Does the actual curvature of spacetime hold energy?

My understanding of GR is that curvature of spacetime reflects the density of energy-matter. Does the curvature itself have energy? Or if energy is assigned to curvature it simply reflects the energy density at that point?

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The gravitational field can indeed be assigned an energy. Unfortunately though whereas for, say, the EM field you can define an energy density at a point ($\bf{E}^2+\bf{B}^2$), for the gravitational field you can't do this. - Whichever way you define the energy in terms of the Christoffel symbols, you run into the problem that you can make them, and hence the energy, vanish at a point be choosing an appropriate frame.

So people have come up with non local energy definitions for the gravitational field- ADM energy, Bondi energy etc. all of which involve integration over spacetime regions.

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I'm a little unsure of terminology here - my training is more mathematical than physical - I think your local means at a point; and non-local means over a small spacetime region. Is that right? – Mozibur Ullah May 13 '13 at 14:41
@MoziburUllah: no, local means a spacetime region - even if it becomes arbitrarily small. – DilithiumMatrix May 13 '13 at 14:53
@Zhermes: yes, thats what I would normally suppose; but twistor59 said non-local. Are you suggesting his terminology is a mistype? – Mozibur Ullah May 13 '13 at 14:56
@MoziburUllah, sorry, the additional criteria is that 'local' refers to a region which resembles flat space-time. – DilithiumMatrix May 13 '13 at 15:02
@zhermes: so his non-local means non-flat or not necessarily flat? – Mozibur Ullah May 13 '13 at 15:03

It is helpful to look at another simpler classical field theory, electromagnetism. Electromagnetism has equations for how electromagnetic fields change in time, they are fairly simple compared to general relativity, namely $$\frac{\partial \vec E}{\partial t}=\frac{1}{\mu_0\epsilon_0}\vec \nabla \times \vec B, \text{ and }\frac{\partial \vec B}{\partial t}=-\vec \nabla \times \vec E.$$

What this means is that it is natural for spacetime to be full of electromagnetic fields but they have to evolve a certain way in time based on what they are right now. And for certain configurations, they don't change in time.

Same with gravity. Instead of an electromagnetic field you have a metric and instead of $$\frac{\partial \vec E}{\partial t}=\frac{1}{\mu_0\epsilon_0}\vec \nabla \times \vec B, \text{ and }\frac{\partial \vec B}{\partial t}=-\vec \nabla \times \vec E$$ telling you how to evolve in a vacuum you have the equation $G=0$ which is a second order (and nonlinear) equation, so isolating something as simple as the time derivative of of the 16 components of the metric isn't easy, and it would be a second derivative anyway.

But the conclusions are the same. It is natural for spacetime to be full of a changing metric even in a vacuum, it just has to evolve in a certain way. Lots of analogous things hold. There are electromagnetic waves that go through the vacuum, there are gravitational waves that go through the vacuum. There is a natural way for metric and electromagnetic fields to evolve through time in a vacuum. And it is natural for them to be there.

But when there is electric current, the electromagnetic field has to evolve differently. It evolves according to $$\frac{\partial \vec E}{\partial t}=\frac{1}{\mu_0\epsilon_0}\vec \nabla \times \vec B-\frac{1}{\epsilon_0}\vec J, \text{ and }\frac{\partial \vec B}{\partial t}=-\vec \nabla \times \vec E.$$

So given a region of current, such as a sheet of current, you can sew together a solution for either side of the sheet that normally wouldn't fit together. The same thing happens with gravity.

You can take a solution to the vacuum equation $G=0$ such as Minkowski space in a ball shaped region of radius 10,000 km. Then you can take another solution to the vacuum equation $G=0$ such as the Schwarzschild solution with parameter $M=M_\oplus$ and cut them along the spherical shell of surface area $4\pi(10,000km)^2$ and sew them together. Normally they wouldn't fit together an still be a solution to $G=0.$ And in fact the computation of $G$ involves some second derivatives of the metric and gives a nonzero $G$ right at that surface. But that is fine because just like electromagnetism could sew things together if there is current there, so can gravity sew together solutions if there is stress-energy there. So you just need to place some energy on the spherical shell between the two regions, just the perfect amount, but thats all it takes.

So the curvature outside the spherical shell exists fine on its own. In fact it is exactly the kind of curvature as to evolve into itself, it is a static field (like if you had a uniform and constant electric field, it can just sit there being itself in empty space) so the curvature outside the shell is capable of evolving into itself. Similarly the curvature on the inside (none) has a metric that is fully capable of evolving into itself. So if the energy can stay put, like a giant shell made out of lego bricks keeping themselves together then everything can sit there forever just fine.

But this doesn't explain where curvature comes from. We just described a situation where the future is like the past which is the same as the present just because it was perfectly set up to do that.

Lets make some curvature!

Start with a spherical shell of matter. Have a Schwarzschild (curved) metric on the region outside the shell and have a Minkowski (uncurved) metric on the region inside the shell. You have to have just the right amount of energy density on the shell to do this, assume you do. But this time, the matter on the thin shell isn't able to hold itself together, so it all falls towards the center. What is different? The shear stress is less and this allows the matter to fall inwards towards the center. As it falls, the region on the outermost surface is forced to evolve in the vacuum way instead of in the there-is-some-energy-density-here way, and we know what that is since the Schwarzschild metric outside is exactly the kind as to make itself, so more Schwarzschild is produced where the outermost surface sued to be. Same with just inside the innermost surface, the new matter there transitions the Minkowski metric to the Schwarzschild metric because it is exactly the right amount of energy density (for that location) to change the metric from one to the other, because that's how much energy density we put there (technically there is a bit more energy since it is falling, but magically that's exactly the amount of energy density you need at the new location to transition between the two). And this works out because the constraint on how the curvature evolves is exactly what you need to be consistent with how matter evolves. Getting that part right is actually what Einstein spent his time doing.

So when matter moves, it does it in a certain way (a way that kinda conserves energy and momentum in a local kind of way) which is exactly what gives the metric in the vacuum around it the freedom to evolve into new vacuum metric in a way as determined by the metric nearby and previously just like electromagnetic fields change based on the previous fields and what is going on nearby. For gravity it changes in a second order way which means its the metric and how the metric changes that is the current "thing" and the vacuum way of evolving is about saying how those rates of change can change. And it is more complicated in that you can't just add two solutions together to get a new solution. But stress and energy just tell spacetime that it can (and must) curve differently than it normally does without stress or energy. Which allows it to transition between different ways it can curve in a vacuum. And allows it to change "in the vacuum way" when the matter moves away and allows it to change to something new (not the vacuum way) when matter comes in.

That is really what is going on. Curvature can happen on its own. Thats why you can orbit something that is far away, the curvature out here was formed long ago when the collapsing parts moved further in than we moved in. So the galaxy left the space out here curved as the galaxy formed, the sun curved the solar system as the sun formed, the earth left the space around the earth curved when the earth formed, and that's why the moon can go around the earth as the earth goes around the sun as the sun orbits the galactic center. It's also why you can feel gravity outside a black hole, the object curved the space outside itself as it collapsed, each bit of infalling matter leaving the region on one side curved differently than on the other ... since that is what matter does. Allows two different but natural curvatures to line up when otherwise they could not.

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