# 212. Weak-Field Gravity from Symmetric Constitutive Closure This appendix develops the weak-field truncation of the gravity interaction used in chapter 13 as far as the static benchmark set. The framework is not meant to stop at weak field. Exact interaction is the actual target. This appendix keeps only the leading weak-field term because the benchmark observables treated here are measured in that regime. The point is not to assume spacetime curvature first and then reinterpret it. The point is to begin from the constitutive closure adopted in chapter 13, derive the corresponding weak-field transport geometry, and then recover from that one geometry the standard leading weak-field observables: - gravitational redshift, - light bending, - Shapiro delay, - perihelion precession. Appendix 215 explains why the light-bending factor of two should be traced more deeply to the two-aspect stress of a null Maxwell probe, and appendix 216 derives the same weak exterior factor directly from the sign-symmetric axial loading of a static toroidal closure. The present appendix does not replace that deeper derivation. It keeps the symmetric constitutive closure as the macroscopic summary used in chapter 13 for the interaction of a probe with a static bounded mass closure and derives its static weak-field consequences. The time-dependent radiative sector is not treated here. ## 212.1 Constitutive Summary Used Here Chapter 13 used the symmetric constitutive modification $$ \varepsilon_{\mathrm{eff}}(r)=\varepsilon_0\bigl(1+2\eta(r)\bigr), \qquad \mu_{\mathrm{eff}}(r)=\mu_0\bigl(1+2\eta(r)\bigr), $$ with $$ \eta(r):=\frac{GM}{rc^2}, \qquad \eta\ll 1. $$ The corresponding local propagation speed is $$ k(r) := \frac{1}{\sqrt{\varepsilon_{\mathrm{eff}}(r)\mu_{\mathrm{eff}}(r)}} = \frac{c}{1+2\eta(r)} = c\bigl(1-2\eta(r)\bigr)+O(\eta^2). $$ So this constitutive summary determines the local transport speed of weak electromagnetic disturbances in the background generated by the central mass. ## 212.2 Determination of the Weak-Field Metric At leading weak-field order in $\eta$, the most general static spherically symmetric isotropic metric can be written as $$ ds^2 = c^2\bigl(1+2A\eta(r)\bigr)\,dt^2 - \bigl(1+2B\eta(r)\bigr)\,\bigl(dr^2+r^2d\Omega^2\bigr) + O(\eta^2), $$ where $A$ and $B$ are constants still to be determined. We now fix them by two conditions. ### 212.2.1 Newtonian Slow-Mode Limit For a slowly moving bounded configuration, write the action $$ S=-mc\int ds. $$ Using $v^2=\dot r^2+r^2\dot\Omega^2$ and dividing by $dt$, the Lagrangian is $$ L = -mc^2 \sqrt{ \bigl(1+2A\eta\bigr) - \bigl(1+2B\eta\bigr)\frac{v^2}{c^2} } +O(\eta^2). $$ Expand to leading order in $\eta$ and $v^2/c^2$: $$ L = -mc^2 \left( 1+A\eta-\frac{v^2}{2c^2} \right) +O\!