List of Figures

View Image Figure 1:
In flat spacetime, the retarded potential at x depends on the particle’s state of motion at the retarded point z(u) on the world line; the advanced potential depends on the state of motion at the advanced point z (v ).
View Image Figure 2:
In curved spacetime, the retarded potential at x depends on the particle’s history before the retarded time u; the advanced potential depends on the particle’s history after the advanced time v.
View Image Figure 3:
In curved spacetime, the singular potential at x depends on the particle’s history during the interval u ≤ τ ≤ v; for the regular potential the relevant interval is − ∞ < τ ≤ v.
View Image Figure 4:
Retarded coordinates of a point x relative to a world line γ. The retarded time u selects a particular null cone, the unit vector Ωa := ˆxa∕r selects a particular generator of this null cone, and the retarded distance r selects a particular point on this generator.
View Image Figure 5:
The base point ′ x, the field point x, and the geodesic segment β that links them. The geodesic is described by parametric relations zμ(λ ) and tμ = dzμ∕dλ is its tangent vector.
View Image Figure 6:
Fermi normal coordinates of a point x relative to a world line γ. The time coordinate t selects a particular point on the word line, and the disk represents the set of spacelike geodesics that intersect γ orthogonally at z(t). The unit vector ωa := ˆxa ∕s selects a particular geodesic among this set, and the spatial distance s selects a particular point on this geodesic.
View Image Figure 7:
Retarded coordinates of a point x relative to a world line γ. The retarded time u selects a particular null cone, the unit vector Ωa := ˆxa∕r selects a particular generator of this null cone, and the retarded distance r selects a particular point on this generator. This figure is identical to Figure 4.
View Image Figure 8:
The retarded, simultaneous, and advanced points on a world line γ. The retarded point x ′ := z(u) is linked to x by a future-directed null geodesic. The simultaneous point ¯x := z(t) is linked to x by a spacelike geodesic that intersects γ orthogonally. The advanced point x ′′ := z(v) is linked to x by a past-directed null geodesic.
View Image Figure 9:
The region within the dashed boundary represents the normal convex neighbourhood of the point x. The world line γ enters the neighbourhood at proper time τ< and exits at proper time τ>. Also shown are the retarded point z (u) and the advanced point z (v ).
View Image Figure 10:
A small body, represented by the black disk, is immersed in a background spacetime. The internal zone is defined by R ≪ 1, while the external zone is defined by R ≫ 𝜀. Since 𝜀 ≪ 1, there exists a buffer region defined by 𝜀 ≪ R ≪ 1. In the buffer region 𝜀∕R and R are both small.
View Image Figure 11:
The spacetime region Ω is bounded by the union of the spacelike surface Σ, the timelike tube Γ, and the null surface 𝒥.