Radiation Reaction in Noncommutative Electrodynamics

Abstract

We study the radiation reaction acting on an accelerating charge moving in noncommutative spacetime and obtain an expression for it. Radiation reaction, due to a nonrelativistic point charge, is found to receive a small noncommutative correction term. The Abraham–Lorentz equation for a point charge in noncommutative spacetime suffers from the preacceleration and the runaway problems. We explore as an application the radiation reaction experienced by a charge which undergoes harmonic oscillations in a noncommutative plane.

APPENDIX A

1.1 CALCULATION OF RADIATION REACTION IN NONCOMMUTATIVE ELECTRODYNAMICS

We have

$$\begin{gathered} \int {{{d}^{3}}x{{\partial }_{0}}} \left( {A_{i}^{{{\text{rad}}}}(x){{J}_{\mu }}(x)F_{{{\text{rad}}}}^{{\mu \nu }}(x)} \right) \\ = \int {{{d}^{3}}x} \frac{\partial }{{\partial {{x}^{0}}}}\left( {A_{i}^{{{\text{rad}}}}(x)F_{{{\text{rad}}}}^{{\mu \nu }}(x)} \right){{J}_{\mu }}(x) \\ + \,\,\int {{{d}^{3}}x} \frac{{\partial {{J}_{\mu }}(x)}}{{\partial {{x}^{0}}}}A_{i}^{{{\text{rad}}}}(x)F_{{{\text{rad}}}}^{{\mu \nu }}(x). \\ \end{gathered} $$

(A.1)

Four vector current density due to a point charge is given by

$$\begin{gathered} {{J}_{\mu }}(x) = ec\int {ds{{{\dot {z}}}_{\mu }}(s)} {{\delta }^{4}}(x - z(s)) \\ = ec{{\left. {\frac{{{{{\dot {z}}}_{\mu }}(s){{\delta }^{3}}(\vec {x} - \vec {z}(s))}}{{{{{\dot {z}}}^{0}}(s)}}} \right|}_{{{{x}^{0}} = {{z}^{0}}(s)}}}. \\ \end{gathered} $$

(A.2)

The first term in the above equation after inserting (A.2) gets simplified as

$$\begin{gathered} \int {{{d}^{3}}x\frac{\partial }{{\partial {{x}^{0}}}}} (A_{i}^{{{\text{rad}}}}(x)F_{{{\text{rad}}}}^{{\mu \nu }}(x)){{J}_{\mu }}(x) \\ = ec\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z){{\left( {\frac{{\partial A_{i}^{{{\text{rad}}}}}}{{\partial {{x}^{0}}}}} \right)}_{{x = z(s)}}} \\ + \,\,ec\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}A_{i}^{{{\text{rad}}}}(z){{\left( {\frac{{\partial F_{{\text{rad}}}^{{\mu \nu }}}}{{\partial {{x}^{0}}}}} \right)}_{{x = z(s)}}}. \\ \end{gathered} $$

(A.3)

The second term in Eq. (A.1) becomes

$$\begin{gathered} \int {{{d}^{3}}x} \frac{{\partial {{J}_{\mu }}(x)}}{{\partial {{x}^{0}}}}(A_{i}^{{{\text{rad}}}}(x)F_{{{\text{rad}}}}^{{\mu \nu }}(x)) \\ = ec\frac{{A_{i}^{{{\text{rad}}}}(z)}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z)\frac{\partial }{{\partial s}}\left( {\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}} \right). \\ \end{gathered} $$

(A.4)

Four-vector potential is defined as

$$A_{\mu }^{{{\text{rad}}}}(x) = e\int {ds} D(x - z(s)){{\dot {z}}_{\mu }}(s).$$

At $x = z(s{\kern 1pt} ')$, we have

$$A_{\mu }^{{{\text{rad}}}}z(s{\kern 1pt} ') = e\int {D(z(s{\kern 1pt} ')} - z(s)){{\dot {z}}_{\mu }}(s)ds.$$

(A.5)

Let $s = s{\kern 1pt} '\,\, + u$ where u is a small parameter.

$$\begin{gathered} {{z}_{\mu }}(s) = {{z}_{\mu }}(s{\kern 1pt} ') + u{{{\dot {z}}}_{\mu }}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{2}{{{\ddot {z}}}_{\mu }}(s{\kern 1pt} ') + \frac{{{{u}^{3}}}}{6}{{{\dddot z}}_{\mu }}(s{\kern 1pt} ') \\ + ...{\kern 1pt} {{{\dot {z}}}_{\mu }}(s) = {{{\dot {z}}}_{\mu }}(s{\kern 1pt} ') + u{{{\ddot {z}}}_{\mu }}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{2}{{{\dddot z}}_{\mu }}(s{\kern 1pt} ') + .... \\ \end{gathered} $$

We know [3]

$$D(z(s{\kern 1pt} ') - z(s)) = \frac{1}{{2\pi }}\frac{{d\delta (u)}}{{du}}.$$

Therefore,

$$A_{\mu }^{{{\text{rad}}}}(z(s{\kern 1pt} ')) = - \frac{e}{{2\pi }}\int {\delta (u)du} ({{\ddot {z}}_{\mu }}(s{\kern 1pt} ') + u{{\dddot z}_{\mu }}(s{\kern 1pt} ') + \ldots ).$$

At $u = 0,\,\,s = s{\kern 1pt} '$. Hence

$$A_{\mu }^{{{\text{rad}}}}(z(s)) = - \frac{e}{{2\pi }}{{\ddot {z}}_{\mu }}(s).$$

