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We've all heard mass tells space how to curve and curved space tells matter how to move. But where does the energy to curve space come from? Likewise where does the energy that curved space uses to push planets around come from? I mean if I tell my son to clean his room, and he does, then I did not provide him the energy to do so.

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The theory of General Relativity works with the energy momentum tensor and one has to work with the mathematics of it in order to really understand what is happening, not handwaving. It is a fact that all cosmological and astrological data follow the general relativity equations, as one can see in this link The cosmological model accepted now is the Big Bang model, based on general realtivity and its constants adjusted with observational values. In this model all the energy of the universe, the one the energy momentum tensor describes, came at the Big Bang singularity:

historyof universe

The expanding universe utilizes this original energy . (Special relativity is part of General relativity, and thus the equivalence of mass with energy is accounted for). As the Big Bang model is quite successful , the answer is : the energy for everything comes from the original singularity. At the planetary and galactic level it is described well with Newtonian mechanics. At the atomic and nuclear the laws of quantum mechanics and special relativity are adequate to describe energy transformations.

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Mass is the same as energy by $E = mc^2$ So the energy that curves space-time is the mass that curves space-time. Then you could ask why does mass curve space? As far as I know that is equivalent to asking where do Einstein's field equations come from. For that one needs a complete and consistent theory quantum gravity which has Einstein's equation in the low energy limit. No one has that.

"We've all heard mass tells space how to curve and curved space tells matter how to move." This poetic but be a bit misleading. Einstein's equations simply describe in detail how space-time changes as a result of changes in matter. There isn't some process or mechanism happening between matter and space-time i.e when space-time changes there is a process that occurs that then can be used to describe what matter will then do.

Lastly, you are using gravitational energy in the newtonian way of looking at gravity. The gravitational potential which ultimately gives gravitational energy in newton's theory is buried in the metric.

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  • $\begingroup$ That may be what energy is but like I said cannot explain how curve space can tell matter how to move. Where did that gravitational energy come from? $\endgroup$ – Bill Alsept Nov 1 '16 at 21:51
  • $\begingroup$ The metric of space-time in the non-relativistic limit produces the gravitational potential which can ultimately be used to arrive at gravitational energy. Secondly, they way we derive gravitational potential energy in newtonian gravity is not valid in the gr case. $\endgroup$ – Amara Nov 1 '16 at 22:02
  • $\begingroup$ That's what I meant by there's never been a good answer. What causes the initial energy? If something is moving away from a massive object how and why does it turn around and head back toward the massive object? $\endgroup$ – Bill Alsept Nov 1 '16 at 22:12
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If we pick the (most common or most useful) example of the Schwarzchild solution, it is simply the unique geometry that solves the Einstein's equations that is both static and spherically symmetric . It is solved in vacuum (which means that the energy density in the part of spacetime we are solving for is zero, while there may be a mass at the origin). This is similar to solving for the electromagnetic field, where you solve Maxwell's equations in vacuum to get the field due to a point charge. The charge itself is not losing any energy by the generation of the electric field. It is a description of interactions and not something that the charge expends energy in creating.

To answer the second part of the question, objects that are not being acted on by an external force (i.e., objects in free fall), follow geodesics: In simpler terms, if I set a ball rolling on a frictionless surface with constant velocity, it will follow a path where governed by the equation $\vec{F}=0$, which would be a straight line. It is the same concept in general relativity, except that the paths are not necessarily 'straight lines', because of the curvature of spacetime itself. (If I set a ball rolling on the earth, it will not shoot off at a tangent to the surface, but roll on the surface). It is important to note that the geodesics are locally straight. In other words, if I look at a small patch, spacetime looks flat, and bodies move along a continuous path. In other words, 'straightness' in general relativity need not correspond to what we intuitively think of as straight in non-relativistic contexts. In the case of orbiting bodies, the paths they follow $\textit{are}$ the paths that are locally straight in their frame of reference.

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  • $\begingroup$ Objects in freefall are being acted on by some external force that is why they are falling. If an object was moving in a perfect straight-line away from the mass what would make it stop, turn around and go back? $\endgroup$ – Bill Alsept Nov 1 '16 at 23:26
  • $\begingroup$ That's the difference between classical Newtonian physics and general relativity. In classical gravity, we see it as an interaction force between masses. But in GR, masses (or energy density in general) curves spacetime itself, and bodies in spacetime follow geodesics which are 'generalizations of the notion of a "straight line" to "curved spaces"'. In other words, gravity isn't seen as a force. Spacetime is curved and objects just follow the paths that correspond to straight lines in the curved space. $\endgroup$ – Gowri Nov 2 '16 at 21:23
  • $\begingroup$ You can see the difference easily if you consider light. With classical physics, light is massless and should not be affected by gravity at all. But in GR, we would expect the curvature of spacetime to affect anything that moves in spacetime, even if it is massless (although massless particles behave a bit differently from massive ones). This is why we have effects like gravitational lensing. $\endgroup$ – Gowri Nov 2 '16 at 21:26
  • $\begingroup$ Everything that happens is an interaction. Even with general relativity you can't explain it without using verbs like follow or curves. Those are both actions. Also the explanation of something following curved space does not explain something that is not moving at all or something moving away from gravity. If an object is not following curved space then what makes it start following curved space? Also how does mass curved space?? $\endgroup$ – Bill Alsept Nov 2 '16 at 22:37

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