\left(\eta\frac{v^2}{c^2},\,\eta^2,\,\frac{v^4}{c^4}\right). $$ Up to the irrelevant additive constant $-mc^2$, this becomes $$ L = \frac{1}{2}mv^2 -mAc^2\eta +O(c^{-2}). $$ Since $$ c^2\eta=\frac{GM}{r}, $$ the effective potential is $$ V(r)=mAc^2\eta=\frac{AGMm}{r}. $$ To recover the Newtonian attractive potential $$ V_{\mathrm{N}}(r)=-\frac{GMm}{r}, $$ we must have $$ A=-1. $$ ### 212.2.2 Null Transport Speed For a radial null path, $ds^2=0$ and $d\Omega=0$, so $$ 0 = c^2\bigl(1+2A\eta\bigr)\,dt^2 - \bigl(1+2B\eta\bigr)\,dr^2. $$ Hence the radial coordinate speed is $$ \frac{dr}{dt} = c\sqrt{\frac{1+2A\eta}{1+2B\eta}} = c\bigl(1+(A-B)\eta\bigr)+O(\eta^2). $$ But the constitutive closure already fixed the transport speed to be $$ k(r)=c\bigl(1-2\eta\bigr)+O(\eta^2). $$ Therefore $$ A-B=-2. $$ Since $A=-1$, it follows that $$ B=1. $$ ### 212.2.3 Resulting Weak-Field Metric The weak-field transport geometry selected by the adopted constitutive closure is therefore $$ ds^2 = c^2\bigl(1-2\eta(r)\bigr)\,dt^2 - \bigl(1+2\eta(r)\bigr)\,\bigl(dr^2+r^2d\Omega^2\bigr) + O(\eta^2). $$ This is exactly the weak-field Schwarzschild metric in isotropic coordinates. So the constitutive summary used in chapter 13 does not merely reproduce light bending. It fixes the full static weak-field metric at leading order. ## 212.3 Gravitational Redshift For a stationary clock at radius $r$, we have $dr=d\Omega=0$, so $$ d\tau = \sqrt{1-2\eta(r)}\,dt = \bigl(1-\eta(r)\bigr)\,dt+O(\eta^2). $$ Thus the proper time of a static localized mode runs more slowly deeper in the potential well. In a static metric, the covector component $p_t$ of a photon is conserved. The frequency measured by a static observer with four-velocity proportional to $\partial_t$ is therefore proportional to $1/\sqrt{g_{tt}}$. Hence $$ \frac{\nu_{\mathrm{obs}}}{\nu_{\mathrm{em}}} = \sqrt{\frac{g_{tt}(r_{\mathrm{em}})}{g_{tt}(r_{\mathrm{obs}})}} = \sqrt{\frac{1-2\eta(r_{\mathrm{em}})}{1-2\eta(r_{\mathrm{obs}})}}. $$ At leading weak-field order, $$ \frac{\Delta\nu}{\nu} := \frac{\nu_{\mathrm{obs}}-\nu_{\mathrm{em}}}{\nu_{\mathrm{em}}} = \eta(r_{\mathrm{obs}})-\eta(r_{\mathrm{em}}) +O(\eta^2). $$ If the observer is far away, $\eta(r_{\mathrm{obs}})\approx 0$, then $$ \frac{\Delta\nu}{\nu} = -\frac{GM}{r_{\mathrm{em}}c^2}. $$ So the signal is redshifted when it climbs out of the central field. ## 212.4 Optical Index, Light Bending, and Shapiro Delay For null propagation in a static isotropic metric, the optical index is $$ n(r) := \sqrt{\frac{1+2\eta(r)}{1-2\eta(r)}} = 1+2\eta(r)+O(\eta^2) = 1+\frac{2GM}{rc^2}+O(\eta^2). $$ This is exactly the index used heuristically in chapter 13, but it is now derived from the same weak-field metric that also yields redshift and orbital precession. ### 212.4.1 Light Bending Take a ray passing the mass with impact parameter $b$. At leading weak-field order, use the straight-line approximation $$ r(z)=\sqrt{b^2+z^2}. $$ The transverse gradient of the refractive index is $$ \partial_b n(r(z)) = \frac{d}{db} \left( 1+\frac{2GM}{c^2\sqrt{b^2+z^2}} \right) = -\frac{2GM\,b}{c^2(b^2+z^2)^{3/2}}. $$ The total bending magnitude is therefore $$ \theta = \int_{-\infty}^{\infty}|\partial_b n|\,dz = \frac{2GM\,b}{c^2} \int_{-\infty}^{\infty}\frac{dz}{(b^2+z^2)^{3/2}}. $$ Since $$ \int_{-\infty}^{\infty}\frac{dz}{(b^2+z^2)^{3/2}} = \frac{2}{b^2}, $$ we obtain $$ \theta = \frac{4GM}{bc^2}. $$ This is the standard weak-field