A point particle carries a scalar charge and moves on a world line described by relations , in which is an arbitrary parameter. The particle generates a scalar potential and a field . The dynamics of the entire system is governed by the action
The field action is given by
Demanding that the total action be stationary under a variation of the field configuration yields the wave equation
The retarded solution to Eq. (17.5) is , where is the retarded Green’s function introduced in Section 14. After substitution of Eq. (17.6) we obtain
We now specialize Eq. (17.9) to a point near the world line; see Figure 9. We let be the normal convex neighbourhood of this point, and we assume that the world line traverses . Let be the value of the proper-time parameter at which enters from the past, and let be its value when the world line leaves . Then Eq. (17.9) can be broken up into the three integrals
The third integration vanishes because is then in the past of , and . For the second integration, is the normal convex neighbourhood of , and the retarded Green’s function can be expressed in the Hadamard form produced in Section 14.2. This gives
and to evaluate this we refer back to Section 10 and let be the retarded point associated with ; these points are related by and is the retarded distance between and the world line. We resume the index convention of Section 10: to tensors at we assign indices , , etc.; to tensors at we assign indices , , etc.; and to tensors at a generic point on the world line we assign indices , , etc.
To perform the first integration we change variables from to , noticing that increases as passes through . The change of on the world line is given by , and we find that the first integral evaluates to with identified with . The second integration is cut off at by the step function, and we obtain our final expression for the retarded potential of a point scalar charge:
When we differentiate the potential of Eq. (17.10) we must keep in mind that a variation in induces a variation in because the new points and must also be linked by a null geodesic – you may refer back to Section 10.2 for a detailed discussion. This means, for example, that the total variation of is . The gradient of the scalar potential is therefore given by
We shall now expand in powers of , and express the results in terms of the retarded coordinates introduced in Section 10. It will be convenient to decompose in the tetrad that is obtained by parallel transport of on the null geodesic that links to ; this construction is detailed in Section 10. The expansion relies on Eq. (10.29) for , Eq. (10.31) for , and we shall need
are frame components of the Ricci tensor evaluated at . We shall also need the expansions
Collecting all these results gives
are frame components of the Riemann tensor evaluated at , and
The gradient of the scalar potential can also be expressed in the Fermi normal coordinates of Section 9. To effect this translation we make the new reference point on the world line. We resume here the notation of Section 11 and assign indices , , …to tensors at . The Fermi normal coordinates are denoted , and we let be the tetrad at that is obtained by parallel transport of on the spacelike geodesic that links to .
Our first task is to decompose in the tetrad , thereby defining and . For this purpose we use Eqs. (11.7), (11.8), (17.17), and (17.18) to obtain
We must still translate these results into the Fermi normal coordinates . For this we involve Eqs. (11.4), (11.5), and (11.6), from which we deduce, for example,
in which all frame components (on the right-hand side of these relations) are now evaluated at ; to obtain the second relation we expressed as since according to Eq. (11.4), .
Collecting these results yields
are frame components of the Ricci tensor, and is the Ricci scalar evaluated at . Finally, we have that
We shall now compute the averages of and over , a two-surface of constant and ; these will represent the mean value of the field at a fixed proper distance away from the world line, as measured in a reference frame that is momentarily comoving with the particle. The two-surface is charted by angles () and it is described, in the Fermi normal coordinates, by the parametric relations ; a canonical choice of parameterization is . Introducing the transformation matrices , we find from Eq. (9.16) that the induced metric on is given by
The averaged fields are defined by
The singular potential
To evaluate the integral of Eq. (17.31) we assume once more that is sufficiently close to that the world line traverses ; refer back to Figure 9. As before we let and be the values of the proper-time parameter at which enters and leaves , respectively. Then Eq. (17.31) can be broken up into the three integrals
The first integration vanishes because is then in the chronological future of , and by Eq. (14.21). Similarly, the third integration vanishes because is then in the chronological past of . For the second integration, is the normal convex neighbourhood of , the singular Green’s function can be expressed in the Hadamard form of Eq. (14.32), and we have
To evaluate these we re-introduce the retarded point and let be the advanced point associated with ; we recall from Section 11.4 that these points are related by and that is the advanced distance between and the world line.
To perform the first integration we change variables from to , noticing that increases as passes through ; the integral evaluates to . We do the same for the second integration, but we notice now that decreases as passes through ; the integral evaluates to . The third integration is restricted to the interval by the step function, and we obtain our final expression for the singular potential of a point scalar charge:
We use the techniques of Section 17.3 to differentiate the potential of Eq. (17.32). We find
We recall first that a relation between retarded and advanced times was worked out in Eq. (11.12), that an expression for the advanced distance was displayed in Eq. (11.13), and that Eqs. (11.14) and (11.15) give expansions for and , respectively.
