Post-Newtonian Limit of Hybrid Metric-Palatini f(R)-Gravity

Abstract

Using the latest most accurate values of post-Newtonian parameters γ and β obtained by MESSENGER we impose restrictions on the recently proposed hybrid f(R)-gravity model in its scalar–tensor representation. We show that the presence of a light scalar field in this theory does not contradict the experimental data based not only on the γ parameter (as was shown earlier), but also on all other PPN parameters. The application of parameterized post-Newtonian formalism to gravitational theories with massive fields is also discussed.

Appendices

POINT-MASS AND PERFECT FLUID PPN METRIC

Point-mass metric [47]:

$$\begin{gathered} {{g}_{{00}}} = - 1 + 2{{\sum\limits_k^{} {\frac{G}{{{{c}^{2}}}}\frac{{{{m}_{k}}}}{{{{r}_{k}}}} - 2\beta \left( {\sum\limits_k^{} {\frac{G}{{{{c}^{2}}}}\frac{{{{m}_{k}}}}{{{{r}_{k}}}}} } \right)} }^{2}} \\ + \,2(1 - 2\beta + {{\zeta }_{2}})\sum\limits_k^{} {\frac{G}{{{{c}^{2}}}}\frac{{{{m}_{k}}}}{{{{r}_{k}}}}\sum\limits_{j \ne k}^{} {\frac{G}{{{{c}^{2}}}}\frac{{{{m}_{j}}}}{{{{r}_{{jk}}}}}} } \\ \, + (2\gamma + 1 + {{\alpha }_{3}} + {{\zeta }_{1}})\sum\limits_k^{} {\frac{G}{{{{c}^{4}}}}\frac{{{{m}_{k}}{v}_{k}^{2}}}{{{{r}_{k}}}}} \\ \, - {{\zeta }_{1}}\sum\limits_k^{} {\frac{G}{{{{c}^{4}}}}\frac{{{{m}_{k}}}}{{r_{k}^{3}}}{{{({{{\mathbf{v}}}_{k}} \cdot {{{\mathbf{r}}}_{k}})}}^{2}} - ({{\alpha }_{1}} - {{\alpha }_{2}} - {{\alpha }_{3}}){{w}^{2}}} \\ \, \times \sum\limits_k^{} {\frac{G}{{{{c}^{4}}}}\frac{{{{m}_{k}}}}{{r_{k}^{3}}}} - {{\alpha }_{2}}\sum\limits_k^{} {\frac{G}{{{{c}^{4}}}}\frac{{{{m}_{k}}}}{{{{r}_{k}}}}} {{({\mathbf{w}} \cdot {{{\mathbf{r}}}_{k}})}^{2}} \\ \end{gathered} $$

$$\, + 2{{\alpha }_{3}} - {{\alpha }_{1}})\sum\limits_k^{} {\frac{G}{{{{c}^{4}}}}\frac{{{{m}_{k}}}}{{{{r}_{k}}}}} ({\mathbf{w}} \cdot {{{\mathbf{v}}}_{k}}),$$

((34))

$$\begin{gathered} {{g}_{{0j}}} = - \frac{1}{2}(4\gamma + 3 + {{\alpha }_{1}} - {{\alpha }_{2}} + {{\zeta }_{1}})\sum\limits_k^{} {\frac{G}{{{{c}^{3}}}}\frac{{{{m}_{k}}{v}_{k}^{j}}}{{{{r}_{k}}}}} \\ - \frac{1}{2}(1 + {{\alpha }_{2}} - {{\zeta }_{1}})\sum\limits_k^{} {\frac{G}{{{{c}^{3}}}}\frac{{{{m}_{k}}}}{{r_{k}^{3}}}({{{\mathbf{v}}}_{k}} \cdot {{{\mathbf{r}}}_{k}})r_{k}^{j}} \\ - \frac{1}{2}({{\alpha }_{1}} - 2{{\alpha }_{2}}){{w}^{j}}\sum\limits_k^{} {\frac{G}{{{{c}^{3}}}}\frac{{{{m}_{k}}}}{{{{r}_{k}}}}} + {{\alpha }_{2}}\sum\limits_k^{} {\frac{G}{{{{c}^{3}}}}\frac{{{{m}_{k}}}}{{r_{k}^{3}}}({\mathbf{w}} \cdot {{{\mathbf{r}}}_{k}})r_{k}^{j},} \\ {{g}_{{ij}}} = \left( {1 + 2\gamma \sum\limits_k^{} {\frac{G}{{{{c}^{3}}}}\frac{{{{m}_{k}}}}{{{{r}_{k}}}}} } \right){{\delta }_{{ij}}}, \\ \end{gathered} $$

here wⁱ is coordinate velocity of PPN coordinate system relative to the mean rest frame of the universe.

