Is the Total Energy of the Universe Really Zero?

[thinkandmull]

I saw a video by a physics writer who said that the beginning of the universe was like digging a hole: the hole and the pile of dirt cancel each other out. This of course means that the ground, so to speak, before the digging was zero. But where is the "hole" once the universe began? I asked some people about this and they quoted Stephen Hawking's A Brief History of Time : "There are something like ten million million million million million million million million million million million million million million (1 with eighty zeroes after it) particles in the region of the universe that we can observe. Where did they all come from? The answer is that, in quantum theory, particles can be created out of energy in the form of particle/antiparticle pairs. But that just raises the question of where the energy came from. The answer is that the total energy of the universe is exactly zero. The matter in the universe is made out of positive energy. However, the matter is all attracting itself by gravity. Two pieces of matter that are close to each other have less energy than the same two pieces a long way apart, because you have to expend energy to separate them against the gravitational force that is pulling them together. Thus, in a sense, the gravitational field has negative energy. In the case of a universe that is approximately uniform in space, one can show that this negative gravitational energy exactly cancels the positive energy represented by the matter. So the total energy of the universe is zero."

When he says the matter of the universe of made of positive energy, is he referring to both the particles and anti-particles he mentioned? Are there two things cancelling the energy of matter: antiparticles AND gravity? How does this all fit together? THANKS!

 

[PeterDonis]

When he says the matter of the universe of made of positive energy, is he referring to both the particles and anti-particles he mentioned?

Yes.

Are there two things cancelling the energy of matter: antiparticles AND gravity?

No. In this viewpoint, negative gravitational potential energy cancels the positive energy of both particles and antiparticles.

You should also be aware that not all physicists share the viewpoint you describe. The view that "negative gravitational potential energy cancels out positive energy in particles and antiparticles" is not the actual physical model, it's an interpretation of what the physical model says. For a different interpretation, see this article by Sean Carroll:

http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[thinkandmull]

So in Hawking's model, how could their even be motion in the world?

 

[PeterDonis]

So in Hawking's model, how could their even be motion in the world?

I don't understand. Why would there not be?

 

[trendal]

Does the expansion of space mean that everything is gaining energy?

When you take two objects and separate them by a distance, the gravitational energy between the two increases. Does the expansion of space result in the same thing?

 

[PeterDonis]

When you take two objects and separate them by a distance, the gravitational energy between the two increases.

Yes, but that energy comes from the work you did in separating the objects. There's no violation of energy conservation.

Does the expansion of space result in the same thing?

Not really. This is one reason why many physicists (and others, like me) don't like the interpretation you describe in your OP; it leads to incorrect inferences like the one you're making here. No work is being done on the galaxies in the universe to "make" them expand. (At least, not if we leave out dark energy; I'll come to that in a bit.)

A better analogy for the expansion of the universe (again leaving out dark energy) following the Big Bang would be a rock that was thrown upward from, say, the surface of the Earth. (We ignore air resistance here--make it the Moon if you like.) The rock starts out with a large initial velocity upward; as it rises, its upward velocity (and hence its kinetic energy) decreases, but its gravitational potential energy (relative to the Earth) increases. The two changes exactly balance, so there is no net change in total energy. And, correspondingly, no work is being done on the rock (after the initial impulse of being thrown).

The Big Bang, in this analogy, corresponds to the initial throwing upward of the rock. In the simple model you describe in your OP, when the initial matter was "created out of nothing" by "borrowing" energy from the gravitational field, it was created, not "at rest", but expanding rapidly--i.e., the total (negative) gravitational potential energy "borrowed" was not just the rest energy of the matter (and antimatter), but its kinetic energy (due to its expanding rapidly) as well.

As I noted above, this model does not include dark energy. For the last few billion years, the expansion of our universe has been accelerating. If only ordinary matter and energy were present, it would be decelerating (like the rock's upward motion in the analogy above); that's why cosmologists talk about dark energy as something different from ordinary matter and energy (since it produces an effect that ordinary matter and energy can't produce). The simple model you describe in your OP cannot account for the effect of dark energy, since that effect amounts to a constant energy density everywhere in the universe, which means expansion increases the total energy. That's another reason why many people don't like the interpretation you describe in your OP.

Of course, the effect of dark energy that I just described sounds exactly like what you were thinking couldn't happen--energy continually getting added to the universe. However, that perception is an artifact of the model you're using. In a different model (like the one Carroll describes in the article I linked to), energy is simply not conserved in this scenario; instead, it changes according to a precise rule (which is ultimately derived from the Einstein Field Equation).

 

[ohwilleke]

As imperfect as the alternative may be, I really dislike Sean Carroll's alternative. In my experience, as a matter of pedagogy, people get very hung up on statements that seem to create loopholes, like "GR doesn't conserve energy". In contrast, if you at least have a book keeping method that imposes an "artificial" limit on available energy, it is much easier to digest because it doesn't have what superficially looks like an obvious and serious problem.

In the same vein, somebody who isn't a mechanic may not notice that there is one belt too few in a car engine for a conventional design, but everybody is going to notice if your tires are flat.

 

[rootone]

'It all adds up to zero' seems to me a better proposition than 'it adds up to some other arbitrary number'.
I think it's personal taste though.

 

[Chronos]

Given GR does not offer a global definition of energy, any effort to claim it is, or is not, conserved under GR is predestined to be logically unsound.

