Detweiler–Whiting Singular and Regular Fields

The Detweiler–Whiting decomposition replaces an ill-defined instruction—evaluate a point particle’s divergent retarded field on its own worldline—with a precise local split. The singular field has the same source and local divergence as the retarded field. Their difference is smooth on the worldline and satisfies the homogeneous linearized field equation. Under the stated first-order hypotheses, that regular field alone determines the scalar, electromagnetic, or gravitational self-force.

Required background. PDEs and Green Functions for Gravitation supplies distributional sources, fundamental solutions, and causal support. Helpful background. The Einstein Equation supplies the nonlinear parent theory; Weak Metric Perturbations and Gauge Freedom supplies linearized gauge transformations; Presymplectic Potentials and Currents supplies the field-space distinction between sourced and homogeneous perturbations; Black-Hole Perturbation Theory supplies the fixed-background wave operators.

Green-function bridge. For a linear differential operator $L$, a Green function is a bidistribution whose image under $L$ is an invariant delta distribution with a declared normalization. “Retarded” means support only when the field point lies in the causal future of the source point. A homogeneous two-point function instead satisfies $LH=0$. Subtracting two inhomogeneous Green functions with the same delta normalization necessarily produces a homogeneous object; smoothness requires the stronger statement that their local singular structures also agree.

Hadamard fields in a convex normal neighborhood

Assume a smooth globally hyperbolic background and a smooth timelike worldline $\gamma$. For the gravitational problem, take the background to be vacuum and work at first order in the small mass. Locally restrict $x$ and $x'$ to a convex normal neighborhood, so a unique geodesic connects them. Synge’s world function $\sigma(x,x')$ is one half the squared geodesic distance; with the site’s $(-,+,+,+)$ signature, $\sigma\lt 0$ for timelike separation and $\sigma\gt 0$ for spacelike separation.

Begin with a massless scalar field,

$$L\Phi \equiv(\Box-\xi R)\Phi =-4\pi q\int_\gamma \delta_4(x,z(\tau))\,d\tau.$$

The delta distribution is normalized by

$$\int f(x)\delta_4(x,x')\,dV_x=f(x')$$

for every smooth compactly supported test function $f$. The retarded and advanced Green functions obey

$$L G_{\pm}(x,x')=-4\pi\delta_4(x,x')$$

and have future or past causal support. Inside the convex normal neighborhood they possess the Hadamard form

$$G_{\pm} =U(x,x')\delta_{\pm}(\sigma) +V(x,x')\theta_{\pm}(-\sigma),$$

where $U$ and $V$ are smooth biscalars. The direct term lies on the null cone; $V$ produces the tail inside it. Transport equations along the unique connecting geodesic determine these coefficients from the operator and geometry. The construction, normalization, and support statements are derived in Poisson, Pound, and Vega 2011, §§ 14.1–14.5, eqs. (14.1)–(14.33).

The local singular Green function is

$$G_{\mathrm S}(x,x') =\frac12U(x,x')\delta(\sigma) -\frac12V(x,x')\theta(\sigma).$$

It is symmetric under $x\leftrightarrow x'$, obeys the same inhomogeneous equation as $G_{\mathrm{ret}}$, and vanishes for chronologically related points. Its support lies on the null cone and at spacelike separation, not throughout the causal past. Define

$$G_{\mathrm R}=G_{\mathrm{ret}}-G_{\mathrm S}.$$

Then $LG_{\mathrm R}=0$. Convolution with the worldline produces

$$\Phi_{\mathrm{ret}} =\Phi_{\mathrm S}+\Phi_{\mathrm R}.$$

Although $G_{\mathrm S}$ contains advanced-point information locally, $\Phi_{\mathrm R}$ on the worldline depends causally on the past once the construction is combined with the retarded solution. It is not the naive half-retarded minus half-advanced field in curved spacetime; that choice would generally retain dependence on the entire future tail.

The scalar, electromagnetic, and gravitational problems

The three theories share the split but not the same operator or force law.

Scalar charge. The field equation is the one above, and the regular force orthogonal to the four-velocity is

$$m a^\mu =q\left(g^{\mu\nu}+u^\mu u^\nu\right) \nabla_\nu\Phi_{\mathrm R}.$$

For a scalar charge the component $u^\mu\nabla_\mu\Phi_{\mathrm R}$ can also change the particle’s effective mass, depending on the matter coupling. It must not be discarded by importing electromagnetic intuition.

