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