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If total energy is conserved just transformed and never newly created, is there a sum of all energies that is constant? Why is it probably not that easy?

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Here is a related question that might be helpful physics.stackexchange.com/questions/2838/… –  Hal Swyers Oct 17 '12 at 1:32
The total energy of the universe is not well defined, so we can't even discuss whether it's constant. physicsforums.com/showthread.php?t=506985 –  Ben Crowell Oct 17 '12 at 5:36
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3 Answers

up vote 8 down vote accepted

No. The universe is dominated by dark energy, which is consistent with a cosmological constant $\Lambda$. In other words, as the universe expands, the energy density stays roughly the same. So the (energy density)*volume is growing exponentially at late times.

Although the total energy is not well defined (as the volume of the universe may be infinite), the fractional rate of growth is certainly nonzero.

You might wonder how the total energy can grow without violating energy conservation. The answer is that in general relativity, we just need $\boldsymbol{\nabla} \cdot \boldsymbol{T} = 0$, so a cosmological constant is perfectly consistent as $\boldsymbol{\nabla} \cdot \Lambda \boldsymbol{g} = 0$

For a nice explanation by Sean Carroll, see http://blogs.discovermagazine.com/cosmicvariance/2010/02/22/energy-is-not-conserved/

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The total energy isn't just undefined because of the possibility that the universe is infinite. It's undefined for the reasons given in juanrga's answer. –  Ben Crowell May 6 '13 at 21:06
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Your question is tagged as general-relativity and cosmology, and as textbooks remark (e.g. Peebles [1]) "there is not a general global energy conservation law in general relativity theory.

Therefore: ”The conclusion, whether we like it or not, is obvious: energy in the universe is not conserved” [2].

[1] Peebles P. J. E., 1993, Principles of Physical Cosmology (Princeton Univ. Press).

[2] Harrison E., 1981, Cosmology ( Cambridge University Press)

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The only thing that prevents us defining a total conserved energy for the entire universe is that if the universe is infinite then the total energy could be infinite or indeterminate.

The statements that say energy is not conserved in general relativity are wrong, irrespective of who says them. You can define energy over any finite volume of space and you can define the flux of energy over the boundary surrounding the volume. The rate at which energy decreases in the volume is equal to the flux of energy across the boundary. This is the the most general way to express energy conservation globally.

All statements to the contrary can be refuted and to avoid arguing around in circles I have done that at length in my write-up at http://vixra.org/abs/1305.0034

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