World line and retarded coordinates

1.5 World line and retarded coordinates

To flesh out the ideas contained in the preceding Section 1.4 I add yet another layer of mathematical formalism and construct a convenient coordinate system to chart a neighbourhood of the particle’s world line. In the next Section 1.6 I will display explicit expressions for the retarded, singular, and radiative fields of a point electric charge.

Let $γ$ be the world line of a point particle in a curved spacetime. It is described by parametric relations $μ z (τ )$ in which $τ$ is proper time. Its tangent vector is $μ μ u = dz ∕dτ$ and its acceleration is $μ μ a = Du ∕dτ$ ; we shall also encounter $μ μ ˙a ≡ Da ∕dτ$ .

On $γ$ we erect an orthonormal basis that consists of the four-velocity $uμ$ and three spatial vectors $eμ a$ labelled by a frame index $a = (1,2,3)$ . These vectors satisfy the relations $gμνuμu ν = − 1$ , $g u μeν= 0 μν a$ , and $g eμeν = δ μν a b ab$ . We take the spatial vectors to be Fermi–Walker transported on the world line: $μ μ De a∕d τ = aau$ , where

are frame components of the acceleration vector; it is easy to show that Fermi–Walker transport preserves the orthonormality of the basis vectors. We shall use the tetrad to decompose various tensors evaluated on the world line. An example was already given in Equation (17

) but we shall also encounter frame components of the Riemann tensor,

as well as frame components of the Ricci tensor,

We shall use $δab = diag(1,1,1)$ and its inverse $ab δ = diag(1,1,1)$ to lower and raise frame indices, respectively.

Consider a point $x$ in a neighbourhood of the world line $γ$ . We assume that $x$ is sufficiently close to the world line that a unique geodesic links $x$ to any neighbouring point $z$ on $γ$ . The two-point function $σ(x,z )$ , known as Synge’s world function [55 ], is numerically equal to half the squared geodesic distance between $z$ and $x$ ; it is positive if $x$ and $z$ are spacelike related, negative if they are timelike related, and $σ(x, z)$ is zero if $x$ and $z$ are linked by a null geodesic. We denote its gradient $∂ σ∕∂zμ$ by $σμ(x,z)$ , and $− σμ$ gives a meaningful notion of a separation vector (pointing from $z$ to $x$ ).

To construct a coordinate system in this neighbourhood we locate the unique point $′ x ≡ z(u)$ on $γ$ which is linked to $x$ by a future-directed null geodesic (this geodesic is directed from $′ x$ to $x$ ); I shall refer to $x′$ as the retarded point associated with $x$ , and $u$ will be called the retarded time. To tensors at $x′$ we assign indices $α ′$ , $β′$ , …; this will distinguish them from tensors at a generic point $z(τ)$ on the world line, to which we have assigned indices $μ$ , $ν$ , …. We have $σ(x,x ′) = 0$ , and $α′ ′ − σ (x, x)$ is a null vector that can be interpreted as the separation between $′ x$ and $x$ .

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.

The retarded coordinates of the point $x$ are $a (u, ˆx )$ , where $a a α′ ˆx = − eα′σ$ are the frame components of the separation vector. They come with a straightforward interpretation (see Figure 4 ). The invariant quantity

is an affine parameter on the null geodesic that links $x$ to $x′$ ; it can be loosely interpreted as the time delay between $x$ and $x ′$ as measured by an observer moving with the particle. This therefore gives a meaningful notion of distance between $x$ and the retarded point, and I shall call $r$ the retarded distance between $x$ and the world line. The unit vector

is constant on the null geodesic that links $x$ to $′ x$ . Because $a Ω$ is a different constant on each null geodesic that emanates from $′ x$ , keeping $u$ fixed and varying $a Ω$ produces a congruence of null geodesics that generate the future light cone of the point $x ′$ (the congruence is hypersurface orthogonal). Each light cone can thus be labelled by its retarded time $u$ , each generator on a given light cone can be labelled by its direction vector $Ωa$ , and each point on a given generator can be labelled by its retarded distance $r$ . We therefore have a good coordinate system in a neighbourhood of $γ$ .

To tensors at $x$ we assign indices $α$ , $β$ , …. These tensors will be decomposed in a tetrad $(eα,eα) 0 a$ that is constructed as follows: Given $x$ we locate its associated retarded point $x′$ on the world line, as well as the null geodesic that links these two points; we then take the tetrad $(uα′,eα′) a$ at $x′$ and parallel transport it to $x$ along the null geodesic to obtain $(eα,eα) 0 a$ .