Tidal Tensor - Newton's Theory

Newton's Theory

In the field theoretic elaboration of Newtonian gravity, the central quantity is the gravitational potential, which obeys the Poisson equation

where is the mass density of any matter present. Note that this equation implies that in a vacuum solution, the potential is simply a harmonic function.

The tidal tensor is given by the traceless part

of the Hessian

where we are using the standard Cartesian chart for E3, with the Euclidean metric tensor

Using standard results in vector calculus, this is readily converted to expressions valid in other coordinate charts, such as the polar spherical chart

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