Forces, Structure, and Limits of Geodesic Motion

Geodesic motion is the monopole, force-free, test-body limit—not a universal equation for every object called “free.” Electromagnetic force, pressure, thrust, spin, finite size, radiation reaction, and a body’s own gravitational field each introduce a distinct correction. Their covariant form and controlling scale determine whether the geodesic approximation is justified.

Required background. Affine and Variational Geodesics supplies affine geodesics, Jacobi fields, and their variational characterization; What the Equivalence Principles Do and Do Not Imply supplies local-inertial reasoning and the scope of universal free fall.

Helpful background. Events, Intervals, and Minkowski Geometry supplies causal types and timelike and null worldlines; Tensor Fields and Natural Operations supplies coordinate-independent tensor operations.

The geodesic benchmark

For a massive representative worldline with unit tangent $u^\mu$, define

$$a^\mu=u^\nu\nabla_\nu u^\mu.$$

Ideal geodesic motion is $a^\mu=0$. If rest mass remains constant, differentiating $u^2=-1$ gives $u_\mu a^\mu=0$, so any admissible four-force $f^\mu=ma^\mu$ is spatial in the instantaneous rest frame.

This benchmark presupposes a fixed background metric and a structureless monopole. A coordinate change can make the connection vanish at one event, but it cannot make a nonzero vector $a^\mu$ vanish. The question is therefore not whether a trajectory looks curved on a coordinate plot; it is whether a covariant acceleration or multipolar correction is small in the regime being modeled.

Applied forces and internal stress

For a charged point particle,

$$ma^\mu=qF^\mu{}_{\nu}u^\nu.$$

Antisymmetry of $F_{\mu\nu}$ makes the right-hand side orthogonal to $u^\mu$, providing a normalization check. This equation reduces to a geodesic when $qF^\mu{}_{\nu}u^\nu=0$. Reall obtains the curved-spacetime Lorentz force by minimal coupling in Reall 2013, §5.1, pp. 49–50, PDF, while Carroll gives the same covariant modification in Carroll 1997, Ch. 3, p. 70, PDF.

Thrust, drag, contact forces, and pressure gradients likewise give nonzero proper acceleration. Fluid elements are not generally geodesic when pressure varies across their rest spaces, even if the fluid as a whole is isolated. Calling such motion “free” because no coordinate force has been written down conceals the physical stress that an accelerometer can detect.

Spatially varying forces also affect comparisons between bodies. For a smooth family of unit timelike tangents $u$ with acceleration $a=\nabla_u u$, choose a connecting field $\xi$ with $[u,\xi]=0$. Torsion freedom gives $\nabla_u\xi=\nabla_\xi u$, and commuting the next two derivatives yields

$$\frac{D^2\xi^\mu}{d\tau^2} =R^\mu{}_{\alpha\beta\gamma}u^\alpha u^\beta\xi^\gamma +\xi^\nu\nabla_\nu a^\mu.$$

The first term is geodesic deviation; the second is the difference in applied acceleration across the family. The connecting field need not remain spatial, so an actual instrument comparison must also specify its rest-space projection. Even in flat spacetime, a nonuniform applied force can produce relative acceleration. A gradiometer infers curvature only after such force gradients and the motion of its reference frame are controlled.

Spin and finite-size multipoles

An extended body occupies a worldtube, not a unique worldline. At monopole order, stress-energy conservation selects geodesic motion. At pole–dipole order, momentum $p^\mu$ and spin $S^{\alpha\beta}$ obey, with the site’s curvature convention,

$$\frac{Dp^\mu}{d\tau} =-\frac12R^\mu{}_{\nu\alpha\beta} u^\nu S^{\alpha\beta}+F^\mu_{\rm ext},$$ $$\frac{DS^{\mu\nu}}{d\tau} =p^\mu u^\nu-p^\nu u^\mu+N^{\mu\nu}_{\rm ext}.$$

A spin-supplementary condition, such as $p_\mu S^{\mu\nu}=0$, is also required to choose a centroid inside the body. Momentum and tangent need not be parallel. At quadrupole order, representative force terms contain curvature gradients,

$$F^\mu_{\rm quad} =-\frac16J^{\alpha\beta\gamma\delta} \nabla^\mu R_{\alpha\beta\gamma\delta}+\cdots,$$

with the precise moments and torque depending on the adopted multipole convention. Steinhoff and Puetzfeld derive the monopole, pole–dipole, and quadrupole hierarchy from stress-energy conservation in Steinhoff and Puetzfeld 2010, §§IV–VI, pp. 3–11.