(A.6)

1.1.1 Calculation of ${{({{\partial }_{\nu }}{\mathbf{A}}_{\mu }^{{{\mathbf{rad}}}})}_{{{\mathbf{x}} = {\mathbf{z}}({\mathbf{s}})}}}$

$$\begin{gathered} {{\partial }_{\nu }}A_{\mu }^{{{\text{rad}}}}(x) = e\frac{\partial }{{\partial {{x}^{\nu }}}}\int {dsD} (x - z(s)){{{\dot {z}}}_{\mu }}(s) \\ = e\int {dsD} (x - z(s))\frac{d}{{ds}}\left[ {\frac{{{{{\dot {z}}}_{\mu }}(s){{{(x - z(s))}}_{\nu }}}}{{{{{(x - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}(s)}}} \right]. \\ \end{gathered} $$

At $x = z(s{\kern 1pt} ')$

$$\begin{gathered} {{\partial }_{\nu }}A_{\mu }^{{{\text{rad}}}}(z(s{\kern 1pt} ')) = e\int {dsD} ((z(s{\kern 1pt} ') \\ - \,\,z(s))\frac{d}{{ds}}\left( {\frac{{{{{\dot {z}}}_{\mu }}(s){{{(z(s{\kern 1pt} ') - z(s))}}_{\nu }}}}{{{{{(z(s{\kern 1pt} ') - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}}}} \right). \\ \end{gathered} $$

Then,

$${{(z(s{\kern 1pt} ') - z(s))}^{\sigma }}{{\dot {z}}_{\sigma }} = - u + o({{u}^{3}}),$$

(A.7)

$$\begin{gathered} {{{\dot {z}}}_{\mu }}(s){{(z(s{\kern 1pt} ') - z(s))}_{\nu }} \\ = - u\left\{ {\mathop {\dot {z}}\nolimits_\mu (s{\kern 1pt} '){{{\dot {z}}}_{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{{}}}}}{2}\mathop {\dot {z}}\nolimits_\mu (s{\kern 1pt} '){{{\ddot {z}}}_{\nu }}(s{\kern 1pt} ') + u{{{\ddot {z}}}_{\mu }}(s{\kern 1pt} '){{{\dot {z}}}_{\nu }}(s{\kern 1pt} ')} \right. \\ + \,\,\frac{{{{u}^{2}}}}{6}{{{\dot {z}}}_{\mu }}(s{\kern 1pt} '){{{\dddot z}}_{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{2}{{{\ddot {z}}}_{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}_{\nu }}(s{\kern 1pt} ') \\ \left. { + \,\,\frac{{{{u}^{2}}}}{2}{{{\dddot z}}_{\mu }}(s{\kern 1pt} '){{{\dot {z}}}_{\nu }}(s{\kern 1pt} ') + ...} \right\}, \\ \end{gathered} $$

(A.8)

$$\begin{gathered} {{({{\partial }_{\nu }}A_{\mu }^{{{\text{rad}}}})}_{{x = z(s)}}} = - \frac{e}{{2\pi }} \\ \times \,\,\left( {\frac{1}{3}{{{\dot {z}}}_{\mu }}(s){{{\dddot z}}_{\nu }}(s) + {{{\ddot {z}}}_{\mu }}(s){{{\ddot {z}}}_{\nu }}(s) + {{{\dddot z}}_{\mu }}(s){{{\dot {z}}}_{\nu }}(s)} \right). \\ \end{gathered} $$

(A.9)

Therefore,

$$\begin{gathered} F_{{{\text{rad}}}}^{{\mu \nu }}(z(s)) = {{\left( {{{\partial }^{\mu }}A_{{{\text{rad}}}}^{\nu } - {{\partial }^{\nu }}A_{{{\text{rad}}}}^{\mu }} \right)}_{{x = z(s)}}} \\ = \frac{e}{{3\pi }}\left( {{{{\dot {z}}}^{\nu }}(s){{{\dddot z}}^{\mu }}(s) - {{{\dddot z}}^{\nu }}(s){{{\dot {z}}}^{\mu }}(s)} \right). \\ \end{gathered} $$

(A.10)

1.1.2 Calculation of the Term $ec\tfrac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s)){{\left( {\tfrac{{\partial A_{i}^{{{\text{rad}}}}}}{{\partial {{x}^{0}}}}} \right)}_{{x = z(s)}}}$

From Eq. (A.9)

$$\begin{gathered} {{({{\partial }_{0}}A_{i}^{{{\text{rad}}}})}_{{x = z(s)}}} = - \frac{e}{{2\pi }} \\ \times \,\,\left( {\frac{1}{3}{{{\dot {z}}}_{i}}(s){{{\dddot z}}_{0}}(s) + {{{\ddot {z}}}_{i}}(s){{{\ddot {z}}}_{0}}(s) + {{{\dddot z}}_{i}}(s){{{\dot {z}}}_{0}}(s)} \right). \\ \end{gathered} $$

