Endpoint-Matched Shapiro Time Delay from Optical Geometry

1. Introduction

In a stationary spacetime, a light-time observable is defined by an emission event, a reception event, and a null-geodesic branch joining them. The coordinate travel time is therefore a time-transfer function of the endpoints, and its derivatives determine the propagation direction and impact parameter [1,2,3]. This endpoint formulation is essential in a lensing configuration, where more than one branch may connect the same events.

Black-hole lensing also provides observables beyond the primary weak-deflection branch. Relativistic images and their differential delays can constrain lens masses and distances, while time-delay and magnification-centroid observables distinguish compact-object geometries [4,5,6]. The strong-field structure is organized geometrically by photon surfaces [7], and complementary image information can be encoded in distortion relations [8]. Plasma and dark-matter environments modify optical propagation [9], whereas eikonal constructions relate unstable null-orbit data to strong lensing, shadows, quasinormal ringing, and grey-body transmission [10]. These developments make it important to distinguish ray-labelled integral differences from delays defined between fixed endpoint events.

Many calculations in black-hole spacetimes instead parameterize a ray by its radial turning point and subtract a reference integral evaluated at the same turning radius. This is a useful intermediate construction, but it is not generally a time-transfer function: the physical and reference angular sweeps differ, so the rays do not share both endpoint events. The distinction is small in a controlled weak-field regime but becomes nonuniform near alignment, where the shift of the reference turning point is enhanced. It is also relevant in nonasymptotically flat geometries, because the reference spacetime and the map between endpoint events must be stated explicitly.

Optical geometry provides a common language for travel time and bending. Null rays in a static spacetime are geodesics of a Riemannian optical metric, and their coordinate travel time is the optical length. The Gauss–Bonnet construction expresses finite-distance deflection through optical curvature and boundary data [11,12,13,14]. In isothermal coordinates the curvature area integral becomes a boundary flux, whereas the first variation of travel time is a line integral of the same conformal perturbation [15,16,17]. These are complementary functionals, but they are not harmonic conjugates.

We develop the endpoint conversion directly from the radial Hamilton–Jacobi characteristic. The exact identity

$$T=W+b\mathrm{\Phi },{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}\mathrm{\Phi }}}=b,$$

(1)

implies an exact nonlinear matching law in ray space and, at first perturbative order,

$$\delta T_{\mathrm{end}}=\delta T_{r_0}-b_{\mathrm{ref}}\delta {\mathrm{\Phi }}_{r_0}.$$

(2)

Here T is the coordinate travel time, $\mathrm{\Phi }$ is the continuous azimuthal sweep on a fixed branch, and b is the impact parameter. The label $r_0$ means that the physical and reference quantities are first evaluated at a common turning radius. We also give the quadratic matching term and explicit metric-perturbation kernels that can be applied coefficient by coefficient in a gravitational effective field theory.

Schwarzschild, Reissner–Nordström (RN), and Kottler spacetimes provide closed-form tests. For Schwarzschild we reconcile the areal-coordinate result with the standard first-post-Minkowskian time-transfer function. For Kottler the reference is de Sitter, not Minkowski. This static-patch subtraction must not be confused with an image-to-image cosmological time delay, in which observer motion, angular-diameter distances, and the lens equation reorganize the apparent $\mathrm{\Lambda }$ dependence [18].

We set $G=c=1$ and use signature $(-,+,+,+)$. The analysis is restricted to a single-pass weak branch with one simple radial turning point and a reference ray connecting the same endpoint events.

