5.1 Motion of a scalar charge

5.1.1 Dynamics of a point scalar charge

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

where is an action functional for a free scalar field in a spacetime with metric , is the action of a free particle moving on a world line in this spacetime, and is an interaction term that couples the field to the particle.

The field action is given by

where the integration is over all of spacetime; the field is coupled to the Ricci scalar by an arbitrary constant . The particle action is
where is the bare mass of the particle and is the differential of proper time along the world line; we use an overdot on to indicate differentiation with respect to the parameter . Finally, the interaction term is given by
Notice that both and are invariant under a reparameterization of the world line.

Demanding that the total action be stationary under a variation of the field configuration yields the wave equation

for the scalar potential, with a charge density defined by
These equations determine the field once the motion of the scalar charge is specified. On the other hand, demanding that the total action be stationary under a variation of the world line yields the equations of motion for the scalar charge,
We have here adopted as the parameter on the world line, and introduced the four-velocity . The dynamical mass that appears in Equation (387) is defined by , which can also be written in differential form as
It should be clear that Equations (387) and (388) are valid only in a formal sense, because the scalar potential obtained from Equations (385) and (386) diverges on the world line. Before we can make sense of these equations we have to analyze the field’s singularity structure near the world line.

5.1.2 Retarded potential near the world line

The retarded solution to Equation (385) is , where is the retarded Green’s function introduced in Section 4.3. After substitution of Equation (386) we obtain

in which gives the description of the world line . Because the retarded Green’s function is defined globally in the entire spacetime, Equation (389) applies to any field point .

We now specialize Equation (389) 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 Equation (389) can be broken down 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 4.3.2. This gives

and to evaluate this we refer back to Section 3.3 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 3.3: 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:

This expression applies to a point sufficiently close to the world line that there exists a nonempty intersection between and .

5.1.3 Field of a scalar charge in retarded coordinates

When we differentiate the potential of Equation (390) 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 3.3.2 for a detailed discussion. This means, for example, that the total variation of is . The gradient of the scalar potential is therefore given by

where the “tail integral” is defined by
In the second form of the definition we integrate from to almost , but we cut the integration short at to avoid the singular behaviour of the retarded Green’s function at . This limiting procedure gives rise to the first form of the definition, with the advantage that the integral need not be broken down into contributions that refer to and its complement, respectively.

We shall now expand in powers of , and express the results in terms of the retarded coordinates introduced in Section 3.3. 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 3.3. Note that throughout this section we set , where is the rotation tensor defined by Equation (138): The tetrad vectors are taken to be Fermi–Walker transported on . The expansion relies on Equation (166) for , Equation (168) for , and we shall need

which follows from Equation (275) and the relation first encountered in Equation (144); recall that

are frame components of the Ricci tensor evaluated at . We shall also need the expansions

and
which follow from Equations (276); recall from Equation (141) that the parallel propagator can be expressed as . And finally, we shall need
a relation that was first established in Equation (278); here is the Ricci scalar evaluated at .

Collecting all these results gives

where are the frame components of the acceleration vector,

are frame components of the Riemann tensor evaluated at , and

are the frame components of the tail integral evaluated at . Equations (397) and (398) show clearly that is singular on the world line: The field diverges as when , and many of the terms that stay bounded in the limit depend on and therefore possess a directional ambiguity at .

5.1.4 Field of a scalar charge in Fermi normal coordinates

The gradient of the scalar potential can also be expressed in the Fermi normal coordinates of Section 3.2. To effect this translation we make the new reference point on the world line. We resume here the notation of Section 3.4 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 Equations (224, 225) and (397, 398) to obtain

and
where all frame components are still evaluated at , except for and which are evaluated at .

We must still translate these results into the Fermi normal coordinates . For this we involve Equations (221, 222, 223), from which we deduce, for example,

and

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 Equation (221), .

Collecting these results yields

In these expressions, are the frame components of the acceleration vector evaluated at , and are frame components of its covariant derivative, are frame components of the Riemann tensor evaluated at ,

are frame components of the Ricci tensor, and is the Ricci scalar evaluated at . Finally, we have that

are the frame components of the tail integral – see Equation (392) – evaluated at .

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 Equation (127) that the induced metric on is given by

where is the metric of the unit two-sphere, and where depends on and the angles . From this we infer that the element of surface area is given by
where is an element of solid angle – in the canonical parameterization, . Integration of Equation (404) produces the total surface area of , and .

The averaged fields are defined by

where the quantities to be integrated are scalar functions of the Fermi normal coordinates. The results
are easy to establish, and we obtain
The averaged field is still singular on the world line. Regardless, we shall take the formal limit of the expressions displayed in Equations (407) and (408). In the limit the tetrad reduces to , and we can reconstruct the field at by invoking the completeness relations . We thus obtain
where the tail integral can be copied from Equation (392),
The tensors appearing in Equation (409) all refer to , which now stands for an arbitrary point on the world line .

