The entropy on apparent horizon
can be defined as follows:
Lastly observe that since [phi] vanishes at [partial derivative][M.sub.0], we must have that [partial derivative][M.sub.0] is an apparent horizon
. However [partial derivative]M is an outermost apparent horizon
, so in fact [partial derivative][M.sub.0] = [partial derivative]M, and hence [M.sub.0] = M.
(iii) Power law correction: in thermodynamics of apparent horizon
in the standard FRW cosmology, the geometric entropy is assumed to be proportional to its horizon area ([S.sub.A] = A/4).
Cao, "Unified first law and the thermodynamics of the apparent horizon
in the FRW universe," Physical Review D, vol.
To study this law, we first find the dynamical apparent horizon
evaluated by the relation
Then the entanglement energy of quantum particles across the apparent horizon
[R.sub.H], which is defined as disturbed vacuum energy due to the presence of a boundary , is missed in the cosmological equations written for the Hubble volume and can be taken into account by introduction of a boundary term.
Considering the Hawking temperature (similar to (12), we ignore the direct correction to the radius in the Hawking temperature, and the changes of numbers of degrees of freedom directly stem from the corrections of the area of the apparent horizon
) T = 1/2[pi][r.sub.A] and E = -([rho] + 3p)[??] with dark energy in the bulk, we obtain
Dynamics of the Cosmical Apparent Horizon
in Eddington-Born-Infeld Gravity
These equations would also be obtainable by modifying the Einstein field equations as [G.sub.[mu][nu]] = 8[pi][T.sub.[mu][nu]](e), in agreement with attributing the Bekenstein entropy to the apparent horizon
(see the paragraph after (16)).
In the Einstein gravity, the Bekenstein-Hawking relation S = A/(4G) defines the horizon entropy, where A = 4[pi][[??].sup.2.sub.A] is the area of the apparent horizon
Then one can show the following relation for the apparent horizon
Here, we are interested in investigating the effects of the displacement vector field on the apparent horizon
entropy of the FRW universe and thus its thermodynamics.