S3U

Science / Forces / Graduate / sfe20

Gravity: Tidal Fields and Local Free Fall

Analyze relative acceleration without confusing a local free-fall frame with the absence of curvature.

All worksheets
5 questions0m 0s

Prerequisites

Multivariable calculus, matrix eigenvalues, Newtonian fields, and the local equivalence principle.

Learn the skill

In Newtonian vacuum outside a point mass, g(r) = -mu r/|r|^3, where mu = GM. For a small separation vector xi, relative acceleration is J xi to first order, where J = (mu/r^3)(3nn^T - I) and n is the radial unit vector. The eigenvalues are 2mu/r^3 radially and -mu/r^3 in each transverse direction.

Worked example

Two particles separated radially by a small outward displacement L have relative outward acceleration approximately 2mu L/r^3: the outer particle is pulled inward less strongly. This is a weak-field, slow-motion local approximation.

Two small masses are at center distances r and r plus L outside a spherical source, with L much less than r. The nearer mass has a stronger inward acceleration, increasing radial separation.
Two small masses are at center distances r and r plus L outside a spherical source, with L much less than r. The nearer mass has a stronger inward acceleration, increasing radial separation.
Question 1 For outward radial separation L much smaller than r, relative radial acceleration is what?
Question 2 For a small transverse displacement y, the transverse relative acceleration is what?
Question 3 What is the trace of J at a vacuum point outside the mass?
Question 4 What can a sufficiently local freely falling frame remove at a chosen event?
Question 5 At fixed small L, doubling r changes the leading radial tidal acceleration by what factor?

Further inquiry

Differentiate the vector field to derive J, diagonalize it in a radial basis, and interpret the trace in vacuum. Compare with geodesic deviation in general relativity, explicitly declaring your curvature-sign convention and the weak-field, slow-motion assumptions needed to recover this result.

Review criteria

  • Keep vector directions and eigenvalue signs explicit.
  • State the first-order separation approximation and vacuum assumption.
  • Distinguish common acceleration from measurable relative tidal acceleration.