The singular potential

is the (unphysical) solution to Equations (385) and (386) that is obtained by adopting the singular Green’s function of Equation (295) instead of the retarded Green’s function. As we shall see, the resulting singular field reproduces the singular behaviour of the retarded solution; the difference, , is smooth on the world line.

To evaluate the integral of Equation (411) 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 Equation (411) can be broken down into the three integrals,

The first integration vanishes because is then in the chronological future of , and by Equation (286). 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 Equation (297), and we have

To evaluate these we re-introduce the retarded point and let be the advanced point associated with ; we recall from Section 3.4.4 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 observe that depends on the state of motion of the scalar charge between the retarded time and the advanced time ; contrary to what was found in Section 5.1.2 for the retarded potential, there is no dependence on the particle’s remote past.

We use the techniques of Section 5.1.3 to differentiate the potential of Equation (412). We find

and we would like to express this as an expansion in powers of . For this we shall rely on results already established in Section 5.1.3, as well as additional expansions that will involve the advanced point . Those we develop now.

We recall first that a relation between retarded and advanced times was worked out in Equation (229), that an expression for the advanced distance was displayed in Equation (230), and that Equations (231) and (232) give expansions for and , respectively.

To derive an expansion for we follow the general method of Section 3.4.4 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 Equation (393), and the derivatives of are given by

and

according to Equations (395) and (276). Combining these results together with Equation (229) for gives

which should be compared with Equation (393). It should be emphasized that in Equation (414) and all equations below, the frame components of the Ricci tensor are evaluated at the retarded point , and not at the advanced point. The preceding computation gives us also an expansion for . This becomes
which should be compared with Equation (395).

We proceed similarly to derive an expansion for . Here we introduce the functions and express as . The leading term was computed in Equation (394), and

follows from Equation (276). Combining these results together with Equation (229) for gives

and this should be compared with Equation (394).

The last expansion we shall need is

which follows at once from Equation (396) and the fact that ; the Ricci scalar is evaluated at the retarded point .

It is now a straightforward (but tedious) matter to substitute these expansions (all of them!) into Equation (413) and obtain the projections of the singular field in the same tetrad that was employed in Section 5.1.3. This gives

in which all frame components are evaluated at the retarded point . Comparison of these expressions with Equations (397) and (398) reveals that the retarded and singular fields share the same singularity structure.

The difference between the retarded field of Equations (397, 398) and the singular field of Equations (418, 419) defines the radiative field . Its tetrad components are

and we see that is a smooth vector field on the world line. There is therefore no obstacle in evaluating the radiative field directly at , where the tetrad becomes . Reconstructing the field at from its frame components, we obtain
where the tail term can be copied from Equation (392),
The tensors appearing in Equation (422) all refer to the retarded point , which now stands for an arbitrary point on the world line .

5.1.6 Equations of motion

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 5.1.3 and 5.1.4, and in Section 5.1.5 it was shown to originate from the singular field ; the radiative field was then shown to be smooth 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. This was worked out at the end of Section 5.1.4, and ignoring terms that disappear in the limit , we obtain

where
is formally a divergent quantity and
is the tail part of the force; all tensors in Equation (424) are evaluated at an arbitrary point on the world line.

Substituting Equations (424) and (426) into Equation (387) gives rise to the equations of motion

for the scalar charge, with denoting the (also formally divergent) dynamical mass of the particle. We see that and combine in Equation (427) to form the particle’s observed mass , which is taken to be finite and to give a true measure of the particle’s inertia. All diverging quantities have thus disappeared into the process of mass renormalization. Substituting Equations (424) and (426) into Equation (388), in which we replace by , returns an expression for the rate of change of the observed mass,
That the observed mass is not 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.

Apart from the term proportional to , the averaged field of Equation (424) has exactly the same form as the radiative field of Equation (422), which we re-express as

The force acting on the point particle can therefore be thought of as originating from the (smooth) radiative field, while the singular field simply contributes to the particle’s inertia. After mass renormalization, Equations (427) and (428) are equivalent to the statements
where we have dropped the superfluous label “obs” on the particle’s observed mass. Another argument in support of the claim that the motion of the particle should be affected by the radiative field only was presented in Section 4.3.5.

The equations of motion displayed in Equations (427) and (428) 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 [253047]. 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 [4726] 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 second order.

We shall therefore write, in the final analysis, the equations of motion in the form

and
where denotes the observed inertial mass of the scalar charge, and where all tensors are evaluated at . We recall that the tail integration must be cut short at to avoid the singular behaviour of the retarded Green’s function at coincidence; this procedure was justified at the beginning of Section 5.1.3. Equations (431) and (432) were first derived by Theodore C. Quinn in 2000 [48]. 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.