# LOG#077. Entropic electrogravity.

Tower Wang, in his paper Coulomb Force as an Entropic Force, deduced Coulomb and Newton laws using the Verlinde approach in D=3+1 dimensions.

He begins with the Reissner-Nordstrom metric in D=4 spacetime:

$ds^2=-f(r)dt^2+\dfrac{1}{f(r)}dr^2+r^2d\Omega^2$

with the function

$f(r)=1-\dfrac{G_NM}{c^2r^2}+\dfrac{G_N^2Q^2}{c^4r^2}$ and $M\geq \vert Q\vert$

He introduces a “geometrized” unit of charge so that Coulomb force between point charges Q and q at large separation is measured with the Newton constant of gravity! That is, he “defines”

$F_{em}=F_C=\dfrac{G_NQq}{r^2}$

and you would recover the traditional Coulomb law of electricity provided you rescale charges according to the prescriptions

$Q\rightarrow \dfrac{Q}{\sqrt{4\pi\varepsilon_0G_N}}$

$q\rightarrow \dfrac{q}{\sqrt{4\pi\varepsilon_0G_N}}$

Remark: Think what the about rescaling means in terms of “natural” units (Planck units or any other “clever natural system of units” you select as fundamental system)

Now, we turn into the Verlinde approach of entropic gravity. By the equipartition theorem, we derive

$Mc^2=\dfrac{1}{2}Nk_BT$

and by the holographic principle, we know that

$N=\dfrac{Ac^3}{G_N\hbar}=\dfrac{A}{L_P^2}$

is the number of bits on the boundar with area A. The equipartion theorem is challenged, because we get

$A_H=\dfrac{4\pi G_N^2}{c^4}\left( M+\sqrt{M^2-Q^2}\right)^2$

and

$T_H=\dfrac{2G_N\hbar}{k_B c A_H}\sqrt{M^2-Q^2}$

Obviously, the equipartition theorem seems to fail as long as $Q\neq 0$! How could we save the entropic intepretation of electricity and the equipartition theorem? The paper by Wang solved that issue in two different approaches and it also connects the entropic approach to D-branes and black hole physics.

The idea to save the equipartition theorem is to generalize it. I will review the two schemes Wang uses in his paper.

In his first approach, the equipartition theorem itself is changed into the next “equipartition” rule:

(1) $\boxed{c^2\sqrt{M^2-Q^2}=\dfrac{1}{2}k_BT}$

This relationship holds on the horizon on the RN black hole, Wang claims. On the event horizon, T will be the Bekenstein-Hawking temperature. Outside the event horizon, T is considered a “generalized” Bekenstein-Hawking temperature on “the holographic screen”. Again, despite the effort of any approach to quantum gravity, nothing is saids about the inner of the event horizon. After all, we are considering thermodynamics, so a microscopic understanding of the BH entropy is not yet available! Furthermore, note that (1) makes sense only if $M\geq \vert Q\vert$.

Now, we can follow Verlinde and imagine a test particle with mass m and charge q close enough to the holographic screen. There is a total mass M and total charge Q. Then,

(2) $F=-T\partial_x S$

where “x” represents the emergent generalized coordinate, perpendicular to the holographic screen, and S is the entropy. Thus, we get

(3) $-\partial_x S=\dfrac{2\pi k_Bc}{\hbar}\dfrac{(Mm-Qq)}{\sqrt{M^2-Q^2}}$

Using the holographic principle, and (1)-(3), we easily obtain

(4) $\boxed{F_{em,g}=-\dfrac{G_N}{r^2}\left(Mm-Qq\right)}$

In the second approach, Wang postulates equipartition and entropy changes separately for gravity and electricity:

(5) $\boxed{Mc^2=\dfrac{1}{2}Nk_BT_g}$ $\boxed{\partial_x S_g=\dfrac{2\pi k_B mc}{\hbar}}$

(6) $\boxed{Qc^2=\dfrac{1}{2}Nk_BT_{em}}$ $\boxed{\partial_x S_{em}=-\dfrac{2\pi k_B q c}{\hbar}}$

Wang’s entropic equipartition for electricity in the 2nd approach follows from (6). He even suggests that the holographic screen and the emergent direction for the electromagnetic force can be different from those involving gravity! It is some kind of “entropic decoupling” I find puzzling, but it works. We invoke again the “generalized equipartition theorem” to the electric (or even magnetic) charge Q and the holographic correspondence match the temperature $T_{em}$ with the average charge per bit, somehow. It is important to realize that, unlike the gravitational case, this claim means that $T_{em}$ can be positive or negative according to the sign of the charge Q! This is weird, and the author accepts it as “bizarre”. Nevertheless, he claims, we can never observe $T_{em}$ directly! Therefore, Coulomb’s law follows from the entropic second approach like the Newton’s law:

$F_{em}=-T_{em}\partial_x S_{em}$

Putting together the entropic gravity and electromagnetism, we can even go further and derive the combined form:

