Pulling it out of its differential notation form, what we really have is
$$[\nabla_i,\nabla_j] = \frac{\partial \Gamma_i}{\partial x^j} + \frac{\partial \Gamma_j}{\partial x^i} + \Gamma_i \Gamma_j$$
Aha, this is where the derivative of the Christoffel symbol enters the game! However, what is missing due to the notation here is that the derivatives are not taken on just the Christoffel symbols, but on some object that is unstated in this expression. Remember those missing indices I mentioned earlier? That are supposed to be contracted with a vector field that the covariant derivative is operating on. In other words, this unexplained rewritten formula is not taking derivatives of the object it's operating on into account, making it very suspect as to its correctness.
Let compare this with Reiku's own provided source,
https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html About halfway down (before eq 3.65), Carroll says "The actual computation is very straightforward. Considering a vector field $$V\scriptstyle \rho$$, we take", followed by a derivation. But note that Carroll gets a different answer than we have here.
If we assume the derivative-of-the-Christoffel-symbol clash is just a notational issue, we're still left with:
- A missing minus sign
- A missing Christoffel symbol pair (and its minus sign)
- A missing second term
In other words, this formula is plain wrong, as demonstrated by Reiku's own source. But, let's assume this is just some tex-mixup, and continue!
and is just the Riemann tensor, but we hope we have specified, it one that follows the spacetime relationship
$$\Delta x\ c \Delta t = \Delta X_i \Delta X_j \geq \ell^2$$.
This is not just the Riemann tensor. Corroll is quite clear that only
part of this expression is the Riemann tensor. Reiku mistakenly dropped various Riemann and all non-Riemann tensor contributions. In other words, this sentence is unambiguously incorrect. The formula looks like an inverse of an earlier formula, but in that case, the inequality is the wrong way around. Let's assume this is (another) typo, and soldier on!
What this unveils is that my initial idea of there being some commutator between the connection and the Christoffel symbols was true - but we also have another term, another commutator arising relativistically. The final mathematical model proposed to pave the way towards a unification of two concepts, a non-trivial spacetime uncertainty principle with the concept of geometry:
$$G_{ij} = \mathbf{R}_{ij} - \frac{1}{2}Rg_{ij} \approx [\nabla_i,\nabla_j] = (\nabla_i\nabla_j - \nabla_j \nabla_i) =\frac{\partial \Gamma_i}{\partial x^j} + \frac{\partial \Gamma_j}{\partial x^i} + \Gamma_i \Gamma_j = \frac{1}{\Delta L\ c \Delta t} \geq \frac{c^3}{G \hbar}$$
Let's focus on the formula. We start with the Einstein tensor, with its indices explicitly stated.
The second term is the definition of the Einstein tensor.
The third term is an approximation that we haven't seen before. We recognize the commutator, but nowhere was that argued to be approximately equal to the Einstein tensor. I can only guess that this follows from the "and is just the Riemann tensor" statement (which is wrong). Worse, what was a tensor now suddenly has two unevaluated covariant derivatives! This is of course mathematically impossible.
The fourth term is just an expansion of the third term.
The fifth term follows from an earlier formula.
The sixth term is straight up non-sense. We've switch from a tensor to a scalar; no indices are present, and they cannot be, as all the terms in the expression are scalar-valued.
The seventh term follows from that totally-no-really-irrelevant formula at the start of the post. Let's just assume it's true, even though it looks like the Heisenberg uncertainty principle and we haven't been doing any quantum mechanics up to this point.
So in conclusion, what do we have? We've approximately related through an inequality the Einstein tensor (with indices noted) to a scalar value. Since a tensor is not a number, let's interpret this the mainstream way (I don't see any alternative?): we aren't talking about the tensor itself, but the value of all the elements in it. What consequences does this have?
Immediately obvious should be the lack of any non-positive values (term seven is a positive constant scalar, and each element is larger than that). If we assume a zero cosmological constant, we can translate the Einstein tensor directly into the stress-energy tensor. (See
https://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form ) All its elements will be positive too. So we end up with a spacetime where even empty space has a minimum energy density. We end up with a constant momentum density and momentum flux, even in empty space. We end up with a space that is always exerting pressure. But how large are these contributions? Well, I cannot tell, as the units don't match up. Term seven evaluates to $$1.896\times 10^{-19}\dfrac{m^8}{s^6}$$, which is no energy density units I've ever heard of. We can only conclude that various constants are missing in the last term of the formula, even though the presence of $$c$$ suggests they were put back in, so we can't trust the value that's given for term seven as it is written.
Note that playing with the cosmological constant won't help you much, as by compensating for one term, you're making the others worse due to the minus sign(s) in the metric.
So what was the goal of all this? Let's look ahead to the first bit of the second post...
So... what is the whole point?
Yes, maybe it can describe some way to capture a particle in its own gravitational field, what is the signifiance? This is a question that will require a longer explanation.
Maybe, maybe not, but the question bares no relation to the derivation we just went through. No particle has been postulated; we're still working with empty spacetime. Additionally, there's nothing that suggests any kind of self-gravitation: all we've done is given the elements of the Einstein-tensor a required minimum value. So this statement doesn't help us here, and we must conclude that part two is about something else, and part one stands on its own.
And as it stands, we've got a huge pile of probable typo's, undefined terms, copy-paste mistakes, and a bunch of incorrect statements. We're missing huge steps in the derivation, and thus it looks like assumptions are made left and right. In the end, we end up with a formula whose derivation is so obscured by all these issues, we can only take it as a postulate or assumption. Especially note how the critical "uncertainty relation" was just assumed to be true in the first place. Since all this post was trying to accomplish was relating that uncertainty to something metric-y, the entire post might just as well have succinctly stated:
"Let's assume: $$G_{ij} = [\nabla_i,\nabla_j] \approx [\nabla_i,\nabla_j] \geq \frac{c^3}{G \hbar}$$."
As the post currently stands, this is fully equivalent to it.