light-bending value. ### 212.4.2 Shapiro Delay For a null path, $$ dt=\frac{n(r)}{c}\,ds. $$ So the coordinate travel time between two points $A$ and $B$ is $$ T = \frac{1}{c}\int_A^B n(r)\,ds = \frac{1}{c}\int_A^B ds + \frac{2GM}{c^3}\int_A^B \frac{ds}{r} +O(\eta^2). $$ The first term is the flat-space travel time. The second is the gravitational delay. Approximate the path by a straight line with impact parameter $b$, coordinate $z$, and endpoints at $z=z_A$ and $z=z_B$. Then $$ r(z)=\sqrt{b^2+z^2}, \qquad ds=dz, $$ so $$ \Delta T = \frac{2GM}{c^3} \int_{z_A}^{z_B}\frac{dz}{\sqrt{b^2+z^2}}. $$ Using $$ \int \frac{dz}{\sqrt{b^2+z^2}} = \ln\!\left(z+\sqrt{b^2+z^2}\right), $$ we obtain $$ \Delta T = \frac{2GM}{c^3} \ln\!\left( \frac{z_B+\sqrt{b^2+z_B^2}} {z_A+\sqrt{b^2+z_A^2}} \right). $$ Writing the endpoint radii as $$ r_A=\sqrt{b^2+z_A^2}, \qquad r_B=\sqrt{b^2+z_B^2}, $$ and the coordinate separation as $$ R=z_B-z_A, $$ this is the standard one-way Shapiro form $$ \Delta T = \frac{2GM}{c^3} \ln\!\left( \frac{r_A+r_B+R}{r_A+r_B-R} \right). $$ ## 212.5 Perihelion Precession The perihelion calculation is easiest in the equivalent areal-radius form of the same weak-field metric. Define $$ R:=r\bigl(1+\eta(r)\bigr)=r+\frac{GM}{c^2}. $$ At leading order in $GM/(Rc^2)$, the metric becomes $$ ds^2 = c^2\left(1-\frac{2GM}{Rc^2}\right)dt^2 - \left(1-\frac{2GM}{Rc^2}\right)^{-1}dR^2 - R^2d\Omega^2 + O(c^{-4}). $$ This is the standard weak-field Schwarzschild form, equivalent to the isotropic metric above at the order being kept. Restrict to equatorial motion, $\theta=\pi/2$, and parameterize the worldline by proper time $\tau$. Write $$ \alpha:=\frac{GM}{c^2}. $$ Then the metric is $$ ds^2 = c^2\left(1-\frac{2\alpha}{R}\right)dt^2 - \left(1-\frac{2\alpha}{R}\right)^{-1}dR^2 - R^2d\phi^2. $$ The two conserved quantities are $$ E := c^2\left(1-\frac{2\alpha}{R}\right)\dot t, $$ $$ h := R^2\dot\phi, $$ where a dot denotes $d/d\tau$. The timelike normalization condition is $$ c^2 = c^2\left(1-\frac{2\alpha}{R}\right)\dot t^2 - \left(1-\frac{2\alpha}{R}\right)^{-1}\dot R^2 - R^2\dot\phi^2. $$ Substitute the conserved quantities: $$ \dot t=\frac{E}{c^2\left(1-\frac{2\alpha}{R}\right)}, \qquad \dot\phi=\frac{h}{R^2}. $$ Then $$ \dot R^2 = \frac{E^2}{c^2} - \left(1-\frac{2\alpha}{R}\right) \left(c^2+\frac{h^2}{R^2}\right). $$ Now set $$ u(\phi):=\frac{1}{R}. $$ Since $$ \dot R=\frac{dR}{d\phi}\dot\phi = \left(-\frac{u'}{u^2}\right)(hu^2) = -hu', $$ where a prime denotes $d/d\phi$, the radial equation becomes $$ h^2u'^2 = \frac{E^2}{c^2} - \bigl(1-2\alpha u\bigr)\bigl(c^2+h^2u^2\bigr). $$ Differentiate with respect to $\phi$: $$ 2h^2u'u'' = 2\alpha u'\bigl(c^2+h^2u^2\bigr) - \bigl(1-2\alpha u\bigr)(2h^2uu'). $$ Divide by $2u'$: $$ h^2u'' = \alpha\bigl(c^2+h^2u^2\bigr) - h^2u\bigl(1-2\alpha u\bigr). $$ Therefore $$ h^2u'' = \alpha c^2 -h^2u +3\alpha h^2u^2, $$ or $$ u''+u = \frac{\alpha c^2}{h^2} +3\alpha u^2. $$ Since $\alpha c^2=GM$, this is $$ u''+u = \frac{GM}{h^2} +\frac{3GM}{c^2}u^2. $$ The Newtonian orbit equation is the same without the final term. Its bound