# Account for radiation energy in the statement "total energy of universe is zero"

I've read in Hawking's book The Theory of Everything that total energy of the Universe is zero. He explained this by saying that if there are two bodies separated by a distance and having some mass, they possess potential energy and the same amount of negative energy with the gravitational field. Thus he concluded that total energy of the Universe is zero. But here I've a question: how do we account for energy which is in the form of radiation?

The total energy of the universe consists of the mass energy of all the matter (both normal and dark), the mass-energy of the radiation and of the dark energy plus the gravitational potential energy of the universe (which is negative). The hypotheses is that this sums to zero. As far as I know, there is no experimental evidence for this.

But there is good evidence for the size of the various mass-energy terms. The best current estimates are that the mass-energy of the universe is:

• 73% -- Dark energy
• 23% -- Dark matter
• 3.6% -- Interstellar gas
• 0.4% -- Stars (includes black holes)
• v. small -- Relativistic neutrinos

This balance changes. As the universe expands, the density of matter (dark and normal) varies as the inverse 3rd power of the size of the universe. (Matter stays the same and the volume expands as the cube of the radius.)

But the density of dark energy stays constant, since it is a property of empty space.

The density of radiation is proportional to the inverse 4th power of the radius, because not only does the increased volume dilute the radiation like it dilutes the matter, but also the radiation gets red-shifted, contributing an extra power of the radius.

(Incidentally, that means that the fraction of the total mass-energy of the universe in radiation decreases with time, and is estimated presently to be 1000:1 matter:radiation.)

So radiation seems simply to be too small to matter.

• Good answer. The different scalings with size of the Universe means that before the Universe was ~50,000 years old, radiation dominated the evolution, then matter, and finally after ~10 billion years dark energy.
– pela
May 29 '18 at 12:42
• Yet this implies that at the universe's origin, the energy equation was dominated by radiation. This is quite important, because the issue is, how can a universe appear out of nothing if energy is required? Hawking's goal was to allow us a way where we need zero net energy to have a universe, but that certainly means we must address the radiation. It's mostly what we must address, in fact. To get light out of nothing, Hawking was claiming you can regard the energy for the light as coming from gravitational potential energy-- of the light. May 29 '18 at 14:33
• @ Ken G : I know of no situation wherein conservation of energy does not apply. Nor does anybody else. May 29 '18 at 15:45
• @John Duffield: Conservation of energy is conserved perfectly only in systems which are perfectly time symmetric. (Noether's Theorem.) This is true to a very high accuracy locally (for some pretty big locales, BTW), but not true globally since spacetime is expanding. (Some people try to make energy be conserved globally by adding in terms for universal gravitational energy, but these are a bit iffy.) No one can say presently anything about either symmetries or energy conservation "before" the universe existed. May 29 '18 at 16:11
• Sure: Where does the energy of redshifted light from distant galaxies go? Where does dark energy come from (remember, that as the universe expands dark energy, a property of empty space, must be created.) May 29 '18 at 18:10

I've read in Hawking's book The Theory of Everything that total energy of the Universe is zero.

It isn't true.

He explained this by saying that if there are two bodies separated by a distance and having some mass, they possess potential energy and the same amount of negative energy with the gravitational field.

That isn't true either. If you let these two bodies fall towards each other, their gravitational potential energy is converted into kinetic energy. Then when the bodies collide some of this kinetic energy gets radiated away, and you're left with a mass deficit. But note that it's only a mass deficit. The two bodies don't disappear and end up as some zero-energy nothing. Besides, gravitational field energy isn't negative, it's positive. That's why Einstein said "the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy".

The point to note is that when you throw a 1kg brick up in the air you do work on it. You give it kinetic energy*. Gravity then converts this kinetic energy into gravitational potential energy, and the brick slows down. Note that this gravitational potential energy is in the brick, not in the gravitational field, or anywhere else. You did work on the brick, and when the brick is at the top of its arc, its mass is greater. Hence when the brick falls to Earth you end up with the mass deficit. Hence when you throw the brick upwards at 11.7 km/s, it's got escape velocity. It leaves the system, taking the original 1kg worth of mass-energy away, along with the mass-equivalence of the 11.7 km/s of kinetic energy.