2. Optical Geometry and Complementary Conformal Functionals

2.1. Static Spherical Geometry

Consider

$$\mathrm{d}{s}^2=-A\left(r\right)\mathrm{d}{t}^2+B\left(r\right)\mathrm{d}{r}^2+C\left(r\right)\mathrm{d}{\mathrm{\Omega }}^2,$$

(3)

with $A>0$ in the static region. On the equatorial plane, the null condition gives

$$\mathrm{d}{t}^2={\gamma }_{ij}\mathrm{d}{x}^i\mathrm{d}{x}^j={\displaystyle \frac{B}{A}}\mathrm{d}{r}^2+{\displaystyle \frac{C}{A}}\mathrm{d}{\varphi }^2.$$

(4)

Thus the coordinate travel time is exactly the optical length,

$$T={\int }_{\gamma }\mathrm{d}{\ell }_{\mathrm{opt}}.$$

(5)

For conserved $E=A\dot{t}$ and $L=C\dot{\varphi }$, define $b=L/E$. The radial time and azimuthal integrands are

$$\begin{array}{cc}\hfill F_t(r;b)& \equiv {\displaystyle \frac{\mathrm{d}t}{\mathrm{d}r}}=\sqrt{{\displaystyle \frac{B}{A}}}{\displaystyle \frac{1}{\sqrt{1-A{b}^2/C}}},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill F_{\varphi }(r;b)& \equiv \left|{\displaystyle \frac{\mathrm{d}\varphi }{\mathrm{d}r}}\right|={\displaystyle \frac{b\sqrt{AB}}{C\sqrt{1-A{b}^2/C}}}.\hfill \end{array}$$

(7)

The turning radius $r_0$ satisfies

$$b^2={\displaystyle \frac{C\left(r_0\right)}{A\left(r_0\right)}}.$$

(8)

Unless $C=r^2$, $r_0$ is a coordinate turning radius rather than an invariant areal distance.

2.2. Isothermal Coordinates

Introduce an isothermal radius $\rho$ by

$${\displaystyle \frac{\mathrm{d}\rho }{\rho }}=\sqrt{{\displaystyle \frac{{\gamma }_{rr}}{{\gamma }_{\varphi \varphi }}}}\mathrm{d}r=\sqrt{{\displaystyle \frac{B}{C}}}\mathrm{d}r.$$

(9)

Then

$$\mathrm{d}{\ell }_{\mathrm{opt}}^2=e^{2\lambda }(\mathrm{d}{\rho }^2+{\rho }^2\mathrm{d}{\varphi }^2),e^{\lambda }={\displaystyle \frac{1}{\rho }}\sqrt{{\displaystyle \frac{C}{A}}}.$$

(10)

In Cartesian coordinates on the conformal plane,

$$\mathcal{K}=-e^{-2\lambda }{\Delta }_0\lambda ,T={\int }_{\overline{\gamma }}{e}^{\lambda }\mathrm{d}{\ell }_0.$$

(11)

Consequently,

$$\underset{D}{\int \int }\mathcal{K}\mathrm{d}S=-{\oint }_{\partial D_0}{\partial }_n\lambda \mathrm{d}{\ell }_0,$$

(12)

while the travel time is a trace integral of $e^{\lambda }$ along the ray. The finite-distance deflection additionally contains the geodesic-curvature and corner terms fixed by the closure of the Gauss–Bonnet domain [12,14,16].

For a one-parameter family represented in a common, endpoint-fixing conformal gauge, write $\lambda ={\lambda }_{\mathrm{ref}}+ϵ\psi$. Fermat stationarity then removes the first-order contribution from the displacement of the reference geodesic, and

$$\delta T=ϵ{\int }_{{\overline{\gamma }}_{\mathrm{ref}}}\psi \mathrm{d}{\ell }_{\mathrm{ref}}+\mathcal{O}\left({ϵ}^2\right).$$

(13)

If the physical and reference isothermal charts are chosen independently, their coordinate identification also moves the endpoint coordinates and the corresponding boundary terms must be retained. The Hamilton–Jacobi construction below instead uses a common $(r,\varphi )$ chart and is independent of this conformal-gauge choice. Subject to this identification, Equations (12) and (13) show that bending and time transfer probe, respectively, normal-derivative and trace data of the same conformal perturbation. No generic harmonic-conjugate relation follows: black-hole optical curvature is distributed through the vacuum optical manifold, so ${\Delta }_0\psi$ need not vanish away from the material lens.

