Can we distinguish between two mass distributions in spacetime having the same effect over a test partlicle Einstein's equation is  $$8πT_{ab}=G_{ab}$$  where the left side contains the stress-energy tensor and the right side contains the Einstein tensor.   Is there exactly one unique stress-energy tensor corresponding to a given spacetime curvature? Or is it possible for one curvature (i.e. one spacetime metric) to be produced by multiple different mass configurations
Is every solution of Einstein field equations unique? Einstein's equation is  $$8 \pi T_{ab} = G_{ab},$$  where the left side contains the stress-energy tensor and the right side contains the Einstein tensor.  Is there exactly one unique stress-energy tensor corresponding to a given spacetime curvature? Or is it possible for one curvature (i.e. one spacetime metric) to be produced by multiple different mass configurations?
Is the harmonic oscillator potential unique in having equally spaced discrete energy levels? I was wondering if the good old quadratic potential was the only potential with equally spaced eigenvalues. Obviously you can construct others, such as a potential that is infinite in some places and quadratic in others, but that's only trivially different. I am not referring to equally spaced as a limiting behavior either, I mean truly integer spaced.  Any ideas? If not, is there a proof for its uniqueness?  If there are other potentials with equally spaced eigenvalues, can one use them as starting points for a free-field QFT? It would be interesting to know if there is a deeper mathematical relation between all of these potentials and whether they could be used to study interacting systems.