APPENDIX E

APPENDIX E: CLASSICAL ORBIT EQUATIONS

The classical limit of Eq. (4.16) is found by substituting \(\phi = \exp \left( {R + i{\hbar ^{ – 1}}I} \right)\) into Eq. (4.16) and keeping only the largest terms in powers of \(\hbar \). The result is the relativistic version of the Hamilton-Jacobi equation, \[0 = – {G^\mu }^\nu \left( {{\partial _\mu }I} \right)\left( {{\partial _\nu }I} \right) + {m^2}{c^2}.\tag{E.1}\] The weak-field limit for G is given by Eq. (5.14) and Eq. (5.15), \[\begin{array}{l}{G^{00}} = {(1 – b{r^{ – 1}})^{ – 1}},\\{G^{rr}} = – {(1 + b{r^{ – 1}})^{ – 1}},\\{G^{\theta \theta }} = – {r^{ – 2}},\\{G^{\phi \phi }} = – {r^{ – 2}}{\sin ^{ – 2}}\theta .\end{array}\tag{E.2}\] With no loss of generality, the orbit can lie in the plane where \(\theta = \pi /2\), and Eq. (E.1) becomes, \[0 = {\left( {1 – b{r^{ – 1}}} \right)^{ – 1}}{c^{ – 2}}{\left( {\partial I/\partial \tau } \right)^2} – {\left( {1 + b{r^{ – 1}}} \right)^{ – 1}}{\left( {{\partial _r}I} \right)^2} – {r^{ – 2}}{\left( {{\partial _\phi }I} \right)^2} – {m^2}{c^2}{\rm{.}}\tag{E.3}\] Hamilton’s principle function, I, is as follows: \[I = E\tau + A\phi + f\left( {r,E,A} \right){\rm{,}}\tag{E.4}\] where E is the conserved total energy of the particle, and A is its conserved angular momentum. There are three conditions: \[\begin{array}{l}\partial I/\partial E = 0 = \tau + \partial f\,/\partial E{\rm{,}}\\\partial I/\partial A = 0 = \phi + \partial f\,/\partial A,\\{\left( {1 + b{r^{ – 1}}} \right)^{ – 1}}{\left( {{\partial _r}f} \right)^2} = {\left( {1 – b{r^{ – 1}}} \right)^{ – 1}}{c^{ – 2}}{E^2} – {A^2}{r^{ – 2}} – {m^2}{c^2}{\rm{.}}\end{array}\tag{E.5}\] Set \(\sigma = {r^{ – 1}}\), and then a series of manipulations gives the orbit equation, \[{\left( {d\sigma /d\phi } \right)^2} = {A^{ – 2}}{\left( {1 + b\sigma } \right)^{ – 1}}[{\left( {1 – b\sigma } \right)^{ – 1}}{c^{ – 2}}{E^2} – {A^2}{\sigma ^2} – {m^2}{c^2}{\rm{]}}{\rm{.}}\tag{E.6}\] Eq. (E.6) is accurate for \(b\sigma < < 1\)—the weak-field condition.
To calculate the weak-field orbit from Eq. (E.6), expand the right-hand side as a power series in \(\sigma \) keeping terms only up to the first order in b. Then take the \(\phi \) derivative to obtain the following: \[{d^2}\sigma /d{\phi ^2} = {R^{ – 1}} – \sigma + \alpha {\sigma ^2}{\rm{,}}\tag{E.7}\] where \(R = {A^2}{\left( {GM{m^2}} \right)^{ – 1}}\) and \(\alpha = {\textstyle{3 \over 2}}b = 3GM{c^{ – 2}}\). An approximate solution of Eq. (E.7) follows: \[{r^{ – 1}} = {R^{ – 1}}\{ 1 + \varepsilon \cos [\phi \left( {1 – \alpha {R^{ – 1}}} \right)]\} {\rm{.}}\tag{E.8}\] Eq. (E.8) is the usual Newtonian orbit but with a small advance of perihelion per cycle. This advance is exactly the same as the famous general relativity prediction. The path of a light ray through a central field is described by Eq. (E.7) as well by taking the limit where \(m = 0\), \[{d^2}\sigma /d{\phi ^2} = – \sigma + \alpha {\sigma ^2}{\rm{,}}\tag{E.9}\] A weak-field solution to Eq. (E.9) yields \[{r^{ – 1}} = {R^{ – 1}}\,\cos \phi + \beta \left( {1 + {{\sin }^2}\phi } \right){\rm{,}}\tag{E.10}\] where \(\beta = {\textstyle{1 \over 3}}\alpha {R^{ – 2}}\) and \(\phi = 0\) corresponds to perihelion (R is the impact parameter of scattering theory for the photon—the radius of the sun for an observation). The asymptotes of the path described by Eq. (E.10) are bent at a relative angle of \(\Delta \phi = {\textstyle{4 \over 3}}\alpha {R^{ – 1}}\)—the same as the famous GR prediction. The general relativity versions of the equations of this appendix are developed in [13].
There is another effect of interest, gravitational red shift. Consider a transverse electromagnetic wave-train moving radially outward in a central gravitational field. The wave-train will consist of 2N peaks within an expanding (thin) shell in a vacuum, The solution of Eq. (4.19) for the vector potential (\({A_\theta }(\tau ,r) = {r^{ – 1}}\sin (\psi )\)) follows, \[G_0^0{c^{ – 2}}{\partial ^2}{A_\theta }/\partial {\tau ^2} = {r^{ – 2}}{\partial _r}({r^2}G_r^r{\partial _r}{A_\theta }),\tag{E.11}\] where \(G_0^0 = {(1 – b{r^{ – 1}})^{ – 1}}\) and \(G_r^r = {(1 + b{r^{ – 1}})^{ – 1}}\). The interpretation of Eq. (E.11) will follow the reasoning of section III(D). The argument of the potential can be defined as follows, \[\psi (\tau ,r) = 2\pi \lambda {(r)^{ – 1}}\xi (\tau ,r){\rm{ ,}}\tag{E.12}\] where \(\xi = r – c\tau \) near the center of the wave train (\(\xi = 0\)). Substitute the potential into Eq, (E.11), and keep those terms which could survive in the limit (\(\xi \to 0\)), \[\begin{array}{l}0 = G_r^r\partial _r^2\psi + ({\partial _r}G_r^r){\partial _r}\psi ,\\{\partial _r}\psi = {\lambda ^{ – 1}} – ({\lambda ^{ – 2}}{\partial _r}\lambda )\xi \to {\lambda ^{ – 1}},\\\partial _r^2\psi = – 2{\lambda ^{ – 2}}{\partial _r}\lambda + order(\xi ) \to – 2{\lambda ^{ – 2}}{\partial _r}\lambda ,\\2{\partial _r}\ln \lambda = {\partial _r}\ln G_r^r,\\\lambda (\infty )/\lambda (R) = {(1 + b{R^{ – 1}})^{1/2}} \to {(1 – b{R^{ – 1}})^{ – 1/2}}{\rm~{ weak~field}}{\rm{.}}\end{array}\tag{E.13}\] Eq. (E.13) is the usual weak-field result. \(\lambda (\infty )\) is the observed wavelength (at an infinite distance from the origin) for a photon that was emitted with wavelength \(\lambda (R)\) at a distance R from the origin. Furthermore, Eq. (E.12) indicates that observers at any distance from the origin will see the wave train moving with velocity c. Following the analysis of III(D), the shell contains a fixed number of photons (photons cannot be created or destroyed in flight through a vacuum). The result of Eq. (E.13) is valid for an individual photon. GR agrees with NG for local weak-field situations, but not for distal or strong field situations.