# Does the amount of gravitational potential energy in the universe increase as it expands?

It seems to me that extra gravitational potential energy is created as the universe expands and the distance between massive objects such as galaxy clusters increases; this implies that energy is not conserved in the universe. Is that right?

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Related question physics.stackexchange.com/q/40983 –  twistor59 Jan 8 '13 at 9:06
I've awarded the bounty to @markovchain - best answer so far is "we don't know" which is perfectly honest if a bit unsatisfying! –  roblev Jan 27 '13 at 17:47
Thanks. I share your dissatisfaction. If it helps, here's a video about dark energy, which is quite fun to watch - youtube.com/watch?v=v6o2bUPdxV0&list=PLED25F943F8D6081C –  markovchain Jan 28 '13 at 1:08

That's actually a tricky question.

The short answer to the title question is yes, it does. But the answer to the follow up question about conservation is, it is still conserved.

In a much simpler universe, what hwlau said would be true - as the gravitational potential energy increases, the kinetic energy decreases. But we do know through the Hubble telescope that this is not the case. As the universe expands, the planets and dust and stars accelerate away from each other.

The answer is only hypothetical for now, which is dark energy. Here's the wikipedia page on it. The problem is, dark energy is very unknown. I'm not sure how this force will be able to counteract gravity. However, dark energy only manifests itself through gravity, and not through the other 3 fundamental forces, so there is definitely some relationship there. Dark energy can possibly negate the increased potential energy through some mechanism, but how it does that exactly is also unknown.

There's little I can give you other than the wikipedia link above, but notice when you open it, how the only explanations available are evidence arguing for dark energy's existence, what properties it should have, and other explanations for the expansion aside from dark energy (very interesting part). There is no explanation as to its mechanism, except that it causes things to accelerate against each other.

It's a good question. There is simply no good, accurate answer at the moment.

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hi, I see that dark energy may or may not exist, but I don't see how this is related to the question of whether the expanding universe increases the overall amount of gravitational potential energy. Your last sentence leaves a lot to the imagination! –  roblev Jan 24 '13 at 2:59
That is true, it does. But only because there really is little known about it. The simple answer is, yes, the amount of gravitational potential energy does increase as the universe expands. But as for the violation of conservation of energy in the universe, dark energy is a possible way to counteract this increase, so very likely, energy is conserved. I'll edit my answer to reflect this. :) –  markovchain Jan 24 '13 at 4:57

Here is an extract from this article on math.ucr.edu that explains why energy sometimes seems to be not "conserved".

I figured it would be better if I copy the entire paragraph and post it as answer (so that others can see) instead of providing the link as comment because you never know when the server might go down and the article would be lost.

Original by Michael Weiss and John Baez.

## Is Energy Conserved in General Relativity?

In special cases, yes. In general — it depends on what you mean by "energy", and what you mean by "conserved".

In flat spacetime (the backdrop for special relativity) you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity) this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any infinitesimal piece of spacetime. The integral form says the same for a finite-sized piece. (This may remind you of the "divergence" and "flux" forms of Gauss's law in electrostatics, or the equation of continuity in fluid dynamics. Hold on to that thought!)

An infinitesimal piece of spacetime "looks flat", while the effects of curvature become evident in a finite piece. (The same holds for curved surfaces in space, of course). GR relates curvature to gravity. Now, even in newtonian physics, you must include gravitational potential energy to get energy conservation. And GR introduces the new phenomenon of gravitational waves; perhaps these carry energy as well? Perhaps we need to include gravitational energy in some fashion, to arrive at a law of energy conservation for finite pieces of spacetime?

Casting about for a mathematical expression of these ideas, physicists came up with something called an energy pseudo-tensor. (In fact, several of 'em!) Now, GR takes pride in treating all coordinate systems equally. Mathematicians invented tensors precisely to meet this sort of demand — if a tensor equation holds in one coordinate system, it holds in all. Pseudo-tensors are not tensors (surprise!), and this alone raises eyebrows in some circles. In GR, one must always guard against mistaking artifacts of a particular coordinate system for real physical effects. (See the FAQ entry on black holes for some examples.)

These pseudo-tensors have some rather strange properties. If you choose the "wrong" coordinates, they are non-zero even in flat empty spacetime. By another choice of coordinates, they can be made zero at any chosen point, even in a spacetime full of gravitational radiation. For these reasons, most physicists who work in general relativity do not believe the pseudo-tensors give a good local definition of energy density, although their integrals are sometimes useful as a measure of total energy.

One other complaint about the pseudo-tensors deserves mention. Einstein argued that all energy has mass, and all mass acts gravitationally. Does "gravitational energy" itself act as a source of gravity? Now, the Einstein field equations are

$$G_{mu,nu} = 8\pi T_{mu,nu}$$

Here $G_{mu,nu}$ is the Einstein curvature tensor, which encodes information about the curvature of spacetime, and $T_{mu,nu}$ is the so-called stress-energy tensor, which we will meet again below. $T_{mu,nu}$ represents the energy due to matter and electromagnetic fields, but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity. On the other hand, the Einstein field equations are non-linear; this implies that gravitational waves interact with each other (unlike light waves in Maxwell's (linear) theory). So one can argue that "gravitational energy" IS a source of gravity. In certain special cases, energy conservation works out with fewer caveats. The two main examples are static spacetimes and asymptotically flat spacetimes.

## Article references:

• Clifford Will, The renaissance of general relativity, in The New Physics (ed. Paul Davies) gives a semi-technical discussion of the controversy over gravitational radiation.

• Wheeler, A Journey into Gravity and Spacetime. Wheeler's try at a "pop-science" treatment of GR. Chapters 6 and 7 are a tour-de-force: Wheeler tries for a non-technical explanation of Cartan's formulation of Einstein's field equation. It might be easier just to read MTW!)

