B Black Hole Entropy: Basis of Logarithm Explored

[gerald V]

TL;DR Summary

Which basis of the logarithm underlies the usual formula for Bekenstein-Hawking entropy?

In textbooks, Bekenstein-Hawking entropy of a black hole is given as the area of the horizon divided by 4 times the Planck length squared. But the corresponding basis of the logarithm and exponantial is not written out explicitly. Rather, one oftenly can see drawings where such elementary area is occupied by one bit of information. Thus, I would conclude that the basis of the logarithm and corresponding exponential is 2. Is this correct?

One can express the black hole entropy in vintage units simply by multiplying by Boltzmann‘s constant. In this constant, is there embedded the change of basis of the logarithm from 2 to Euler’s number $e$?

Thank you very much in advance

 

[PeterDonis]

the corresponding basis of the logarithm and exponantial is not written out explicitly.

That's because (a) nobody knows what it is, since we don't have a microphysical model of black holes that tells us what states we are coarse-graining over to obtain the Bekenstein-Hawking entropy, and (b) it doesn't really matter anyway in practical terms since the entropies obtained are so huge compared to the entropies of ordinary objects like stars with similar masses.

By convention, the base of logarithms is taken to be $e$ if it is not given explicitly. But, as I just noted, nobody really knows in this case.

one oftenly can see drawings where such elementary area is occupied by one bit of information.

Nobody really knows whether that is true either. To know it, we would need to have a microphysical model of black holes, which, as above, we don't have.

One can express the black hole entropy in vintage units simply by multiplying by Boltzmann‘s constant. In this constant, is there embedded the change of basis of the logarithm

No. Boltzmann's constant is just a conversion factor between energy units and temperature units.

 

[Demystifier]

In thermodynamic discussions, the entropy is always defined with the basis $e$. That's because the other related quantities are also defined with the $e$ basis, e.g. the Boltzmann factor $e^{-\beta E}$. The basis $2$, on the other hand, is more convenient when one is more interested in the information content than in the thermodynamic one.

 

[vanhees71]

Well, the natural logarithm deals with nature, the log dualis with IT...

 

[geshel]

The basis in that reference is the natural logarithm. For the equivalent in bits, see the introduction to Bousso’s The Holographic Principle here:

https://arxiv.org/abs/hep-th/0203101

 

[geshel]

There is also a paper out there suggesting a new reference unit, instead of the Planck length. Such that an area of one such square unit has a limit of one bit. The linear unit is $$\frac{l_p}{\sqrt{\ln(2)}}$$

 

[patguy]

Hi, in fact the "exact value of the entropy of a black hole comes throught the temperature defined by Hawking. Bekenstein never calculates the exact value. This entropy is build because the area of the black hole can only increase but the exact formula is not an agreement with the principal of statistical physics (kb log W). Look for the error ? Regards

 

[patguy]

Hi, I forget why 4 for the number of states in a Planck cellular ? Why not 1, 10 , 40. Regards

 

[romsofia]

Is there a physical significance to logarithm bases in GR? Why not just change the base using basic log properties?

 

[PeterDonis]

There is also a paper out there

What paper? Please give a reference.

 

[PeterDonis]

the "exact value of the entropy of a black hole comes throught the temperature defined by Hawking.

How? Please give a reference.

This entropy is build because the area of the black hole can only increase

Not if Hawking radiation exists.

the exact formula is not an agreement with the principal of statistical physics (kb log W).

What formula are you talking about? Please give a reference.

 

[geshel]

What paper? Please give a reference.

Yeah, should have looked it up before posting. I also misquoted it. The proposed value is twice what I wrote above, however looking at the paper again quickly there is something I'm missing. Anyway here is the paper:

http://www.lsv.fr/~dowek/Publi/planck.pdf