Einstein Probabilistic Units/Black Holes: Difference between revisions

Latest revision as of 01:32, 17 January 2024

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Black Hole

A black hole can be defined by the Schwarzschild radius

r_{s} = \frac{2 G M_{b h}}{c^{2}}

Expressing this using Einstein Probabilistic Units

r_{b h} = 2 B M_{b h}

Solving for B give us

B_{b h} = \frac{r_{b h}}{2 M_{b h}}

The following table shows some properties of Black Hole using Einstein's Probabilistic Units (EPU); Template:Center top

Property	Equation	EPU
Mass	$M_{s} = 2 \frac{1}{r_{s}} \frac{G}{c^{2}}$	$M_{s} = \frac{2 c}{A_{s} B_{s}}$	$M_{s} = \frac{c}{B_{0}} \frac{A_{s}}{A_{0}^{2}}$
Schwarzschild radius	$r_{s} = 2 M_{s} \frac{G}{c^{2}}$	$r_{s} = 2 M_{s} B_{0}$	$B_{0} = \frac{G}{c^{2}} = \frac{1}{2} B_{s}$
Area	$A_{s} = 16 π M_{s}^{2} \frac{G^{2}}{c^{4}}$	$A_{s} = 16 π M_{s}^{2} B_{0}^{2}$	$B_{0}^{2} = \frac{1}{16 π} B_{s}^{2}$
Density	$ρ_{s} = \frac{3}{32 π} \frac{c^{6}}{M_{s}^{2} G^{3}}$	$ρ_{s} = \frac{3}{32 π} \frac{1}{M_{s}^{2}} \frac{1}{B_{0}^{3}}$	$ρ_{s} = \frac{3}{8 π} \frac{1}{B_{0}} \frac{1}{r_{s}^{2}}$
Entropy	$S_{B H} = \frac{1}{4} k_{B} \frac{A_{s}}{l_{P}^{2}}$	$S_{B H} = \frac{1}{4} k_{B} \frac{B_{s}}{B_{0}}$	$B_{0} = \frac{1}{4} k_{B} \frac{B_{s}}{S_{B H}}$
Temperature	$T = \frac{1}{8 π} \frac{1}{M_{s}} \frac{ℏ c}{k_{B}} \frac{c^{2}}{G}$

Template:Center bottom

Entropy

The Bekenstein–Hawking formula for black-hole entropy is proportional to the area of its event horizon A.

$S_{B H} = \frac{k_{B} A_{s}}{4 l_{P}^{2}} = \frac{k_{B} c^{3} A_{s}}{4 G ℏ} \frac{J}{K}$

Using Einstein's Probabilistic units to express Bekenstein–Hawking formula;

$S_{B H} = k_{B} \frac{c^{3}}{ℏ} \frac{A_{s}}{c^{2}} \frac{c^{2}}{G} = k_{B} \frac{c^{3}}{ℏ} \frac{1}{υ_{s}^{2}} \frac{1}{B_{0}} = k_{B} \frac{B_{s}}{B_{0}} = C o n s t a n t \frac{1}{υ_{s}^{2}} = - k_{B} < l n W >$

$S_{B H} = k_{B} l n (W) = k_{B} \frac{B_{s}}{B_{0}} = k_{B} \frac{A_{0}^{2}}{A_{S}^{2}}$

$W = e^{S / k_{B}} = e^{B_{s} / B_{0}}$

Hawking radiation temperature

$T_{H} = \frac{ℏ c^{3}}{8 π G M k_{B}}$

$T_{H} = \frac{ℏ c^{3}}{8 π μ_{M} k_{B}} = \frac{ℏ c}{8 π k_{B}} \times \frac{m^{2}}{s^{2}} \frac{s^{2}}{m^{3}} = \frac{1}{8 π k_{B}} \frac{Q^{2}}{m}$

$T_{H} = \frac{1}{8 π} \frac{ℏ c}{M} \frac{1}{B_{0}} \frac{1}{k_{B}} = \frac{1}{8 π} \frac{μ}{B_{0}} \frac{1}{k_{B}}$

$B_{0} = \frac{1}{8 π} \frac{μ_{M}}{T_{H}} \frac{1}{k_{B}}$

$B = \frac{h c}{(T_{H} k_{B}) M_{⊙}}$