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Monday, March 23, 2015

Derivation MpRp=4LM #4

Where it's all going - towards consciousness...

This part is how to derive the mass.radius relationship for a black hole.  We are going to approach this in a multi-step approach, with each step getting closer and closer to the correct answer.

First, we are going to calculate the mass of a spherical volume of the fabric of spacetime, T:

$$V_S={4 \over 3} \pi R_S^3$$
$$M_S={4 \over 3} \pi TR_S^3$$

$M_S=$Schwarzschild Mass of black hole
$V_S=$Schwarzschild Volume of black hole
$R_S=$Schwarzschild Radius of black hole

$V_P=$Planck Spherical Unit (PSU) Volume

$T = {Planck Mass \over {Planck Spherical Volume}}= {M_P \over V_P} = {M_P \over {{4 \over 3}\pi{\left(L_P \over 2 \right)}^3}} = {6M_P \over {\pi L_P^3}}$

$$M_S={6M_P \over \pi L_P^3}{4 \over 3} \pi R_S^3$$
$$M_S=8M_P {\left(Rs \over L_P \right)}^3$$

OK, I'm still free-styling, so let's check and see if we are on the right track.  Up to now, we should be calculating a Scpherical Schwarzschild Mass ($M_S$) of Schwarzschild radius ($R_S$). Now, what we gotta do is pretty simple, however, I am making it look hard. What we got to do now is plug in for the Planck Mass ($M_P$) and the Planck Length ($L_P$) and see what this equation for $M_S$ comes out to be and then check to see if it matches the Schwarzschild solution to Einstein's field equations.

I'm letting it all hang out now, deriving it real time, showing possible mistakes.  What if this next step does not result in a correct answer?  What then?  What possibly could go wrong with this approach?

Good Evening, and hopefully I will finish this post with an update later this evening, thus, accomplishing QED or whatever fancy latin phrase that would be appropo.

$$M_P=\sqrt {\hbar c \over G}$$
$$L_P=\sqrt {\hbar G \over c^3}$$

$${M_P \over L_P^3} = {c^5 \over {\hbar G^2}}$$

$R_S^3={{ \hbar M_S  G^2} \over {8 c^5}} \leftarrow $ Incorrect. $ \eta $ must be included.

Things are not quite coming out like expected. The Schwarzschild solution, for comparison is:
$$R_S={2GM_S \over c^2}$$  

...Surfer's board needs recharged... will return tomorrow with corrections, updates, and completions...

(Hey, it's just like in the comic books, sometimes the super hero goes down, but not out...)

The Surfer, OM-IV
-----------------------------------------------UPDATE 3/31/2015:
The information theory part was left out therefore the approach as mentioned in Derivation MpRp=4LM §#3 is needed. That'll teach me for rushing and doing it real time in a blog on the internet. Well, at least I can fix this mess up in the upcoming Derivation MpRp=4LM #5