I’ve had an interesting set of responses to my post “Of Nothing”, so I can’t resist a sequel. (I’m not really taking a position on who “won” the debate. I don’t really care. I’m just commenting on some of the issues raised.) Here, I’ll respond to a few more of Krauss’ comments on nothing and cosmology, this time from this article in the Wall Street Journal:
There are remarkable, testable arguments that provide firmer empirical evidence of the possibility that our universe arose from nothing. … the existence of dark energy and a flat universe has profound implications for those of us who suspected the universe might arise from nothing. Why? Because if you add up the total energy of a flat universe, the result is precisely zero. How can this be? When you include the effects of gravity, energy comes in two forms. Mass corresponds to positive energy, but the gravitational attraction between massive objects can correspond to negative energy. If the positive energy and the negative gravitational energy of the universe cancel out, we end up in a flat universe. Think about it: If our universe arose spontaneously from nothing at all, one might predict that its total energy should be zero. And when we measure the total energy of the universe, which could have been anything, the answer turns out to be the only one consistent with this possibility. Coincidence? Maybe. But data like this coming in from our revolutionary new tools promise to turn much of what is now metaphysics into physics.
If you read the whole article, you quickly discover that Krauss simply doesn’t understand the question “why is there something rather than nothing?” – see my last post for more on that. Here, I will raise two issues with the quote above.
The Energy of the Universe
It is not true that if the universe is flat then it has zero total energy. This idea relies on a Newtonian calculation of the gravitational potential energy of an expanding universe. However, such a calculation simply cannot be performed in general relativity (and even the Newtonian calculation is suspect because the Poisson equation has no solution for a uniform unbounded fluid. See Rindler 1977, pg 199). There is simply no way to calculate the total mass-energy of a general spacetime, and the energy of the gravitational field cannot be calculated in a coordinate invariant way. John Baez and Sean Carroll have excellent discussions of this point, as does Hartle in his GR textbook: “Conserved quantities … cannot be expected in a general spacetime that has no special symmetries”. Note that energy conservation is a consequence of time-translation symmetry, which an expanding universe (and its Robertson-Walker spacetime) does not possess. Here’s Gravitation by Misner, Thorne and Wheeler (MTW):
When evaluating the 4-momentum and angular momentum of a localised system, one must apply the flux integrals only in asymptotically Minkowskian coordinates. If such coordinates do not exist (spacetime is not flat at infinity), one must completely abandon the flux integrals, and any quantities that rely on them for definition: the total mass, momentum, and angular momentum of the gravitating source … ‘Total mass-energy’ is a limited concept, useful only when one adopts a limited viewpoint that ignores cosmology. (Pg 463)
The Robertson-Walker metric is an example of a spacetime that is not asymptotically flat, and thus attempting to calculate its total mass-energy will “easily and unavoidably produce nonsense” (MTW). (Don’t confuse the two different uses of the word flat above. Krauss is referring to the space curvature of an expanding universe, while MTW are referring to spacetime curvature. A spatially-flat expanding universe is not spacetime-flat.)
The point of the above passage is that we cannot in general add up the local contributions of the mass-energy of the universe to produce a total mass-energy. When it comes to gravitational energy, it gets even worse – we can’t even define it locally:
To ask for the amount of electromagnetic energy and momentum in an element of 3-volume makes sense. First, there is one and only one formula for this quantity. Second, and more important, this energy-momentum in principle “has weight”. It curves space. It serves as a source term on the righthand side of Einstein’s field equations. It produces a relative geodesic deviation … It is observable. Not one of these properties does “local gravitational energy-momentum” possess. There is no unique formula for it, but a multitude of quite distinct formulas … Moreover, “local gravitational energy-momentum” has no weight. It does not curve space. It does not serve as a source term … It does not produce any relative geodesic deviation … It is not observable. Anybody who looks for a magic formula for “local gravitational energy-momentum” is looking for the right answer to the wrong question. (MTW, pg 467)
The equivalence principle says that one can always find a local coordinate system in which gravity, and thus any concept of gravitational energy, vanishes. It is not that we do not know how to formulate gravitational potential energy in GR, or that we don’t know how to total up the mass-energy of a given spacetime. Such calculations, which were perfectly sensible within Newtonian gravity, are simply meaningless in the context of general relativity.
