Abstract:
The holographic bound has been extended to the different theory of gravities such as scalar-tensor gravity and $f(R)$ gravity according to the Noether charge definition of the entropy for a black hole surface. We have introduced some popular examples of the flat FRW cosmology in order to investigate holographic bound in scalar-tensor and $f(R)$ gravity. Using the holographic bound, we put an additional constraint on the scalar-tensor gravity and $f(R)$ gravity parameters. We also discuss about the transformation from Jordan frame to Einstein frame.

Abstract:
Why are there no fundamental scalar fields actually observed in physics today? Scalars are the simplest fields, but once we go beyond Galilean-Newtonian physics they appear only in speculations, as possible determinants of the gravitational constants in the so-called Scalar-Tensor theories in non-quantum physics, and as Higgs particles, dilatons, etc., in quantum physics. Actually, scalar fields have had a long and controversial life in gravity theories, with a history of deaths and resurrections. This paper presents a brief overview of this history.

Abstract:
We consider scalar field entanglement entropy generated with black hole of (sub)planck mass scale thus implying the unitary evolution of gravity. The dependence on the dimension of the Hilbert space for degrees of freedom located behind the horizon is taken into account. The obtained results contain polylogarithmic terms.

Abstract:
Scalar field contribution to entropy is studied in arbitrary D dimensional one parameter rotating spacetime by semiclassical method. By introducing the zenithal angle dependent cutoff parameter, the generalized area law is derived. The non-rotating limit can be taken smoothly and it yields known results. The derived area law is then applied to the Banados-Teitelboim-Zanelli (BTZ) black hole in (2+1) dimension and the Kerr-Newman black hole in (3+1) dimension. The generalized area law is reconfirmed by the Euclidean path integral method for the quantized scalar field. The scalar field mass contribution is discussed briefly.

Abstract:
Black holes arising in the context of scalar-tensor gravity theories, where the scalar field is non-minimally coupled to the curvature term, have zero surface gravity. Hence, it is generally stated that their Hawking temperature is zero, irrespectivelly of their gravitational and scalar charges. The proper analysis of the Hawking temperature requires to study the propagation of quantum fields in the space-time determined by these objects. We study scalar fields in the vicinity of the horizon of these black holes. It is shown that the scalar modes do not form an orthonormal set. Hence, the Hilbert space is ill-definite in this case, and no notion of temperature can be extracted for such objects.

Abstract:
In a viscous Bianchi type I metric of the Kasner form, it is well known that it is not possible to describe an anisotropic physical model of the universe, which satisfies the second law of thermodynamics and the dominant energy condition (DEC) in Einstein's theory of gravity. We examine this problem in scalar-tensor theories of gravity. In this theory we show that it is possible to describe the growth of entropy, keeping the thermodynamics and the dominant energy condition.

Abstract:
A perturbative analysis shows that black holes do not remember the value of the scalar field $\phi$ at the time they formed if $\phi$ changes in tensor-scalar cosmology. Moreover, even when the black hole mass in the Einstein frame is approximately unaffected by the changing of $\phi$, in the Jordan-Fierz frame the mass increases. This mass increase requires a reanalysis of the evaporation of primordial black holes in tensor-scalar cosmology. It also implies that there could have been a significant magnification of the (Jordan-Fierz frame) mass of primordial black holes.

Abstract:
We discuss the Kretschmann, Chern-Pontryagin and Euler invariants among the second order scalar invariants of the Riemann tensor in any spacetime in the Newman-Penrose formalism and in the framework of gravitoelectromagnetism, using the Kerr-Newman geometry as an example. An analogy with electromagnetic invariants leads to the definition of regions of gravitoelectric or gravitomagnetic dominance.

Abstract:
The Chern-Simons lagrangian density in the space of metrics of a 3-dimensional manifold M is not invariant under the action of diffeomorphisms on M. However, its Euler-Lagrange operator can be identified with the Cotton tensor, which is invariant under diffeomorphims. As the lagrangian is not invariant, Noether Theorem cannot be applied to obtain conserved currents. We show that it is possible to obtain an equivariant conserved current for the Cotton tensor by using the first equivariant Pontryagin form on the bundle of metrics. Finally we define a hamiltonian current which gives the contribution of the Chern-Simons term to the black hole entropy, energy and angular momentum.

Abstract:
In the paper, hep-th/0501055 (R.G. Cai and S.P. Kim, JHEP {\bf 0502}, 050 (2005)), it is shown that by applying the first law of thermodynamics to the apparent horizon of an FRW universe and assuming the geometric entropy given by a quarter of the apparent horizon area, one can derive the Friedmann equations describing the dynamics of the universe with any spatial curvature; using the entropy formula for the static spherically symmetric black holes in Gauss-Bonnet gravity and in more general Lovelock gravity, where the entropy is not proportional to the horizon area, one can also obtain the corresponding Friedmann equations in each gravity. In this note we extend the study of hep-th/0501055 to the cases of scalar-tensor gravity and $f(R)$ gravity, and discuss the implication of results.