3. Endpoint Primitives and Hamilton–Jacobi Matching

3.1. Fixed-Turning-Point Boundary Primitive

For a single-pass ray from S to R through $r_0$,

$$\begin{array}{cc}\hfill T\left(r_0\right)& =\sum _{e=S,R}{\int }_{r_0}^{r_e}{F}_t(r;b\left(r_0\right))\mathrm{d}r,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill \mathrm{\Phi }\left(r_0\right)& =\sum _{e=S,R}{\int }_{r_0}^{r_e}{F}_{\varphi }(r;b\left(r_0\right))\mathrm{d}r.\hfill \end{array}$$

(15)

If $\mathcal{T}(r;r_0)$ is an antiderivative of $F_t$ at fixed $r_0$, then

$$T\left(r_0\right)=\sum _{e=S,R}[\mathcal{T}(r_e;r_0)-\mathcal{T}(r_0;r_0)].$$

(16)

This is an endpoint representation once the radial integration has been carried out. It does not by itself enforce that a reference ray at the same $r_0$ reaches the same angular endpoints.

Figure 1 illustrates the distinction. The physical ray has turning point P and radius $r_0$. The endpoint-matched reference ray generally has a different closest approach $\overline{P}$ and radius ${\overline{r}}_0$.

3.2. Radial Characteristic

Define the radial Hamilton–Jacobi characteristic

$$\begin{array}{cc}\hfill W\left(b\right)& =\sum _{e=S,R}{\int }_{r_0\left(b\right)}^{r_e}{F}_W(r;b)\mathrm{d}r,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill F_W& =\sqrt{{\displaystyle \frac{B}{A}}}\sqrt{1-{\displaystyle \frac{A{b}^2}{C}}}.\hfill \end{array}$$

(18)

Because $F_W(r_0;b)=0$, differentiation of the lower limit gives no contribution. Directly,

$${\partial }_b{F}_W=-F_{\varphi },F_t=F_W+b{F}_{\varphi }.$$

(19)

Therefore

$$\mathrm{\Phi }=-{\partial }_{b}W,T=W+b\mathrm{\Phi }.$$

(20)

Differentiating along a fixed pair of endpoint radii yields

$${\displaystyle \frac{\mathrm{d}T}{\mathrm{d}b}}=b{\displaystyle \frac{\mathrm{d}\mathrm{\Phi }}{\mathrm{d}b}},{\displaystyle \frac{\mathrm{d}T}{\mathrm{d}\mathrm{\Phi }}}=b.$$

(21)

This is the relevant conjugacy: travel time is the generating function for the endpoint angular separation. Equivalent derivative relations underlie the time-transfer-function and eikonal approaches of Refs. [1,2].

3.3. From Common $R_0$ to Common Endpoints

Let $T,\mathrm{\Phi }$ refer to a perturbed geometry and $T_{\mathrm{ref}},{\mathrm{\Phi }}_{\mathrm{ref}}$ to a chosen reference geometry, identified in the same $(r,\varphi )$ chart. At a common $r_0$, define

$$\delta T_{r_0}=T\left(r_0\right)-T_{\mathrm{ref}}\left(r_0\right),\delta {\mathrm{\Phi }}_{r_0}=\mathrm{\Phi }\left(r_0\right)-{\mathrm{\Phi }}_{\mathrm{ref}}\left(r_0\right).$$

(22)

The endpoint-matched reference turning radius ${\overline{r}}_0$ is determined by

$${\mathrm{\Phi }}_{\mathrm{ref}}\left({\overline{r}}_0\right)=\mathrm{\Phi }\left(r_0\right).$$

(23)

