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In classical electromagnetics, static electric and magnetic fields in a given volume in a vacuum are completely determined by measuring the fields on the surface that bounds the volume. The fields at the surface can be probed by measuring both the force and torque on a spinning charge.

Does a similar situation hold for gravitational fields including 'orthogonal' fields field induced by relativistic motion? Can the gravitational properties of a volume of space be determined from measurements made with a spinning mass just on the surface enclosing that volume? Do such concepts generalize to the properties of spacetime itself rather than just describing the gravitational fields that may be contained?

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    $\begingroup$ "In classical electromagnetics, static electric and magnetic fields in a given volume in a vacuum are completely determined by measuring the fields on the surface that bounds the volume." can you please provide a reference? I'm sure this is true in electrostatics, but not so sure about Maxwell electrodynamics. $\endgroup$ Commented Nov 20, 2020 at 1:32
  • $\begingroup$ @ Prof. Legolasov Yes, I'm confining this to static electric and static magnetic fields. I also am not so sure about changing fields. $\endgroup$
    – Roger Wood
    Commented Nov 20, 2020 at 4:40

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It is a generic feature of differential equations in second order partial derivatives that knowing the values of fields on a closed boundary establishes uniquely the solution in the bulk region. Some caveats are:

  1. Some equations are degenerate and admit multiple bulk solutions, hence invalidating the claim above. In physics, such situations are usually treated in the following manner: the different bulk solutions are interpreted as the same physical configuration; and they must be related by gauge transformations. Therefore, the validity of the claim is restored when we take gauge symmetry in account.
  2. For regions with nontrivial topology, it is possible to have a finite number of degrees of freedom that aren't fixed by boundary values. For the case of a sphere, this doesn't happen, so we won't bother with this caveat.

This is why what you said is true for electrostatics. When the fields don't change in time, the mathematical problem we're solving is a differential equation in second order partial derivatives in 3 dimensions. Hence, values on a 2d spherical boundary are enough to reconstruct the solution, modulo the caveats above.

For fields that change in time, this simplification differential equation in second order partial derivatives in 4 dimensions, and we need to fix the values on a 3-dimensional boundary with a topology of the 3-sphere.

General Relativity can be brought into a similar form (differential equation in second order partial derivatives in 4 dimensions), see e.g. the ADM formalism for reference.

Hence, the same results are true in General Relativity.

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  • $\begingroup$ @ Prof. Legolasov thanks, that's very helpful, though I don't really have the background to follow the ADM reference. Am I correct in saying that, for the static case (for example, stationary in the vicinity of a rotating black hole) it is sufficient to measure the force (3 dimensions) and torque (3 dimensions) on a small spinning proof mass taken over the surface (2 dimensions) of the volume of interest (for a total 8 dimensions). $\endgroup$
    – Roger Wood
    Commented Nov 22, 2020 at 3:28
  • $\begingroup$ @ Prof. Legolasov The 'electro-' and 'magneto-' gravitational fields in the volume are both constrained by the Laplace eqn so have 2 dimensions each and the volume has 3 dimensions for a total of 7. I'm still a bit puzzled about how many variables I need to completely define a piece of space-time - even a static one $\endgroup$
    – Roger Wood
    Commented Nov 22, 2020 at 3:30
  • $\begingroup$ @RogerWood forces and torques are approximations that don’t make sense in the full general relativity, but there are certainly quantities you can measure on the surface to reconstruct the bulk solution. The ADM formalism gives you exactly those. They turn out to be the induced spatial metric components on the boundary. $\endgroup$ Commented Nov 22, 2020 at 3:30

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