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 radiative 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 a a Ω ≡ ˆx ∕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 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 a a Ω ≡ ˆx ∕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 black hole, represented by the black disk, is immersed in a background spacetime. The internal zone extends from r = 0 to r = ri ≪ ℛ, while the external zone extends from r = re ≫ m to r = ∞. When m ≪ ℛ there exists a buffer zone that extends from r = re to r = ri. In the buffer zone m ∕r and r∕ℛ are both small.