The Gravitational Tidal Tensor ($T$), derived from the negative second spatial derivatives of the gravitational potential ( $\phi$ ), describes the differential acceleration experienced by two adjacent particles in a gravitational field. This tensor is fundamental to understanding tidal effects, as it relates the change in acceleration (da) linearly to the particle separation vector ($d x$) via $d a^i=T_j^i d x^j$. Notably, the tensor is symmetric and, for a spherical mass distribution, its components $T_j^i= G M\left[\frac{3 x^i x^j}{r^j}-\frac{\delta_{j i}}{r^3}\right]$ reveal the dual nature of tidal forces: the off-diagonal terms are responsible for the shearing and stretching effects lateral to the mass center, while the diagonal terms govern the radial compression and stretching along the line of centers.
Properties and Physical Effects of the Gravitational Tidal Tensor#audio
The tensor T derived in this problem describes the tidal effect, which is the differential acceleration experienced by two closely separated particles in a gravitational field.
Relation to Potential: The tidal tensor is directly related to the second spatial derivatives of the gravitational potential ( $\phi$ ).
$$ T_j^i=-\frac{\partial^2 \phi}{\partial x^j \partial x^i} $$
This shows that tidal effects depend not on the strength of the field itself ( $-\nabla \phi$ ), but on how the field changes (its gradient).
Differential Acceleration: The relationship $d a^i=T_j^i d x^j$ means that the change in acceleration ( $d \vec{a}$ ) is a linear function of the separation vector ( $d \vec{x}$ ). This linearity is a result of the first-order Taylor approximation used for small displacements.
Symmetry of the Tensor: Because the order of differentiation for the potential does not matter $\left(\frac{\partial^2 \phi}{\partial x^i \partial x^i}=\frac{\partial^2 \phi}{\partial x^i \partial x^j}\right)$, the tidal tensor $T$ is symmetric ( $T_j^i=T_i^j$ ).
Tidal Force for Spherical Masses: For the specific case of a spherical mass distribution (like the Earth), the components of the tensor show the characteristic nature of tidal forces:
$$ T_j^i=G M\left[\frac{3 x^i x^j}{r^5}-\frac{\delta_{i j}}{r^3}\right] $$
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