Therefore,

$$\begin{gathered} ec\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s)){{({{\partial }_{0}}A_{i}^{{{\text{rad}}}})}_{{x = z(s)}}} \\ = - \frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}\frac{2}{3}\frac{{{{e}^{2}}c}}{{2\pi }}\left( {{{{\dot {z}}}^{\nu }}(s){{{\dddot z}}^{\mu }}(s) - {{{\dddot z}}^{\nu }}(s){{{\dot {z}}}^{\mu }}(s)} \right) \\ \times \,\,\frac{e}{{2\pi }}\left( {\frac{1}{3}{{{\dot {z}}}_{i}}(s){{{\dddot z}}_{0}}(s) + {{{\ddot {z}}}_{i}}(s){{{\ddot {z}}}_{0}}(s) + {{{\dddot z}}_{i}}(s){{{\dot {z}}}_{0}}(s)} \right). \\ \end{gathered} $$

We know that

$$\begin{gathered} {{{\dot {z}}}^{\mu }}(s){{{\dot {z}}}^{\mu }}(s) = 1,\,\,\,\,{{{\dot {z}}}^{\mu }}(s){{{\ddot {z}}}^{\mu }}(s) = 0, \\ {{{\dot {z}}}^{\mu }}(s){{{\dddot z}}^{\mu }}(s) + {{{\ddot {z}}}_{\mu }}(s){{{\ddot {z}}}^{\mu }}(s) = 0, \\ {{{\dot {z}}}^{\mu }}(s){{{\dddot z}}^{\mu }}(s) = - {{{\ddot {z}}}^{2}}(s). \\ \end{gathered} $$

Then,

$$\begin{gathered} ec\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s)){{({{\partial }_{0}}A_{i}^{{{\text{rad}}}})}_{{x = z(s)}}} \\ = \frac{2}{3}\frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}}\left( {{{{\dddot z}}^{\nu }}(s) + {{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}^{2}}(s)} \right)\frac{1}{{{{{\dot {z}}}^{0}}(s)}} \\ \times \,\,\left( {\frac{1}{3}{{{\dot {z}}}_{i}}(s){{{\dddot z}}_{0}}(s) + {{{\ddot {z}}}_{i}}(s){{{\ddot {z}}}_{0}}(s) + {{{\dddot z}}_{i}}(s){{{\dot {z}}}_{0}}(s)} \right). \\ \end{gathered} $$

(A.11)

1.1.3 Calculation of the Term $ec\tfrac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}A_{i}^{{{\text{rad}}}}(z(s)){{({{\partial }_{0}}F_{{{\text{rad}}}}^{{\mu \nu }}(x))}_{{x = z(s)}}}$

We have

$${{\partial }_{0}}F_{{{\text{rad}}}}^{{\mu \nu }}(x) = \frac{\partial }{{\partial {{x}^{0}}}}({{\partial }^{\mu }}{{A}^{\nu }}(x) - {{\partial }^{\nu }}{{A}^{\mu }}(x)),$$

(A.12)

where

$$\begin{gathered} \frac{\partial }{{\partial {{x}^{0}}}}({{\partial }^{\nu }}{{A}^{\mu }}(x)) = e\int {ds} \frac{{\partial D(x - z(s))}}{{\partial {{x}^{0}}}}\frac{d}{{ds}} \\ \times \,\,\left[ {\frac{{{{{\dot {z}}}^{\mu }}(s){{{(x - z(s))}}^{\nu }}}}{{{{{(x - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}(s)}}} \right] + e\int {ds} D(x - z(s)) \\ \times \,\,\frac{d}{{ds}}\frac{\partial }{{\partial {{x}^{0}}}}\left[ {\frac{{{{{\dot {z}}}^{\mu }}(s){{{(x - z(s))}}^{\nu }}}}{{{{{(x - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}(s)}}} \right]. \\ \end{gathered} $$

(A.13)

The first term of the above Eq. (A.13) is

$$\begin{gathered} e\int {ds} \frac{{\partial D(x - z(s))}}{{\partial {{x}^{0}}}}\frac{d}{{ds}}\left[ {\frac{{{{{\dot {z}}}^{\mu }}(s){{{(x - z(s))}}^{\nu }}}}{{{{{(x - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}}}} \right] \\ = e\int {ds} D(x - z(s))\frac{d}{{ds}}\left\{ {\frac{{{{{(x - z(s))}}^{0}}}}{{{{{(x - z(s))}}^{\rho }}{{{\dot {z}}}_{\rho }}(s)}}} \right. \\ \times \,\,\left. {\frac{d}{{ds}}\left[ {\frac{{{{{\dot {z}}}^{\mu }}(s){{{(x - z(s))}}^{\nu }}}}{{{{{(x - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}(s)}}} \right]} \right\}. \\ \end{gathered} $$

At $x = z(s{\kern 1pt} ')$ the above equation becomes:

$$\begin{gathered} = e\int {ds} D(z(s{\kern 1pt} ') - z(s))\left\{ {\frac{d}{{ds}}\left( {\frac{{{{{(z(s{\kern 1pt} ') - z(s))}}^{0}}}}{{{{{(z(s{\kern 1pt} ') - z(s))}}^{\rho }}{{{\dot {z}}}_{\rho }}(s)}}} \right)} \right. \\ \left. { \times \,\,\frac{d}{{ds}}\left[ {\frac{{{{{\dot {z}}}^{\mu }}(s){{{(z(s{\kern 1pt} ') - z(s))}}^{\nu }}}}{{{{{(z(s{\kern 1pt} ') - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}(s)}}} \right]} \right\} + e\int {ds} D(z(s{\kern 1pt} ') \\ - \,\,z(s))\left\{ {\frac{{{{{(z(s{\kern 1pt} ') - z(s))}}^{0}}}}{{{{{(z(s{\kern 1pt} ') - z(s))}}^{\rho }}{{{\dot {z}}}_{\rho }}(s)}}} \right. \\ \left. { \times \,\,\frac{{{{d}^{2}}}}{{d{{s}^{2}}}}\left[ {\frac{{{{{\dot {z}}}^{\mu }}(s){{{(z(s{\kern 1pt} ') - z(s))}}^{\nu }}}}{{{{{(z(s{\kern 1pt} ') - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}(s)}}} \right]} \right\}. \\ \end{gathered} $$

