[ExI] First Picture of a Black Hole!

[Stuart LaForge]

John Clark wrote:

>> Einstein says there is a point of infinite density at the center  
>> although Quantum Mechanics disagrees. Nobody knows who's right.
>
>> Probably quantum mechanics.
>
> That is certainly the conventional approach. People, including  
> Einstein, have been monkeying around with General Relativity for a  
> century trying to make it fit in with Quantum Mechanics and all have  
> failed. And meanwhile unmodified General Relativity has passed with  
> flying colors every experimental test thrown at it even the ones in  
> recent years involving super intense gravitational fields. Maybe  
> it's time to try things from the other direction and leave General  
> Relativity alone and start monkeying around with Quantum Mechanics  
> to make it compatible with General Relativity.

Ok, so let's assume that the GR singularity is a literal point mass  
and lets forget about density for a minute. Let's model its position  
as a Dirac delta function. Since it is at an exact location in space  
that Einstein gives us, our uncertainty in the singularity's position  
is minimized. However, Heisenberg says this should result in us having  
maximum uncertainty of the singularity's momentum, that is to say its  
mass, trajectory, and speed. It could be headed away from us or toward  
us and we wouldn't know until it was too late. The most massive things  
in the universe can also be the sneakiest, how's that for irony?

Another way to think about it is that the Fourier transform of the  
Dirac delta function (i.e. position) in the time domain to the  
frequency domain yields a single frequency sine wave as the momentum  
probability amplitude which means that it could have any number of  
possible momenta with equal probability and we have no way of knowing  
which momentum it actuality has.

I will leave it up to you whether you allow Heisenberg to divide by  
zero the way you allowed Einstein to. But the problem is that all  
black holes and causal cells are finite in size regardless of the size  
of the universe as a whole. That is to say that even if the universe  
is infinite, our patch of causally-connected space-time is not.  
Therefore nothing we can measure should ever be allowed to be infinite  
in our calculations.


>> My argument is mathematical and goes as follows: Dividing a finite  
>> mass by zero volume is not an infinite density, it  is a  
>> mathematically UNDEFINED density.
>
> Mathematics is a language, the best language ever found for  
> describing the workings of nature but there is no reason to think  
> it's perfect. Nature is the way it is and doesn't care if humans  
> have defined something or not.

This is precisely why I pay more attention to the measured value of  
dark energy than the nonsense the quantum field theorists calculated  
using infinity in their calculations.

> And besides, Einstein's theory may produce nonsense at one point  
> (the center of a Black Hole) but Quantum Mechanics produces nonsense  
> at every point, it says every volume in a vacuum has an infinite  
> matter/energy density (or if one makes the assumption unsupported by  
> evidence that spacetime is quantized then not infinite but *only*  
> 10^120 times the density of lead) and thus produces a infinite  
> gravitational field (or a gravitational field 10^120 stronger that  
> what lead can produce).

The whole problem rests on the fact that we let quantum field  
theorists use infinity instead of forcing them to use discrete math  
with finite boundary conditions for what is by definition a discrete  
phenomenon. You can't let the wave functions of quantum harmonic  
oscillators or particles or anything else take on infinite values  
because our causal cell is not infinite in extent nor is the  
observable universe.

Allow me to demonstrate how to "fix" the quantum vacuum energy calculation.

First we treat the entire causal cell as a large but not infinite  
collection of quantum oscillators that can assume many different  
vibrational modes but NOT an infinite number of vibrational modes  
since our causal cell is finite in both space and time.

That is to say that our causal cell is 13.8 billion years old. That is  
approximately Tu = 4.45*10^17 seconds old. So we set the fundamental  
frequency as Fmin = 1/Tu or 2.30*10^-18 Hertz. The intuition here is  
that the lowest possible frequency oscillator that can possibly be  
measured is one that has only vibrated half-way since the beginning of  
time, i.e. the Big Bang.

Similarly there is a maximum frequency to a quantum oscillator within  
our causal and that

Now according to QM, the equation for the nth energy state of the  
quantum harmonic oscillator is E(n) = (h*Fmin/2)(n+1/2) with n=0 being  
the ground state and n only taking integer values. h is Planck's  
constant and F is the frequency as usual.

The next step is to realize that the only possible frequencies that  
the quantum harmonic oscillators can vibrate at are whole number  
multiples of the fundamental frequency. This is the whole meaning of  
the term "harmonic". That is to say that the number of possible  
frequencies are finite and bounded both from above as well as below.  
In other words, there is a maximum frequency that a quantum oscillator  
can vibrate at and that is the Planck frequency or one cycle per  
Planck time or Fmax = 1/Tp = 1.85*10^43 Hertz.

That means that the highest harmonic of the fundamental frequency that  
a quantum oscillator can attain is n_max = Fmax/Fmin = 8.08*10^60.

Now to figure out what the vacuum energy of our causal cell is  
requires one final assumption which is that no two quantum harmonic  
oscillators in our causal cell can vibrate with the exact same  
frequency or have the same value of "n".

Now we can calculate the total vacuum energy of our causal cell as

E_vac = E(0) + E(1) + E(2) + . . . + E(n_max) = (h*Fmin/2)*(n_max+1)^2  
= 2.48*10^70 Joules or dividing by c^2 gives 2.76*10^53 kilograms.

This is pretty darn close to current estimates of the mass of the  
observable universe, which makes sense as since the mass of the  
universe is dominated by dark energy s you can see here:

https://www.quora.com/What-is-the-mass-of-the-universe?redirected_qid=21252687

And all I used were the observed age of the universe, the Planck time,  
and the equation for the quantum harmonic oscillator.

Stuart LaForge