Newtonian Tidal Stretch Tensor

The Newtonian tidal tensor describes how gravity varies near a reference point. When an object spans a finite distance in a gravitational field, the varying acceleration across its body produces tidal forces that stretch it radially and compress it transversely.

Jacobian of Gravity

For a point mass, the gravitational acceleration at position vector x is g(x) = -GM * x / |x|³. The tidal stretch tensor T is the Jacobian derivative of g(x), which equivalently is the negative Hessian of the gravitational potential.

At a reference position (r, 0, 0), the tidal tensor takes the diagonal form T = (GM/r³) diag(2, -1, -1). The radial eigenvalue is 2GM/r³, representing stretching, while the two transverse eigenvalues are -GM/r³, representing compression. Away from the source mass, the sum of these eigenvalues (the trace) is exactly zero, consistent with Laplace's equation in empty space.

The linear approximation for the relative acceleration of a nearby parcel at displacement δ is T * δ. This approximation closely tracks the exact difference g(r + δ) - g(r) for small offsets, but diverging terms appear at second-order cross-components.

Calculating Linear Approximation Error

Consider a reference radius of r = 2, GM = 1, and a radial offset fraction of 0.05. The reference position is (2, 0, 0), and the radial displacement is dx = 0.05 * 2 = 0.1, dy = 0.

The reference acceleration is g(2) = -1 * 2 / 2³ = -0.250000000000000 in the x-direction.
The parcel's exact position is (2.1, 0, 0), yielding exact acceleration g(2.1) = -1 / 2.1² ≈ -0.226757369614512 in the x-direction.
The exact relative acceleration is thus -0.226757369614512 - (-0.250000000000000) = 0.023242630385488.
The tidal radial eigenvalue is 2 * 1 / 2³ = 0.25. The linear tidal acceleration is 0.25 * 0.1 = 0.025.
The magnitude of the difference (linearization error) is |0.023242630385488 - 0.025| = 0.001757369614512.

Exploring the Tidal Forces

Use the interactive model to observe tidal forces across varying radii, offsets, and directions.

Set the Offset fraction to a small value (e.g., 0.01) to verify that the linear approximation very closely matches the exact difference vector.
Increase the Offset fraction (e.g., to 0.1) and notice how the exact relative acceleration diverges from the linear approximation, causing a growing linearization error.
Switch the Direction from Radial to Transverse. Observe that the relative acceleration vector flips its orientation to reflect transverse compression (negative eigenvalue) instead of radial stretching.
Compare the Radial tidal eigenvalue with the Transverse tidal eigenvalue to see that the stretching force per unit length is twice as strong as the compressional force.

Assumptions and Limitations

This module considers only static Newtonian gravity from a single point mass. It does not include the self-gravity of the displaced parcel, general relativistic corrections (where the tidal tensor relates to the Riemann curvature tensor), or orbital Coriolis forces (as found in the Hill approximation).

Primary References

For more details on tidal tensors and their role in general relativity versus Newtonian mechanics, refer to the literature on the Newtonian limit of Einstein's field equations.

https://arxiv.org/abs/gr-qc/0612140

Newtonian Tidal Stretch Tensor