LOG#115. Bohr’s legacy (III).

Dedicated to Niels Bohr

and his atomic model

(1913-2013)

3rd part:

From gravatoms to dark matter

500px-GalacticRotation2.svg

Gravatoms

Imagine a proton an an electron were bound together in a hydrogen atom by gravitational forces and not by electric forces. We have two interesting problems to solve here:

1st. Find the formula for the spectrum (energy levels) of such a gravitational atom (or gravatom), and the radius of the ground state for the lowest level in this gravitational Bohr atom/gravatom.

2nd. Find the numerical value of the Bohr radius for the gravitational atom, the “rydberg”, and the “largest” energy separation between the energy levels found in the previous calculation.

We will take the values of the following fundamental constants:

\hbar=1\mbox{.}06\cdot 10^{-34}Js, the reduced Planck constant.

m_p=1\mbox{.}67\cdot 10^{-27}kg, the proton mass.

m_e=9\mbox{.}11\cdot 10^{-31}kg, the electron mass.

G_N=6\mbox{.}67\cdot 10^{-11}Nm^2/kg^2, the gravitational Newton constant.

Let R be the radius of any electron orbit. The gravitational force between the electron and the proton is equal to:

(1) F_g=G_N\dfrac{m_pm_e}{R^2}

The centripetal force is necessary to keep the electron in any circular orbit. According to the gravatom hypothesis, it yields the value of the gravitational force (the electric force is neglected):

(2) F_c=\dfrac{mv^2}{R}

(3) F_c=F_g\leftrightarrow \boxed{\dfrac{mv^2}{R}=G_N\dfrac{m_pm_e}{R^2}}

Using the hypothesis of the Bohr atomic model in this point, i.e., that “the allowed orbits are those for whihc the electron’s orbital angular momentum about the nucleus is an integral multiple of \hbar“, we get

(4) L=m_evR=n\hbar \forall n=1,2,\ldots,\infty

Then,

(5) v=\dfrac{n\hbar}{m_eR} and v^2=\dfrac{n^2\hbar^2}{m_e^2R^2}

From (3), we obtain

(6) \boxed{v^2=G_N\dfrac{m_p}{R}}

Comparing (5) with (6), we deduce that

(7) G_N\dfrac{m_p}{R}=\dfrac{n^2\hbar^2}{m_e^2R^2}

and thus

(8) \boxed{R_n=R(n)=n^2\dfrac{\hbar^2}{G_Nm_pm_e^2}}

This is the gravatom equivalent of Bohr radius in the common Bohr model for the hydrogen atom. To get the spectrum, we recall that total energy is the sum of kinetic and potential energy:

E=T+U=\dfrac{1}{2}m_ev^2-G_N\dfrac{m_pm_e}{R}

Using the value we obtained in (5), by direct substitution, we have

(9) E=\dfrac{1}{2}m_ev^2-G_N\dfrac{m_pm_e}{R}=-G_N\dfrac{m_pm_e}{2R}

and then

(10) E=-\dfrac{G_Nm_em_p}{2}\dfrac{G_Nm_pm_e^2}{n^2\hbar^2}

and so the spectrum of this gravatom is given by

(11) \boxed{E_n=E(n)=-G_N^2\dfrac{m_p^2m_e^3}{2n^2\hbar^2}}

For n=1 (the ground state), we have the analogue of the Bohr radius in the gravatom to be:

R_1=\dfrac{\hbar^2}{G_Nm_pm_e^2}=1\mbox{.}20\cdot 10^{29}m

For comparison, the radius of the known Universe is about R_U=4\mbox{.}4\cdot 10^{26}m. Therefore, R(gravatom)>R_U!!!!!! R_1 is very huge because gravitational forces are much much weaker than electrostatic forces! Moreover, the energy in the ground state n=1 for this gravatom is:

E_1=-G_N^2\dfrac{m_p^2m_e^2}{2\hbar^2}=-4\mbox{.}23\cdot 10^{-97}J

The energy separation between this and the next gravitational level would be about 1-1/4=3/4 this quantity in absolute value, i.e.,

\Delta E=\vert E_2-E_1\vert =3\mbox{.}18\cdot 10^{-97}J=1\mbox{.}99\cdot 10^{-78}eV

This really tiny energy separation is beyond any current possible measurement. Therefore, we can not measure energy splittings in “gravatoms” with known techniques. Of course, gravatoms are a “toy-model” or hypothetical systems (bubble Universes?).

Remark (I): The quantization of angular momentum provided the above gravatom spectrum. It is likely that a full Quantum Gravity theory provides additional corrections to the quantum potential, just in the same way that QED introduces logarithmic (vacuum polarization) corrections and others (due to relativity or additional quantum effects).

Remark (II): Variations in the above quantization rules can modify the spectrum.

Remark (III): In theories with extra dimensions, G_N is changed by a higher value G_N^{eff} as a function of the compactification radius. So, the effect of large enough extra dimensions could be noticed as “dark matter” if it is “big enough”. Can you estimate how large could the compactification radius be in such a way that the separation between n=1 and n=2 for the gravatom could be measured with current technology? Hint: you need to know what is the tiniest energy separation we can measure with current experimental devices.

Remark (IV): In  Verlinde’s entropic approach to gravity, extra corrections arise due to the change of the functional entropy we choose. It can be  due to extra dimensions and the (stringy) Generalized Uncertainty Principle as well.

Gravatoms and Dark Matter: a missing link

I will end this thread of 3 posts devoted to Bohr’s centenary model to recall a connection between atomic physics and the famous Dark Matter problem! The calculations I performed above (and which anyone with a solid, yet elementary, ground knowledge in physics can do) reveals a surprising link between microscopic gravity and the dark matter problem. I mean, the problem of gravatoms can be matched to the problem of dark matter if we substitute the proton mass by the mass of a galaxy! It is not an unlikely option that the whole Dark Matter problem shows to be related to a right infrared/long scale modified gravitational theory induced by quantum gravity. Of course, this claim is quite an statement! I work on this path since months ago…Even when MOND (MOdified Newtonian Dynamics) or MOG (MOdified Gravity) have been seen as controversial since Milgrom’s and Moffat’s pioneer works, I believe it is yet to come its “to be or not to be” biggest test. Yes, even when some measurements like the Bullet Cluster observations and current simulations of galaxy formation requires a component of dark matter, I firmly believe (similarly, I think, to V. Rubin’s opinion) that if the current and the next generation of experiments trying to discover the “dark matter particle/family of particles” fails, we should take this option more seriously than some people are able to accept at current time.

May the Bohr model and gravatoms be with you!


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!


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 }}