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Interpret the orbit in the Schwarzschild metric as an unforced circular timelike geodesic and choose its plane as $\theta =\pi /2$. Put $f=1-2\mu /r$. The radial geodesic equation at constant radius uses the Christoffel symbolsand gives the Angular velocity of a circular Schwarzschild geodesicThe proper-time normalization then yieldsOver one revolution, the Proper period of a circular Schwarzschild geodesic is thereforeBoth the gravitational and orbital kinematic contributions to time dilation are included. Circular timelike geodesics exist for $r>3\mu$; they are stable for $r>6\mu$ and unstable for $3\mu <r<6\mu$, with marginal stability at $6\mu$.

If “circular orbit” instead means a forced trajectory with an externally chosen angular velocity, the radius alone does not fix the answer. Its elapsed time is $\Delta \tau =\Delta t\sqrt{1-2\mu /r-r^2{\Omega }^2/c^2}$, or $2\pi \sqrt{1-2\mu /r-r^2{\Omega }^2/c^2}/|\Omega |$ per revolution. The geodesic interpretation supplies the missing angular velocity.

Normalize the five-dimensional tangent as in part (c), and let $s=\sqrt{1-{\xi }^2}>0$. Since $u=c\tau$, one has $d\tau /d\lambda =s/c$, soSubstitution into the corrected equation in part (d) givesThis is the Lorentz force form in four-dimensional Minkowski spacetime. The effective charge-to-mass coefficient is proportional to the conserved fifth-direction momentum: in $c=1$ units and a potential normalization identifying $A$ with the electromagnetic four-potential, $q/m=\xi /\sqrt{1-{\xi }^2}$. The factor converting the geometrical $A$ to an SI potential fixes the physical charge units. Opposite signs of $\xi$ give opposite effective charges, and $\xi =0$ gives a neutral inertial projection. Antisymmetry gives $d({\eta }_{ab}{u}^a{u}^b)/d\tau =0$.

This is a Kaluza-Klein theory mechanism: a higher-dimensional geodesic appears as forced motion after projection. With the explicitly supplied $+---$ base signature, the positive fibre term gives a second positive direction in the five-dimensional metric. The derivation is a valid geometrical reduction of that metric; a conventional single-time spacetime would require a spacelike extra direction and corresponding sign changes. If one instead retains the sign printed in part (d), the inferred effective charge reverses, but that version is not the geodesic equation of the metric supplied here.

For a tensor cosmological perturbation $h_{ij}$, the time geodesic equation in a local orthonormal tetrad gives $d\log ϵ/d\eta =-{\dot{h}}_{ij}{e}^i{e}^j/2$. The photon comoving energy removes the background redshift; the dot is a partial time derivative, whereas the energy derivative follows the ray.

For the metric in vector cosmological perturbation, an orthonormal tetrad is $E_0=a^{-1}{\partial }_{\eta }$, $E_i=a^{-1}(B_i{\partial }_{\eta }+{\partial }_i)$ to first order. A photon has $p^0=(ϵ/a^2)(1+B_i{e}^i)$ and $p^i=(ϵ/a^2)e^i$, so the covariant component is $p_0=-ϵ+O(B^2)$. The covariant geodesic equation gives $d{p}_0/d\eta =(2p^0{)}^{-1}{\partial }_{\eta }{g}_{\alpha \beta }{p}^{\alpha }{p}^{\beta }=ϵe^i{\dot{B}}_i$. The derivative of the scale factor cancels by the null vector condition, proving the redshift formula.

For the relativistic particle action $S=(m/2)\int [g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }/e-e]d\lambda$, varying the worldline einbein gives $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-e^2$. On the positive-einbein branch the proper time satisfies $ds=ed\lambda$. A reparametrization with ${\lambda }^{\prime }=s$ sets the transformed einbein to one, making the geodesic equation affinely parametrized. This gauge choice is made after retaining the einbein mass shell constraint; fixing the action first would discard that equation. If the parameter interval is held fixed at both ends, its integrated proper length is a modulus and cannot be arbitrarily changed by endpoint-preserving transformations.

An affine parameter on a timelike geodesic can be scaled to proper time. With metric signature $(+---)$ this fixes the tangent-vector norm to $c^2$. A further constant rescaling preserves the affine geodesic equation but changes that norm; conserved energy and angular-momentum constants must be rescaled consistently. The unit term in a proper-time-normalized radial energy equation is consequently a normalization choice, not a property of every affine parameter.

For $d{s}^2=Ad{t}^2-d{r}^2/A-r^{2}d{\Omega }^2$, $A=1-2GM/r-\Lambda r^2/3$, and $c=1$, a radial timelike geodesic has ${\dot{r}}^2=k^2-A$. The radial geodesic equation gives $\ddot{r}=-A^{\prime }/2=-GM/r^2+\Lambda r/3$, including at a turning point. This is a second derivative of the areal radius with respect to proper time; the geodesic's four-acceleration still vanishes. In the slow-motion weak-field limit, restoring units gives $d^{2}r/d{t}^2=-GM/r^2+\Lambda c^{2}r/3$.

A spacelike geodesic has positive tangent norm in signature $(-+++)$ and satisfies the geodesic equation with an affine parameter. Metric compatibility preserves that norm along the curve. In Minkowski spacetime these are straight lines whose spatial tangent is larger in magnitude than the time component; their two ends approach spacelike infinity in a conformal completion.

For a photon in $d{s}^2=a^2[-d{\tau }^2+({\delta }_{ij}+h_{ij})d{x}^{i}d{x}^j]$, the null geodesic condition cancels the homogeneous $2\mathcal{H}{p}^0$ term in the derivative of $q=a^2{p}^0$. The remaining first-order geodesic equation gives the displayed momentum change. Its direction is constant at zeroth order, so first-order redshift and brightness sources can be evaluated on the unperturbed ray; direction deflection multiplied by an already first-order anisotropy is second order.

For a connected Riemannian manifold, two-point homogeneity is equivalent to the isometry group being transitive on the unit tangent bundle. One direction follows by taking short equal-length radial geodesic segments: the Gauss lemma identifies their distance, and injectivity of the exponential map identifies the initial directions after the endpoints are matched. Conversely, unit tangent transitivity implies point homogeneity, hence completeness. The Hopf-Rinow theorem supplies minimizing geodesics for arbitrary equal-distance pairs. Matching their initial unit tangent vectors and using uniqueness of the geodesic equation matches their other endpoints.