Generalized Bekenstein–Hawking system: logarithmic correction
Subenoy Chakraborty
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Department of Mathematics, Jadavpur University
, Kolkata,
India
The present work is a generalization of the recent work [arXiv.1206.1420] on the modified Hawking temperature on the event horizon. Here the Hawking temperature is generalized by multiplying the modified Hawking temperature by a variable parameter representing the ratio of the growth rate of the apparent horizon to that of event horizon. It is found that both the first and the generalized second law of thermodynamics are valid on the event horizon for any fluid distribution. Subsequently, the Bekenstein entropy is modified on the event horizon and the thermodynamical laws are examined. Finally, an interpretation of the parameters involved is presented.
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In black hole physics a semi-classical description shows that a
black hole behaves as a black body emitting thermal radiation
with temperature (known as the Hawking temperature) and
entropy (known as the Bekenstein entropy) proportional to
the surface gravity at the horizon and area of the horizon
[1,2], respectively. Further, this Hawking temperature and
the Bekenstein entropy are related to the mass of the black
hole through the first law of thermodynamics [3]. Due to
this relationship between the physical parameters (namely,
entropy and temperature) and the geometry of the horizon,
there is natural speculation about the relationship between
black hole thermodynamics and the Einstein field equations.
A first step in this direction was put forward by Jacobson
[4], who derived the Einstein field equations from the first
law of thermodynamics: Q = T dS for all locally Rindler
causal horizons with Q and T as the energy flux and Unruh
temperature measured by an accelerated observer just inside
the horizon. Subsequently, Padmanabhan [5,6] on the other
side was able to derive the first law of thermodynamics on
a e-mail: ;
the horizon starting from the Einstein equations for a general
static spherically symmetric space-time.
This idea of equivalence between Einstein field equations
and the thermodynamical laws has been extended in the
context of cosmology. Usually, the universe bounded by the
apparent horizon is assumed to be a thermodynamical system
with Hawking temperature and the entropy as
SA =
where RA is the radius of the apparent horizon. It was shown
that the first law of thermodynamics on the apparent
horizon and the Friedmann equations are equivalent [7].
Subsequently, this equivalent idea was extended to higher
dimensional space-time, namely gravity theory with a Gauss
Bonnet term and Lovelock gravity theory [710]. It is
presumed that such an inherent relationship between the
thermodynamics at the apparent horizon and the Einstein field
equations may lead to some clue on the properties of dark
energy.
Although the cosmological event horizon does not exist
in the usual standard big bang cosmology, in the
perspective of the recent observations [1116], the universe is in an
accelerating phase dominated by dark energy (d < 1/3)
and the event horizon distinct from the apparent horizon.
By defining the entropy and temperature on the event
horizon similar to those for the apparent horizon (given above)
Wang et al. [17] showed that both the first and the second law
of thermodynamics break down on the cosmological event
horizon. They justified it arguing that the first law is
applicable to nearby states of local thermodynamic equilibrium,
while the event horizon reflects the global features of
spacetime. As a result, the thermodynamical parameters on the
non-equilibrium configuration of the event horizon may not
be as simple as on the apparent horizon. Further, they
speculated that the region bounded by the apparent horizon may be
this normal space is
(x ) = hi j (x )i R j R = 1 (H 2 + k/a2)R2
where k = 0, 1 stands for flat, closed or open model of the
universe. The Friedmann equations are
H 2 + k/a2 =
H k/a2 = 4 G( + p)
RA =
TA =
taken as the Bekenstein system, i.e., the Bekenstein entropy
or mass bound, S < 2 RE , and the entropy or area bound,
S < A/4, are satisfied in this region. Now due to universality
of the Bekenstein bounds and as all gravitationally stable
special regions with weak self-gravity should satisfy the above
Bekenstein bounds, the corresponding thermodynamical
system is termed a Bekenstein system. Further, due to the radius
of the event horizon being larger than the apparent horizon,
Wang et al. [17] termed the universe bounded by the event
horizon a non-Bekenstein system.
In the recent past there were published a series of works
[1823] investigating the validity of the generalized second
law of thermodynamics of the universe bounded by the event
horizon for Einstein gravity [18,19] and in other gravity
theories [1821] and for different fluid systems [18,19,22,23]
(including dark energy [19,22,23]). In these works the
validity of the first law of thermodynamics on the event horizon
was assumed and it was possible to show the validity of the (...truncated)