Derivation and Calculation of the Gravitational Tidal Tensor

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.

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Properties and Physical Effects of the Gravitational Tidal Tensor#audio

Key takeaways

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:

$$ T_j^i=G M\left[\frac{3 x^i x^j}{r^5}-\frac{\delta_{i j}}{r^3}\right] $$

Off-Diagonal Terms $\left(\propto x^i x^j\right)$ : These terms cause particles separated laterally (e.g., in the $x y$-plane) to be pulled toward the center or pushed apart, responsible for the shearing or stretching effect.
Diagonal Terms $\left(\alpha 3\left(x^i\right)^2 / r^5-1 / r^3\right)$ : These describe the radial compression and stretching effects along the axis pointing toward the mass center.

✍️Mathematical Proof

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Black Holes to Ocean Tides Decoding the Gravitational Tidal Tensor

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