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Wikipedia / Tidal tensor

In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the tidal tensor represents:

  1. tidal accelerations of a cloud of (electrically neutral, nonspinning) test particles

  2. tidal stresses in a small object immersed in an ambient gravitational field

The tidal tensor describes the relative acceleration due to gravity between two test masses separated by an infinitesimal direction.

Tidal tensor for a spherical body

The most common example of tides is the tidal force around a spherical body (e.g., a planet or a moon). Here we compute the tidal tensor for the gravitational field outside an isolated spherically symmetric massive object. According to Newton's gravitational law, the acceleration a at a distance r from a central mass m is

(to simplify the math, in the following derivations we use the convention of setting the gravitational constant G to one. To calculate the differential accelerations, the results are to be multiplied by G.)

Let us adopt the frame in polar coordinates for our three-dimensional Euclidean space, which are given the subscripts 1, 2, and 3 respectively.

We will directly compute each component of the tidal tensor, expressed in this frame. First, compare the gravitational forces on two nearby objects lying on the same radial line at distances from the central body differing by a distance h:

Because in discussing tensors we are dealing with multilinear algebra.

but displaced by an (infinitesimal) distance h direction. Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to the sphere which has magnitude

.

This is the Coulomb form characteristic of spherically symmetric central force fields in Newtonian physics.

Hessian formulation

In the more general case where the mass is not a single spherically symmetric central object, the tidal tensor can be derived from the gravitational potential, which obeys the Poisson equation:

is the Laplace operator. 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

Spherically symmetric field

, which gives

, and then invoke the spherical symmetry.

In General Relativity

In general relativity, the tidal tensor is generalized by the Riemann curvature tensor of the curvature tensor.

See also

References