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Does, for instance, energy emit gravitational force on nearby masses, or energies, and thereby increase the overall energy of the system? (I am ignorant of field theory and would appreciate being pointed in the right direction).

If this is not true, it seems you could increase the gravitational energy between the moon and earth by converting energy to mass on earth, which is absurd as it would break the conservation of energy.

On the other hand if this is true would not the gravitational energy between the moon and earth induce additional mass-energy which then induced more gravitational force? Furthermore, which direction would this additional force be i.e. where would this added mass-energy be localized?

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The field equation of GR equates the curvature of spacetime to the stress-energy tensor of matter and radiation, plus a cosmological constant (“dark energy”). The stress-energy tensor includes electro-weak and nuclear effects, but surprisingly not gravity itself. The gravitational potential energy is instead hidden inside the nonlinear expression for curvature in a subtle way. Einstein did not view gravity as one more classical field just like the electromagnetic field, which is the only field we can honestly claim to understand. The treatment of gravitational potential energy leads to some interesting paradoxes in cosmology.

One way to unravel the math is to ask what linearized (a.k.a. weak field) GR omits. To a first approximation, gravity couples to stress-energy just as electromagnetism couples to charge-current. The only obvious difference is that gravity has a tensor potential, whereas electromagnetism has a vector potential.

The electromagnetic contribution to the stress-energy tensor is simply quadratic in the field strength, e.g., the energy density being ${{T}^{00}}=\tfrac{1}{8\pi }({{E}^{2}}+{{B}^{2}})$. The divergence of this contribution to the stress-energy tensor matches the work and impulse predicted by the Lorentz force law: ${{\partial }_{\mu }}{{T}^{\mu \nu }}={{F}^{\mu \nu }}{{J}_{\mu }}$ .

One runs into trouble when trying to construct a corresponding stress-energy tensor for gravity. It can’t simply be quadratic because T now replaces J. (The LHS can’t be quadratic if the RHS is cubic.) In fact, the gravitational contribution to stress-energy can’t be a polynomial function of the field strength. There is really no substitute for going to the fully and frightfully nonlinear field equation.

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