Cosmological horizon entropy and generalized second law for flat Friedmann universe
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Regular Article - Theoretical Physics
Cosmological horizon entropy and generalized second law for flat Friedmann universe Titus K. Mathewa , Aiswarya R, Vidya K. Soman Department of Physics, Cochin University of Science and Technology, Kochi, India
Received: 14 June 2013 / Revised: 28 September 2013 © Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013
Abstract We investigate the generalized second law (GSL) and the constraints imposed by it for two types of Friedmann universes. The first one is the Friedmann universe with radiation and a positive cosmological constant, and the second one consists of non-relativistic matter and a positive cosmological constant. The time evolution of the event horizon entropy and the entropy of the contents within the horizon are studied by obtaining the Hubble parameter. It is shown that the GSL constrains the temperature of both the radiation and matter of the Friedmann universe. It is also shown that, even though the net entropy of the radiation (or matter) is decreasing at sufficiently large times as the universe expands, it exhibits an increase during the early times when the universe is decelerating. That is, the entropy of the radiation within the comoving volume is decreasing only when the universe is undergoing an accelerated expansion.
1 Introduction Bekenstein and Hawking have showed that the entropy of black holes is proportional to the area of their event horizon [1–3]. In units of C = 1, G = 1, k = 1 and = 1, the black hole entropy is given as S=
Ah 4
(1)
where Ah is the area of event horizon of the black hole. Hawking has shown that the black hole can evaporate by emitting radiation, consequently its event horizon area decreases. He had also shown that the event horizon of the black hole posses temperature, which is inversely proportional to its mass or proportional to its surface gravity. During the process of evaporation the entropy of the black hole will decrease. But due to the emitted radiation, the entropy a e-mail:
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of the surrounding universe will increase. Hence the second law of thermodynamics was modified in such a way that the entropy of the black hole plus the entropy of the exterior environment of the black hole will never decrease; this is called the generalized second law (GSL), which can be represented as d (Sevn + Sb ) ≥ 0 dt
(2)
where Senv is the entropy of environment exterior to the black hole and Sb is the entropy of the black hole. The thermodynamic properties of the event horizon were shown to exist on a more basic level [4, 5], by recasting the Einstein field equation for a spherically symmetric space time in the form of the first law of thermodynamics. In Refs. [7, 8] one can find investigations of the applicability of the first law of thermodynamics to the cosmological event horizon. Jacobson [6] showed that Einstein’s field equations are equivalent to the thermodynamical equation of state of the space time. In cosmology the counter part of black hole horizon is the cosmological event horizon.
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