Electric charge. In Lorenz gauge, $\nabla_\mu A^\mu=0$, the vector potential satisfies

$$\left(\Box\delta_\mu{}^\nu-R_\mu{}^\nu\right)A_\nu =-4\pi j_\mu.$$

The vector Green function contains a parallel propagator in its delta normalization, and its Hadamard coefficients are bitensors. Subtraction gives a smooth homogeneous $A_\mu^{\mathrm R}$, and

$$m a^\mu =q F_{\mathrm R}^{\mu\nu}u_\nu, \qquad F_{\mu\nu}^{\mathrm R} =2\nabla_{[\mu}A_{\nu]}^{\mathrm R}.$$

Current conservation and Lorenz gauge are part of the hypotheses. The vector construction is not obtained by attaching a free index to the scalar result; the curvature coupling matters.

Point mass. On a smooth vacuum background, introduce the trace-reversed Lorenz-gauge perturbation $\bar h_{\mu\nu}$. Its linear operator has the schematic form

$$E_{\mu\nu}[\bar h] =\Box\bar h_{\mu\nu} +2R_{\mu}{}^\rho{}_{\nu}{}^\sigma\bar h_{\rho\sigma} =-16\pi G T_{\mu\nu}.$$

Here $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12g_{\mu\nu}g^{\rho\sigma}h_{\rho\sigma}$ and $\nabla^\nu\bar h_{\mu\nu}=0$; $h_{\mu\nu}$ includes the small mass factor rather than denoting its expansion coefficient. The tensor Green function carries parallel-propagated symmetrized index pairs. The same Hadamard argument yields

$$h_{\mu\nu}^{\mathrm{ret}} =h_{\mu\nu}^{\mathrm S} +h_{\mu\nu}^{\mathrm R}, \qquad E_{\mu\nu}[\bar h^{\mathrm R}]=0,$$

with $h_{\mu\nu}^{\mathrm R}$ smooth on $\gamma$ in a sufficiently regular gauge. To first order in the mass ratio, a nonspinning monopolar body obeys

$$a^\mu =-\frac12 \left(g^{\mu\nu}+u^\mu u^\nu\right) \left( 2h_{\nu\rho;\sigma}^{\mathrm R} -h_{\rho\sigma;\nu}^{\mathrm R} \right)u^\rho u^\sigma.$$

The Lorenz constraint and stress-energy conservation require a geodesic source at strict first order. A self-consistently accelerated source is consistent only order by order: the divergence produced by its first-order acceleration is of the next perturbative order and must be supplied by the higher-order field equations.

Equivalently, it is geodesic in $g_{\mu\nu}+h_{\mu\nu}^{\mathrm R}$ through the retained order. Detweiler and Whiting state the scalar decomposition and extend it to electromagnetic and gravitational fields in Detweiler and Whiting 2003, §§ III–IV and VI, especially eqs. (17)–(19), (24), and (39)–(41).

Four claims and the logic connecting them

The argument is clearest when four claims are kept separate.

First, the retarded field splits. This is a definition once $G_{\mathrm S}$ exists: $\Psi_{\mathrm R}=\Psi_{\mathrm{ret}}-\Psi_{\mathrm S}$ for the relevant scalar, vector, or tensor field. The singular field has the same distributional source as the retarded field.

Second, the regular field is smooth and homogeneous. Homogeneity follows algebraically from equal source normalization. Smoothness follows because the Hadamard singular terms cancel in a convex normal neighborhood. It is a local analytical result; simply subtracting arbitrary inhomogeneous solutions would not guarantee it.

Third, motion can be written in an effective field or metric. For scalar and electromagnetic charges, the regular field enters the ordinary force law. For gravity, matched small-body expansions show that the representative worldline is geodesic in $g+h^{\mathrm R}$ at first order. This is stronger than assigning a finite value to a divergent point-particle expression.

Fourth, the singular field exerts no direct force. In the original point-particle presentation this was the Detweiler–Whiting prescription, motivated by the singular field’s local symmetry and support. Later small-body matching gives the gravitational statement a derivational foundation: the singular piece matches the body’s local multipolar field, while the smooth regular piece supplies the external tidal field that accelerates the center of mass. The distinction between the original postulate and later justification is emphasized in Poisson, Pound, and Vega 2011, §§ 22.5 and 24.2, especially eqs. (22.80)–(22.84).

