Motivated by email from Philippe H.
There's this socalled Theory of Everything
which hopes to explain the universe and all its interactions and forces and ...
>Don't tell me! You have such a theory. Am I right?
No, you're not right.
However, I was reading this blog and the authors begin with the claim that:
The amount of energy is proportional to its volume.
That got me thinking.
>That was your first mistake.
Suppose that, as Findlay & Haenggi suggest, Energy is proportional to Volume.
That is, an Energy E would be associated with a Volume V, and:
[1] E = k V for some constant k.
Since Energy has the dimensions of M (L^{2} / T^{2})
where E = Mass, L = Length and T = Time
and Volume has the dimensions of L^{3}, then:
[2] E / V = k = M (L^{2} / T^{2}) / L^{3}
However, speed^{2} has the dimensions of (L^{2} / T^{2}) and mass density has the dimensions of M /L^{3}, so we can write:
[3] E / V = ρ c^{2}
where ρ is some massdensity and c is some speed.
Now we have the constant k, eh?
>Some density? What density? What speed?
I thought you'd never ask.
We should pick something that's sexy, like
[4] ρ = density of the mass within the Volume and c is the speed of light.
>You're kidding, right? The speed of light is HUGE! Every bit of volume has that much energy? Remember E = m c^{2}?
Well, let's see. The density of the universe is roughly 10^{30} gm/cm^{3}
and the speed of light is about 3x10^{10} cm/sec so we'd get something like:
[5] E / V = 9x10^{10} gm/cm/sec^{2} or ergs.
>Is that a lot?
Okay, 1 calorie is about 4.2x10^{7} ergs so ...
>So an erg is about ... uh, let's see ...
The energy associated with a cubic centimetre is roughly 2 x 10^{17} calories ... and that's pretty small.
Notice that we don't distinguish between Volume and Energy.
>So all volume has energy?
Well, all volume that contains some mass, as measured by ρ.
Indeed, we might posit the existence of "invisible" matter in every volume, so every volume is associated with some energy.
>Huh? Invisible?
Yes. Matter that can't be observed directly, but only assumed. Let's call this ...
>How about dark matter?
Excellent!
 Consider a mass m_{1} residing within some Volume V_{1}.
 We assume it absorbs energy from the surrounding volume, so that surrounding volume might diminish.
>If the Energy decreases in the surrounding volume, then that volume must decrease. Right?
Could be.
 Consider a second volume within which lies a second mass m_{2} within Volume V_{2}.
 Each absorbs energy from the surrounding volume ... hence from each other.
 Maybe the volume of space between the two masses diminishes.
>They get closer together?
Could be. And that's called ...
>Gravitational attraction?
Excellent!
>But if the Energy is constantly being absorbed by m_{1}, then doesn't V_{1} continually increase?
Ah, yes ... because Energy is equivalent to Volume, eh? One one goes up, so must the other.
But let's look again at [3]. If E increases, it could be because ρ increases, not V.
>If ρ increases then the mass m_{1} increases. Is that what happens? The mass gets continually heavier?
The density of matter within V_{1} could increase without m_{1} increasing. It could be that "other" matter enters V_{1}.
>Don't tell me! Dark matter?
Why not? Isn't it fun to speculate?
>No! Maybe you've heard of Newton. He said that gravitational acceleration is inversely proportional to the square of the distance separating two masses.
Aha! And you want to know how we'd get that, eh?
Okay, let's consider the distance between the two masses. We'll call it R.
 Mass m_{2} lies on the surface of a sphere or radius R with mass m_{1} as centre.
 The Energy being absorbed by m_{1} passes through the surface of this sphere
with surface area 4πR^{2} (on its way to m_{1}).
 The density of Energy being absorbed, at the location of this surface, is E / 4πR^{2} energyunits per unit area.
 It's this Energydensity which generates the gravitational acceleration ... so the acceleration is inversely proportional to R^{2}.
See how it works?
>Do you really believe this stuff?
Is that relevant? It's fun! And it makes you think.
>Okay, so material masses attract Energy in the form of dark matter. That'd mean that dark matter congregates about material masses, right?
Why not? We might imagine every mass having a cloud of dark matter about it.
Indeed, we might imagine a distribution of dark matter about the mass ... with an energydensity which decreases as we move away from the mass.
>An inverse square law distribution!
That'd do it.
 
Note that we may write introduce a 4dimensional spacetime universe, with dimensions identified by
[6a] x, y, z and τ = ct
... where c is the speed of light so τ = ct has the dimensions of length.
An elemental volume in spacetime would be measured by
[6b] dv = dx dy dz dτ = c dx dy dz dt
In order to incorporate spacetime, we could change [3] for an elemental spacetime volume dv and associated energy dE ... like so:
[7a] dE / dv = ρ c^{2}
... where ρ measures the mass density within the volume dv.
Or, to put it differently:
[7b] dE = ρ dv c^{2} = ρ dxdydz c^{3} dt = m c^{3} dt
... where m is the mass within the volume dv.
>Haven't you heard of Einstein? He says: E = m c^{2}?
Einstein? Never heard of him.
If you'd like to discuss the ramifications of the theory, please write to the authors at:
the blog
