# Black-hole (Planck)

Planck-unit Black-hole universe

A Planck-unit Black-hole universe defines the universe superstructure in terms of discrete Planck-units and is applicable to Simulation Hypothesis programming that use digital time where time is measured in discrete units of Planck time (the universe simulation clock-rate).

A base structure, a discrete and indivisible Planck micro black-hole is assigned the Planck-units , forming the scaffolding for particle substructures.

Universe time tage is defined by incremental additions of these Planck micro black-holes to the sum Planck Black-hole universe, as such for any chosen unit of simulation (Planck) time tage, there can be calculated the equivalent cosmic microwave background parameters .

In this table tage = 0.428 1061 units of Planck time;

cosmic microwave background parameters; calculated vs observed
Parameter Calculated Observed
Age (billions of years) 14.624 13.8
Age (units of Planck time) 0.428 1061
Mass density 0.21 x 10-26 kg.m-3 0.24 x 10-26 kg.m-3
Radiation energy density 0.417 x 10-13 kg.m-1.s-2 0.417 x 10-13 kg.m-1.s-2
Hubble constant 66.86 km/s/Mpc 67 (ESA's Planck satellite 2013)
CMB temperature 2.727K 2.7255K
CMB peak frequency 160.2GHz 160.2GHz
Entropy CEH 2.3 x 10122kB 2.6 x 10122kB
Casimir length 0.42mm

#### Mass density

For each increment to time tage, to the Planck Black-hole is added a Planck micro black-hole comprising a unit of Planck time tp, Planck mass mP and Planck (spherical) volume (Planck length = lp), such that we can then calculate the mass, volume and so mass density of the sum Black-hole for any chosen unit of time, (tsec if the age of the Black-hole is measured in seconds).

$t_{p}={\frac {2l_{p}}{c}}$
$mass:\;m_{universe}=2t_{age}m_{P}$
$volume:\;v_{universe}={\frac {4\pi r^{3}}{3}}\;\;\;(r=4l_{p}t_{age}=2ct_{sec})$
${\frac {m_{universe}}{v_{universe}}}={2t_{age}m_{P}}.\;{\frac {3}{4\pi {(4l_{p}t_{age})}^{3}}}={\frac {3m_{P}}{128\pi t_{age}^{2}l_{p}^{3}}}\;({\frac {kg}{m^{3}}})$

Gravitation constant G as Planck units;

$G={\frac {c^{2}l_{p}}{m_{P}}}$
${\frac {m_{universe}}{v_{universe}}}={\frac {3}{32\pi t_{sec}^{2}G}}$

From the Friedman equation; replacing p with the above mass density formula, √(λ) reduces to the radius of the universe;

$\lambda ={\frac {3c^{2}}{8\pi Gp}}=4c^{2}t_{sec}^{2}$
${\sqrt {\lambda }}=radius\;r=2ct_{sec}\;(m)$

#### Temperature

Measured in terms of Planck temperature = TP;

$T_{bh}={\frac {T_{P}}{8\pi {\sqrt {t_{age}}}}}\;(K)$

The mass/volume formula uses tage2, the temperature formula uses √(tage). We may therefore eliminate the age variable tage and combine both formulas into a single constant of proportionality that resembles the radiation density constant.

$T_{p}={\frac {m_{P}c^{2}}{k_{B}}}={\sqrt {\frac {hc^{5}}{2\pi G{k_{B}}^{2}}}}$
${\frac {m_{bh}}{v_{bh}T_{bh}^{4}}}={\frac {2^{5}3\pi ^{3}m_{P}}{l_{p}^{3}T_{P}^{4}}}={\frac {2^{8}3\pi ^{6}k_{B}^{4}}{h^{3}c^{5}}}$

From Stefan Boltzmann constant σSB

$\sigma _{SB}={\frac {2\pi ^{5}k_{B}^{4}}{15h^{3}c^{2}}}$
${\frac {4\sigma _{SB}}{c}}.T_{bh}^{4}={\frac {c^{2}}{1440\pi }}.{\frac {m_{bh}}{v_{bh}}}$

#### Casimir formula

The Casimir force per unit area for idealized, perfectly conducting plates with vacuum between them; F = force, A = plate area, dc 2 lp = distance between plates in units of Planck length

${-F_{c}}{A}={\frac {\pi hc}{480{(d_{c}2l_{p})}^{4}}}$

if dc = 2 π √tage then the Casimir force equates to the radiation energy density.

${\frac {-F_{c}}{A}}={\frac {c^{2}}{1440\pi }}.{\frac {m_{bh}}{v_{bh}}}$

A radiation energy density pressure of 1Pa gives tage = 0.8743 1054 tp (2987 years), Casimir length = 189.89nm and temperature TBH = 6034 K.

#### Hubble constant

1 Mpc = 3.08567758 x 1022.

$H={\frac {1Mpc}{t_{sec}}}$

#### Black body peak frequency

${\frac {xe^{x}}{e^{x}-1}}-3=0,x=2.821439372...$
$f_{peak}={\frac {k_{B}T_{bh}x}{h}}={\frac {x}{8\pi ^{2}{\sqrt {t_{age}}}t_{p}}}$

#### Entropy

$S_{BH}=4\pi t_{age}^{2}k_{B}$

#### Cosmological constant

Riess and Perlmutter using Type 1a supernovae to show that the universe is accelerating. This discovery provided the first direct evidence that Ω is non-zero giving the cosmological constant as ~ 1071 years;

$t_{end}\sim 1.7x10^{-121}\sim 0.588x10^{121}$  units of Planck time;

This remarkable discovery has highlighted the question of why Ω has this unusually small value. So far, no explanations have been offered for the proximity of Ω to 1/tuniv2 ~ 1.6 x 10-122, where tuniv ~ 8 x 1060 is the present expansion age of the universe in Planck time units. Attempts to explain why Ω ~ 1/tuniv2 have relied upon ensembles of possible universes, in which all possible values of Ω are found  .

The maximum temperature Tmax would be when tage = 1. What is of equal importance is the minimum possible temperature Tmin - that temperature 1 Planck unit above absolute zero, this temperature would signify the limit of expansion (the black hole could expand no further). For example, taking the inverse of Planck temperature;

$T_{min}\sim {\frac {1}{T_{max}}}\sim {\frac {8\pi }{T_{P}}}\sim 0.177\;10^{-30}\;K$

This then gives us a value for the final age in units of Planck time (about 0.35 x 1073 yrs);

$t_{end}=T_{max}^{4}\sim 1.014\;10^{123}$

The mid way point (Tuniverse = 1K) would be when (about 108.77 billion years);

$t_{u}=T_{max}^{2}\sim 3.18\;10^{61}$

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