(7) $\boxed{F_g+F_{em}=-T_g\partial_x S_g-T_{em}S_{em}=-G_N\dfrac{(Mm-Qq)}{r^2}}$

Evindently, the second approach reveals itself to be more flexible and to have more general application than the first approach. The reason is obvious: the approach one only works whenever $M\geq \vert Q\vert$ and when the distribution of the Newtonian potential matches the distributionof the Coulomb potential. It suggests that we should be able to “guess” the approach one from the second approach. And it show to be the case. Introduce the temperatures:

(8) $T^2=T^2_g-T^2_{em}$

(9) $T\partial_x S=T_g\partial_x S_g-T_{em}\partial_x S_{em}$

In (8), Wang claims, likely only the temperature T is observable while $T_{em}$ would be never seen! That is quite a claim! Moreover, and for consistency, $T_g$ would not be observable if $T_{em}\neq 0$. However, the combined value T would be “observable”. It reminds somehow to the spacetime interval in special relativity, where only combinations in the form $x^2-c^2t^2$ are meaningul, while a solitary assignment of “x” and “t” would be generally meaningless to correctly place a spacetime “event”. In fact, the above temperature rule is also known in the D-brane picture of Black Holes! Basically, left and right movers for the temperature are introduced:

$T_L=\dfrac{2}{\pi r}\sqrt{\dfrac{N_L}{Q_1Q_5}}$

$T_R=\dfrac{2}{\pi r}\sqrt{\dfrac{N_R}{Q_1Q_5}}$

Thus, we could match these two equations with (8), if $T_g=T_L$ and $T=T_R$, so

$T^2_{em}=T_L^2-T_R^2=\left(\dfrac{2}{\pi r}\right)^2\dfrac{(N_L-N_R)}{Q_1Q_5}=\left(\dfrac{2}{\pi r_e}\right)^2$

and where $r_e$ is the horizon radius of the near extremal black hole. In fact, if you generalize (6) to five dimensions, you can recover this precise result.

The final part of the paper faces the reproduction of the Maxwell’s field equation “a la Jacobson”. Jacboson showed long ago that Einstein’s field equations follows from thermodynamics in a clever way! Holographic screens correspond to equipotential surfaces, according to the Verlinde approach, so it seems natural to define the gravitational temperature by the gradient of the Newton potential:

$k_B T_g=\dfrac{\hbar}{2\pi c}\nabla \Phi_g$

Indeed, and I don’t know if the author realizes it too, the above equation is essentially a “disguised” form of the Unruh temperature:

$T_U=\dfrac{\hbar g}{2\pi k_B c}$

Obviously, the two equations match if $g=\nabla \Phi_g$!

We can go further and generalize the holographic principle into a differential form

$dN=\dfrac{c^3}{G\hbar}dA$

from which the Poisson equation from gravity follows naturally using the entropic arguments! Can we do the same for electromagnetism? It seems yes! We can define the electromagnetic analogue of the above equation for gravity:

$k_B T_{em}=-\dfrac{\hbar}{2\pi c}\nabla \Phi_{em}$

and where again we should use the same “geometrized” units Wang uses in the beginning of the paper for electric charges. That is, we rescale the electromagnetic potential in the following way:

$\Phi_{em}\rightarrow \dfrac{\Phi_{em}}{\sqrt{4\pi\varepsilon_0 G_N}}$

and using the integral analogue of the equipartition theorem

$\displaystyle{Qc^2=\dfrac{k_B}{2}\oint_{\partial V}T_{em}dV}$

we obtain with the aid of the Gauss theorem the charge

$\displaystyle{Q=-\dfrac{1}{4\pi G_N}\int_V \nabla^2\Phi_{em}dV}$

and thus the Poisson equation is recovered

$\nabla^2 \Phi_{em}=-4\pi G_N \rho_{em}$

or equivalently, in usual units of charge

$\nabla^2 \Phi_{em}=-\dfrac{1}{\varepsilon_0}\rho_{em}$

The Jacobson trick also works. Suppose a time-like Killing vector $\xi^\mu$, then the covariant Poisson equation will be

$\rho_{em}=\xi_\mu j^\mu$

$\nabla^2 \Phi_{em}=\dfrac{1}{\sqrt{-g}}\xi_\mu \xi_\nu \left( \sqrt{-g}F^{\mu \nu}\right)$

and then

$\boxed{\dfrac{1}{\sqrt{-g}}\xi_\mu\xi_\nu\left( \sqrt{-g}F^{\mu\nu}\right)=-\dfrac{1}{\varepsilon_0}\xi_\mu j^\mu}$

The covariant form of the Maxwell equations! One half of them, indeed! The remaining equations can be also obtained following a variant of Jacobson’s trick, but it is left as an exercise for the reader ;).

Even if it is a fiction…May the (entropic) Force be with you!