Green’s functions in curved spacetime

1.4 Green’s functions in curved spacetime

In a curved spacetime with metric $gαβ$ the wave equation for the vector potential becomes

where $αβ □ = g ∇ α∇ β$ is the covariant wave operator and $R αβ$ is the spacetime’s Ricci tensor; the Lorenz gauge conditions becomes $∇ αA α = 0$ , and $∇ α$ denotes covariant differentiation. Retarded and advanced Green’s functions can be defined for this equation, and solutions to Equation (13

) take the same form as in Equations (6

) and (7

), except that $dV ′$ now stands for $∘ -----′- 4 ′ − g(x )d x$ .

The causal structure of the Green’s functions is richer in curved spacetime: While in flat spacetime the retarded Green’s function has support only on the future light cone of $x′$ , in curved spacetime its support extends inside the light cone as well; $G α+β′(x, x′)$ is therefore nonzero when $x ∈ I+ (x ′)$ , which denotes the chronological future of $x′$ . This property reflects the fact that in curved spacetime, electromagnetic waves propagate not just at the speed of light, but at all speeds smaller than or equal to the speed of light; the delay is caused by an interaction between the radiation and the spacetime curvature. A direct implication of this property is that the retarded potential at $x$ is now generated by the point charge during its entire history prior to the retarded time $u$ associated with $x$ : The potential depends on the particle’s state of motion for all times $τ ≤ u$ (see Figure 2 ).

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 $v$ .

Similar statements can be made about the advanced Green’s function and the advanced solution to the wave equation. While in flat spacetime the advanced Green’s function has support only on the past light cone of $′ x$ , in curved spacetime its support extends inside the light cone, and $α ′ G −β′(x,x )$ is nonzero when $− ′ x ∈ I (x)$ , which denotes the chronological past of $′ x$ . This implies that the advanced potential at $x$ is generated by the point charge during its entire future history following the advanced time $v$ associated with $x$ : The potential depends on the particle’s state of motion for all times $τ ≥ v$ .

The physically relevant solution to Equation (13 ) is obviously the retarded potential $α A ret(x)$ , and as in flat spacetime, this diverges on the world line. The cause of this singular behaviour is still the pointlike nature of the source, and the presence of spacetime curvature does not change the fact that the potential diverges at the position of the particle. Once more this behaviour makes it difficult to figure out how the retarded field is supposed to act on the particle and determine its motion. As in flat spacetime we shall attempt to decompose the retarded solution into a singular part that exerts no force, and a smooth radiative part that produces the entire self-force.

To decompose the retarded Green’s function into singular and radiative parts is not a straightforward task in curved spacetime. The flat-spacetime definition for the singular Green’s function, Equation (9 ), cannot be adopted without modification: While the combination half-retarded plus half-advanced Green’s functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Equation (13 ), this candidate for the singular Green’s function would produce a self-force with an unacceptable dependence on the particle’s future history. For suppose that we made this choice. Then the radiative Green’s function would be given by the combination half-retarded minus half-advanced Green’s functions, just as in flat spacetime. The resulting radiative potential would satisfy the homogeneous wave equation, and it would be smooth on the world line, but it would also depend on the particle’s entire history, both past (through the retarded Green’s function) and future (through the advanced Green’s function). More precisely stated, we would find that the radiative potential at $x$ depends on the particle’s state of motion at all times $τ$ outside the interval $u < τ < v$ ; in the limit where $x$ approaches the world line, this interval shrinks to nothing, and we would find that the radiative potential is generated by the complete history of the particle. A self-force constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and radiative Green’s functions.

The proper definitions were identified by Detweiler and Whiting [23 ], who proposed the following generalization to Equation (9 ):

The two-point function $H αβ′(x,x′)$ is introduced specifically to cure the pathology described in the preceding paragraph. It is symmetric in its indices and arguments, so that $GS ′(x,x′) αβ$ will be also (since the retarded and advanced Green’s functions are still linked by a reciprocity relation); and it is a solution to the homogeneous wave equation, $α ′ α γ ′ □H β′(x,x ) − R γ(x)H β′(x,x ) = 0$ , so that the singular, retarded, and advanced Green’s functions will all satisfy the same wave equation. Furthermore, and this is its key property, the two-point function is defined to agree with the advanced Green’s function when $x$ is in the chronological past of $x′$ : $H α ′(x,x′) = G α ′(x,x ′) β − β$ when $x ∈ I− (x′)$ . This ensures that $G α′(x,x′) Sβ$ vanishes when $x$ is in the chronological past of $′ x$ . In fact, reciprocity implies that $α ′ H β′(x,x )$ will also agree with the retarded Green’s function when $x$ is in the chronological future of $′ x$ , and it follows that the symmetric Green’s function vanishes also when $x$ is in the chronological future of $x ′$ .

The potential $Aα(x ) S$ constructed from the singular Green’s function can now be seen to depend on the particle’s state of motion at times $τ$ restricted to the interval $u ≤ τ ≤ v$ (see Figure 3 ). Because this potential satisfies Equation (13 ), it is just as singular as the retarded potential in the vicinity of the world line. And because the singular Green’s function is symmetric in its arguments, the singular potential can be shown to exert no force on the charged particle. (This requires a lengthy analysis that will be presented in the bulk of the paper.)

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 $− ∞ < τ ≤ v$ .

The Detweiler–Whiting [23 ] definition for the radiative Green’s function is then

The potential $A αR(x)$ constructed from this depends on the particle’s state of motion at all times $τ$ prior to the advanced time $v$ : $τ ≤ v$ . Because this potential satisfies the homogeneous wave equation, it is well behaved on the world line and its action on the point charge is well defined. And because the singular potential $α AS (x )$ can be shown to exert no force on the particle, we conclude that $α A R(x)$ alone is responsible for the self-force.

From the radiative potential we form an electromagnetic field tensor $F Rαβ = ∇ αARβ − ∇βARα$ , and the curved-spacetime generalization to Equation (4 ) is

where $uμ = dzμ∕d τ$ is again the charge’s four-velocity, but $aμ = Du μ∕dτ$ is now its covariant acceleration.