On a branch where ${\mathrm{\Phi }}_{\mathrm{ref}}$ is monotonic, Equation (21) gives the exact reference-time shift

$$T_{\mathrm{ref}}\left({\overline{r}}_0\right)-T_{\mathrm{ref}}\left(r_0\right)={\int }_{{\mathrm{\Phi }}_{\mathrm{ref}}\left(r_0\right)}^{\mathrm{\Phi }\left(r_0\right)}{b}_{\mathrm{ref}}\left({\mathrm{\Phi }}^{\prime }\right)\mathrm{d}{\mathrm{\Phi }}^{\prime }.$$

(24)

Consequently, the exact endpoint conversion is

$$\delta T_{\mathrm{end}}=\delta T_{r_0}-{\int }_{{\mathrm{\Phi }}_{\mathrm{ref}}\left(r_0\right)}^{\mathrm{\Phi }\left(r_0\right)}{b}_{\mathrm{ref}}\left({\mathrm{\Phi }}^{\prime }\right)\mathrm{d}{\mathrm{\Phi }}^{\prime }.$$

(25)

This form is equivalent to solving Equation (23), but it makes the canonical structure explicit. Expanding the ray-space integral gives

$$\begin{array}{cc}\hfill \delta T_{\mathrm{end}}=& \delta T_{r_0}-b_{\mathrm{ref}}\delta {\mathrm{\Phi }}_{r_0}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\mathrm{d}{b}_{\mathrm{ref}}}{\mathrm{d}{\mathrm{\Phi }}_{\mathrm{ref}}}}\right)}_{r_0}{\left(\delta {\mathrm{\Phi }}_{r_0}\right)}^2+\mathcal{O}\left({\left(\delta {\mathrm{\Phi }}_{r_0}\right)}^3\right).\hfill \end{array}$$

(26)

Equation (2) is the first-order truncation.

For equal endpoint radii R in a flat reference, ${\mathrm{\Phi }}_{\mathrm{ref}}=2arccos(r_0/R)$ and $b_{\mathrm{ref}}\left(\mathrm{\Phi }\right)=Rcos(\mathrm{\Phi }/2)$. Equation (25) then reduces to

$$\begin{array}{cc}\hfill \delta T_{\mathrm{end}}=\delta T_{r_0}-2R[& \text{sin}{\displaystyle \frac{{\mathrm{\Phi }}_{\mathrm{ref}}+\delta {\mathrm{\Phi }}_{r_0}}{2}}\hfill \\ & -\text{sin}{\displaystyle \frac{{\mathrm{\Phi }}_{\mathrm{ref}}}{2}}],\hfill \end{array}$$

(27)

which remains useful when the linearized shift of the reference turning point is inaccurate but the same weak reference branch still exists.

3.4. First-order Metric-Perturbation Kernels

The matching formula can be combined with a general perturbation of the static spherical metric. Write

$$A=\overline{A}+ϵa,B=\overline{B}+ϵ\beta ,C=\overline{C}+ϵc,$$

(28)

where barred quantities define the reference geometry. At fixed $r_0$, introduce

$$\begin{array}{cccc}\hfill \overline{D}\left(r\right)& =1-{\displaystyle \frac{\overline{A}\left(r\right){\overline{b}}^2}{\overline{C}\left(r\right)}},\hfill & \hfill {\overline{b}}^2& ={\displaystyle \frac{\overline{C}\left(r_0\right)}{\overline{A}\left(r_0\right)}},\hfill \\ \hfill {\eta }_0& ={\displaystyle \frac{c\left(r_0\right)}{\overline{C}\left(r_0\right)}}-{\displaystyle \frac{a\left(r_0\right)}{\overline{A}\left(r_0\right)}},\hfill & \hfill \mathrm{\Xi }\left(r\right)& ={\displaystyle \frac{a}{\overline{A}}}-{\displaystyle \frac{c}{\overline{C}}}+{\eta }_0.\hfill \end{array}$$