Perfect fluid metric [57]:

$$\begin{gathered} {{g}_{{00}}} = - 1 + 2\frac{1}{{{{c}^{2}}}}U - 2\beta \frac{1}{{{{c}^{4}}}}{{U}^{2}} \\ \, + (2\gamma + 1 + {{\alpha }_{3}} + {{\zeta }_{1}} - 2\xi )\frac{1}{{{{c}^{4}}}}{{\Phi }_{1}} \\ \, - 2(2\beta - 1 - {{\zeta }_{2}} - \xi )\frac{1}{{{{c}^{4}}}}{{\Phi }_{2}} + 2(1 + {{\zeta }_{3}})\frac{1}{{{{c}^{4}}}}{{\Phi }_{3}} \\ \end{gathered} $$

$$\begin{gathered} \, + \frac{1}{{{{c}^{4}}}}{{\Phi }^{{{\text{PF}}}}} + 2(3\gamma + 3{{\zeta }_{4}} - 2\xi )\frac{1}{{{{c}^{4}}}}{{\Phi }_{4}} \\ \, - ({{\zeta }_{1}} - 2\xi )\frac{1}{{{{c}^{4}}}}{{\Phi }_{6}} - 2\xi \frac{1}{{{{c}^{4}}}}{{\Phi }_{W}}, \\ \end{gathered} $$

((35))

$$\begin{gathered} {{g}_{{0j}}} = - \frac{1}{2}(4\gamma + 3 + {{\alpha }_{1}} - {{\alpha }_{2}} + {{\zeta }_{1}} - 2\xi )\frac{1}{{{{c}^{3}}}}{{V}_{j}} \\ - \frac{1}{2}(1 + {{\alpha }_{2}} - {{\zeta }_{1}} + 2\xi )\frac{1}{{{{c}^{3}}}}{{W}_{j}} + \frac{1}{{{{c}^{3}}}}\Phi _{j}^{{{\text{PF}}}}, \\ \end{gathered} $$

$${{g}_{{ij}}} = \left( {1 + 2\gamma \frac{1}{{{{c}^{2}}}}U} \right){{\delta }_{{ij}}},$$

where PPN potentials are represented

$$\begin{gathered} U = \int {G\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} d{\mathbf{r}}{\kern 1pt} ',\quad {{\Phi }_{1}} = \int {G\frac{{\rho {\kern 1pt} '{v}{\kern 1pt} {{'}^{2}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} d{\mathbf{r}}{\kern 1pt} ', \\ {{\Phi }_{2}} = \int {G\frac{{\rho {\kern 1pt} 'U{\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} d{\mathbf{r}}{\kern 1pt} ',\quad {{\Phi }_{3}} = \int {G\frac{{\rho {\kern 1pt} '\Pi {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} d{\mathbf{r}}{\kern 1pt} ', \\ {{\Phi }_{4}} = \int {G\frac{{p{\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} d{\mathbf{r}}{\kern 1pt} ',\quad {{V}_{j}} = \int {G\frac{{\rho {{{v}}_{j}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} d{\mathbf{r}}{\kern 1pt} ', \\ \end{gathered} $$

$$\begin{gathered} {{\Phi }_{6}} = \int {G\rho {\kern 1pt} '} {v}_{j}^{'}{v}_{k}^{'}\frac{{{{{(r - r{\kern 1pt} ')}}^{j}}{{{(r - r{\kern 1pt} ')}}^{k}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{{{\text{|}}}^{3}}}}d{\mathbf{r}}{\kern 1pt} ', \\ \Phi _{j}^{{{\text{PF}}}} = - \frac{1}{2}{{\alpha }_{1}}{{w}_{j}}U + {{\alpha }_{2}}{{w}^{i}}{{U}_{{ij}}}, \\ \end{gathered} $$

((36))

$$\begin{gathered} {{\Phi }_{W}} = \int {{{G}^{2}}\rho {\kern 1pt} '\rho {\kern 1pt} ''\frac{{{{{(r - r{\kern 1pt} ')}}_{j}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{{{\text{|}}}^{3}}}}} \left[ {\frac{{{{{(r{\kern 1pt} '\, - r{\kern 1pt} '')}}^{j}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}} - \frac{{{{{(r - r{\kern 1pt} '')}}^{j}}}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}} \right]d{\mathbf{r}}{\kern 1pt} '{\kern 1pt} d{\mathbf{r}}{\kern 1pt} '', \\ {{W}_{j}} = \int {G\frac{{\rho {\kern 1pt} '{\kern 1pt} {\mathbf{v}}{\kern 1pt} '({\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '){{{(r - r{\kern 1pt} '{\kern 1pt} )}}_{j}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{{{\text{|}}}^{3}}}}d{\mathbf{r}}{\kern 1pt} ',} \\ {{U}_{{ij}}} = \int {G\frac{{\rho {\kern 1pt} '{\kern 1pt} {\mathbf{v}}{\kern 1pt} '{{{(r - r{\kern 1pt} '{\kern 1pt} )}}_{i}}{{{(r - r{\kern 1pt} '{\kern 1pt} )}}_{j}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}d{\mathbf{r}}{\kern 1pt} ',} \\ {{\Phi }^{{{\text{PF}}}}} = ({{\alpha }_{3}} - {{\alpha }_{1}}){{w}^{2}}U + {{\alpha }_{2}}{{w}^{j}}{{w}^{i}}{{U}_{{ij}}} + (2{{\alpha }_{3}} - {{\alpha }_{1}}){{w}^{j}}{{V}_{j}}. \\ \end{gathered} $$

