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Thermogravitation Mohammed Alzain Email ; [email protected] Abstract ; In attempt to create a complete mathematical description to the theory of entropic gravity, I will try to represent the interaction between spacetime curvature and the entropic force through assumption that the entropy is a vector physical quantity, which originally motivated by the thermodynamics of black hole, in this way we will see clearly how gravity is an entropic force and connected with electromagnetism . 1 Gravity as an entropic force ; Starting from Bekensteinβs entropy formula that itβs proportional to the horizon area then the entropy variation (dS) associated with a piece of the horizon (dA) satisfies ; ππ = Ξ© ππ΄ (1) This assumption would affect the values of the entropy in different directions , motivated by this formula , let us postulate that the entropy is a state vector ; π(x,t) = Ξ³ π(kx β wt) (2) Where ( S(x,t) ) entropy at a given point in space and time , (Ξ³) maximum entropy , Bekenstein argued that black holes are maximum entropy objects ,when black holes absorb some photons in a pure state described by a wave function , they re-emit new photons in a thermal mixed state described by entropy , so the classical looking to entropy as a scalar is not valid . Itβs clear that entropy is proportional to mass-energy (M) ; ππ = πΏπβπ (3) so entropy is proportional to the gravitational field ; π=Ξ΅πΈ (4) (Ξ΅) constant . Then the entropy changing creates a gravitational field ; and this phenomenon is described by the equation ; Ξ΅ π»× πΈ = βΌ ππβππ₯ (5) Now we can derive a unique relation between stress-energy tensor and entropy which is the core of this paper ; from equation(5) we find ; π = βΌ (π»× π) ΟβΞ΅ (6) ( f ) force density , ( Ο ) mass-energy surface density . and from equation (3) ; π = βΌ (π»× π)× π Ο±βΞ΅ (7) ( Ο± ) temperature surface density . From curl identity ; ½ π»π2 = (π»× π)× π + (π». π) π (8) π = ((π». π) π βΌ ½ π»π2 ) Ο±βΞ΅ (9) Finally the stress-energy tensor that describes the interaction between the entropic force and the density and flux of energy and momentum in spacetime ; πΞΌΞ½ = ( πµ πΞ½ βΌ ½ Ξ΄µΞ½ π2 ) Ο±βΞ΅ (10) ( Ξ΄µΞ½ ) the Kronecker delta , and from The Einstein field equation ; πΊµΞ½ = ΞΊ πµΞ½ (11) ( πΊµΞ½ ) the Einstein tensor; this equation represents the interaction between spacetime curvature and the entropy and the result that gravity is an emergent phenomenon from physical systems tendency to increase their entropy. 2 The derivation of Maxwellβs equations ; To unify the electromagnetic force with gravity, from the entropic origin , I will assume that the electromagnetic field intensity is depend on the distance of the displacement of matter by an entropic force , then ; π(πΈ+ππ΅)=ππ₯ 1 (12) (e,c,k) constants ;the electron charge, the speed of light, and the rate constant . (E , B) the electric and magnetic fields. (x) the distance , so ; π₯2 + π¦2 + π§2 = π2 π‘2 + Ξ±2 (πΈ2 + π2 π΅2) (13) Where ( Ξ± = π/π ) , this equation show us how the electromagnetic field is connected with space and time as fifth dimension , and this extension of spacetime to five dimensions directly leads to the Maxwell equations as in KaluzaβKlein theory . References ; [1] J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333. [2] J.M. Bardeen, B. Carter and S.W. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161. [3] S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199. [4] E.P. Verlinde, On the Origin of Gravity and the Laws of Newton, JHEP 04, 29 (2011). 2