(29)

The turning-point condition gives $\delta b^2/{\overline{b}}^2={\eta }_0$ and $\delta D=-(1-\overline{D})\mathrm{\Xi }$. Varying Equations (6) and (7) yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta F_t}{{\overline{F}}_t}}& ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\beta }{\overline{B}}}-{\displaystyle \frac{a}{\overline{A}}}\right)+{\displaystyle \frac{1-\overline{D}}{2\overline{D}}}\mathrm{\Xi },\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta F_{\varphi }}{{\overline{F}}_{\varphi }}}& ={\displaystyle \frac{{\eta }_0}{2}}+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{a}{\overline{A}}}+{\displaystyle \frac{\beta }{\overline{B}}}\right)-{\displaystyle \frac{c}{\overline{C}}}+{\displaystyle \frac{1-\overline{D}}{2\overline{D}}}\mathrm{\Xi }.\hfill \end{array}$$

(31)

Although the last terms contain ${\overline{D}}^{-1}$, they are regular at a simple turning point because $\mathrm{\Xi }\left(r_0\right)=0$. Integrating $\delta F_t$ and $\delta F_{\varphi }$ on both radial legs gives $\delta T_{r_0}$ and $\delta {\mathrm{\Phi }}_{r_0}$, after which Equation (26) performs the endpoint conversion. These kernels apply directly to metric corrections organized by Wilson coefficients; no new geodesic variation is required at first order.

For a Minkowski reference in an areal polar chart, the endpoint-matched kernel takes a particularly simple form. Set

$$A=1+ϵa\left(r\right),B=1+ϵ\beta \left(r\right),C=r^2+ϵc\left(r\right),$$

(32)

and parameterize each straight reference leg by $z\in [0,s_e]$, with $r^2=r_0^2+z^2$. Fermat stationarity gives directly

$$\delta T_{\mathrm{end}}^{\left(1\right)}={\displaystyle \frac{ϵ}{2}}\sum _{e=S,R}{\int }_0^{s_e}\left[\beta \left(r\right){\displaystyle \frac{z^2}{r^2}}-a\left(r\right)+c\left(r\right){\displaystyle \frac{r_0^2}{r^4}}\right]\mathrm{d}z.$$

(33)

Here $r_0$ is the zeroth-order Euclidean closest approach; replacing it by the physical turning radius changes Equation (33) only at second order. This line kernel is equivalent to integrating Equations (30) and (31) and applying the linear term in Equation (26). It supplies a direct route from a static spherical EFT metric to its first-order time-transfer coefficient.

The first-order conversion is local in ray space. It requires the matched reference turning radius to remain perturbatively close to $r_0$,

$$|\delta {\mathrm{\Phi }}_{r_0}|\ll r_0\left|{\displaystyle \frac{\partial {\mathrm{\Phi }}_{\mathrm{ref}}}{\partial r_0}}\right|.$$

(34)

For flat or de Sitter reference rays with $C=r^2$, $\partial {\mathrm{\Phi }}_{\mathrm{ref}}/\partial r_0=-{\sum }_e{s}_e^{-1}$. In a flat reference, $\mathrm{d}{b}_{\mathrm{ref}}/\mathrm{d}{\mathrm{\Phi }}_{\mathrm{ref}}=-{\left({\sum }_e{s}_e^{-1}\right)}^{-1}.$ For equal distant endpoints in Schwarzschild, Equation (34) reduces parametrically to $MR/r_0^2\ll 1$. The limits $R/r_0\to \infty$ and weak endpoint matching are therefore nonuniform near perfect alignment.

The matching is branch dependent. In a lensing configuration there may be several physical rays, and a weak-field reference branch may cease to exist when the continuous angular sweep crosses its limiting value [3]. Near alignment or on multiple-image branches, one should solve Equation (23) without linearization or impose the appropriate lens equation. The closed formulas below apply locally on a single weak, single-pass branch.

