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Alright, so: Let's start by taking the equation for the Bekenstein-Hawking Boundary Entropy/Bekenstein–Hawking black hole entropy, which happens to saturate the Bekenstein bound. That supposedly has some implications that lead to people wondering about holographic universes, though while the universe having a possibly-infinite tree of nested black holes, the boundaries of which represent space time regions, sounds pretty interesting in and of itself, that stuff goes over my head a bit and it's not what I'm going to focus on in this entry anyway.
I'm just going to play around with a model universe where the Bekenstein Bound is instead the Bekenstein Equation and see what I get from that! Also, I'm going to be using Planck units, partly for simplicity in general, but also because the Bekenstein bound (as presented in the Wikipedia article, anyway) doesn't seem to prescribe a particular unit of entropy/information for determining, say, how many bits of information you get by multiplying a plank energy times a sphere of plank length radius, just a general proportion.
So yeah, we doing dumb unit conversion with a presumption of a Bekenstein Equation relation of [entropy]=[area]/4=4*[Pi]*[mass]2. Taking just [area]/4=4*[Pi]*[mass]2. Now let's look at vacuum energy! Let's ignore its actual value for now, but just look at how it's in the form of [energy]/[length]3. Now, area is [length]2, so we can rewrite the preceding space-mass equation as [length]2/4=4*[Pi]*[mass]2. Further, in Planck units, the mass energy equation is just [energy]=[mass], so we can further write it as [length]2/4=4*[Pi]*[energy]2.
So, let's simplify for length first. Multiply both sides of [length]2/4=4*[Pi]*[energy]2 by four, get [length]2=16*[Pi]*[energy]2. Now take the square root, of both, so you have [length]=sqrt(16*[Pi]*[energy]2). I'm not entirely sure, but if I understand the communitivities involved correctly, this means we get [length]= Plugging in 4*sqrt([Pi])*[energy] as a replacement for length in the [energy]/[length]3 unit value, we have [energy]/(4*sqrt([Pi])*[energy])3. Checking this expression simplifier, it seems that x/x3=x-2. If I'm again understanding communitvitiy right, we can change this to be [energy]/(4*sqrt([Pi]))3*[energy]3, and then to 1/64*[Pi]3/2*[energy]-2.
Trusting the expression simplifier again, it says 1/64*X^(3/2)*Y^-2 evaluates to (1/64)X3/2Y-2, therefore 1/64*[Pi]3/2*[energy]-2 evaluates to (1/64)Pi3/2Y-2. Actually, I should see about just plugging in some stuff from earlier to see if it matches up. The expression simplifier says that (16*X*Y^2)^0.5=4YX0.5, which looks much nicer than what I was working with before, and Y/(16*X*Y^2)^3 to... oh dear... (1/4096)Y-5X-3?
Well, I was going to go ahead and do a similar thing for the other side by simplifying and substituting energy, and then using that and the fact that both resulting values equal [energy]/[length]3 to arrive at either the same energy/mass to area equation as before, or to a different one that had maybe a different balance of exponents for length or something. Now, though, I'm not really sure what to do to make sure I'm doing my math consistently. I guess either check with someone else, or try finding some math analysis software that can help me keep track of what I'm doing better?
I'm just going to play around with a model universe where the Bekenstein Bound is instead the Bekenstein Equation and see what I get from that! Also, I'm going to be using Planck units, partly for simplicity in general, but also because the Bekenstein bound (as presented in the Wikipedia article, anyway) doesn't seem to prescribe a particular unit of entropy/information for determining, say, how many bits of information you get by multiplying a plank energy times a sphere of plank length radius, just a general proportion.
So yeah, we doing dumb unit conversion with a presumption of a Bekenstein Equation relation of [entropy]=[area]/4=4*[Pi]*[mass]2. Taking just [area]/4=4*[Pi]*[mass]2. Now let's look at vacuum energy! Let's ignore its actual value for now, but just look at how it's in the form of [energy]/[length]3. Now, area is [length]2, so we can rewrite the preceding space-mass equation as [length]2/4=4*[Pi]*[mass]2. Further, in Planck units, the mass energy equation is just [energy]=[mass], so we can further write it as [length]2/4=4*[Pi]*[energy]2.
So, let's simplify for length first. Multiply both sides of [length]2/4=4*[Pi]*[energy]2 by four, get [length]2=16*[Pi]*[energy]2. Now take the square root, of both, so you have [length]=sqrt(16*[Pi]*[energy]2). I'm not entirely sure, but if I understand the communitivities involved correctly, this means we get [length]= Plugging in 4*sqrt([Pi])*[energy] as a replacement for length in the [energy]/[length]3 unit value, we have [energy]/(4*sqrt([Pi])*[energy])3. Checking this expression simplifier, it seems that x/x3=x-2. If I'm again understanding communitvitiy right, we can change this to be [energy]/(4*sqrt([Pi]))3*[energy]3, and then to 1/64*[Pi]3/2*[energy]-2.
Trusting the expression simplifier again, it says 1/64*X^(3/2)*Y^-2 evaluates to (1/64)X3/2Y-2, therefore 1/64*[Pi]3/2*[energy]-2 evaluates to (1/64)Pi3/2Y-2. Actually, I should see about just plugging in some stuff from earlier to see if it matches up. The expression simplifier says that (16*X*Y^2)^0.5=4YX0.5, which looks much nicer than what I was working with before, and Y/(16*X*Y^2)^3 to... oh dear... (1/4096)Y-5X-3?
Well, I was going to go ahead and do a similar thing for the other side by simplifying and substituting energy, and then using that and the fact that both resulting values equal [energy]/[length]3 to arrive at either the same energy/mass to area equation as before, or to a different one that had maybe a different balance of exponents for length or something. Now, though, I'm not really sure what to do to make sure I'm doing my math consistently. I guess either check with someone else, or try finding some math analysis software that can help me keep track of what I'm doing better?
