Tidal Force Tensor

Back to Physics World
Back to Classical Mechanics

Consider a Cartesian coordinate system that is in free-fall in a gravitational field. Define F such that the potential energy function, V, has the value V = m_pF where m_p is the passive gravitational mass of the particle. The inertial mass of the particle is expressed as m_i. Then since V is defined such that the gravitational force, F_g, is the gradient of V then it follows that

g is the acceleration of the particle due to gravity. According to the Principle of Equivalence m_p = m_i = m. Therefore the masses of the particle cancel out to yield

The difference in gravitational acceleration of a test particle, dg, relative to a reference point in free-fall, is then found to be

The tidal force, F_T, is defined as

The quantity t_jk are the components of the tidal force tensor and have the values

The tidal force tensor is a Cartesian, hence the name. That t_jk is a Cartesian tensor is readily shown by showing that it transforms under an orthogonal transformation. Start by taking the derivative of the scalar F and using the chain rule to obtain

where a is matrix representing an orthogonal transformation and

are the elements of the transpose of a. Taking the derivative of Eq. (6) with respect to gives

Substituting the expression for the components of the tidal force tensor we get

Thus the quantities t_mn are the covariant components of a Cartesian tensor.

Back to Classical Mechanics
Back to Physics World