The conclusion is therefore not “a symmetric field can never exert a force.” It is that, under the stated perturbative and matching hypotheses, the locally singular sourced piece is absorbed into the body’s multipolar description, and the regular homogeneous piece produces the direct first-order self-force.

Gauge regularity and domain of the statement

For gravity, both the split and the coordinate acceleration depend on gauge. If

$$h_{\mu\nu}\mapsto h_{\mu\nu}+2\nabla_{(\mu}\xi_{\nu)},$$

then the representative worldline also moves. A gauge vector that is smooth enough on $\gamma$ maps a smooth regular field to another smooth regular field and transforms the self-force by the corresponding deviation term. A singular or discontinuous gauge vector can destroy the assumed local differentiability. Radiation gauges with strings or jumps therefore require a specially defined completion and averaging prescription; the Lorenz-gauge formula cannot be copied unchanged.

The Hadamard formula itself is local. Beyond a convex normal neighborhood, multiple null geodesics, caustics, and global topology complicate the Green function. A globally defined Detweiler–Whiting pair may exist, but local coefficients alone do not prove the required global extension. Moreover, Poisson, Pound, and Vega explicitly note in a static Schwarzschild example that a locally motivated pair can have the essential “singular sourced plus regular homogeneous” properties without its identification with the global Detweiler–Whiting pair being proved Poisson, Pound, and Vega 2011, § 1.10, after eq. (1.56). Essential local properties and global identity are distinct claims.

Other limitations are equally important. Nonvacuum gravitational backgrounds introduce Ricci and matter couplings. At second order, quadratic products of first-order singular fields are more singular than distributions can naively multiply, so the linear decomposition is insufficient. Spin and finite size add multipolar forces. The regular field is smooth only to the order and differentiability supplied by the exact singular subtraction; a finite-order puncture leaves a residual field of finite, not infinite, differentiability.

Checks that discriminate a correct split

A proposed singular field should pass four tests. Applying the linear operator must reproduce the exact delta-normalized source. Its local expansion must match every divergent and direction-dependent term needed for the desired derivative of the residual field. The difference from the retarded field must solve the homogeneous equation and approach a unique smooth limit on the worldline. Finally, an observable derived from the regular field must agree across admissible gauges after asymptotic time and mass normalization are matched.

These tests rule out common false substitutes. Subtracting only the Coulomb $1/s$ term can leave a direction-dependent derivative. Half-retarded plus half-advanced has the wrong curved-spacetime tail structure. A numerical puncture is normally only a finite local approximation to $h^{\mathrm S}$, not the exact singular field. Those practical approximations are developed on Mode-Sum Regularization and Effective-Source and Puncture Methods.

Exercise

Let $\Phi_{\mathrm{ret}}$ and $\Phi_{\mathrm S}$ satisfy the same scalar equation $L\Phi=-4\pi\rho$, and suppose their local expansions near the worldline are

$$\Phi_{\mathrm{ret}} =\frac{q}{s}+a+b_i n^i+O(s), \qquad \Phi_{\mathrm S} =\frac{q}{s}+b_i n^i+O(s),$$

where $s$ is spatial geodesic distance and $n^i$ its direction. Determine the equation obeyed by $\Phi_{\mathrm R}=\Phi_{\mathrm{ret}}-\Phi_{\mathrm S}$ and its worldline limit. Why would subtracting only $q/s$ be insufficient for a force calculation?

Solution

Linearity gives

$$L\Phi_{\mathrm R} =L\Phi_{\mathrm{ret}}-L\Phi_{\mathrm S} =0.$$

The local subtraction yields

$$\Phi_{\mathrm R}=a+O(s),$$

so the residual has the unique direction-independent limit $a$ on the worldline. If one removed only $q/s$, the remainder would contain $b_i n^i$. That term has no unique limit as $s\to0$ from different directions, and its derivative is even more singular. A self-force depends on a derivative of the regular field, so the singular approximation must be carried to sufficiently high local order, not merely through the leading Coulomb divergence.

References

• Detweiler, Steven, and Bernard F. Whiting. “Self-Force via a Green’s Function Decomposition.” Physical Review D 67, 024025 (2003). doi:10.1103/PhysRevD.67.024025.
• Poisson, Eric, Adam Pound, and Ian Vega. “The Motion of Point Particles in Curved Spacetime.” Living Reviews in Relativity 14, 7 (2011). doi:10.12942/lrr-2011-7.