Here “PF”-potentials are responsible for preferred frames effects.

THE PERFECT FLUID METRIC OF THE HYBRID f(R)-GRAVITY

The perfect fluid metric of hybrid f(R)-gravity:

$$\begin{gathered} {{g}_{{00}}} = - 1 + \frac{{{{k}^{2}}}}{{4\pi (1 + {{\phi }_{0}}){{c}^{2}}}}\int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \\ \times \,\,\left( {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right)d{\mathbf{r}}{\kern 1pt} '\, - \frac{{{{k}^{4}}}}{{32{{\pi }^{2}}{{{(1 + {{\phi }_{0}})}}^{2}}{{c}^{4}}}} \\ \end{gathered} $$

$$\begin{gathered} \times \int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \left( {1\, - \,\frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right)\frac{{\rho {\kern 1pt} ''}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}\left( {1\, - \,\frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}}} \right)d{\mathbf{r}}{\kern 1pt} ''{\kern 1pt} d{\mathbf{r}}{\kern 1pt} ' \\ + \frac{{{{k}^{4}}}}{{32{{\pi }^{2}}{{{(1 + {{\phi }_{0}})}}^{2}}{{c}^{4}}}}\frac{{{{\phi }_{0}}(1 + {{\phi }_{0}})}}{{18}} \\ \end{gathered} $$

$$\begin{gathered} \, \times \int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} {{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}\frac{{\rho {\kern 1pt} ''}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}}d{\mathbf{r}}{\kern 1pt} ''{\kern 1pt} d{\mathbf{r}}{\kern 1pt} ' \\ + \frac{{{{k}^{2}}}}{{4\pi (1 + {{\phi }_{0}}){{c}^{4}}}}\int {\frac{{\Pi {\kern 1pt} '{\kern 1pt} \rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}\left( {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right)d{\mathbf{r}}{\kern 1pt} '} \\ \end{gathered} $$

$$\begin{gathered} \, + \frac{{{{k}^{2}}}}{{2\pi (1 + {{\phi }_{0}}){{c}^{4}}}}\int {\frac{{\rho {\kern 1pt} '{\kern 1pt} {v}{\kern 1pt} {{'}^{2}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} d{\mathbf{r}}{\kern 1pt} '\, + \frac{{3{{k}^{2}}}}{{4\pi (1 + {{\phi }_{0}}){{c}^{4}}}} \\ \, \times \int {\frac{{p{\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \left( {1 + \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right)d{\mathbf{r}}{\kern 1pt} ' \\ \end{gathered} $$

$$\begin{gathered} + \frac{{{{k}^{2}}}}{{32\pi (1 + {{\phi }_{0}}){{c}^{4}}}}\int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \left[ {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right]d{\mathbf{r}}{\kern 1pt} ' \\ \, \times \int {\frac{{\hat {\rho }}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}} \left( {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}}} \right)d{\mathbf{\hat {r}}} \\ \end{gathered} $$

$$\begin{gathered} + \frac{{3{{k}^{4}}}}{{32{{\pi }^{2}}{{{(1 + {{\phi }_{0}})}}^{2}}{{c}^{4}}}}\int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \left[ {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right]d{\mathbf{r}}{\kern 1pt} ' \\ \, \times \int {\frac{{\hat {\rho }}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}} \left( {1 + \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}}} \right)d{\mathbf{\hat {r}}} \\ \end{gathered} $$

$$\begin{gathered} - \frac{{{{k}^{4}}}}{{16{{\pi }^{2}}{{{(1 + {{\phi }_{0}})}}^{2}}{{c}^{4}}}}\int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \left[ {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right]d{\mathbf{r}}{\kern 1pt} ' \\ \, \times \int {\frac{{\hat {\rho }}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}} \left( {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}}} \right)d{\mathbf{\hat {r}}} \\ \end{gathered} $$

