Download Thermogravitation

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 Bekenstein’s entropy formula that it’s proportional to the horizon area then the entropy variation
(dS) associated with a piece of the horizon (dA) satisfies ;
𝑑𝑆 = Ω 𝑑𝐴
(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 ;
𝑆(x,t) = γ 𝑒(kx – wt)
(2)
Where ( S(x,t) ) entropy at a given point in space and time , (γ) 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 .
It’s clear that entropy is proportional to mass-energy (M) ;
𝑑𝑆 = 𝛿𝑀∕𝑇
(3)
so entropy is proportional to the gravitational field ;
𝑆=ε𝐸
(4)
(ε) constant . Then the entropy changing creates a gravitational field ; and this phenomenon is described by the
equation ;
ε 𝛻× 𝐸 = ⎼ 𝜕𝑆∕𝜕𝑥
(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 ;
𝑓 = ⎼ (𝛻× 𝑆) ρ∕ε
(6)
( f ) force density , ( ρ ) mass-energy surface density . and from equation (3) ;
𝑓 = ⎼ (𝛻× 𝑆)× 𝑆 ϱ∕ε
(7)
( ϱ ) temperature surface density . From curl identity ;
½ 𝛻𝑆2 = (𝛻× 𝑆)× 𝑆 + (𝛻. 𝑆) 𝑆
(8)
𝑓 = ((𝛻. 𝑆) 𝑆 ⎼ ½ 𝛻𝑆2 ) ϱ∕ε
(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 ;
𝑇μν = ( 𝑆µ 𝑆ν ⎼ ½ δµν 𝑆2 ) ϱ∕ε
(10)
( δµν ) the Kronecker delta , and from The Einstein field equation ;
𝐺µν = κ 𝑇µν
(11)
( 𝐺µν ) 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 Maxwell’s 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 ;
𝑒(𝐸+𝑐𝐵)=𝑘𝑥
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 ;
𝑥2 + 𝑦2 + 𝑧2 = 𝑐2 𝑡2 + α2 (𝐸2 + 𝑐2 𝐵2)
(13)
Where ( α = 𝑒/𝑘 ) , 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
Kaluza–Klein 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