(A.14)

The first term in Eq. (A.14) can be expressed as

$$\begin{gathered} = \frac{e}{{2\pi }} \\ \times \,\,\int {du} \frac{{d\delta (u)}}{{du}}\left\{ {\left[ {\frac{d}{{du}}({{{\dot {z}}}^{0}}(s{\kern 1pt} ')} \right.} \right.\left. { + \frac{u}{2}{{{\ddot {z}}}^{0}}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{6}{{{\dddot z}}^{0}}(s{\kern 1pt} ')....)} \right] \\ \times \,\,\frac{d}{{du}}\left( {{{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') + \frac{u}{2}{{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ')} \right. \\ + \,\,u{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{6}{{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dddot z}}^{\nu }}(s{\kern 1pt} ') \\ + \,\,\frac{{{{u}^{2}}}}{2}{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{2}{{{\dddot z}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') \\ \left. {\left. { + \,\,\frac{{{{u}^{3}}}}{6}{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dddot z}}^{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{3}}}}{4}{{{\dddot z}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ')...} \right)} \right\} \\ = - \frac{e}{{2\pi }}\int {du} \delta (u)\left( {\frac{{{{d}^{2}}P(u)}}{{d{{u}^{2}}}}\frac{{dQ(u)}}{{du}} + \frac{{dP(u)}}{{du}}\frac{{{{d}^{2}}Q(u)}}{{d{{u}^{2}}}}} \right), \\ \end{gathered} $$

(A.15)

where

$$\begin{gathered} P(u) = ({{{\dot {z}}}^{0}}(s{\kern 1pt} ') + \frac{u}{2}{{{\ddot {z}}}^{0}}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{6}{{{\dddot z}}^{0}}(s{\kern 1pt} ')....), \\ Q(u) = {{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') + \frac{u}{2}{{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ') \\ + \,\,u{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{6}{{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dddot z}}^{\nu }}(s{\kern 1pt} ') \\ + \,\,\frac{{{{u}^{2}}}}{2}{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{2}}}}{2}{{{\dddot z}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') \\ + \,\,\frac{{{{u}^{3}}}}{6}{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dddot z}}^{\nu }}(s{\kern 1pt} ') + \frac{{{{u}^{3}}}}{4}{{{\dddot z}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ') \ldots \\ \end{gathered} $$

(A.16)

The second term in Eq. (A.14) can be expressed as

$$ = - \frac{e}{{2\pi }}\int {du} \delta (u)\left( {\frac{{dP(u)}}{{du}}\frac{{{{d}^{2}}Q(u)}}{{d{{u}^{2}}}} + P(u)\frac{{{{d}^{3}}Q(u)}}{{d{{u}^{3}}}}} \right).$$

(A.17)

Now, Eq. (A.14) becomes:

$$\begin{gathered} = - \frac{e}{{2\pi }}\int {du} \delta (u)\left\{ {\frac{{{{d}^{2}}P(u)}}{{d{{u}^{2}}}}\frac{{dQ(u)}}{{du}}} \right. \\ \left. { + \,\,2\frac{{dP(u)}}{{du}}\frac{{{{d}^{2}}Q(u)}}{{d{{u}^{2}}}} + P(u)\frac{{{{d}^{3}}Q(u)}}{{d{{u}^{3}}}}} \right\} \\ = - \frac{e}{{2\pi }}\left[ {\frac{{{{{\dddot z}}^{0}}(s)}}{3}\left( {\frac{{{{{\dot {z}}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s)}}{2} + {{{\ddot {z}}}^{\mu }}(s){{{\dot {z}}}^{\nu }}(s)} \right)} \right. \\ + \,\,{{{\ddot {z}}}^{0}}(s) \\ \times \,\,\left\{ {{{{\dddot z}}^{\nu }}(s)\frac{{{{{\dot {z}}}^{\mu }}(s)}}{3}(s) + {{{\ddot {z}}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s) + {{{\dddot z}}^{\mu }}(s){{{\dot {z}}}^{\nu }}(s)} \right\} \\ + \,\,{{{\dot {z}}}^{0}}(s)\left. {\left\{ {{{{\ddot {z}}}^{\mu }}(s){{{\dddot z}}^{\nu }}(s) + \frac{3}{2}{{{\dddot z}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s)} \right\}} \right]. \\ \end{gathered} $$

(A.18)

