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The Bekenstein-Hawking entropy of a black hole contains the Planck surface.

Mead's discussion of the gravitational microscope yields the Planck length as the length measurement limit.

Is there any system where the Planck volume plays a role? (The Planck volume is, arguably, the size of the universe after one Planck time; it is also said to be the smallest possible black hole. But these are quite speculative situations.)

Do more concrete examples of physical systems exist, in which the Planck volume arises or plays a role?

Or again:

Is there any equation for a physical law that contains, after simplification, the Planck volume, i.e., the cube of the Planck length?

And more:

Exactly why is there no equation with the Planck volume?

Added:

  • Is the reason that the ratio between macroscopic volume and Planck volume cannot arise in an equation at all?

  • Or is the reason that the ratio between macroscopic volume and Planck volume must disappear (in the limit) for quantum theory and general relativity to make any sense?

  • Or is the reason that holography is valid in nature?

  • Or is the reason that volume is not built from many smallest volumes, in contrast to area and length?

  • In short: is it because volume is not quantized?

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    $\begingroup$ You can substitute in place of Plank surface $$3 \frac {V_P}{\ell_P}$$, but I suspect you have in mind a "natural" Plank unit being used, and not some complex substitute. Btw, The Bekenstein-Hawking entropy can be expressed as number of micro black holes needed $N$ to get the same entropy of macro black hole, so variables used in original equation does not necessarily mean that they are crucial in that equation. $\endgroup$ Dec 10, 2022 at 12:50
  • $\begingroup$ True, but do micro black holes exist? They decay after a few Planck times ... $\endgroup$
    – KlausK
    Dec 10, 2022 at 13:34
  • $\begingroup$ I don't know, it is just a theoretical concept at the moment. However, short lifetime of a particle doesn't say anything about proving or disproving it's existence. For example, $Z^0$ bozon lives just for $\approx 0.1~ys=10^{-25}~s$ tiny period. It was theorized at 1968 and discovered in CERN at 1983 after 20 years ! (As a side note - laser principles were theorized by Einstein in 1917, but a first laser was built just in 1960, after 43 years ! Should we stop theorizing even if we are not sure if phenomena exists ? Confirmations are always late.) $\endgroup$ Dec 10, 2022 at 18:08
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    $\begingroup$ Of course you can theorize as much as you want! But t does not answer the question ... Thank you nevertheless! $\endgroup$
    – KlausK
    Dec 10, 2022 at 20:23
  • $\begingroup$ Since there is no universally accepted quantum theory of gravity, I don't think a question of the form "is physical quantity $X$ relevant in quantum gravity" is well defined, since our current state of knowledge could (in fact, will) change. $\endgroup$
    – Andrew
    Dec 10, 2022 at 22:58

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Obviously: $$ A_{\text{P}} = \ell_{\text{P}}^2 = (\ell_{\text{P}}^3)^{\frac{2}{3}} \equiv V_{\text{P}}^{2/3}. $$ So any expression that involves the Planck area could be expressed in terms of the Planck volume instead.

But notice that the internal volume of a black hole is hard to define, since there's no static observers inside. Only the external area has a meaning. So I don't see why you would prefer to have equations with the Planck volume, which doesn't have a well defined physical meaning.

One situation which I know the Planck volume may have some meaning is the Einstein-Cartan theory with torsion (or ECSK theory). In that theory, the Dirac equation have an extra "contact interaction" term. The Planck volume $V_{\text{P}}$ doesn't shows explicitely in that term, but the Dirac field density enters the interaction term and imposes a limit to the energy density. You may want to check the literature on that subject, but be aware that the mathematics are extremely heavy when you add torsion to General Relativity. It is quite disgusting!

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    $\begingroup$ In the question I asked for Planck volume after simplification, so your equation does not count... But I will read about ECSK Theory, thank you. $\endgroup$
    – KlausK
    Dec 10, 2022 at 20:19
  • $\begingroup$ ECKS theory seems to have no relation to experimental data. The Dirac equation is changed. So it looks as if the suspicion that the Planck volume (after simplification) does not arise in any equation still holds. $\endgroup$
    – KlausK
    Dec 12, 2022 at 5:37

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