A spin on Shannon’s Missing Information

The Universe‘s Shannon Missing Information content, is its “angular momentum” in units of the Planck’s constant, h being that dimensionless ratio of its inertial mass-energy to its “Planck energy” [hf] in which the Hubble parameter acts as a frequency. Consequently Boltzmann shows us what we know that the entropy of the universe increases with falling temperature. Further simple dimensional analysis and use of Dirac’s large number hypothesis shows how this definition of the Universe’s entropy relates to a decay in Newton’s gravitational “constant”, G.

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From the same simple dimensional analysis that delivers the Planck length, l_p=(\frac{hG}{c^3})^{1/2} one can easily determine the Mass of the Universe by correctly postulating that a power law expression comprising the speed of light, c Newton’s Gravitational, G and Hubble’s constant, H_0 govern its grand inertia, M\sim c^\alpha G^\beta H^\gamma. Homogeneity of units constrains the mass to be some multiple (twice as much if one solves for the Friedmann Walker metric using the full chicanery of General Relativity) of

M=\frac{c^3}{GH_0}.

Simple dimensional analysis also shows that for a galaxy with a flat rotation curve has a characteristic acceleration

g_c=\kappa\frac{v^4_f}{GM_b}

where v_f is the rotation speed for a purely Keplerian orbital system, M_b is the baryonic mass, and G is Newton’s constant. M_b is determined empirically from the ratio \alpha=M/L as luminosity is deemed a good proxy for Baryonic mass distribution. The quality of the proxy \alpha varying between o.5 and 0.7 for galactic discs and bulges so that the proportionality factor \kappa lies between 1.2 and 1.6 when including the disk geometry of rotating galaxies.

The observation that more distant galaxies are receding faster than nearer ones is captured in the Hubble expansion formulae H_0=\dot{R}/R, for a Ricci curvature Scalar parameter, R. Noting that expansion scales as some polynomial in t, R(t)\propto t^n we have \dot{R}\sim nt^{n-1} so that Hubble parameter scales as inverse time, H_0 \sim \frac{nt^{n-1}}{t^n}\sim t^{-1}.

Dirac noted two numbers, \mathcal{D}_1, \mathcal{D}_2 that in natural units in which c=h=e=1 are of similar order in magnitude, 10^{40}:

\mathcal{D}_1=\frac{c/H_0}{e^2/m_ec^2}=\frac{m_ec^3}{e^2}\frac{1}{H_0}[\sim kgm^3s^{-4}A^{-2}];

\mathcal{D}_2=\frac{e^2}{G}\frac{1}{m_em_p}[\sim A^2 kg^{-1}m^{-3}s^{4}A].

\mathcal{D}_1 being roughly the ratio of the radius of the universe to that of the electron, while \mathcal{D}_2 is the ratio of the electrostatic to gravitational attraction of the electron to the proton. Here m_e and m_p are the mass of an electron and proton respectively. A third number, \mathcal D_3, dimensionless in both standard and natural units is thus:

\mathcal D_3=\mathcal D_1\mathcal D_2=\frac{1}{m_p}\frac{c^3}{GH_0}\sim 10^{80}.

Given our mass of the universe formula above, we see that \mathcal{D}_3 is a count of the number of baryons (protons) in the universe, as

\mathcal{D}_3=\frac{M}{m_p}.

Consider now the ratio of the inertial mass-energy of the universe, Mc^2 to its “Planck energy”, hH_0. That is, a dimensionless number independent of Newton’s Gravitational constant, G that we shall refer to as Shannon’s Information content of the universe, N

N=\frac{Mc^2}{hH_0}.

We note that as the dimension of Planck’s constant is h\sim kgm^2s^{-1}, being the units of angular momentum, \hat{L}\sim M{\bf g}\times {\bf r}, the Information content, N of the universe can be thought as the Universe’s angular momentum, \hat{L}_M= \frac{Mc^2}{H_0} in units of the Planck’s constant:

N=\frac{\hat{L}_M}{h}.

Now Boltzmann’s entropy formula in terms of Shannon information, N and Boltzmann constant, k_B reads,

S=Nk_Bln 2.

Substituting for N with our quantised angular momentum expression we have

S=(\frac{Mc^2}{H_0})\frac{1}{h}k_Bln2.

We think of Boltzmann constant in terms that k_BT is the amount of heat required to increase the thermodynamic entropy of a system, in natural units, by one “nat”. With the following quantised energy unit equality, k_BT=hf and on our grand Universal scale this reads as k_BT=hH_0. We have then that the entropy of the universe increases as its temperature falls T^{-1},

S\sim (Mc^2)\frac{k_B}{hH_0}\sim Mc^2\frac{1}{T}.

Making clear now the explicit dependence on H_0, S\sim (k_B\frac{Mc^2}{h})\frac{1}{H_0}, we note the inverse time fall-off of the Hubble parameter, H\sim t^{-1}, such that S\sim H_0^{-1} so that,

\dot{S}\sim -H_0^{-2}.

That is, we have that the fractional increase in the Universe’s entropy being a function of the Hubble parameter drops over time according to

\frac{\dot{S}}{S}\sim\frac{-H_0^{-2}}{H_0^{-1}}\sim-H_0^{-1}\sim -t.

If we return to Dirac’s first two natural dimensionless numbers, and do as he postulated that it is more than coincidental that \mathcal{D}_1\sim \mathcal{D}_2 and equate them to give

G\sim ec^3 (\frac{e^3}{m_e^2m_pc^6})H_0.

If we do not equate them we note that given H_0\mathcal{D}_1= G\mathcal{D}_2 our expression for Shannon Information reads thus

N=\frac{Mc^2}{hG}\frac{\mathcal{D}_1}{\mathcal{D}_2}.

If we do insist on parity, we can now rewrite our entropy expression in terms of a time changing G, \frac{\dot{G}}{G}\sim -\frac{t^{-2}}{t^{-1}}=-t^{-1} so that the entropy of the universe is

S= (\frac{k_Be}{hc})[\frac{e^3}{m_e^2m_p}]G^{-1},

being proportional to the reciprocal of Newton’s gravitational “parameter”, with a proportionality constant being a ratio of the four fundamental (microscopically inertial) constants k_B,h,c,e multiplied by the charge to mass ratio of a protium (hydrogen-1) anion.

Entropy of Universe

References

Cosmologies with Variable Gravitational Constant, J. V. Narlikar, Foundations of Physics, Vol. 13, No. 3, 1983ID 809695http://repository.iucaa.in:8080/jspui/bitstream/11007/1584/1/124A_1983.pdf

Geometry of the Universe and Its Relation to Entropy and Information, I. Haranas and I Gkigkitzi, Advances in Astronomy, Volume 2013 |Article ID 809695, https://www.hindawi.com/journals/aa/2013/809695/

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