Point-Particle Actions and Reparameterization Invariance

A structureless relativistic body is represented at long distances by a worldline, but the parameter used to label points on that curve is not physical. Reparameterization invariance forces the leading action to depend on the geometric proper length and yields geodesic motion. The same symmetry then organizes every allowed finite-size, spin, and dissipative correction.

Required background. The Gibbons-Hawking-York Boundary Term supplies the gravitational variational problem, and Weak Metric Perturbations and Gauge Freedom supplies the expansion about a background metric.

Helpful background. The Einstein Equation fixes the bulk field law, Presymplectic Potentials and Currents clarifies variational boundary terms and generators, and Black-Hole Perturbation Theory supplies response data used when the effective theory is matched.

The invariant worldline action

Let $z^\mu(\lambda)$ be a timelike curve with arbitrary monotonically increasing parameter $\lambda$. The leading action is

$$S_{\mathrm{pp}} =-m\int d\lambda\, \sqrt{-g_{\mu\nu}(z)\dot z^\mu\dot z^\nu}, \qquad \dot z^\mu=\frac{dz^\mu}{d\lambda}.$$

Under $\lambda\mapsto\lambda'(\lambda)$, the velocity and measure transform inversely, so the integrand is invariant. Equivalently,

$$S_{\mathrm{pp}}=-m\int d\tau,$$

where $d\tau^2=-g_{\mu\nu}dz^\mu dz^\nu$. The coefficient $m$ is a Wilson coefficient: at this order it is the mass measured by matching the long-distance field of the compact object.

Vary the worldline while holding its endpoints fixed. Writing

$$L=-m\sqrt{-\dot z^2},$$

the Euler–Lagrange equation is

$$\frac{d}{d\lambda} \left( \frac{m g_{\mu\nu}\dot z^\nu}{\sqrt{-\dot z^2}} \right) -\frac{m}{2\sqrt{-\dot z^2}} \partial_\mu g_{\alpha\beta}\dot z^\alpha\dot z^\beta=0.$$

Choosing proper time after variation, so $-\dot z^2=1$, gives

$$u^\nu\nabla_\nu u^\mu=0.$$

This is the independent normalization check: the leading worldline action reproduces geodesic motion without choosing a preferred parameter. The derivation and its role in gravitational EFT are given in Porto 2016, §§6.1–6.2, pp. 27–30.

Einbein form and the mass shell

An auxiliary einbein $e(\lambda)$ makes the gauge structure more transparent:

$$S[z,e] =\frac12\int d\lambda \left( e^{-1}g_{\mu\nu}\dot z^\mu\dot z^\nu -em^2 \right).$$

Varying $e$ gives

$$g_{\mu\nu}\dot z^\mu\dot z^\nu+e^2m^2=0.$$

Substitution returns the square-root action, up to the branch chosen for a future-directed timelike curve. Canonical momentum is

$$p_\mu=e^{-1}g_{\mu\nu}\dot z^\nu,$$

and the constraint becomes

$$g^{\mu\nu}p_\mu p_\nu+m^2=0.$$

The einbein Hamiltonian is $H_\lambda=e(p^2+m^2)/2$, with the additional constraint that the momentum conjugate to $e$ vanishes. It therefore vanishes on the constraint surface, but its derivatives still generate evolution in $\lambda$. Coordinate-time evolution arises after a gauge choice such as $z^0=t$. Gauge fixing before variation is allowed only if the missing constraint is retained. Otherwise the reduced theory can lose the mass shell.

For example, in flat spacetime $p^2=-p_0^2+\mathbf p^2$. Choosing the future-directed branch gives $p_0=-\sqrt{m^2+\mathbf p^2}$; the reduced action is $\int dt\,[\mathbf p\cdot\dot{\mathbf z}-H_t]$ with $H_t=-p_0=\sqrt{m^2+\mathbf p^2}$. Thus a vanishing parameter Hamiltonian does not imply a vanishing physical energy.

The massless limit is smooth in the einbein form but singular in $-m\int d\tau$. This is a useful reminder that algebraically equivalent actions can have different convenient domains, even when they describe the same massive trajectories.