$$\begin{gathered} + \frac{{{{k}^{4}}}}{{16{{\pi }^{2}}{{{(1 + {{\phi }_{0}})}}^{2}}{{c}^{4}}}}\frac{{{{\phi }_{0}}(1 + {{\phi }_{0}})}}{{18}}\int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} {{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}d{\mathbf{r}}{\kern 1pt} ' \\ \, \times \int {\frac{{\hat {\rho }}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}} {{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}}d{\mathbf{\hat {r}}} \\ \end{gathered} $$

$$\begin{gathered} + \frac{{(7{{\phi }_{0}} + 1){{k}^{4}}{{\phi }_{0}}}}{{2304{{\pi }^{3}}{{{(1 + {{\phi }_{0}})}}^{2}}{{c}^{4}}}}m_{\varphi }^{2}\int {\frac{{{{e}^{{ - {{m}_{\varphi }}|{\mathbf{r}} - {\mathbf{r}}'|}}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}d{\mathbf{r}}{\kern 1pt} '} \\ \times \left( {\int {\frac{{\hat {\rho }}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}}\frac{{\rho {\kern 1pt} ''}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}}d{\mathbf{\hat {r}}}d{\mathbf{r}}{\kern 1pt} ''} } \right) \\ \end{gathered} $$

((37))

$$\begin{gathered} + \frac{{{{k}^{4}}{{\phi }_{0}}}}{{192{{\pi }^{3}}{{{(1 + {{\phi }_{0}})}}^{2}}{{c}^{4}}}}{{m}_{\varphi }}\int {\frac{{{{e}^{{ - {{m}_{\varphi }}|{\mathbf{r}} - {\mathbf{r}}'|}}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}d{\mathbf{r}}{\kern 1pt} '} \\ \times \left( {\int {\frac{{\hat {\rho }}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}}\frac{{\rho {\kern 1pt} ''}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}}d{\mathbf{\hat {r}}}d{\mathbf{r}}{\kern 1pt} ''} } \right) \\ \end{gathered} $$

$$\begin{gathered} + \frac{{{{k}^{4}}\phi _{0}^{2}}}{{1728{{\pi }^{3}}{{c}^{4}}}}\left[ {V{\kern 1pt} ''\, - \frac{{{{\phi }_{0}}}}{2}V{\kern 1pt} '''} \right]\int {\frac{{{{e}^{{ - {{m}_{\varphi }}|{\mathbf{r}} - {\mathbf{r}}'|}}}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}d{\mathbf{r}}{\kern 1pt} '} \\ \times \left( {\int {\frac{{\hat {\rho }}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{\hat {r}}}{\text{|}}}}}\frac{{\rho {\kern 1pt} ''}}{{{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}}{\kern 1pt} '\, - {\mathbf{r}}{\kern 1pt} ''{\text{|}}}}}d{\mathbf{\hat {r}}}d{\mathbf{r}}{\kern 1pt} ''} } \right), \\ \end{gathered} $$

$$\begin{gathered} {{g}_{{0i}}} = - \frac{{{{k}^{2}}}}{{4\pi (1 + {{\phi }_{0}}){{c}^{3}}}}\int {\frac{{\rho {\kern 1pt} '{\kern 1pt} {v}_{i}^{'}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \left( {1 + \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right)d{\mathbf{r}}{\kern 1pt} ' \\ - \frac{{3{{k}^{2}}}}{{16\pi (1 + {{\phi }_{0}}){{c}^{3}}}}\int {\frac{{\rho {\kern 1pt} '{\kern 1pt} {v}_{i}^{'}}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}} \left( {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right)d{\mathbf{r}}{\kern 1pt} ' \\ - \frac{{{{k}^{2}}}}{{16\pi (1 + {{\phi }_{0}}){{c}^{3}}}}\int {\frac{{\rho {\kern 1pt} '{\kern 1pt} x_{i}^{'}({\mathbf{v}}{\kern 1pt} '\, \cdot {\mathbf{r}}{\kern 1pt} ')}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{{{\text{|}}}^{3}}}}\left[ {1 - \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right]} d{\mathbf{r}}, \\ \end{gathered} $$

$$\begin{gathered} {{g}_{{ij}}} = {{\delta }_{{ij}}}\left( {1 + \frac{{{{k}^{2}}}}{{4\pi (1 + {{\phi }_{0}}){{c}^{2}}}}} \right. \\ \left. {\, \times \int {\frac{{\rho {\kern 1pt} '}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}\left( {1 + \frac{{{{\phi }_{0}}}}{3}{{e}^{{ - {{m}_{\varphi }}{\text{|}}{\mathbf{r}} - {\mathbf{r}}{\kern 1pt} '{\text{|}}}}}} \right)d{\mathbf{r}}{\kern 1pt} '} } \right). \\ \end{gathered} $$