthogonal to the spherical symmetry. Using null coordinates (l, m) the above metric can be written as as future pointing null vectors.
For any horizon (having area radius R) the surface gravity is defined as [25] or in explicit form Now the total energy inside the horizon is a purely geometric quantity, related to the structure of the space-time and to the Einstein's equations [22]. According to Misner and sharp [18][19][20][21][22][23], the total energy is given by which on simplification gives According to Hayward [18][19][20][21], the Unified first law is nothing but the rearrangement of the Einstein equations. In the above, A and V stand for the area and volume bounded by the horizon, the work density is regarded as the work done by a change of the horizon and the energy-supply term determines the total energy flow (i.e. δQ = Aψ) through the horizon.
We now introduce the Kodama vector for the present FRW model. It is defined as [27,29] K a = ǫ ab ∇ b R where ǫ ab is the usual Levi-civita tensor in the 2D radial-temporal plane (i.e. normal to the spherical symmetry). For the present homogeneous and isotropic FRW model and Note that Kodama vector is very similar to the Killing vector ∂ and Hence we have and Aψ = 2πR 2 (ρ + p) {−HRdt + adr} Also, from eq. (8) we obtain We shall now show that by projecting the UFL along the Kodama vector gives the second Friedmann equation, in general.
For the above one forms using the scalar product with the Kodama vector we have Now, and Thus, projecting UFL along the Kodama vector gives, which is nothing but the second Friedmann equation on any arbitrary horizon.
We shall now show that the first Friedmann equation can also be obtained from the Unified first law by projecting it along a vector orthogonal to the Kodama vector namely Clearly the vector U µ lies on the radial-temporal plane and it has the following properties : (i) The vector may be space-like, time-like or null depending on R.
(ii) It is divergence-free in nature (i.e. ∇ µ U µ = 0) and there is a current associated with the vector U µ given by the relation ξ µ = G µν U ν . Clearly, the vector ξ µ is conserved i.e.
The scalar product of the individual one-form terms on both sides of the UFL with U µ gives and Hence projecting the UFL along U µ and after some algebra we obtain the first Friedmann equation i.e.
Further, it has been shown in the literature that the first law of BH thermodynamics can be obtained by projecting the UFL along the trapping horizon [18][19][20][21][22] i.e.
where z is a vector tangential to the trapping horizon.
We shall now show that the situation is not so easy in case of event horizon (EH). The area radius of the EH is given by (Note that the improper integral converges for accelerating phase of the Universe).
The normal vector to the null hypersurface R − a ∞ t dt a = 0 is given by n a = (−1, a, 0, 0), a null vector.
From the property of the null vector, n a is also tangential to the (null) event horizon hypersurface. Then one can easily see that the clausius relation i.e. eq. (27) is not satisfied for the event horizon. Thus the claim [18][19][20][21][22] of obtaining the first law of thermodynamics by projecting the UFL along the tangent is only true for trapping horizon, not for any other horizon.
Note that, the relation (21) gives the rate of energy across the horizon. Thus the energy flux across the event horizon during infinitesimal time dt is dQ = 4πHR 3 E (ρ + p)dt (29) or using the second Friedmann equation has been used in the last equality.
Recently, a notion of generalized Hawking temperature [17] (see eq. (1)) has been introduced on the event horizon for the validity of the thermodynamical laws. So in the present context using the first choice in eq. (1) we have Thus we obtain the clausius relation δQ = T

(G)
E dS E on the event horizon, by projecting the UFL along the Kodama vector on the horizon. It is interesting to note that the present approach to obtain the clausius relation (i.e. the first law of thermodynamics) from the Unified first law is a general prescription and it holds in any horizon even in the trapping horizon. So we have the following conclusions :   Finally, we redefine the surface gravity motivated by Rindler observer. We have seen that at the local Rindler causal horizon the Unruh temperature is proportional to the acceleration of the free falling observer. So in analogy with the Unruh temperature we assume that the surface gravity should be proportional to the acceleration of the model i.e.
where k 0 is a dimensionless constant of proportionality and R is introduced on dimensional ground. Then using Einstein equations, the redefined Hawking temperature becomes By introducing vector