Coupling the worldline to gravity

The full long-distance action begins

$$S_{\mathrm{eff}} =\frac{1}{16\pi G} \int d^4x\,\sqrt{-g}\,R +\sum_A S_{\mathrm{pp}}[z_A,g] +S_{\mathrm{finite\ size}} +S_{\mathrm{diss}}.$$

Varying the metric yields the point-particle stress tensor

$$T^{\mu\nu}(x) =\frac{m}{\sqrt{-g}} \int d\tau\,u^\mu u^\nu \delta^{(4)}(x-z(\tau)).$$

Its distributional conservation is equivalent to geodesic motion at leading order. The delta function is not a claim that the physical body is literally structureless. It is the first term of a scale expansion valid when the external wavelength and curvature radius are much larger than the body size.

In a binary, integrating out gravitational modes with different scalings produces conservative potentials, radiation, reaction, and scale-dependent coefficients. Gauge fixing for the gravitational field and gauge fixing for the worldline parameter are distinct operations; both must leave invariant observables unchanged.

Which corrections are allowed

Worldline operators are classified by general covariance, reparameterization invariance, internal symmetries, parity, and power counting. For a nonspinning object, the first curvature terms include

$$S_{\mathrm{finite\ size}} =\frac12\int d\tau \left( C_E E_{\mu\nu}E^{\mu\nu} +C_B B_{\mu\nu}B^{\mu\nu} +\cdots \right),$$

where

$$E_{\mu\nu}=C_{\mu\alpha\nu\beta}u^\alpha u^\beta$$

is the electric Weyl tensor and $B_{\mu\nu}$ is its magnetic counterpart with a stated dual convention. The coefficients $C_E$ and $C_B$ encode object response and are fixed by matching, not by symmetry.

Some lower-dimensional curvature operators are redundant in vacuum or removable by field redefinitions. Redundancy means that on-shell observables are unchanged after coefficients and variables are transformed consistently; it does not mean that one can delete an operator from one calculation and compare off-shell potentials term by term.

The foundational scale separation and operator classification are developed in Goldberger and Rothstein 2006, §§II–IV. A body of radius $R$ in external curvature scale $\mathcal R$ admits this local expansion only for $R/\mathcal R\ll1$ and below internal resonances. Near tidal disruption or a mode resonance, additional degrees of freedom must be retained.

Reparameterization identities and checks

Homogeneity of the square-root Lagrangian in $\dot z^\mu$ implies

$$\dot z^\mu\frac{\partial L}{\partial\dot z^\mu}=L.$$

Hence the canonical Hamiltonian $p_\mu\dot z^\mu-L$ vanishes before gauge fixing. This is not zero physical energy; it is the signature of a constrained parameter-invariant system. After choosing an asymptotic time coordinate, its conjugate momentum supplies the physical energy.

Three checks should accompany any worldline action:

1. Parameter check: the integrand and measure combine to a scalar under $\lambda$ redefinitions.
2. Constraint check: gauge fixing preserves the mass shell or its corrected analogue.
3. Matching check: a representative observable, such as low-frequency scattering or tidal response, fixes each nonminimal coefficient.

The leading action fails for an extended body when $R/\mathcal R$ is not small, for strong internal excitation, or when a multipolar truncation cannot resolve the desired accuracy. It also omits self-force regularization: a singular field evaluated on the same worldline requires a separate prescription rather than naive substitution.

Exercises

Show directly that the square-root action is invariant under an orientation-preserving reparameterization $\lambda\mapsto\lambda'$.

Solution

The velocity transforms as

$$\frac{dz^\mu}{d\lambda'} =\frac{d\lambda}{d\lambda'}\dot z^\mu.$$

Therefore the square root acquires the positive factor $d\lambda/d\lambda'$, while $d\lambda'=(d\lambda'/d\lambda)d\lambda$. The factors cancel, leaving $d\lambda\sqrt{-\dot z^2}=d\tau$ unchanged.

Vary the einbein action with respect to $e$ and eliminate it. Which sign choice reproduces a future-directed massive worldline?

Solution

The constraint gives

$$e=\frac{\sqrt{-\dot z^2}}{m}$$

for positive $m$ and an orientation in which increasing $\lambda$ follows the future-directed curve. Then $p^\mu=e^{-1}\dot z^\mu$ is future directed and substitution gives $S=-m\int d\lambda\sqrt{-\dot z^2}$. At fixed oriented curve, choosing $e\lt0$ instead reverses the momentum relative to its tangent and changes the sign of the square-root action. It is not an allowed orientation-preserving reparameterization of the positive-energy branch.

References

• Goldberger, Walter D., and Ira Z. Rothstein. “An Effective Field Theory of Gravity for Extended Objects.” Physical Review D 73 (2006): 104029. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.