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Is the total information available in the universe compatible with the entropy we would find on the surface of an hypothetical black hole whose radius is the visible universe one ($\sim14$ Gpc)?

We know that the entropy of a BH is proportional to its surface. I would love to discuss the possibility that we are actually living inside a BH, just in a toy model... Of course keeping into account the actual BBR at 2,7 K. Is a totally insane direction?

Pietro Oliva
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  • "I would love to discuss the possibility that we are actually living inside a BH" See https://physics.stackexchange.com/questions/23118/are-we-inside-a-black-hole/52280#52280 – D. Halsey Jan 05 '19 at 00:23
  • Looks like it was already discussed in a way. But wat about the calculation of the surface entropy (Bekenstein-Hawking style)? has anyone attempted to check if this is close to a system at 3 K of our size? moreover i find several fundamental properties of the fileds (expecially in terms of geometry, e.g. the properties of laplacian that select the most "boring" functions) going toward the direction of some singularity as external shell, like said in your link. – Pietro Oliva Jan 05 '19 at 00:31
  • If the universe is inside a black hole, why are galaxies accelerating away from each other rather than falling toward the singularity as things inside black holes do? (Answer: The universe is not inside a black hole.) – G. Smith Jan 05 '19 at 03:15
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    FWIW, a Schwarzschild BH with a Hawking radiation temperature of 2.725K (the same as the CMBR) would have a radius of 66.87 microns. – PM 2Ring Jan 05 '19 at 10:03

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I won't comment on the "actually living inside a BH" idea, but here are some numeric comparisons, using entropy to quantify "information," and interpreting "radius" as "Schwarzschild radius."

According to [1], the entropy of the observable universe (excluding black holes) is dominated by the cosmic microwave background photons. Based on this, and using $$ 1\text{ parsec}\sim 3\times 10^{16}\text{ meters}, \tag{1} $$ we can very roughly estimate the entropy of the observable universe to be $$ S_\text{CMB}\sim \left(\frac{14\times 10^{9}\text{ parsecs}}{1\text{ millimeter}}\right)^3 \sim 10^{89}. \tag{2} $$ Inside the parentheses, the numerator is the radius of the observable universe, and the denominator is the wavelength of a typical CMB photon [2]. This is only a very crude estimate, of course.

For comparison, the Bekenstein-Hawking entropy associated with a black hole [3] is $$ S_\text{BH} = \frac{c^3}{\hbar G}\,\frac{A}{4} = \frac{A}{4L^2} \tag{3} $$ where $A$ is the area of the horizon with Schwarzschild radius $R$, and $$ L=\sqrt{\hbar G/c^3}\sim 1.6\times 10^{-35}\text{ meter} $$ is the Planck length. According to equation (3), the entropy of a solar-mass black hole ($R\approx 3$ km) is $\sim 10^{77}$, and the entropy of a black hole with Schwarzschild radius $$ R\sim 14\times 10^9\text{ parsecs} \sim 4\times 10^{26}\text{ meters} $$ would be $$ S_\text{BH} \sim 10^{123}. $$ According to these estimates, the entropy of the observable universe, $\sim 10^{89}$, is much less than the entropy of a universe-sized black hole, $\sim 10^{123}$. I am practicing the art of understatement.

Reference:

[1] Section 4.5 in Harlow (2014), "Jerusalem Lectures on Black Holes and Quantum Information," http://arxiv.org/abs/1409.1231

[2] "CMB Spectrum," https://asd.gsfc.nasa.gov/archive/arcade/cmb_spectrum.html

[3] Sections I and II in Bousso (2002), "The holographic principle", https://arxiv.org/abs/hep-th/0203101

Chiral Anomaly
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    Does the radius of a black hole look the same from the outside than from the inside? inside a black hole the radial direction becomes the time direction, right? –  Jan 05 '19 at 16:02
  • @Wolphramjonny Excellent comment. The $R$ here refers to the Schwarzschild radius, which is simply a particular constant in the Schwarzschild metric. It's name includes the word "radius" because the area of the event horizon (which is coordinate-independent) is equal to $4\pi R^2$. However, the usual concept of "radius" is not appropriate in this context, for exactly the reason you stated. I took the liberty of interpreting the word "radius" in the OP as "Schwarzschild radius," because otherwise I don't know what it would mean. I edited the answer to acknowledge this assumption. – Chiral Anomaly Jan 05 '19 at 16:11
  • @Wolphramjonny yes indeed, not inside the black hole itself but right beyond its event horizon. – Pietro Oliva Jan 07 '19 at 17:07