 

[PeterDonis]

GR does not offer a global definition of energy

The issue is more complicated than that. The model described in the OP (which, as I've pointed out, only works in the absence of dark energy) makes use of the fact that, when expressed in appropriate coordinates, the dynamics of a matter-dominated FRW universe are identical to the dynamics of an object thrown upward and then moving freely in a Newtonian gravitational field. Closed universe = object thrown upward at less than escape velocity; critical density universe = object thrown upward at exactly escape velocity; open universe = object thrown upward at greater than escape velocity. So, mathematically, it's perfectly possible to use this correspondence to define a negative "gravitational potential energy" for the universe such that the total energy--kinetic energy of matter + gravitational potential energy--is a constant of the motion, which can be adjusted to be zero by appropriate normalization.

The question is how this mathematical device is to be physically interpreted. One school of thought (the one described in the OP) says that interpreting the mathematical "total energy" I just described as the physical "total energy of the universe" is perfectly sound. The other school of thought (the one I prefer, and which is described in Carroll's article) says that the mathematical "total energy" doesn't tell us anything meaningful physically (particularly when we recognize that it doesn't work in the presence of dark energy); the physical meaning is in the Einstein Field Equation and the local conservation law it satisfies (covariant divergence = 0). Since both schools of thought agree on all the physical predictions, the dispute isn't really about physics, it's about terminology and "philosophy".

 

[thinkandmull]

The reason I wondered how there could be motion in Hawking's version of a "zero energy universe" is that I am imagining it as if the energies are acting against each other and cancelling each other out. If two equal forces are pressing against each other, there is no motion. I must be misunderstanding what it means to say the "gravitational energy of two objects increases when you separate them". Any help?

 

[PeterDonis]

The reason I wondered how there could be motion in Hawking's version of a "zero energy universe" is that I am imagining it as if the energies are acting against each other and cancelling each other out.

They aren't. The "energies" are just a bookkeeping device; they don't mean anything is "acting against each other". That's a big reason why this model can be misleading, and hence why people like me don't like it.

I must be misunderstanding what it means to say the "gravitational energy of two objects increases when you separate them".

It means that you can define a "gravitational potential energy" which increases as two objects separate. Again, it's a bookkeeping device; by itself it tells you nothing about what's happening physically.

For example, consider the rock being thrown upward that I discussed earlier. Suppose you are standing at the top of a tower, and I am at the bottom. I throw a rock up to you with just enough velocity that it comes to rest just as it reaches you. Then I put a second identical rock on a dumbwaiter and use a rope and pulley to lift it up to you. When the two rocks reach you, both of them have increased their gravitational potential energy by exactly the same amount (relative to the Earth), and both of their kinetic energies are the same (they're at rest relative to you). So just from knowing the energies involved, there's no way to know how the rocks got there.

It's true that the two rocks do have something in common; I had to impart a force to both of them to get them up to you. But now imagine that there is a tunnel at the foot of the tower that goes all the way through the Earth, and a third rock that has come from the other side of the Earth and is free-falling through the tunnel with just enough upward velocity that it comes to rest at the top of the tower, where you are, when the other two rocks reach you. Now you have three rocks, all with the same energy, and no way of telling, just from the rocks, how they got there; and one of them didn't have to have any force imparted to it at all, it was free-falling the whole time.

 

[Mordred]

That's a good description, I like it.

 

[thinkandmull]

Let's take the example of Earth and Saturn, both circling the sun. Stephen Hawking's principle would mean that the Earth and the sun together have less energy than Saturn and the sun together because the latter two are further apart. How is gravity the cause the this? I want to understand where Hawking was going with this..

 

[PeterDonis]

Stephen Hawking's principle would mean that the Earth and the sun together have less energy than Saturn and the sun together because the latter two are further apart.

No, that's not what Hawking's principle says. Gravitational potential energy is not the only kind of energy. Saturn has more mass than the Earth, so even if Saturn were at the same distance from the Sun as the Earth, Saturn + Sun would have more total energy than Earth + Sun. For a fair comparison, you would have to take Sun + Earth, vs. Sun + a planet exactly identical to Earth in all respects except that it is in Saturn's orbit.

Also, objects in orbit are moving, so they have kinetic energy as well as gravitational potential energy and energy due to their rest mass. A planet identical to Earth except that it was in Saturn's orbit would be moving more slowly than Earth, so it would have less kinetic energy; that would offset part of the gain in gravitational potential energy (for ideal Keplerian orbits, half of it, when you run the numbers) due to increasing distance from the Sun.

Basically, if you try to focus solely on gravitational potential energy, you're going to miss things and get an incorrect understanding. You need to look at the whole picture. Also, as I said before, "gravitational potential energy" is really a bookkeeping device in itself. To understand what's going on, you need to look at the actual dynamics: how things are moving and how they got that way.

I want to understand where Hawking was going with this..

Hawking's idea that you described in the OP is a speculation about how quantum gravity might deal with the beginning of the universe. It makes use of the idea that negative "gravitational potential energy" can cancel out positive energy contained in particles and their motions, which can be useful in other contexts as well, but the overall idea is still a speculation.

 

[thinkandmull]

How does "gravitational potential energy", which gives objects further away more energy, become "negative"?

 

[PeterDonis]

How does "gravitational potential energy", which gives objects further away more energy, become "negative"?

It's negative relative to the potential energy of two objects at rest at infinite separation. The "zero" point of gravitational potential energy is arbitrary; you can put it wherever you want for the problem you're trying to solve. If you're trying to lift objects from the surface of the Earth, the potential energy at the surface of the Earth is a convenient zero point, so objects above the surface have positive potential energy. When you're studying bound orbits, such as in the solar system, putting the zero point at rest at infinite separation is more useful, because it means the potential energy of an object at any finite radius is negative. That makes it easier to calculate whether a given object is in a bound orbit or not.