Thus he concluded that total energy of the Universe is zero.

It's popscience rubbish I'm afraid. Hawking was always coming out with things like that, and few would criticise him because of the wheelchair. Those that did tended not to get any publicity.

But here I've a question: how do we account for energy which is in the form of radiation?

By measuring the CMBR. There isn't much radiation compared to other stuff. The universe review dark energy article says radiation comprises only about 0.005% of the mass-energy of the universe.

* Whilst conservation of p=mv momentum means there was an effect on the Earth, it's very slight. The KE=½mv² kinetic energy is not shared equally because the Earth didn't move in any detectable fashion.

• Potential energy is a highly technical topic in GR, and there are ways to make it formal which could be interpreted in the way Hawking does. It's not "popscience rubbish," but its significance is often doubted by experts. In contrast, your own position seems internally inconsistent, as you at once claim it is wrong that the energy is zero, but it is right that the energy is conserved! A better understanding may come from a paper on this at prespacetime.com/index.php/pst/article/view/81, though I am not expert enough to be able to tell if it suffices to put the question to bed. May 30 '18 at 14:17
• @Ken G : it isn't a highly technical topic when you throw a brick up in the air. Energy is conserved. Please note that I didn't say the energy of the universe was ever zero. I've read Philip Gibbs' work on conservation of energy in GR before, it's on viXra: vixra.org/abs/1008.0051. Whilst I concur that energy is conserved, I dislike his reasoning. See the Tamara Davis article in Scientific American: Is the Universe leaking energy?. May 30 '18 at 15:59
• To say "energy is conserved" is nothing more than an exercise in bookkeeping. There's not a "thing" called energy, there's a mathematical effort to create a quantity that is conserved. We succeed when throwing a brick, we have a tougher time in GR but under some conditions we can, with enough effort, succeed there too. It's on us, we do it. Likewise, that's what Hawking is doing-- it was on him to try to find a way to make the energy zero. To say "the energy isn't zero" is to fail to understand the process of inventing energy. May 30 '18 at 20:32
• @Ken G : there is a thing called energy. That's why the mass of a body is a measure of its energy content. Energy can be neither created nor destroyed. There are no perpetual motion machines. We have no evidence of any way by which energy can be invented, so understanding the process is extremely questionable. May 31 '18 at 11:44
• If you have an equation of motion that is integrable to produce a constant, you have a useful bookkeeping result. To say it's a "thing" is just a kind of metaphysical attitude, which is fine, but one must not make the error of thinking that "thing" has now transcended the mathematics that motivated it. When you change the mathematics, like using General Relativity instead of Newton's laws, you should expect to need a new "thing"-- and it's up to you to decide what that new "thing" is. That's what Hawking did when he said the energy was zero. Jun 1 '18 at 14:26

It must be the same situation as if you had bits of mass flying off the surface of the two masses in the simple model. It would not take much kinetic energy, at the huge $mc^2$ scale, to get those bits to fly off to infinity in a Newtonian potential energy, but you'd have a huge $mc^2$ lost from your system. So you must also have a similar loss in the general relativistic gravitational potential energy, if the energy was zero originally.

So the lost radiation must also represent a significant loss of potential energy, not just the tiny gravitational redshift. I don't know how Hawking is counting potential energy to get the energy to be zero, but it can't be the simple Newtonian potential. Remember that the zero in a potential energy scale is not usually of physical consequence, because it's only changes in potential that matter, but to get the total energy to be zero, he must have some way to get a relevant scale. Much more formal mathematical arguments are available, such as http://www.prespacetime.com/index.php/pst/article/view/81, but even these are debated by experts, who each attribute different degrees of significance to the ability to make the total energy zero. It is not at all clear that total energy is a relevant physical quantity in general relativity, but for those who wish to make it so, as Hawking does, it may indeed be possible.

Also note that the tiny energy in the radiation today is not really relevant to this question, because the question is about originating a universe. When the universe originated, radiation was extremely important, so the question does need to be addressed for anyone who wishes the energy to be zero. The real issue appears to be, whether or not that is an important goal in the first place.