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!

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LOG#031. Entropic Gravity (II).

We will generalize the entropic gravity approach to include higher dimensions in this post. The keypoint from this theory of entropic gravity, according to Erik Verlinde, is that gravity does not exist as “fundamental” force and it is a derived concept. Entropy is the fundamental object somehow. And it can be generalized to a a d-dimensional world as follows.

The entropic force is defined as:

\boxed{F=-\dfrac{\Delta U}{\Delta x}=-T\dfrac{\Delta S}{\Delta x}}

The entropic force is a force resulting from the tendency of a system to increase its entropy. Since \Delta S>0 the sign of the force (whether repulsive or attractive) is determined by how we take the definition of \Delta x as it is related to the system in question.

An arbitrary mass distribution M induces a holographic screen \Sigma  at some distance R that has encoded on it gravitational information. Today, we will consider the situation in d spatial dimensions.Using the holographic principle, the screen owns all physical information contained within its volume in bits on the screen whose number N is given by:

\boxed{N=\dfrac{A_\Sigma (R)}{l_p^{d-1}}}

This condition implies the quantization of the hyperspherical surface, where the hyperarea (from the hypersphere) is defined as:

\boxed{A_\Sigma=\dfrac{2\pi^{d/2}}{\Gamma \left(\frac{d}{2}\right)}R^{d-1}}

By the equipartition principle:

\boxed{E=Mc^2=\dfrac{N}{2}k_BT}

Therefore,

k_BT=\dfrac{2Mc^2l_p^{d-1}}{A_\Sigma}

The entropy shift due to some displacement is:

\Delta S=2\pi k_B \dfrac{\Delta x}{\bar{\lambda}}=2\pi k_B mc\dfrac{\Delta x}{\hbar}

Plugging the expression for the temperature and the entropy into the entropic force equation, we get:

F=-T\dfrac{\Delta S}{\Delta x}=-\dfrac{2Mc^2}{k_B}\dfrac{2\pi k_B mc}{\hbar}\dfrac{l_p^{d-1}}{A_\Sigma}=-\dfrac{4\pi Mmc^3l_p^{d-1}}{\hbar A_\Sigma}

and thus we finally get

F=-\dfrac{2\pi^{1-\frac{d}{2}}\Gamma \left(\frac{d}{2}\right) l_p^{d-1}Mmc^3}{\hbar R^{d-1}}=-G_d\dfrac{Mm}{R^{d-1}}

i.e.,

\boxed{F=-G_d\dfrac{Mm}{R^{d-1}}}\leftrightarrow \boxed{F=-\dfrac{2\pi^{1-\frac{d}{2}}\Gamma \left(\frac{d}{2}\right) l_p^{d-1}Mmc^3}{\hbar R^{d-1}}}

where we have defined the gravitational constant in d dimensions to be

\boxed{G_d\equiv \dfrac{2\pi^{1-\frac{d}{2}}\Gamma \left(\frac{d}{2}\right) l_p^{d-1}c^3}{\hbar }}


LOG#030. Entropic Gravity (I).

In 2010, Erik Verlinde made himself famous once again. Erik Verlinde is a theoretical physicist who has made some contributions to String Theory. In particular, the so-called Verlinde formula. However, this time was not apparently a contribution related to string theory. He guessed a way to derive both the Newton’s second law and the Newton’s law of gravity. He received a prize time later, and some critical voices against his approach were raised.

I will review in this post his deductions.

A. Newton’s second law.

There are some hypothesis to begin with:

1st. Entropic force ansatz. Forces aren’t really fundamental, they are derived from some entropy functional. More precisely, forces are ”entropy fluxes”. Mathematically speaking:

F\Delta x=T\Delta S

or

\boxed{F=T\dfrac{\Delta S}{\Delta x}}

2nd. Acceleration has a temperature. Equivalently, this is the well known Unruh’s effect from QFT in curved spacetime. Any particle that is accelerated is equivalent to some thermal system. This parallels the Hawking’s effect in black hole physics as well. It stands mysterious for me yet, since indeed, the temperature is relative to some vacuum or rest system.

T=\dfrac{\hbar a}{2\pi k_B c}

3rd. Holographic principle.   A variant of the holographic principle is postulated to hold. The idea is that a particle separated certain distance from a “holographic screen” has an entropy shift:

\Delta S = 2\pi k_B \dfrac{mc}{\hbar} \Delta x

Then, plugging the holographic entropy and the Unruh’s temperature into the entropic force ansatz, we get easily

F= 2\pi k_B \dfrac{mc}{\hbar} \dfrac{\hbar a}{2\pi k_B c}

i.e. we get the Newton’s second law of Dynamics

F=ma

B. Newton’s gravity.  In order to get the Newton’s law of gravitation, we have to modify a bit the auxiliary hypothesis but yet we conserve the core approach.

1st. The entropic force ansatz. Again,

\boxed{F=T\dfrac{\Delta S}{\Delta x}}

2nd. Holographic principle. Again,

\Delta S = 2\pi k_B \dfrac{mc}{\hbar} \Delta x

3rd. Equipartition principle of relativistic energy.   Temperature is obtained at the statistical level when you distribute N quanta in thermal equilibrium, and they equal the relativistic energy formula. Equivalently,

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

4th. Microscopical degrees of freedom and minimal length ( or area quantization). The number of allowed microscopical quanta or microstates can not exceed and must match in the extreme case the ration of the area available and the square of Planck’s length ( or some other squared fundamental length). In other words, the number of bits can not overcome the area of a ball in Planck’s units. EQuivalently, the (hyper)area must be quantized (through a number N). Mathematically speaking,

N=\dfrac{A}{l_{p}^{2}}

Then, we plug the hypothesis 2 and 3 into 1, to have:

F=\dfrac{2Mc^2}{k_B N}2\pi k_B \dfrac{mc}{\hbar}

and now we use 4, in order to get

F=\dfrac{2Mc^2}{k_B A}l_{p}^{2} 2\pi k_B \dfrac{mc}{\hbar}

that is

F=4\pi Mm \dfrac{l_{p}^{2}c^3}{\hbar A}

And now, recalling that in 3 spatial dimensions, a ball has an area A=4\pi r^2 and that the Newton’s constant of gravity is indeed given as function of Planck’s length as G=\dfrac{l_{p}^{2}c^3}{\hbar}, we have what we wanted to derive

F=4\pi Mm \dfrac{G}{4\pi r^2}

i.e. the Newton’s gravitational law has been derived from the entropic force too

F=G\dfrac{Mm}{r^2}

It is done. Is it just a trick or something deeper is behing all this stuff? Nobody knows for sure…People think he is probably wrong, but there are a whole line of research opened from his works. It is quite remarkable his approach is quite general and he suggests that every fundamental force is “entropic” or “emergent”, i.e., also electromagnetic fields, or even Yang-Mills fields could be entropic according to this approach.

Is he right? Time will tell…Some doubts arise from the fact that he has only derived the “temporal” components of Einstein Field Equations for gravity (a.k.a. newtonian gravitation) but, indeed, he and other physicists have been able to derive the remaining components as well. Perhaps, the strongest critics comes from neutron interferometry results. However, theoretical ideas like this one, like extra dimensions, could be saved by some clever argument.