4. Example Geometries

For the examples, $C=r^2$ and $B=A^{-1}$. Write

$$s_e\equiv \sqrt{r_e^2-r_0^2},{\theta }_e\equiv arccos{\displaystyle \frac{r_0}{r_e}},u_e\equiv \sqrt{{\displaystyle \frac{r_e-r_0}{r_e+r_0}}}.$$

(35)

All results retain the exact dependence on $r_S,r_R,r_0$ at the stated order in M, $Q^2$, and $\mathrm{\Lambda }$.

4.1. Schwarzschild

For $A=1-2M/r$, the turning-point-regular primitive through $\mathcal{O}\left(M\right)$ is

$$\mathcal{T}\left(r\right)=\sqrt{r^2-r_0^2}+2Mln{\displaystyle \frac{r+\sqrt{r^2-r_0^2}}{r_0}}+M\sqrt{{\displaystyle \frac{r-r_0}{r+r_0}}}.$$

(36)

Subtracting the flat primitive at the same $r_0$ gives

$$\delta T_{r_0}^{\mathrm{Schw}}=M\sum _{e=S,R}\left[2ln{\displaystyle \frac{r_e+s_e}{r_0}}+u_e\right].$$

(37)

The corresponding angular excess is

$$\delta {\mathrm{\Phi }}_{r_0}^{\mathrm{Schw}}={\displaystyle \frac{M}{r_0}}\sum _{e=S,R}\left({\displaystyle \frac{s_e}{r_e}}+u_e\right).$$

(38)

Since $b_{\mathrm{ref}}=r_0$ in flat space, Equation (2) yields

$$\delta T_{\mathrm{end}}^{\mathrm{Schw}}=M\sum _{e=S,R}\left[2ln{\displaystyle \frac{r_e+s_e}{r_0}}-{\displaystyle \frac{s_e}{r_e}}\right].$$

(39)

Equation (39) agrees with the standard first-post-Minkowskian time-transfer function. To see the coordinate conversion explicitly, let ${\mathrm{\Phi }}_0={\mathrm{\Phi }}_{\mathrm{ref}}\left(r_0\right)$ and define the Euclidean endpoint separation

$$R_{SR}=\sqrt{r_S^2+r_R^2-2r_S{r}_{R}cos{\mathrm{\Phi }}_0}=s_S+s_R.$$

(40)

In isotropic Schwarzschild coordinates, ${\rho }_e=r_e-M+\mathcal{O}\left(M^2\right)$. The familiar first-order result is

$$\begin{array}{cc}\hfill T_{1\mathrm{PM}}=& R_{SR}({\rho }_S,{\rho }_R,{\mathrm{\Phi }}_0)\hfill \\ \hfill & +2Mln{\displaystyle \frac{{\rho }_S+{\rho }_R+R_{SR}}{{\rho }_S+{\rho }_R-R_{SR}}}+\mathcal{O}\left(M^2\right).\hfill \end{array}$$

(41)

At fixed angular endpoints,

$$R_{SR}({\rho }_S,{\rho }_R,{\mathrm{\Phi }}_0)-R_{SR}(r_S,r_R,{\mathrm{\Phi }}_0)=-M\sum _e{\displaystyle \frac{s_e}{r_e}},$$

while

$$ln{\displaystyle \frac{r_S+r_R+R_{SR}}{r_S+r_R-R_{SR}}}=\sum _{e}ln{\displaystyle \frac{r_e+s_e}{r_0}}.$$

Their sum is Equation (39). The nonlogarithmic term is therefore the radial-coordinate conversion required when the reference endpoints are identified in Schwarzschild areal coordinates, not a disagreement with the usual time-transfer formula [1,2].

