The Poincaré recurrence time of certain systems is the time for them to revert to a state almost identical to their current state. The system should satisfy the following properties:[1]
- All the particles in the system are bound to a finite volume.
- The system has a finite number of possible states.
The universe might not satisfy these properties.
Don Page, physicist at the University of Alberta, Canada, has estimated the Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the mass \(M\) equal to mass of observable universe using the equation \(t_{poincare}=e^{e^{4 \pi \times M^2}} \approx 10^{10^{10^{10^{2.08}}}}\) where time and mass \(M\) are expressed in planck units. This number is so large that the estimate remains the same whether measuring time in Planck times, years, millennia, or any other time units of the same exponential order.[2]
Page also estimated a Poincaré recurrence time of a Linde-type super-inflationary universe at \(10^{10^{10^{10^{10^{1.1}}}}}\) years, which Page claimed to be, to his knowledge, the longest finite length of time ever explicitly calculated by any physicist.[3][4]
12 years later Don Page gave estimation of universe size as at least \(10^{10^{10^{122}}}\) Mpc.[5] Considering this estimation, the Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the mass \(M\) equal to mass of the universe (observable or not) is equal to \(t_{poincare}=e^{e^{4 \pi \times M^2}} \approx 10^{10^{10^{10^{10^{122}}}}}\) Planck times, millenia, or whatever.
Approximations[]
| Notation | Approximation |
|---|---|
| Fast-growing hierarchy | \(f^5_2(399)\) |
Sources[]
- ↑ Physics Stackexchange Forum
- ↑ Page, Don. Information Loss in Black Holes and/or Conscious Beings?. Retrieved 2017-02-28.
- ↑ How to Get a Googolplex
- ↑ The longest time
- ↑ Page, Don. Susskind's Challenge to the Hartle-Hawking No-Boundary Proposal and Possible Resolutions. Retrieved 2017-02-28.
See also[]
Sagan's number · Avogadro's number · Eddington number · Planck units · Promaxima · Poincaré recurrence time · Universe size