At $x = z(s{\kern 1pt} ')$ the second term in Eq. (A.13) is

$$\begin{gathered} e\int {ds} D(z(s{\kern 1pt} ') - z(s))\frac{d}{{ds}}\frac{\partial }{{\partial {{z}^{0}}(s{\kern 1pt} ')}} \\ \times \,\,\left[ {\frac{{{{{\dot {z}}}^{\mu }}(s){{{(z(s{\kern 1pt} ') - z(s))}}^{\nu }}}}{{{{{(z(s{\kern 1pt} ') - z(s))}}^{\sigma }}{{{\dot {z}}}_{\sigma }}(s)}}} \right] \\ = - \frac{e}{{2\pi }}\int {du} \frac{1}{{{{{\dot {z}}}^{0}}(s{\kern 1pt} ')}}\delta (u)\frac{{{{d}^{2}}}}{{d{{u}^{2}}}} \\ \times \,\,\left( {{{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ') + {{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') + \frac{u}{2}{{{\dot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dddot z}}^{\nu }}(s{\kern 1pt} ')} \right. \\ + \,\,\frac{{3u}}{2}{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ') + u{{{\dddot z}}^{\mu }}(s{\kern 1pt} '){{{\dot {z}}}^{\nu }}(s{\kern 1pt} ') \\ \left. { + \,\,\frac{{2{{u}^{2}}}}{3}{{{\ddot {z}}}^{\mu }}(s{\kern 1pt} '){{{\dddot z}}^{\nu }}(s{\kern 1pt} ') + {{u}^{2}}{{{\dddot z}}^{\mu }}(s{\kern 1pt} '){{{\ddot {z}}}^{\nu }}(s{\kern 1pt} ')} \right) \\ = - \frac{e}{{2\pi }}\frac{1}{{{{{\dot {z}}}^{0}}(s)}}\left( {\frac{4}{3}{{{\dddot z}}^{\nu }}(s){{{\ddot {z}}}^{\nu }}(s) + 2{{{\dddot z}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s)} \right). \\ \end{gathered} $$

(A.19)

By substituting Eqs. (A.18) and (A.19) in Eq. (A.13), we have,

$$\begin{gathered} \frac{\partial }{{\partial {{x}^{0}}}}{{({{\partial }^{\nu }}{{A}^{\mu }}(x))}_{{x = z(s)}}} = - \frac{e}{{2\pi }} \\ \times \,\,\left[ {\frac{1}{3}{{{\dddot z}}^{0}}(s)\left\{ {\frac{1}{2}{{{\dot {z}}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s) + {{{\ddot {z}}}^{\mu }}(s){{{\dot {z}}}^{\nu }}(s)} \right\}} \right. \\ + \,\,{{{\ddot {z}}}^{0}}(s)\left\{ {\frac{1}{3}{{{\dddot z}}^{\nu }}(s){{{\dot {z}}}^{\mu }}(s) + {{{\ddot {z}}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s) + {{{\dddot z}}^{\mu }}(s){{{\dot {z}}}^{\nu }}(s)} \right\} \\ + \,\,{{{\dot {z}}}^{0}}(s)\left\{ {{{{\ddot {z}}}^{\mu }}(s){{{\dddot z}}^{\nu }}(s) + \frac{3}{2}{{{\dddot z}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s)} \right\} \\ \left. { + \,\,\frac{1}{{{{{\dot {z}}}^{0}}(s)}}\left\{ {\frac{4}{3}{{{\dddot z}}^{\nu }}(s){{{\ddot {z}}}^{\mu }}(s) + 2{{{\dddot z}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s)} \right\}} \right]. \\ \end{gathered} $$

(A.20)

Then

$$\begin{gathered} {{({{\partial }_{0}}{{F}^{{\mu \nu }}}(x))}_{{x = z(s)}}} = \tfrac{e}{{2\pi }}\left[ {\tfrac{{{{{\dddot z}}^{0}}(s)}}{6}\left( {{{{\ddot {z}}}^{\mu }}(s){{{\dot {z}}}^{\nu }}(s) - {{{\ddot {z}}}^{\nu }}(s){{{\dot {z}}}^{\mu }}(s)} \right)} \right. \\ + \,\,\tfrac{2}{3}{{{\ddot {z}}}^{0}}(s)\left( {{{{\dddot z}}^{\mu }}(s){{{\dot {z}}}^{\nu }}(s) - {{{\dddot z}}^{\nu }}(s){{{\dot {z}}}^{\mu }}(s)} \right) \\ \left. { + \,\,\left( {{{{\dddot z}}^{\mu }}(s){{{\ddot {z}}}^{\nu }}(s) - {{{\dddot z}}^{\nu }}(s){{{\ddot {z}}}^{\mu }}(s)} \right)\left( {\tfrac{1}{2}{{{\dot {z}}}^{0}}(s) + \tfrac{2}{3}\tfrac{1}{{{{{\dot {z}}}^{0}}(s)}}} \right)} \right]. \\ \end{gathered} $$

Therefore,

$$\begin{gathered} ec\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}A_{i}^{{{\text{rad}}}}(z(s)){{({{\partial }_{0}}F_{{{\text{rad}}}}^{{\mu \nu }}(x))}_{{x = z(s)}}} = \frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s)}}{{{{{\dot {z}}}^{0}}(s)}} \\ \times \,\,\left[ {\frac{{{{{\dddot z}}^{0}}(s){{{\ddot {z}}}^{\nu }}(s)}}{6}} \right. + \frac{2}{3}{{{\ddot {z}}}^{0}}(s)\left( {{{{\dddot z}}^{\nu }}(s) + {{{\ddot {z}}}^{2}}(s){{{\dot {z}}}^{\nu }}(s)} \right) \\ \left. { + \,\,{{{\ddot {z}}}^{2}}(s){{{\ddot {z}}}^{\nu }}(s)\left( {\frac{1}{2}{{{\dot {z}}}^{0}}(s) + \frac{2}{3}\frac{1}{{{{{\dot {z}}}^{0}}(s)}}} \right)} \right]. \\ \end{gathered} $$

(A.21)

