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If (M,g) is a riemannian manifold, I define the Laplace-Einstein equations:
μric(g)+μ′Δ(ric(g))=λg+λ′Δ(g)
where ric is the Ricci curvature and Δ is the Laplacian.
Have we black holes solutions of the Laplace-Einstein equations?
Is there any physical motivation for this choice of equations? The equations contain 4th-order derivatives of the metric; one motivation for Einstein's equations was to have equations with at most second order derivatives of the metric.
As for black hole solutions: Why don't you start out like Karl Schwarzschild and try to derive a spherically symmetric solution?
It may be interesting to note that with the Laplace-Einstein equations, the Einstein tensor ric(g)−λg can be a proper vector of the Laplacian operator.
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