Hi all, I was talking to my professor about the black hole information paradox, and all of this got me to thinking about something: Why do you think the conservation of energy is so vital to the existence of the universe? Suppose we didn't conserve energy, but actually lost it in very small amounts with every encounter. What would happen? It seems to me that this would imply the slow die off of the universe. Now if energy was never conserved in the first place, how could we conceive of events such as the big bang every occurring in nature? Wouldn't this imply that we would need loads and loads of energy to create such an explosion? If energy is not conserved, where would nature pool this energy from? also, I am very aware of the empirical evidence which supports the opposite of the initial assumption I wish to make in the above question. Try and suspend the urge to rely on that. I hope this isn't too 'out there'... This more of a curious wandering than any real desire to make any serious claim. Enough of that though, what do you think?
This is a little too far out there, especially the stuff about the big bang. It's so out there it doesn't even make any sense. (And the Big Bang was not an explosion in space)
If energy wasn't a conserved quantity, then the laws of physics wouldn't be time-invariant. The laws of physics yesterday would be different than they are today.
It's best to first talk about (flat) Minkowski spacetime. In this case, if the system is invariant under time translations, then energy is conserved. We could also use a model of a system where energy is not conserved, for example, if we include friction. But (I think) that it is believed that all real systems really do conserve energy. For example, in the system with friction, energy is actually transferred as heat, not just lost. Now, if we talk about general relativity, energy is not globally conserved. But also, in general relativity, spacetime is locally flat, so we can say that energy is conserved locally, but not globally. So, from the idea of local energy conservation, we get some differential equations that constrain how the matter and radiation can move around in spacetime. (These equations are similar to the differential form of Maxwell's equations. In fact, Maxwell's equations with covariant derivatives give us the equations for local energy and momentum conservation of the electromagnetic field). So anyway, energy is not globally conserved in the universe. And I'm not sure what you were asking about the big bang... we can't really say anything about the very very early universe, since we would need a theory of quantum gravity. And after that, there was a very high energy density. And now there is a much lower energy density. But this is OK, since energy is not globally conserved, as I said before.
What do you mean that energy is not globally conserved? When you say that spacetime is locally flat, do you mean that I can think of it as a non-distorted Cartesian coordinate system?
yeah, in general relativity, there is no 'total energy of the universe' to be kept constant. And yes, in the limit of 'spacetime events that are very close to you', you can label them using the Cartesian coordinate system in the usual way that you would in a flat spacetime.
It was an event where our current knowledge of physics is not useful any more. After this event, there was spacetime, with space rapidly expanding. You cannot even define a global energy in a meaningful, unique way.
Ah I see! So the claim is that we are not wholly sure that the laws of physics which governed the Big Bang are necessarily the same ones which govern out current universe (as far as we know). I'd say that is a fair enough assumption. Can we assume nothing about that event then? I suppose we could say that its effects were 1) our current universe and 2) that it is expanding. What reason do we have to say that the laws of physics which governed that event are not necessarily the ones which governed every single event after it? As to defining a meaningful definition of global energy, if we could, would that mean we would have to figure out a way to relate all forms of energy to each other? I suppose there would be no way to really measure this quantity... When you say there is no meaningful way to define it, is this what you mean?
In general, it is expected that the same laws of physics are true everywhere - we just have no idea how those laws look in case of conditions at/close to the big bang. That does not help. In our universe, you cannot define a global "now", or "future". So how can you speak about the energy "now", or "in the future"? If you cannot define that, you cannot look how some quantity changes.
That the Big Bang was an explosion in space is a widespread misconception. Thinking of the Big Bang as an explosion in space would lead one to think that one could identify a point in space where the Big Bang happened. That's not what the cosmic microwave background tells us. The Big Bang happened everywhere. A much better (but still somewhat simplistic) point of view is that the Big Bang was an explosion of space rather than an explosion in space.
This page is pretty good, although maybe a little technical http://en.wikipedia.org/wiki/Mass_in_general_relativity It says that since the Poincare group is a finite parameter continuous group, Noether's theorem defines a (scalar) conserved energy for the system. But since general relativity is a diffeomorphism invariant theory, it has an infinite parameter continuous group of symmetries, and because of this, we can't use Noether's theorem to guarantee us a concept of conserved energy. But in the special case where the system has a timelike killing vector field, there is an associated 'conserved energy' called the Komar mass. But as I said, this is a special case, and our universe is not one of those special cases.
This inference is not correct. It is true that if the system is described by Lagrangian and if this Lagrangian is time-indepedent, the energy is conserved. But the equivalent statement to this is IF the energy is not conserved, then EITHER the Lagrangian depends on time, OR the system is not described by Lagrangian. If the second option happens, it may happen that the laws will still be formulated in statements that do not change in time. For example, the Newton law of viscosity is formulated as time-independent statement, but still leads to loss of mechanical energy. This is because this law does not admit Lagrangian formulation.