To derive an expansion for we follow the general method of Section 11.4 and define a function of the proper-time parameter on . We have that
where overdots indicate differentiation with respect to , and where . The leading term was worked out in Eq. (17.13), and the derivatives of are given by
according to Eqs. (17.15) and (14.11). Combining these results together with Eq. (11.12) for gives
We proceed similarly to derive an expansion for . Here we introduce the functions and express as . The leading term was computed in Eq. (17.14), and
follows from Eq. (14.11). Combining these results together with Eq. (11.12) for gives
The last expansion we shall need is
It is now a straightforward (but tedious) matter to substitute these expansions (all of them!) into Eq. (17.33) and obtain the projections of the singular field in the same tetrad that was employed in Section 17.3. This gives
The difference between the retarded field of Eqs. (17.17), (17.18) and the singular field of Eqs. (17.38), (17.39) defines the regular field . Its frame components are
The retarded field of a point scalar charge is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and affect its motion. The field’s singularity structure was analyzed in Sections 17.3 and 17.4, and in Section 17.5 it was shown to originate from the singular field ; the regular field was then shown to be regular on the world line.
To make sense of the retarded field’s action on the particle we temporarily model the scalar charge not as a point particle, but as a small hollow shell that appears spherical when observed in a reference frame that is momentarily comoving with the particle; the shell’s radius is in Fermi normal coordinates, and it is independent of the angles contained in the unit vector . The net force acting at proper time on this hollow shell is the average of over the surface of the shell. Assuming that the field on the shell is equal to the field of a point particle evaluated at , and ignoring terms that disappear in the limit , we obtain from Eq. (17.29)
Substituting Eqs. (17.44) and (17.46) into Eq. (17.7) gives rise to the equations of motionnot conserved is a remarkable property of the dynamics of a scalar charge in a curved spacetime. Physically, this corresponds to the fact that in a spacetime with a time-dependent metric, a scalar charge radiates monopole waves and the radiated energy comes at the expense of the particle’s inertial mass.
We must confess that the derivation of the equations of motion outlined above returns the wrong expression for the self-energy of a spherical shell of scalar charge. We obtained , while the correct expression is ; we are wrong by a factor of . We believe that this discrepancy originates in a previously stated assumption, that the field on the shell (as produced by the shell itself) is equal to the field of a point particle evaluated at . We believe that this assumption is in fact wrong, and that a calculation of the field actually produced by a spherical shell would return the correct expression for . We also believe, however, that except for the diverging terms that determine , the difference between the shell’s field and the particle’s field should vanish in the limit . Our conclusion is therefore that while our expression for is admittedly incorrect, the statement of the equations of motion is reliable.
Apart from the term proportional to , the averaged field of Eq. (17.44) has exactly the same form as the regular field of Eq. (17.42), which we re-express as
The equations of motion displayed in Eqs. (17.47) and (17.48) are third-order differential equations for the functions . It is well known that such a system of equations admits many unphysical solutions, such as runaway situations in which the particle’s acceleration increases exponentially with , even in the absence of any external force [56, 101]. And indeed, our equations of motion do not yet incorporate an external force which presumably is mostly responsible for the particle’s acceleration. Both defects can be cured in one stroke. We shall take the point of view, the only admissible one in a classical treatment, that a point particle is merely an idealization for an extended object whose internal structure – the details of its charge distribution – can be considered to be irrelevant. This view automatically implies that our equations are meant to provide only an approximate description of the object’s motion. It can then be shown [112, 70] that within the context of this approximation, it is consistent to replace, on the right-hand side of the equations of motion, any occurrence of the acceleration vector by , where is the external force acting on the particle. Because is a prescribed quantity, differentiation of the external force does not produce higher derivatives of the functions , and the equations of motion are properly of the second order.
We shall strengthen this conclusion in Part V of the review, when we consider the motion of an extended body in a curved external spacetime. While the discussion there will concern the gravitational self-force, many of the lessons learned in Part V apply just as well to the case of a scalar (or electric) charge. And the main lesson is this: It is natural – indeed it is an imperative – to view an equation of motion such as Eq. (17.47) as an expansion of the acceleration in powers of , and it is therefore appropriate – indeed imperative – to insert the zeroth-order expression for within the term of order . The resulting expression for the acceleration is then valid up to correction terms of order . Omitting these error terms, we shall write, in final analysis, the equations of motion in the form. In his paper Quinn also establishes that the total work done by the scalar self-force matches the amount of energy radiated away by the particle.
Living Rev. Relativity 14, (2011), 7
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