APPENDIX J: SOURCES FOR EQ. (4.22)
I will concentrate on a non-relativistic scalar boson source defined by Eq. (4.16) in this appendix. For that case, the LDM will be as follows for cartesian coordinates,
\[\begin{array}{l}
\mathcal{L}_M = \mathcal{L}_M^{\rm T} – \mathcal{L}_M^{\rm V},\\
\mathcal{L}_M^{\rm T} = \mathcal{N}\,{G^{\alpha \beta }}{\partial _\alpha }\phi (x){\partial _\beta }\phi {(x)^ * },\\
\mathcal{L}_M^{\rm V} = \mathcal{N}L^{ – 2}\phi (x)\phi {(x)^ * },\\
L^{ – 1} = mc/\hbar {\rm{ ,}}
\end{array}\tag{J.1}\]
where \(\mathcal{N}\) is a normalization constant. \({\mathcal{L}_M}\) has dimensions of energy density divided by \(c\). The energy density for this non-relativistic particle is dominated by \(c{\mathcal{L}_M}^{\rm{V}}\), and the following results,
\[\int {{d^3}} x\,\mathcal{N}L^{ – 2}{\left| {\phi (x)} \right|^2} = mc.\tag{J.2}\]
The Schrodinger wave function comes from \(\phi \left( {{x_0},{\bf{x}}} \right) = \psi \left( {\tau ,{\bf{x}}} \right)\,\exp \left( {i{x_0}{L^{ – 1}}} \right)\), where \({x^0} = c\tau \). The weak field source for Eq. (4.22) is then as follows,
\[{J_{00}}(q,\tau ,{\bf{x}}) = – {Q_{GC}}(q,E){\kappa _q}\{ \mathcal{N}{L^{ – 2}}{\left| {\psi (\tau ,{\bf{x}})} \right|^2}\} ,\tag{J.3}\]
where the other components of \({J_{\mu \nu }}\) are negligible. The mass of the “particle” could have any value—an electron, a stellar mass object, or even a galaxy. The classical limit for a source (being treated as a point) is given by using Eq. (J.2) and Eq. (J.3),
\[\{ \mathcal{N}L^{ – 2}{\left| {\psi (\tau ,{\bf{x}})} \right|^2}\} \to mc{\delta ^3}({\bf{x}} – \mathbf{X}(\tau )),\tag{J.4}\]
where \({\delta ^3}\) is the three-dimensional Dirac delta function, and \({\rm{X(}}\tau {\rm{)}}\) is the vector position of the point source as a function of physical time. Thus, Eq. (4.22) for a point source in cartesian coordinates is given by,
\[{\nabla ^2}{v_{00}}(q,\tau ,{\bf{x}}) – L_{gq}^{ – 2}{v_{00}}(q,\tau ,{\bf{x}}) = 8\pi Gm{c^{ – 2}}{Q_{GC}}(q,E){Z_{gq}}{\delta ^3}({\bf{x}} – \mathbf{X}(\tau )),\tag{J.5}\]
where G is the Newtonian gravitational constant. \({Q_{GC}}(q,E)\) is defined in section IV(D) and \({Z_{gq}}\) is defined in section V(B). The \({S_0}\) and \({N_0}\) terms of Eq. (4.22) cancel out the time derivatives of \({v_{00}}\) and the time derivatives of other components will be ignorable for most problems of interest. An exception would be the generation of strong gravitational waves from matter in accelerated paths.
Eq. (J.5) is linear so a collection of N point sources with different positions will generate a gravitational field which is the sum of the individual solutions of Eq. (J.5).