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As long as I tried to find out what the total positive amount of energy is in the universe, the only answer I got was 'the total energy in the universe zero', which I do appreciate, but that's not what I am looking for. My question is: what is the total approximated amount of positive energy, in all forms - mass energy or kinetic energy - in the universe, assuming it's not infinite, or, at least, in the observable universe?

I need this information because I want to work out the highest number it is theoretically possible for a machine exploiting approximately all the energy in the universe to count up to until it faces the hypothetical death of the universe (false vacuum or, to take the upper bound, the heat death), using the Margolus–Levitin theorem, the approximate value of 10^100 years till the universe 'dies' and the total positive energy in the universe.

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I will answer this as a question on the different possible ways you could answer what is the universe's usable energy.

Well, the energy of the universe is all positive energy, if you consider the energy in the stress energy tensor and the dark energy. The dark energy has negative pressure, but positive energy. I say below something about what it is if you include gravitational energy also.

Note: you can calculate numbers and get to answers below. Whether that makes sense as the source of energy for your theorem I won't speculate.

So if you want the total energy now (in the cosmological comoving time now) just calculate the critical energy density. We're pretty close to that, if you count the dark density and both normal and dark matter. And then multiply by the volume of the observable universe. Since you're probably going to be comoving that is your available total energy.

The numbers are easy enough. Critical density is about $10^{-26}$ kg/$m^3$. This includes the mass equiv. The radius is 46 Gly, and volume you can compute to be about $10^{80}$ $m^3$. The total mass is then about $10^{54}$ kg. Multiple by $c^2$ and you get about $10^{70}$ joules. See some of the numbers from which you can calculate and verify at https://en.wikipedia.org/wiki/Observable_universe

However, the energy is not conserved. Over time the observable universe grows, and the density of matter energy decreases, except for the dark energy density which remains the same. That's 68% of the total density, or in order of magnitude the same as the total energy density. That density is about the same, and the universe becomes mostly a deSitter universe, expanding exponentially, which keeps growing to infinity. So total energy (which is all then dark energy) is infinite. In a comoving frame (remember the energy depends on the coordinate frame, if you change it it changes since the universe has no timelike symmetry). Also, although the observable universe at any one time is finite, the total universe is infinite, and so it is a not a bounded compact spacetime nor an asymptotically (I space or conformally) flat spacetime, and total energy is not conserved like it would be in an asymptotically flat spacetime. So in this view it just grows without limit. You can calculate it at $10^{100}$ years, it's about $10^{340}$ joules, order of magnitude assuming linear growth [I multiplied the $10^{70}$ joules now (roughly) by a factor of $10^{90}$ cubed, for a quick ROM]. This is a lower limit since the growth is exponential, so it may be closer to $10^{10^{340}}$ joules.

That would be one story. But dark energy is something we still don't understand. Another story would say most thinking leads to it probably being some kind of vacuum energy. Over time it is probable that that vacuum energy, could decay (say from false ground state), create matter energy, and then maybe form some other universe or bubbles of them. Whatever happens after our unknown dark energy starts decaying it really is impossible to predict now. This is highly speculative.

If you meant energy w/o the dark energy your number is above, as of now. Over time it goes to 0.

Either way, good luck with your Margolus-Levitin theorem. I know nothing of it. But the energy of the universe, depending on which energies you think you can use, are easily calculated as above. The less well defined in general relativity, the gravitational energy, you ignore unless you want to get to a total of 0, if you take the Einstein and the stress energy tensors. Or depending on how things fall out, you can get anywhere from 0 to about $10^{10^{340}}$ joules.

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  • $\begingroup$ Thank you for the numbers and the explanation; however, I got a question: since, as you'd pointed out, the dark energy should remain constant, and you assumed the number 10^70 joules would grow, does that mean it does not include dark energy? $\endgroup$ – Max May 21 '17 at 11:41
  • $\begingroup$ @Max. No. Careful with the definitions, that's the thing. The dark energy DENSITY is constant, BUT as more space is created when the universe expands the TOTAL ENERGY increases. DARK ENERGY density is joules per cube meter. Just a density. It is the same in each cubic meter. But as the universe expands there are more cubic meters, so the TOTAL DARKNESS ENERGY increases proportionally to the volume. And the volume increses as the cube of the radius (what is normally stated as the size of the universe, though sometimes people use the diameter, twice the radius). Sorry, easy to confuse the two. $\endgroup$ – Bob Bee May 21 '17 at 19:11
  • $\begingroup$ Sorry, I did miss the word 'density', but that means, then, that 10^340 is the ultimate lower bound; thanks! $\endgroup$ – Max May 22 '17 at 17:12
  • $\begingroup$ Ten to the ten to the 340 power. Big number $\endgroup$ – Bob Bee May 22 '17 at 17:31

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