In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the tidal tensor represents
Contents
- tidal accelerations of a cloud of (electrically neutral, nonspinning) test particles,
- tidal stresses in a small object immersed in an ambient gravitational field.
Newton's theory
In the field theoretic elaboration of Newtonian gravity, the central quantity is the gravitational potential
where
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
As an example, we compute the tidal tensor for the vacuum field outside an isolated spherically symmetric massive object in two different ways.
Let us adopt the frame obtained from the polar spherical chart for our three-dimensional Euclidean space:
We will directly compute the tidal tensor, expressed in this frame, by elementary means, as follows. First, compare the gravitational forces on two nearby observers lying on the same radial line:
Because in discussing tensors we are dealing with multilinear algebra, we retain only first order terms, so
By using the small angle approximation, we have ignored all terms of order
Next, let us plug the gravitational potential
After a rotation of our frame, which is adapted to the polar spherical coordinates, this expression agrees with our previous result. (The easiest way to see this is probably to set y,z to zero so that the off-diagonal terms vanish and
General relativity
In general relativity, the tidal tensor is identified with the electrogravitic tensor, which is one piece of the Bel decomposition of the Riemann tensor.