# If universe expands without limit but dark/vacuum energy density remains the same, then as space time expands, more of that energy is created?

And if it is the case, does it mean that as universe expansion has no limit the energy that can be created is infinite and therefore there is infinite potential energy?

• – user4552
Mar 16, 2019 at 21:21

General relativity has local conservation of energy but no global conservation of energy. (That is, there is no global, scalar, conserved measure of energy.) GR doesn't have any way of defining the total energy of the universe, or the total energy of any region of space that is large compared to the scale set by the curvature of spacetime.

The simplest way of incorporating dark energy into a cosmological model is by using the cosmological constant $$\Lambda$$. Locally, we have conservation of energy if and only if $$\Lambda$$ is constant, but $$\Lambda$$ can be nonzero.

So basically the answer to your question is that there is no nontechnical answer, but the technical answer is that there is no violation of any principle that makes sense to state in GR.

Yes, more energy is created. Energy is not nominally conserved in General Relativity. The cosmological solution that includes the dark energy and all we know has the dark energy total energy always increasing proportionally to radius cube, i.e., the volume).

Because it is repulsive (the diagonal of the dark energy contribution of the Einstein tensor is ($\rho$, -$\rho$, -$\rho$, -$\rho$), where $\rho$ = -p = $\Lambda$, with the latter being the cosmological constant, $\rho$ the density and p the pressure of the equivalent perfect fluid model of the cosmological constant. It is repulsive gravity, and the universe's acceleration keeps increasing, with the universe radius expanding exponentially when dark energy dominates.

The argument that energy is conserved is somewhat specious. In General Relativity is comes from the Einstein tensor = mass energy tensor + dark energy tensor, or label it as G = T + DE. And then rewriting it as T + DE - G = 0. Then they interpret this as meaning the total energy, including the mass energy + the dark energy - the gravitational energy (or plus, it depends on how one counts) is zero. So as dark energy grows it goes into a higher gravitational energy (ignoring mass and normal particles and radiation energy, but it also can be counted).

The argument for zero energy winds up being of some speculative and multiverse value ('create a universe out of nothing'), but it has not helped in determining an entity that one can treat physically and does any more than Einstein's equations do. As Motl say in his blog, having a conservation law for an entity that is zero just says that zero doesn't change, and it has not been found useful in defining for instance a Hamiltonian to quantize gravity. See Motl's blog on this at http://motls.blogspot.com/2010/08/why-and-how-energy-is-not-conserved-in.html

• Thanks! So one last thing, if universe expands without limit and more dark vacuum and gravitational energy are created, then can infinite of these energies be created and therefore there's infinite potential energy? Nov 19, 2016 at 10:23
• Gravitational energy is not a covariant entity and potential energy does not exist in general relativistic gravity. But it does look like the currently understood model of the universe is that it is likely going to continue expanding and more total dark energy will be created. Nov 20, 2016 at 2:19
• I see...and would i be correct if i say then that energy is (potentially) infinite (if infinite of it can be created as universe expansion has not limit)? Nov 20, 2016 at 10:18
• I guess, but take it with a grain of salt. We still are not sure what dark energy is, some people think of it as the energy of vacuum, and if space becomes infinite they'd agree. But I'd take the whole question of dark energy, other than the fact that it keeps of increasing with the space co-moving volume, that it has negative pressure somehow (some fields can exhibit that, we haven't found the field for it), and more measurements to be done on it, as an open research question. he infinite spacetime is not certain, it is within the error bounds that it could still be very large but finite. Nov 21, 2016 at 6:19
• The material about T + DE - G = 0 seems just wrong to me. These are all energy densities, not energies. There is nothing speculative or controversial about the fact that in GR, energy-momentum is locally but not globally conserved. There is no reason to invoke any idea about zero total energy of the universe, which is not an idea that can even be clearly formulated in GR.
– user4552
Mar 16, 2019 at 21:13

I wrote a post on stack exchange on the expansion of the universe and energy, so I will not repeat the derivation. In this you will see that this depends on the total energy being zero. In a more general relativistic setting from the stress-energy tensor the mass-energy is $\rho~+~wp$ for $\rho$ the vacuum energy density and $p$ the pressure. For $w~=~-1$ this is zero and the pressure term is negative. As a result positive vacuum energy density can generate a repulsive force or negative pressure outwards.

The case with $w~=~-1$ results in the generation of no net energy. It is also the case that connects pretty well with a Newtonian approach. There are some subtle reasons why Newtonian mechanics works this well. Further, when the total energy in the Newtonian case is zero it corresponds to the general relativistic $w~=~-1$ case.