1.1.4 Calculation of the Term $ec\tfrac{{A_{i}^{{{\text{rad}}}}(z(s))}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s))\tfrac{\partial }{{\partial s}}\left( {\tfrac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}} \right)$

We have

$$\begin{gathered} \frac{{A_{i}^{{{\text{rad}}}}(z(s))}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s))\frac{\partial }{{\partial s}}\left[ {\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}} \right] = \frac{{{{{\ddot {z}}}_{\mu }}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}} \\ \times \,\,F_{{rad}}^{{\mu \nu }}(z(s))A_{i}^{{{\text{rad}}}}(z(s)) - \frac{{{{{\dot {z}}}_{\mu }}{{{\ddot {z}}}^{0}}}}{{{{{({{{\dot {z}}}^{0}})}}^{3}}}}F_{{{\text{rad}}}}^{{\mu \nu }}(z)A_{i}^{{{\text{rad}}}}(z). \\ \end{gathered} $$

Then,

$$\begin{gathered} \frac{{{{{\ddot {z}}}_{\mu }}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s))A_{i}^{{{\text{rad}}}}(z(s)) = - \frac{2}{3}\frac{{{{e}^{2}}}}{{4{{\pi }^{2}}}} \\ \times \,\,\frac{{{{{\ddot {z}}}_{i}}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}}\left( {{{{\dot {z}}}^{\mu }}(s){{{\ddot {z}}}_{\mu }}(s){{{\dddot z}}^{\nu }}(s) - {{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}_{\mu }}(s){{{\dddot z}}^{\mu }}(s)} \right) \\ = \frac{2}{3}\frac{{{{e}^{2}}}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}}{{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}_{\mu }}(s){{{\dddot z}}^{\mu }}(s) \\ \end{gathered} $$

(A.22)

and

$$\begin{gathered} \frac{{{{{\ddot {z}}}^{0}}(s){{{\dot {z}}}_{\mu }}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{3}}}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s))A_{i}^{{{\text{rad}}}}(z(s)) \\ = \frac{2}{3}\frac{{{{e}^{2}}}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s){{{\ddot {z}}}^{0}}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{3}}}}\left( {{{{\dddot z}}^{\nu }}(s) + {{{\ddot {z}}}^{2}}(s){{{\dot {z}}}^{\nu }}(s)} \right). \\ \end{gathered} $$

(A.23)

Therefore,

$$\begin{gathered} ec\frac{{A_{i}^{{{\text{rad}}}}(z(s))}}{{{{{\dot {z}}}^{0}}(s)}}F_{{{\text{rad}}}}^{{\mu \nu }}(z(s))\frac{\partial }{{\partial s}}\left( {\frac{{{{{\dot {z}}}_{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}} \right) \\ = \frac{2}{3}\frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}}{{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}_{\mu }}(s){{{\dddot z}}^{\mu }}(s) \\ - \,\,\frac{2}{3}\frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s){{{\ddot {z}}}^{0}}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{3}}}}\left( {{{{\dddot z}}^{\nu }}(s) + {{{\ddot {z}}}^{2}}(s){{{\dot {z}}}^{\nu }}(s)} \right). \\ \end{gathered} $$

(A.24)

From Eqs. (A.11) and (A.21),

$$\begin{gathered} \int {\frac{\partial }{{\partial {{x}^{0}}}}} (A_{i}^{{{\text{rad}}}}F_{{{\text{rad}}}}^{{\mu \nu }}){{J}_{\mu }}{{d}^{3}}x = \frac{2}{3}\frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}} \\ \times \,\,\left( {{{{\dddot z}}^{\nu }}(s) + {{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}^{2}}(s)} \right)\frac{1}{{{{{\dot {z}}}^{0}}(s)}} \\ \times \,\,\left( {\frac{1}{3}{{{\dot {z}}}_{i}}(s){{{\dddot z}}_{0}}(s) + {{{\ddot {z}}}_{i}}(s){{{\ddot {z}}}_{0}}(s) + {{{\dddot z}}_{i}}(s){{{\dot {z}}}_{0}}(s)} \right) \\ + \,\,\frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s)}}{{{{{\dot {z}}}^{0}}(s)}}\left[ {\frac{{{{{\dddot z}}^{0}}(s){{{\ddot {z}}}^{\nu }}(s)}}{6}} \right. \\ + \,\,\frac{2}{3}{{{\ddot {z}}}^{0}}(s)\left( {{{{\dddot z}}^{\nu }}(s) + {{{\ddot {z}}}^{2}}(s){{{\dot {z}}}^{\nu }}(s)} \right) \\ \left. { + \,\,{{{\ddot {z}}}^{2}}(s){{{\ddot {z}}}^{\nu }}(s)\left( {\frac{1}{2}{{{\dot {z}}}^{0}}(s) + \frac{2}{3}\frac{1}{{{{{\dot {z}}}^{0}}(s)}}} \right)} \right]. \\ \end{gathered} $$

(A.25)

From Eqs. (A.22) and (A.23), we have

$$\begin{gathered} \int {\frac{{\partial {{J}_{\mu }}}}{{\partial {{x}^{0}}}}} (A_{i}^{{{\text{rad}}}}F_{{{\text{rad}}}}^{{\mu \nu }}){{d}^{3}}x \\ = \frac{2}{3}\frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}}{{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}_{\mu }}(s){{{\dddot z}}^{\mu }}(s) \\ - \,\,\frac{2}{3}\frac{{{{e}^{3}}c}}{{4{{\pi }^{2}}}}\frac{{{{{\ddot {z}}}_{i}}(s){{{\ddot {z}}}^{0}}(s)}}{{{{{({{{\dot {z}}}^{0}}(s))}}^{3}}}}\left( {{{{\dddot z}}^{\nu }}(s) + {{{\ddot {z}}}^{2}}(s){{{\dot {z}}}^{\nu }}(s)} \right). \\ \end{gathered} $$