In Schwarzschild areal coordinates, the two far-field forms are

$$\begin{array}{cc}\hfill \delta T_{r_0}^{\mathrm{Schw}}& =2Mln{\displaystyle \frac{4r_S{r}_R}{r_0^2}}+2M+\dots ,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill \delta T_{\mathrm{end}}^{\mathrm{Schw}}& =2Mln{\displaystyle \frac{4r_S{r}_R}{r_0^2}}-2M+\dots .\hfill \end{array}$$

(43)

The constants are not invariant observables: they depend on the radial coordinate and on the reference matching. Differential arrival times between two rays received by the same observer remove common convention-dependent terms. Equations (42) and (43) are coefficient limits of the finite-distance expressions; the endpoint result must also satisfy Equation (34) and cannot be extrapolated to exact alignment at fixed M.

Figure 2 compares the two definitions for $r_S=r_R=R$. Both have the same leading logarithm, but the finite matching term separates them. With $x=r_0/R$, the endpoint coefficient obeys

$${\displaystyle \frac{\delta T_{\mathrm{end}}^{\mathrm{Schw}}}{M}}-\left(2ln{\displaystyle \frac{4}{x^2}}-2\right)=-{\displaystyle \frac{x^4}{8}}+\mathcal{O}\left(x^6\right),$$

(44)

so the quadratic endpoint correction cancels exactly.

4.2. Reissner–Nordstr Öm

For $A=1-2M/r+Q^2/r^2$, expand independently in $M/r_0$ and $Q^2/r_0^2$. The formulas below isolate the coefficient linear in $Q^2$; terms of order $M{Q}^2$ and $Q^4$ are not included. At fixed $r_0$, the charge contribution has integrand and primitive

$$F_t^{\left(Q^2\right)}=-{\displaystyle \frac{3Q^2}{2r\sqrt{r^2-r_0^2}}},\delta {\mathcal{T}}^{\left(Q^2\right)}=-{\displaystyle \frac{3Q^2}{2r_0}}arccos{\displaystyle \frac{r_0}{r}}.$$

(45)

Thus

$$\delta T_{r_0}^{\left(Q^2\right)}=-{\displaystyle \frac{3Q^2}{2r_0}}\sum _{e=S,R}{\theta }_e.$$

(46)

The fixed-$r_0$ angular correction is

$$\delta {\mathrm{\Phi }}_{r_0}^{\left(Q^2\right)}=-{\displaystyle \frac{Q^2}{r_0^2}}\sum _{e=S,R}\left({\displaystyle \frac{3{\theta }_e}{4}}+{\displaystyle \frac{sin2{\theta }_e}{8}}\right).$$

(47)

Endpoint matching therefore gives

$$\delta T_{\mathrm{end}}^{\left(Q^2\right)}={\displaystyle \frac{Q^2}{r_0}}\sum _{e=S,R}\left(-{\displaystyle \frac{3{\theta }_e}{4}}+{\displaystyle \frac{sin2{\theta }_e}{8}}\right).$$

(48)

At the displayed order, charge decreases the coordinate light-time coefficient relative to Schwarzschild in both conventions. For $r_S,r_R\gg r_0$,

$$\delta T_{r_0}^{\left(Q^2\right)}\to -{\displaystyle \frac{3\pi Q^2}{2r_0}},\delta T_{\mathrm{end}}^{\left(Q^2\right)}\to -{\displaystyle \frac{3\pi Q^2}{4r_0}}.$$

(49)

The factor-of-two difference is entirely the endpoint-matching term, not a discrepancy in the radial primitive.

Figure 3. Magnitude of the coefficient linear in $Q^2$ for $r_S=r_R=R$. The ordinate is $-r_0\delta T^{\left(Q^2\right)}/Q^2$. The correction vanishes as $r_0\to R$. The horizontal lines are the far-field limits $3\pi /2$ for common-$r_0$ subtraction and $3\pi /4$ for matched endpoints.

