Organization of this review

1.11 Organization of this review

The main body of the review begins in Section 2 with a description of the general theory of bitensors, the name designating tensorial functions of two points in spacetime. I introduce Synge’s world function $′ σ (x, x )$ and its derivatives in Section 2.1, the parallel propagator $α ′ g α′(x, x)$ in Section 2.3, and the van Vleck determinant $Δ (x,x′)$ in Section 2.5. An important portion of the theory (covered in Sections 2.2 and 2.4) is concerned with the expansion of bitensors when $x$ is very close to $x′$ ; expansions such as those displayed in Equations (23

) and (24

) are based on these techniques. The presentation in Section 2 borrows heavily from Synge’s book [55

] and the article by DeWitt and Brehme [24

]. These two sources use different conventions for the Riemann tensor, and I have adopted Synge’s conventions (which agree with those of Misner, Thorne, and Wheeler [40

]). The reader is therefore warned that formulae derived in Section 2 may look superficially different from what can be found in DeWitt and Brehme.

In Section 3 I introduce a number of coordinate systems that play an important role in later parts of the review. As a warmup exercise I first construct (in Section 3.1) Riemann normal coordinates in a neighbourhood of a reference point $x′$ . I then move on (in Section 3.2) to Fermi normal coordinates [36 ], which are defined in a neighbourhood of a world line $γ$ . The retarded coordinates, which are also based at a world line and which were briefly introduced in Section 1.5, are covered systematically in Section 3.3. The relationship between Fermi and retarded coordinates is worked out in Section 3.4, which also locates the advanced point $z(v)$ associated with a field point $x$ . The presentation in Section 3 borrows heavily from Synge’s book [55 ]. In fact, I am much indebted to Synge for initiating the construction of retarded coordinates in a neighbourhood of a world line. I have implemented his program quite differently (Synge was interested in a large neighbourhood of the world line in a weakly curved spacetime, while I am interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally his.

In Section 4 I review the theory of Green’s functions for (scalar, vectorial, and tensorial) wave equations in curved spacetime. I begin in Section 4.1 with a pedagogical introduction to the retarded and advanced Green’s functions for a massive scalar field in flat spacetime; in this simple context the all-important Hadamard decomposition [28 ] of the Green’s function into “light-cone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in Section 4.2 along with its restrictions on the past and future null cones of a reference point $′ x$ . The retarded, advanced, singular, and radiative Green’s functions for the scalar wave equation are introduced in Section 4.3. In Sections 4.4 and 4.5 I cover the vectorial and tensorial wave equations, respectively. The presentation in Section 4 is based partly on the paper by DeWitt and Brehme [24 ], but it is inspired mostly by Friedlander’s book [27 ]. The reader should be warned that in one important aspect, my notation differs from the notation of DeWitt and Brehme: While they denote the tail part of the Green’s function by $− v(x, x′)$ , I have taken the liberty of eliminating the silly minus sign and I call it instead $+V (x,x ′)$ . The reader should also note that all my Green’s functions are normalized in the same way, with a factor of $− 4π$ multiplying a four-dimensional Dirac functional of the right-hand side of the wave equation. (The gravitational Green’s function is sometimes normalized with a $− 16 π$ on the right-hand side.)

In Section 5 I compute the retarded, singular, and radiative fields associated with a point scalar charge (Section 5.1), a point electric charge (Section 5.2), and a point mass (Section 5.3). I provide two different derivations for each of the equations of motion. The first type of derivation was outlined previously: I follow Detweiler and Whiting [23 ] and postulate that only the radiative field exerts a force on the particle. In the second type of derivation I take guidance from Quinn and Wald [49 ] and postulate that the net force exerted on a point particle is given by an average of the retarded field over a surface of constant proper distance orthogonal to the world line – this rest-frame average is easily carried out in Fermi normal coordinates. The averaged field is still infinite on the world line, but the divergence points in the direction of the acceleration vector and it can thus be removed by mass renormalization. Such calculations show that while the singular field does not affect the motion of the particle, it nonetheless contributes to its inertia. In Section 5.4 I present an alternative derivation of the MiSaTaQuWa equations of motion based on the method of matched asymptotic expansions [35 , 32 , 58 , 19 , 1 , 20 ]; the derivation applies to a small nonrotating black hole instead of a point mass. The ideas behind this derivation were contained in the original paper by Mino, Sasaki, and Tanaka [39 ], but the implementation given here, which involves the retarded coordinates of Section 3.3 and displays explicitly the transformation between external and internal coordinates, is original work.

Concluding remarks are presented in Section 5.5. Throughout this review I use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [40 ].