Apologies for subjecting you to such long quotes, but I’ve heard the claim that the total energy of the universe is zero too many times to let it be. The total mass-energy of the universe is not well-defined; even if it was, it wouldn’t be conserved in Robertson-Walker spacetime; and gravitational energy is not a meaningful concept in GR.
Nothing has no energy
My second problem with Krauss’ quote above is this claim: “If our universe arose spontaneously from nothing at all, one might predict that its total energy should be zero”. In my last post I argued that the idea that the universe came from nothing is simply nonsense. Nothing has no properties, and the ability to spontaneously create a universe is a rather spectacular property.
But suppose the universe could come from nothing. Would it follow that it should have zero energy? This claim seems plausible – what other energy could nothing have? My first problem is that having an energy, even if that energy is zero, is a property. It makes sense to say that some physical system (say, a collapsing cloud of gas) could have zero energy, but other things – say, the C major scale – simply do not have the property we call energy. There is a difference between “having an energy, and that energy is zero” and “not possessing the property of having an energy”. The C major scale does not have zero energy; it does not have the property at all, just as it has no colour, mass or taste. Thus, “having zero energy” is a property, and anything possessing a property cannot be nothing.
The English language is partially to blame here. The phrase “has no energy” is ambiguous. It could mean “having an energy of zero” or “not possessing the property of having an energy”. Nothing, which has no properties, has no energy in the latter sense. A universe that came from nothing would not involve a transition from a state with zero energy to another state with zero energy.
There is second problem lurking here, which we will tease out with a bit of mathematical formalism. The symbol represents the probability of some event A given the information represented by B. For example, A could be “I win this hand of poker” and B is “I have just been dealt a royal flush”. Actually, we would need B to also specify the entire scenario, including the rules of poker, the number of players etc. The probability of A depends on B. If B were “I have just been dealt (2C, 3D, 4H, 6H, 9C)” then the probability of A would be significantly lower. We will call the place in occupied by A the outcome slot, and the slot occupied by B the given slot. The given slot controls the outcome slot.
Now, suppose that Krauss were correct, and that we could predict that a universe which came out of nothing would have zero energy. In this case, the outcome slot would be occupied by A = “Universe with zero energy”. But what would be in the given slot? Nothing. Not anything. Remember: nothing is not a type of something. There is not a kind of thing out there called “nothing” whose properties we can place in the given slot to see what outcomes are possible. There is literally not anything in the the given slot. Nothing is given. We would be calculating .
If you have a sneaking suspicion that such a calculation is nonsense, then you’re starting to see things my way. But we can go a step further. Suppose that , where p is positive number. Let’s now ask: what is , where C = “Universe with positive energy”? Ordinarily, we would appeal to the information in the given slot to answer this question. But nothing is given. There is literally no possible reason why should not also be some positive number. There is nothing, not anything, to control what outcomes are impossible or possible, improbable or certain. We must conclude that, if a universe with zero energy could come out of nothing, then so could a universe with positive energy. Similarly, a universe with negative energy could come out of nothing.
It gets worse, as you must have realised by now that C could be anything at all. No possible reason can be given for denying the possibility that a polar bear could come out of nothing. When we calculate the probability that a radioactive nucleus will emit an alpha particle, we appeal to the laws of physics and the properties of the nucleus in question. When we calculate that the probability of the nucleus emitting a polar bear is zero, we do so on the basis of the information in the given slot. If there is nothing in the given slot, then there is nothing to control the outcome slot. We can put any outcome we like in there, and calculate a non-zero probability that it will appear out of nothing.
Worse still, if something could come out of nothing at the beginning of the universe, then something could come out of nothing now. If polar bears need no necessary or sufficient conditions to spontaneously appear out of nothing, then the conditions in this room right now fit the bill perfectly. Further, there no reason why the probability of such an appearance should not be one. Remember: nothing is given! There is literally no possible reason for polar bears to fail to appear in this room right now.
We can then make the following prediction: if a universe with zero energy can appear out of nothing, then polar bears should be appearing out of nothing in this room right now. I would call this a clear case of reductio ad absurdum.
All this to say: if something can some out of nothing, then anything and everything can and should come out of nothing at all times and places. This, then, is the empirical evidence we would need in order to believe that the universe could come out of nothing.