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It is well known that the method of images is a useful tool for solving electrostatics problems. I was wondering why this technique is not applied when considering newtonian gravity?

Obviously there is no "negative mass" to correspond to a negative charge in electromagnetism, but surely the unphysical nature of negative mass could be ignored and considered a mathematical trick to solve a given problem?

The classic example for the method of images is the point charge near an infinite conducting plane, surely there is a way to apply a similar method to calculate the gravitational field between a point mass and an infinite thin plane?

Some preliminary research online has resulted in no sources on this idea so any references for/against this would be great.

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  • $\begingroup$ Method of images is used because the charge induces some charge on the infinite plane sheet which make the potential of the system difficult to calculate. charge can be induced because the electrons of sheet moves freely and they are attracted by charge. mass can be induced? $\endgroup$ – Paul Jan 31 '15 at 18:34
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While rare, there are a few uses of the method of images to gravitational problems. As lurscher says, the problem is finding equipotential surfaces. In most problems, such a surface doesn't exist, and hence the scare use of the method of images in GR.

One class of problems for which it does applies are the so-called Dirichlet problems. Suppose one was interested in solving for the metric in some region, with specified boundary conditions on the boundary surface. This is not usually what is done--usually the entire spacetime is solved for. For the case of Dirichlet boundary conditions (requiring the metric to approach some specified value on the boundary surface), image charges can be useful. In this case the image charges could correspond to image black holes, for example.

However, this is somewhat of an exotic problem, and I've only seen perhaps one or two examples where image charges have been used.

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  • $\begingroup$ Can you give any reference?and what are the difficulties in calculating the potential directly than to use method of images? $\endgroup$ – Paul Feb 1 '15 at 1:42
  • $\begingroup$ I googled and did not find any example, $\endgroup$ – Paul Feb 1 '15 at 1:50
  • $\begingroup$ The one example I know of is actually unpublished research from a fellow grad student. A google search found some other examples, here's one: arxiv.org/abs/1405.3197. As far as difficulties go, I would imagine the main difficulty is that now the equations are much more complicated than for electrostatics problems. But of course, the method of images only applies in very specific settings. $\endgroup$ – Surgical Commander Feb 1 '15 at 8:10
  • $\begingroup$ Any physical systems where such a boundary condition would be imposed would be interesting even if exotic, as the induced mass surface charge at the nearest point between the surface and the positive mass should be predominantly negative energy $\endgroup$ – lurscher Feb 2 '15 at 15:53
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The method of images works on the electrostatic case because the axis of symmetry of the mirror charges induces an equipotential line that is equivalent to the infinite conductor surface. In gravitational physics, there are no known instances of a physical surface that is at the same potential in the gravitational field.

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