solution is $$ u_0(\phi) = \frac{GM}{h^2}\bigl(1+e\cos\phi\bigr). $$ Now write $$ u=u_0+u_1, $$ where $u_1$ is first order in $GM/c^2$. Substituting into $$ u''+u = \frac{GM}{h^2} + \frac{3GM}{c^2}u^2 $$ and retaining only first order in $GM/c^2$ gives $$ u_1''+u_1 = \frac{3GM}{c^2}u_0^2. $$ Since $$ u_0^2 = \left(\frac{GM}{h^2}\right)^2 \bigl(1+2e\cos\phi+e^2\cos^2\phi\bigr) $$ and $$ \cos^2\phi=\frac{1+\cos 2\phi}{2}, $$ the forcing becomes $$ u_1''+u_1 = \frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2 \left( 1+\frac{e^2}{2}+2e\cos\phi+\frac{e^2}{2}\cos 2\phi \right). $$ The constant term and the $\cos 2\phi$ term change the detailed shape of the orbit but do not accumulate a secular phase shift. The resonant term $$ 2e\,\frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2\cos\phi $$ does. To isolate that secular part, solve $$ y''+y=A\cos\phi. $$ A direct check shows that $$ y_p(\phi)=\frac{A}{2}\phi\sin\phi $$ is a particular solution, because $$ \left(\frac{A}{2}\phi\sin\phi\right)'' + \frac{A}{2}\phi\sin\phi = A\cos\phi. $$ Therefore the resonant contribution to $u_1$ is $$ u_{1,\mathrm{res}}(\phi) = \frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2 e\,\phi\sin\phi. $$ Now compare with a precessing ellipse $$ \frac{GM}{h^2} \Bigl(1+e\cos((1-\delta)\phi)\Bigr), \qquad \delta\ll 1. $$ Using $$ \cos((1-\delta)\phi) = \cos\phi+\delta\,\phi\sin\phi+O(\delta^2), $$ the secular correction produced by a precession $\delta$ is $$ \frac{GM}{h^2}e\,\delta\,\phi\sin\phi. $$ Matching this with $u_{1,\mathrm{res}}$ gives $$ \frac{GM}{h^2}e\,\delta = \frac{3GM}{c^2}\left(\frac{GM}{h^2}\right)^2 e, $$ so $$ \delta = \frac{3G^2M^2}{h^2c^2}. $$ For a Newtonian ellipse, $$ h^2=GMa(1-e^2), $$ therefore $$ \delta = \frac{3GM}{a(1-e^2)c^2}. $$ After one orbit, the perihelion advance is $$ \Delta\omega = 2\pi\delta = \frac{6\pi GM}{a(1-e^2)c^2}. $$ This is the standard weak-field perihelion precession. ## 212.6 Summary Starting from the symmetric constitutive summary $$ \varepsilon_{\mathrm{eff}}=\varepsilon_0(1+2\eta), \qquad \mu_{\mathrm{eff}}=\mu_0(1+2\eta), \qquad \eta=\frac{GM}{rc^2}, $$ the local transport speed is $$ k(r)=c\bigl(1-2\eta(r)\bigr)+O(\eta^2). $$ Combining that transport speed with the Newtonian slow-mode limit determines the unique leading weak-field static isotropic metric $$ ds^2 = c^2(1-2\eta)\,dt^2 - (1+2\eta)\,(dr^2+r^2d\Omega^2) + O(\eta^2). $$ From that one weak-field closure follow: - gravitational redshift, - light bending $$ \theta=\frac{4GM}{bc^2}, $$ - one-way Shapiro delay $$ \Delta T = \frac{2GM}{c^3} \ln\!\left( \frac{r_A+r_B+R}{r_A+r_B-R} \right), $$ - perihelion advance $$ \Delta\omega = \frac{6\pi GM}{a(1-e^2)c^2}. $$ So within the symmetric constitutive summary used here, chapter 13 no longer rests on light bending alone. The full static weak-field benchmark set is recovered from one and the same transport geometry. Appendices 215 and 216 then isolate the deeper factor-of-two point: a null electromagnetic probe carries two equal stress channels, and a static toroidal closure samples both through its axial line.