If $L$ is body size and $\mathcal R$ a common scale for curvature and its variation, spin and quadrupole effects are estimated by ratios such as $\lvert S\rvert/(m\mathcal R)$ and $(L/\mathcal R)^2$. When those scales differ, the displayed forces give the safer estimates: $|R|\,|S|/m$ and $|\nabla R|\,|J|/m$ must be small compared with the acceleration accuracy required. Small size relative to a curvature radius alone does not bound a large curvature gradient. Strong internal gravity does not automatically invalidate a small-body expansion, but it demands a derivation that controls the body’s self-field rather than treating it as an elementary point.

Self-force and radiation reaction

A mass perturbs the geometry through which it moves. In a small-mass-ratio expansion $g_{\mu\nu}=g^{(0)}_{\mu\nu}+h_{\mu\nu}$, the leading worldline is a geodesic of $g^{(0)}_{\mu\nu}$. At the next order, the suitably regular part of the body’s field changes the motion; dissipative pieces account for radiation reaction, while conservative pieces shift orbital relations. The retarded field is singular on an ideal point source, so simply inserting it into a force law is not defined.

Poisson, Pound, and Vega explain the singular–regular split, gravitational self-force equation, and small-body justification in Poisson, Pound, and Vega 2011, §§1.9, 19.6, and 20–22.6, pp. 15–17 and 115–138, PDF. A perturbative displacement of a worldline can be gauge dependent; conclusions should be expressed through a specified gauge or invariant observable. Detailed self-force and two-body schemes lie beyond this chapter.

Choosing the controlled approximation

The useful hierarchy is summarized below. Each row names a physical omission, not a coordinate artifact.

Effect	Leading covariant signature	Typical control
Applied force	$ma^\mu=f^\mu_{\rm ext}$	force-induced acceleration small on the problem’s scale
Spin	$R^\mu{}_{\nu\alpha\beta}u^\nu S^{\alpha\beta}$	$\lvert S\rvert/(m\mathcal R)\ll1$
Finite size	$J\cdot\nabla R$ and higher multipoles	moments and curvature derivatives controlled across the body
Self-field	regularized field acting on the representative curve	perturbative mass or charge ratio small
Radiation reaction	history-dependent conservative and dissipative response	radiation-reaction time long compared with local motion

Before using a geodesic, state which effects are neglected and compare their estimated acceleration or phase accumulation with the required accuracy. A correction can be instantaneously small yet accumulate over a long inspiral. Conversely, a large coordinate acceleration does not by itself signal failure if the covariant acceleration remains zero.

Exercise

Choose the first controlled model beyond—or including—geodesic motion in each case: (a) a neutral, negligibly spinning capsule with $L/\mathcal R\ll1$ observed for only a few dynamical times; (b) a compact gyroscope for which $\lvert S\rvert/(m\mathcal R)$ is comparable to the requested fractional accuracy; (c) a nonspinning extended body for which $(L/\mathcal R)^2$ is comparable to that accuracy; (d) a small mass on a very long inspiral, where the instantaneous mass-ratio correction is small but its accumulated phase is not.

Solution

(a) The geodesic monopole model is controlled, provided applied forces and the capsule’s self-field are also below tolerance. (b) Retain pole–dipole spin-curvature coupling and specify a spin-supplementary condition. (c) Retain quadrupole and other finite-size terms, with constitutive moments rather than size alone. (d) Use a self-force or consistently matched two-body expansion: a small instantaneous acceleration is insufficient when it accumulates over many cycles. In every case, the comparison is with the required observable accuracy, not merely with unity.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997), Ch. 3, printed pp. 54–97. arXiv.
• Poisson, Eric, Adam Pound, and Ian Vega. “The Motion of Point Particles in Curved Spacetime.” Living Reviews in Relativity 14, article 7 (2011). DOI; arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), § 5.1, printed pp. 49–50. PDF.
• Steinhoff, Jan, and Dirk Puetzfeld. “Multipolar Equations of Motion for Extended Test Bodies in General Relativity.” Physical Review D 81 (2010): 044019. DOI; arXiv.