# If empty space has energy, and space is expanding, is this energy equally distributed as space expands?

The cosmological constant (dark energy) is often described in terms of empty space having a non 0 energy value and this energy being the source of the accelerated expansion of the universe. If space is expanding, what does this imply for energy conservation? Is this energy conserved by being more thinly distributed as space expands? Does this question make any sense or am I just crazy?

The cosmological constant is a constant energy density per unit volume of space, so as the universe expands this does indeed create energy as it creates new space. In this sense conservation of energy is violated.

Actually this is less surprising than you might think. Conservation of energy is linked to a symmery called time shift symmetry by Noether's theorem. However this symmetry is violated by the expanding universe so there's no reason why energy should be conserved (though you should note that the symmetry is strongly broken only on vast scales and conservation of energy is an excellent approximation locally).

But, but, but ...

There is a long and sometimes heated debate about whether energy really is or isn't conserved in an expanding universe. See for example this article by a member of this site. Basically the argument comes down to how you define the total energy - in General Relativity this is not a trivial question.

• "though you should note that the symmetry is strongly broken only on vast scales and conservation of energy still applies locally". That's simply impossible (also mathematically). – bright magus Jul 12 '14 at 10:41
• But, but, but: [ insert rant about Noether's 2nd theorem, general covariance and the gravitational stress-energy pseudo-tensor ] – Christoph Jul 12 '14 at 13:12
• @Christoph: indeed :-) – John Rennie Jul 12 '14 at 13:30
• Yeah, I fully agree - the more stuff there is the less it's obvious there is nothing behind ... (except there is more energy although there isn't) – bright magus Jul 12 '14 at 16:37

Actually, energy is often not Conserved in general Relativity. For are more in deep explanation see:

http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

But just notice that Dark Energy might not necessarily end up being the cosmological constant, but a new force field, so its behavior might differ from that of an actual cosmological constant (which would increase total energy as you suggest in your question)

Dark energy due to a cosmological constant does not get diluted by metric expansion of space. However, this does not violate energy conservation as the increase in energy will be cancelled by gravitational potential energy. The problem with general relativity is (some would say arguably) not energy conservation, but energy localization:

In a rotating frame of reference, energy conservation only holds if you include the contribution of the centrifugal potential. Analogously, in general relativity, energy conservation only holds if you include the contribution of the gravitational potential. In both cases, you cannot associate a meaningful energy density with these contributions. General relativity just comes with the additional complication that you cannot transform away gravitational energy by going to a globally inertial frame of reference.

Note that there are ways to separate gravity from inertia if you go beyond general relativity (eg the teleparallel approach or bimetric theories) and get a gravitational energy density this way, but energy conservation will only hold if you include inertial effects.

• "However, this does not violate energy conservation as the increase in energy will be cancelled by gravitational potential energy." I do not quite understand. Could you explain? – bright magus Jul 12 '14 at 11:03
• @brightmagus: Noether's second theorem applies to general relativity and yields a 'trivial' law of energy conservation, where the contribution due to the cosmological constant gets cancelled by the negative potential energy of the gravitational field; this law of conservation of energy can be expressed in terms of a gravitational stress-energy pseudo-tensor, but there's no tensorial expression for gravitational stress-energy as general relativity does not differentiate between gravity and inertia – Christoph Jul 12 '14 at 11:14
• You actually restated the original quote with this: "where the contribution due to the cosmological constant gets canceled by the negative potential energy of the gravitational field", again apparently not explaining what you mean. OK, I will make a wild guess. Is this to say that because universe is larger (and infinity is further away ...) then the force of gravity of a given body must extend further away, which means the potential energy near the source (assumed to be negative) must change? Or else ... – bright magus Jul 12 '14 at 11:28
• @brightmagus, it's hard to argue with forces in GR: if you think of dark energy due to a cc as some fluid, you have to wonder why gravitational attraction acts repulsively if you choose the right equation of state (negative pressure), without any gradients being present; I'm unsure what kind of explanation you're looking for beyond "that's what Noether's theorem tells us"; essentially, energy conservation boils down to the first Friedmann equation $\rho+\rho_\Lambda-3\dot a^2/8\pi Ga^2=0$, which many physicists are uncomfortable with, claiming it's not a proper conservation law – Christoph Jul 12 '14 at 12:07
• "you have to wonder why gravitational attraction acts repulsively if you choose the right equation of state (negative pressure), without any gradients being present" I don't have to - it's just bad (unphysical) maths. Anyway, I was hoping for a physical explanation, and not just mathematics. But ... forget it ... – bright magus Jul 12 '14 at 12:15