(A.26)

Radiation reaction becomes

$$\begin{gathered} G_{{NC}}^{\nu } = - \frac{2}{3}\frac{{{{e}^{2}}}}{{4\pi }}\frac{1}{{{{{\dot {z}}}^{0}}(s)}}\left( {{{{\dddot z}}^{\nu }}(s)\, + \,{{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}^{2}}(s)} \right)\, + \,{{\theta }^{{0i}}}\frac{{{{e}^{3}}}}{{16{{\pi }^{2}}}} \\ \times \,\left\{ {\frac{{{{{\ddot {z}}}^{i}}(s)}}{{{{{\dot {z}}}^{0}}(s)}}\left[ {\frac{2}{3}{{{\ddot {z}}}^{0}}(s)\left( {{{{\dddot z}}^{\nu }}(s)\, + \,{{{\ddot {z}}}^{2}}(s){{{\dot {z}}}^{\nu }}(s)} \right)} \right.} \right. \\ \times \,\left( {\frac{1}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}} - 2} \right)\, - \,\frac{1}{6}{{{\dddot z}}^{0}}(s){{{\ddot {z}}}^{\nu }}(s) \\ - \,{{{\ddot {z}}}^{2}}(s){{{\ddot {z}}}^{\nu }}(s)\left( {\frac{1}{2}{{{\dot {z}}}^{0}}(s)\, + \,\frac{2}{3}\frac{1}{{{{{\dot {z}}}^{0}}(s)}}} \right) \\ \left. { - \,\frac{{{{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}_{\mu }}(s){{{\dddot z}}^{\mu }}(s)}}{{{{{\dot {z}}}^{0}}(s)}}} \right]\, - \,\frac{2}{3}\left( {{{{\dddot z}}^{\nu }}(s)\, + \,{{{\dot {z}}}^{\nu }}(s){{{\ddot {z}}}^{2}}(s)} \right) \\ \times \,\left. {\frac{1}{{{{{\dot {z}}}^{0}}(s)}}\left( {\frac{1}{3}{{{\dot {z}}}^{i}}(s){{z}_{0}}(s)\, + \,{{{\dddot z}}^{i}}(s){{{\dot {z}}}_{0}}(s)} \right)} \right\}, \\ \end{gathered} $$

(A.27)

where

$$\begin{gathered} \frac{{{{{\ddot {z}}}^{i}}(s){{{\ddot {z}}}^{0}}(s)}}{{{{{\dot {z}}}^{0}}(s)}}\left( {{{z}^{j}}(s) + {{{\dot {z}}}^{j}}(s){{{\ddot {z}}}^{2}}(s)} \right)\left( {\frac{1}{{{{{({{{\dot {z}}}^{0}}(s))}}^{2}}}} - 2} \right) \\ = - \frac{1}{\gamma }\frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}\left( {{{{\dot {v}}}_{i}} + \frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{v}_{i}}(\vec {v} \cdot \dot {\vec {v}})} \right)\frac{{{{\gamma }^{4}}}}{{{{c}^{3}}}}(\vec {v} \cdot \dot {\vec {v}})\frac{{{{\gamma }^{3}}}}{{{{c}^{3}}}} \\ \times \,\,\left( {{{{\ddot {v}}}_{j}} + \frac{{3{{\gamma }^{2}}}}{{{{c}^{2}}}}{{{\dot {v}}}_{j}}(\vec {v} \cdot \dot {\vec {v}}) + \frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{v}_{j}}(\vec {v} \cdot \ddot {\vec {v}})} \right. \\ \left. { + \,\,\frac{{3{{\gamma }^{4}}}}{{{{c}^{4}}}}{{{(\vec {v} \cdot \dot {\vec {v}})}}^{2}}{{v}_{j}}} \right)\left( {1 + \frac{{{{v}^{2}}}}{{{{c}^{2}}}}} \right) \\ = - \frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\dot {v}}}_{i}}(\vec {v} \cdot \dot {\vec {v}}){{{\ddot {v}}}_{j}} - \frac{{{{\gamma }^{8}}}}{{{{c}^{{10}}}}}{{v}^{2}}{{{\dot {v}}}_{i}}(\vec {v} \cdot \dot {\vec {v}}){{{\ddot {v}}}_{j}}, \\ \end{gathered} $$

(A.28)

with,

$$\begin{gathered} \frac{{{{{\ddot {z}}}^{i}}(s)}}{{{{{\dot {z}}}^{0}}(s)}}{{{\ddot {z}}}^{j}}{{z}^{0}}(s) = \frac{1}{\gamma }\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}\left( {{{{\dot {v}}}_{i}} + \frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{v}_{i}}(\vec {v} \cdot \dot {\vec {v}})} \right) \\ \times \,\,\left( {{{{\dot {v}}}_{j}} + \frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{v}_{j}}(\vec {v} \cdot \dot {\vec {v}})} \right) \\ \times \,\,\left( {(\vec {v} \cdot \ddot {\vec {v}}) + {{{\dot {v}}}^{2}} + 4\frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{{(\vec {v} \cdot \dot {\vec {v}})}}^{2}}} \right) \\ = \frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\dot {v}}}_{i}}\left( {(\vec {v} \cdot \ddot {\vec {v}}) + {{{\dot {v}}}^{2}}} \right){{{\dot {v}}}_{j}} \\ \end{gathered} $$

