Later, we will compare Eq. (2-26) with the equations of motion for a particle
moving in electromagnetic [3], scalar [6] and gravitational fields [4, 5] in curved
spacetime. Then, it would be more convenient to write Eq. (2-26) in the alternative
signature convention (-1, +1, +1, +1) to be consistent with the other equations of
motion in sign, namely
ma = ev bbin + 2e2 (a '2a). (2-27)
2.2 Dewitt and Brehme: Electromagnetic Radiation Damping in
Curved Spacetime
2.2.1 Bi-tensors
As Dirac's work on the classical radiating electron in Section 2.1 was developed
under Lorentz invariance throughout, Dewitt and Brehme's curved-spacetime
generalization of Dirac's is carried out under general covariance throughout. This
covariant generalization involves non-locality questions, and it is essential to
introduce bi-tensors, which are a generalization of ordinary tensors. A bi-tensor is
a set of functions of two spacetime points, each member of which transforms under
a coordinate transformation like an ordinary local tensor, with the difference that
the transformation indices do not all refer to the same point, but rather to the two
separate points. The simplest example of a bi-tensor is the product of two local
vectors, Aa(x) and Bb'(z), taken at different spacetime points, x and z with the
indices a and b' running from 0 to 3:
Cab(X,z ) =Aa(x) Bb' (z). (2-28)
Here the convention is that the usual, non-primed indices are alv--,v- to be associ-
ated with the point x, while the primed indices are ahv--, to be associated with
the point z. Then the coordinates of the points themselves are expressed as x' and
zb