Gravitation And Black Hole Thermodynamics

The introduction of general theory of relativity has enabled people to understand gravitation.Even people began to use the general relativity to study cosmology,but with the development of astronomical observations,it was found that Einstein’s theory was not imaginarily perfect.For example,the cosmological explosion and dark energy problems are not well explained,and some physicists and researchers began to find other gravitational models based on Einstein’s gravitational theory,such as f(R)gravity,scalar field theory,Massive Gravity,Ei BI gravity,and even other quantum gravity models.All of them can explain to some extent the difficulties that general relativity brings in modern astronomy and cosmology.In addition,the discovery of black hole thermodynamics has made us realize that gravitation and thermodynamics may have a profound relationship.In recent years,with the development of gravitational models and quantum gravity,some physicists have discovered that the entropy of black holes is not simple proportional to area of horizon in some specific cases and some corrections have been added.In this paper,we will study the modified gravity and black hole thermodynamics.On one hand,we studied gravity’s rainbow,in which the metric could be treated as a conformal rescaling of the original metric.By using the conformal transformation technique,we get a specific form of modified Newton’s constant and cosmological constant in gravity’s rainbow,which implies that the total vacuum energy is dependent on probe energy.Moreover,the result shows that the Einstein gravity’s rainbow could be described by an energy-dependent f(E,?R)gravity.At last,we study the f(R)gravity,when the gravity’s rainbow is considered,it can also be described as another energy-dependent?f(E,?R)gravity.One the other hand,Einstein’s equation could be interpreted as the first law of thermodynamic near the spherically symmetric horizon.Through recalling the Einstein gravity with a more general static spherical symmetric metric,we find that the entropy would have a correction in Einstein gravity.By using this method,we investigate the Eddington-inspired Born-Infeld(Ei BI)gravity.Without matter field,we can also derive the first law in Ei BI gravity.With an electromagnetic field,as the field equations have a more general spherically symmetric solution in Ei BI gravity,we find that correction of the entropy could be generalized to Ei BI gravity.Furthermore,we point out that the Einstein gravity and Ei BI gravity might be equivalent on the event horizon.At last,under Ei BI gravity with the electromagnetic field,a specific corrected entropy of black hole is given.