What is the relationship between spacetime and energy?

[jameseg]

Hi everyone I was in philosophy class today and my professor was going over what he said was a paradox with E=mc^2. He said that as defined energy is the ability to do work and that it does work on matter and in turn the work done on the matter creates energy in a cyclic process. The professor argued that there had to be a starting point and he used this as a way to suggest that the metaphysical may be a useful explanation for the start of the energy mass cycle. I disagree with the metaphysical being a viable explanation but i didnt know what to say that would just be open to him calling my logic regressive. I know about the singularity and the theory that branes collide to cause singularities and the inflationary theory. could anyone shed some light on what they would have said to my professor, this "paradox" is weighing heavily on my mind. Thanks to anyone who responds.

 

[Parlyne]

Your philosophy professor ought to know better than to base metaphysical arguments on physics that he clearly doesn't understand.

First, there is nothing about $E = mc^2$ that implies either the creation or the destruction of energy. All that the equation says is that mass is a form of energy. In fact, mass can be annihilated in such a way as to do useful work on something else. In that case, the energy which started out as mass will end up as kinetic energy, or heat, or some other form of energy; but, the total amount of energy will remain the same.

So, if your professor want to argue that there is a paradox, it can't have anything to do with Einstein's equation. In fact, it sounds like he's trying to argue something of the form "Energy exists; therefor, God." However, if the topic overall is the existence of energy in the universe at large, there is a big subtlety that needs to be addressed.

Energy is an observer-dependent quantity. In fact, this shouldn't be too much of a surprise. If you were drifting in space right next to a teapot, after you finished chastising Bertrand Russell for leaving it drifting about the solar system, you would note that it's total energy was its mass (or, rest energy). If, however, your professor drifted by (at a constant speed) it would appear to him that both you and the teapot were drifting by (due to the principle of relativity); and, he would claim that both you and the teapot have kinetic energy as well as mass. Since mass is frame-independent and kinetic energy is always positive, your professor must see the teapot as having more energy than you see it having.

Now, classically, or even in special relativity, frame dependence isn't a problem for discussing energy on a global scale. You just have to specify your frame and you have no problem.

The real issue comes about in general relativity, where frames of reference are only valid locally, as spacetime is both curved and dynamic. This means that coordinates defined by an inertial observer are unlikely to correspond to inertial behavior more than a short distance from the observer. Since this makes it questionable at best to add the energies of well-separated objects, it is not obvious that there even could be a globally valid definition of energy except in special cases.

Given this difficulty, your professor needs to justify his assumption that there is a non-zero net energy in the universe before we can even accept a generous reading of his premise.

Beyond that, of course, there's the additional problem that a recourse to the metaphysical requires the additional presumption that a physical explanation is not possible; which, of course, it seems he has also not justified.

 

[Chalnoth]

A first cause argument, eh?

Well, here's one for you: you can produce all the energy in matter you want in General Relativity without doing any work. Here's how you do it. The gravitational potential energy, which we see in what is known as the Hamiltonian formulation of General Relativity, is always negative. If you create a closed universe, the gravitational potential energy turns out to exactly cancel out the energy in the matter fields, and you end up with a universe that has total energy identically equal to zero, even if that universe ends up looking just like ours, complete with stars, galaxies, and people making bad arguments.

This sort of universe can, for instance, be started from a random quantum vacuum fluctuation. We haven't worked out all of the details of this sort of thing yet, but there's not any reason to believe it can't happen as a necessary consequence of quantum mechanics coupled with gravity.

Anyway, on a more philosophical point, to my mind the more sophisticated apologists moved on from the "first cause" argument a long time ago, and now pursue "prime cause" arguments. The idea with a prime cause argument is not so much that there must be a terminating sequence of causes into the past, but rather that for everything that happens in our universe, there is something that causes that sort of thing to happen, eventually terminating at a "prime cause". That is to say, if a new region of space-time can be born through a random quantum vacuum fluctuation, then that sort of fluctuation exists only as a consequence of physical laws. Those physical laws themselves must also exist as a consequence of something else, eventually terminating at some "prime cause".

What undercuts the intent of most people making this sort of argument is that this prime cause could be nothing more than a single principle like, "all mathematical structures have real existence." See, for example, this paper by Max Tegmark discussing this hypothesis:
http://arxiv.org/abs/0704.0646

I do hope your professor was arguing, however, just for a place for philosophy within science, not for a god. Because there is definitely a place for philosophy within science. The above principle, for example, is a metaphysical principle. But your metaphysics had better be chained to reality through experiment and observation, or you're pretty much guaranteeing that you'll get it wrong.

 

[jameseg]

Thank you both for the clarification.

 

[brainstorm]

Energy is an observer-dependent quantity. In fact, this shouldn't be too much of a surprise. If you were drifting in space right next to a teapot, after you finished chastising Bertrand Russell for leaving it drifting about the solar system, you would note that it's total energy was its mass (or, rest energy). If, however, your professor drifted by (at a constant speed) it would appear to him that both you and the teapot were drifting by (due to the principle of relativity); and, he would claim that both you and the teapot have kinetic energy as well as mass. Since mass is frame-independent and kinetic energy is always positive, your professor must see the teapot as having more energy than you see it having.

Now, classically, or even in special relativity, frame dependence isn't a problem for discussing energy on a global scale. You just have to specify your frame and you have no problem.

The real issue comes about in general relativity, where frames of reference are only valid locally, as spacetime is both curved and dynamic. This means that coordinates defined by an inertial observer are unlikely to correspond to inertial behavior more than a short distance from the observer. Since this makes it questionable at best to add the energies of well-separated objects, it is not obvious that there even could be a globally valid definition of energy except in special cases.

Given this difficulty, your professor needs to justify his assumption that there is a non-zero net energy in the universe before we can even accept a generous reading of his premise.

People always talk about frame-dependence but it seems to me that the only reason any object has spacetime to move through is because of kinetic energy among other particles of matter that spread them out and gravitation that lumps them together away from each other. Without that spreading and contracting to create in-between distances, there would be no spacetime between destinations to traverse. So why shouldn't it be said that spacetime itself is a product of energy and that any object traversing that spacetime is dissipating energy into it? In other words, just as a star or other radiating object loses energy by means of emission, a moving object can be said to be losing energy to motion, no? I know you're going to say that an object moving only due to its inertial momentum is not expending energy, but it is insofar as its position within some gravitational field is changing, therefore it is losing potential energy (although I suppose it could be gaining it if it is decelerating out of a gravity well below escape velocity).

Am I looking at this wrong? Is not every position in spacetime subject to specific gravitational influences such that any object occupies a position in the gravitational topography of spacetime? As such, wouldn't ever object have a certain potential energy of position/altitude relative to surrounding gravity-emitters?