(A.29)

and

$$\begin{gathered} \frac{{{{{\ddot {z}}}^{i}}(s)}}{{{{{\dot {z}}}^{0}}(s)}}{{{\ddot {z}}}^{2}}(s){{{\ddot {z}}}^{j}}(s)\left( {\frac{{{{z}^{0}}(s)}}{2} + \frac{2}{{3{{{\dot {z}}}^{0}}(s)}}} \right) \\ = \frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}\left( {{{{\dot {v}}}_{i}} + \frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{v}_{i}}(\vec {v} \cdot \dot {\vec {v}})} \right)\frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}\left( {{{{\dot {v}}}_{j}} + \frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{v}_{j}}(\vec {v} \cdot \dot {\vec {v}})} \right) \\ \times \,\,\left( { - \frac{{{{\gamma }^{4}}}}{{{{c}^{4}}}}} \right)\left( {{{{\dot {v}}}^{2}} + 2\frac{{{{\gamma }^{2}}}}{{{{c}^{2}}}}{{{(\vec {v} \cdot \dot {\vec {v}})}}^{2}} + \frac{{{{\gamma }^{4}}}}{{{{c}^{4}}}}{{v}^{2}}{{{(\vec {v} \cdot \dot {\vec {v}})}}^{2}}} \right. \\ \left. { - \,\,\frac{{{{\gamma }^{4}}}}{{{{c}^{2}}}}{{{(\vec {v} \cdot \dot {\vec {v}})}}^{2}}} \right)\left( {\frac{1}{2} + \frac{2}{3}\left( {1 - \frac{{{{v}^{2}}}}{{{{c}^{2}}}}} \right)} \right) \\ = - \frac{{7{{\gamma }^{8}}}}{{6{{c}^{8}}}}{{{\dot {v}}}_{i}}{{{\dot {v}}}^{2}}{{{\dot {v}}}_{j}} + \frac{{2{{\gamma }^{8}}}}{{3{{c}^{{10}}}}}{{{\dot {v}}}_{i}}{{{\dot {v}}}^{2}}{{{\dot {v}}}_{j}}{{v}^{2}}. \\ \end{gathered} $$

(A.30)

Here,

$$\begin{gathered} \frac{{{{{\ddot {z}}}^{i}}}}{{{{{({{{\dot {z}}}^{0}})}}^{2}}}}{{{\dot {z}}}^{j}}{{{\ddot {z}}}_{\mu }}{{z}^{\mu }} = - \frac{{{{\gamma }^{6}}}}{{{{c}^{6}}}}{{{\dot {v}}}_{i}}{{v}_{j}}\dot {\vec {v}} \cdot \ddot {\vec {v}} - 3\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\dot {v}}}_{i}}{{v}_{j}}{{{\dot {v}}}^{2}}\vec {v} \cdot \dot {\vec {v}} \\ - \,\,\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\dot {v}}}_{i}}{{v}^{j}}\vec {v} \cdot \dot {\vec {v}}\left( {{{{\dot {v}}}^{2}} + \vec {v} \cdot \ddot {\vec {v}}} \right) \\ - \,\,\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\dot {v}}}^{i}}{{v}^{j}}\left( {\vec {v} \cdot \dot {\vec {v}}} \right)\left( {\vec {v} \cdot \ddot {\vec {v}}} \right) \\ - \,\,\frac{{{{\gamma }^{8}}}}{{{{c}^{{10}}}}}{{v}^{i}}{{v}^{j}}\left( {\vec {v} \cdot \dot {\vec {v}}} \right)\left( {\dot {\vec {v}} \cdot \ddot {\vec {v}}} \right), \\ \end{gathered} $$

(A.31)

and

$$\begin{gathered} \frac{1}{{{{{\dot {z}}}^{0}}}}\left( {{{z}^{i}}{{{\dot {z}}}^{0}} + \frac{1}{3}{{{\dot {z}}}^{i}}{{z}^{0}}} \right)\left( {{{z}^{j}} + {{{\dot {z}}}^{j}}\mathop {\ddot {z}}\nolimits^2 } \right) \\ = \frac{{{{\gamma }^{6}}}}{{{{c}^{6}}}}{{{\ddot {v}}}_{i}}{{{\ddot {v}}}_{j}} + 3\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\ddot {v}}}_{i}}{{{\dot {v}}}_{j}}\left( {\vec {v} \cdot \dot {\vec {v}}} \right) \\ + \,\,\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\ddot {v}}}_{i}}{{v}_{j}}\left( {\vec {v} \cdot \ddot {\vec {v}}} \right) + 3\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{{\dot {v}}}_{i}}{{{\ddot {v}}}_{j}}\left( {\vec {v} \cdot \dot {\vec {v}}} \right) \\ + \,\,\frac{4}{3}\frac{{{{\gamma }^{8}}}}{{{{c}^{8}}}}{{v}_{i}}{{{\ddot {v}}}_{j}}\left( {{{{\dot {v}}}^{2}} + \vec {v} \cdot \ddot {\vec {v}}} \right). \\ \end{gathered} $$

(A.32)