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I'm curious as to the equations necessary for finding a total energy of 0 (or, I suppose, the energy density of empty space due to quantum fluctuations) in a flat Friedmann universe such as ours.

The FLRW metric is as follows:
$ \mathrm{d}\mathbf{\Sigma}^2 = \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\mathbf{\Omega}^2 $

And solving the formula the formula: $$\mathrm{d}\mathbf{\Sigma}^2 = \mathrm{d}r^2 + S_k(r)^2 \, \mathrm{d}\mathbf{\Omega}^2$$ For a $S_k(r)$ where $k=0$ gives us $r$. Adding in the scale factor, we get a simplified equation: $$d^2(t) = a^2(t)(x^2 + y^2 + z^2)$$ But this has nothing to do with the energy of the system, only the curverature and the distance between points at a given time $t$. I'd like to understand why a flat universe has a cumulative energy of zero in a mathematical equation. I understand that the gravitational potential energy is the counteractive energy to matter, but I'm looking for a mathematical equation that shows this and I'm wondering if it is truly absolute zero, since the $\text{Uncertainty Principle}$ tells us zero is impossible for an oscillating system: $$ \Delta E\cdot \Delta t \ge {\hbar\over 2} $$

Qmechanic
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The total energy of the universe is a vexed issue since different commentators have different views about what the concept means. See the question Total energy of the Universe for a sampling of the various viewpoints. If you Google for zero energy universe you'll find several papers purporting to show that the total energy is zero. However since their results depend on the assumptions they make at the start treat their conclusions with care - because a paper appears on the Arxiv does not necessarily make it authoritative.

You might want to read Phil Gibbs' views on the subject, though note that another prominent member of this site, Luboš Motl, heartily disagrees.

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John Rennie
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