• Taylor and Wheeler, Spacetime Physics.

• Goldstein, Classical Mechanics.

• Arnold, Mathematical Methods in Classical Mechanics.

• Misner, Thorne, and Wheeler (MTW), Gravitation, chapters 7, 20, and 25

• Wald, General Relativity, Appendix E. This has the Hamiltonian formalism and a bit about deparametrizing, and chapter 11 discusses energy in asymptotically flat spacetimes.

• H. A. Buchdahl, Seventeen Simple Lectures on General Relativity Theory Lecture 15 derives the energy-loss formula for the binary star, and criticizes the derivation.

• Sachs and Wu, General Relativity for Mathematicians, chapter 3.

• John Stewart, Advanced General Relativity. Chapter 3 (Asymptopia) shows just how careful one has to be in asymptotically flat spacetimes to recover energy conservation. Stewart also discusses the Bondi-Sachs mass, another contender for "energy".

• Damour, in 300 Years of Gravitation (ed. Hawking and Israel). Damour heads the "Paris group", which has been active in the theory of gravitational radiation.

• Penrose and Rindler, Spinors and Spacetime, vol II, chapter 9. The Bondi-Sachs mass generalized.

• J. David Brown and James York Jr., Quasilocal energy in general relativity, in Mathematical Aspects of Classical Field Theory.

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I think your reasoning is partially correct.

Yes, the Universe is expanding. But, the energy is conserved. How?

From the second law of thermodynamics, $$dG = dH-TdS$$ Taking our universe as a closed system. And, our universe does not exchange heat with the surroundings (well, there are no one to exchange).

So, $$dH=0$$

As our universe is expanding, $dS$ increases. Therefore, $-TdS$ decreases. Hence, $dG$ decreases. The energy for the universe to do a work (can be seen as potential energy) is lost in expansion. As the potential energy of the universe is decreasing, it can be seen as Gravitational Potential Energy + Internal Energy (of Masses) + Electro Magnetic Radiation. The mass is conserved (for large objects (principle of momentum during collisions)), Radiation remains constant. The major change occurs in Gravitational Potential Energy. And, it decreases as $dG>0$

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OK but your argument just states that an expanding universe is a closed system... I can just as well state that it is not a closed system and this thermodynamics argument doesn't apply. –  roblev Jan 24 '13 at 3:09
And going back to basics (Newtonian mechanics) if the masses are unchanged, the distance between them increases, and the gravitational constant is unchanged then the total GPE increases. You are arguing total GPE decreases, how do you explain this basic break with Newtonian mechanics? –  roblev Jan 24 '13 at 3:15
Our universe is not in thermodynamical equilibrium, as a matter of fact. –  Alexey Bobrick Jan 26 '13 at 11:55
@roblev As the distance between masses increases, the potential energy between them decreases. E=Gmm/r –  Fr34K Jan 28 '13 at 14:10
@Fr34K I think you are missing a minus sign, wikipedia shows the formula negated (see General formula) en.wikipedia.org/wiki/Potential_energy –  roblev Jan 29 '13 at 1:53
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The problem with this question is that gravitational potential energy between massive objects is a Newtonian concept but the question of energy conservation in cosmology can only be discussed properly in terms in general relativity.

The general answer is that energy is always conserved if you take into account the energy in the gravitational field as well as the matter and radiation fields. Dark energy can also be accounted for.

The full treatment is long and gets technical so I refer to my article at http://vixra.org/abs/1305.0034

The answer in the Physics FAQ above is incorrect and the points raised are treated in my article

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quoting from your paper: "This means that as the universe expands new energy is being created out of no where. " This is/should be unacceptable in physics. Read this pdf A self-similar model of the Universe unveils the nature of dark energy. DE is an artifact in the process of measuring. –  Helder Velez May 6 '13 at 10:55
@Helder Valez You misunderstood the structure of the article. The numbered texts written in bold are paraphrasing incorrect assertions made by other people and are refuted by the text in italics. –  Philip Gibbs May 6 '13 at 11:58
my comment was addressed to all that in the last decades use free lunches in Physics. Since you depart from false premises then your article is also wrong. You will understand if you take some of your time to read the paper I've linked. –  Helder Velez May 6 '13 at 20:13

Your reasoning is not right. Energy can still conserved even expansion occurs.

Considering the simplest case that there are only two massive particles moving away from each other. As the distance increase, the potential energy increase while the kinetic energy decrease. Hence, the speed is slowing down, but they are still moving away from each other (expanding).

For large among of particles, you can also think of some kind of explosion driving them away from each other. The expansion is slowing down as time pass, but they are still expanding. There is no need for "creation of extra potential energy". To claim that the energy is not conserved, you need more evidences such as the speed distribution, etc.

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OK but why would the kinetic energy or speed of the two massive particles decrease just because the universe expanded, increasing the distance between them? The speed (measured in meters per second) shouldn't change because of expansion. –  roblev Jan 8 '13 at 13:07
The massive objects are still acting on each other gravitationally. –  Deiwin Jan 9 '13 at 11:14
I.e. the potential energy needs to come from somewhere and it comes out of the kinetic energy of those 2 objects. (I know this is a circular statement, as it relies on the conversation of energy, but it's simply a way to look at what's happening) –  Deiwin Jan 9 '13 at 11:19
yes the massive objects act on each other gravitationally, but somehow the universe expanding needs to "boost" the gravitational interaction to make the objects slow down more than if the universe was not expanding. Doesn't seem obvious to me. –  roblev Jan 9 '13 at 12:53
As bodies move away from each other due to the expansion of space, they are accelerating apart, not decelerating (that is, the more they move apart, the faster they move apart). Look up Hubble's law. –  Kyle Jan 23 '13 at 19:03