Figure 3. Magnitude of the coefficient linear in $Q^2$ for $r_S=r_R=R$. The ordinate is $-r_0\delta T^{\left(Q^2\right)}/Q^2$. The correction vanishes as $r_0\to R$. The horizontal lines are the far-field limits $3\pi /2$ for common-$r_0$ subtraction and $3\pi /4$ for matched endpoints.

4.3. Kottler Geometry

For

$$A=1-{\displaystyle \frac{2M}{r}}-{\displaystyle \frac{\mathrm{\Lambda }{r}^2}{3}},$$

(50)

de Sitter space is the natural background reference in the static patch [19]. Set $H^2=\mathrm{\Lambda }/3$. The exact de Sitter time primitive is

$${\mathcal{T}}_{\mathrm{dS}}(r;r_0)={\displaystyle \frac{1}{H}}artanh\left({\displaystyle \frac{H\sqrt{r^2-r_0^2}}{\sqrt{1-H^2{r}_0^2}}}\right),$$

(51)

with $r<H^{-1}$. Its small-$\mathrm{\Lambda }$ expansion is

$${\mathcal{T}}_{\mathrm{dS}}=\sqrt{r^2-r_0^2}+{\displaystyle \frac{\mathrm{\Lambda }}{18}}(2r^2+r_0^2)\sqrt{r^2-r_0^2}+\mathcal{O}\left({\mathrm{\Lambda }}^2\right).$$

(52)

At common $r_0$, the lens-induced delay relative to de Sitter is

$$\begin{array}{cc}\hfill \delta T_{r_0}^{\mathrm{lens}}=& \delta T_{r_0}^{\mathrm{Schw}}+{\displaystyle \frac{M\mathrm{\Lambda }}{6}}\sum _{e=S,R}(4r_e^2+4r_e{r}_0+r_0^2)u_e.\hfill \end{array}$$

(53)

Terms quadratic in M and terms proportional to $M{\mathrm{\Lambda }}^2$ are omitted. The total common-$r_0$ excess relative to Minkowski additionally contains

$$\delta T_{r_0}^{\mathrm{dS}}=\sum _{e=S,R}\left[{\mathcal{T}}_{\mathrm{dS}}(r_e;r_0)-s_e\right],$$

(54)

which is a background time rather than a lens signal.

The orbit radicand satisfies

$$A\left(r_0\right)r^2-A\left(r\right)r_0^2=r^2-r_0^2-2M\left({\displaystyle \frac{r^2}{r_0}}-{\displaystyle \frac{r_0^2}{r}}\right),$$

(55)

so the Kottler angular sweep parametrized by $r_0$ is exactly independent of $\mathrm{\Lambda }$. The fixed-$r_0$ angular excess relative to de Sitter is therefore Equation (38) at order M. However,

$$b_{\mathrm{dS}}={\displaystyle \frac{r_0}{\sqrt{1-\mathrm{\Lambda }{r}_0^2/3}}}=r_0\left(1+{\displaystyle \frac{\mathrm{\Lambda }{r}_0^2}{6}}\right)+\mathcal{O}\left({\mathrm{\Lambda }}^2\right).$$

(56)

Applying Equation (2) gives

$$\delta T_{\mathrm{end}}^{\mathrm{lens}}=\delta T_{\mathrm{end}}^{\mathrm{Schw}}+{\displaystyle \frac{M\mathrm{\Lambda }}{6}}\sum _{e=S,R}{\displaystyle \frac{(4r_e^2-r_0^2)s_e}{r_e}}.$$

(57)

Reference subtraction and endpoint matching are distinct operations. A total delay relative to Minkowski and a lens delay relative to de Sitter must each be defined using endpoint events matched within the corresponding reference problem.

Equation (57) is a static-coordinate time-transfer building block. It does not establish an independent observable effect of $\mathrm{\Lambda }$ in a cosmological lensing delay. In an image-to-image measurement with comoving source and observer, the lens equation, observer motion, proper-time conversion, and angular-diameter distances must be expanded together. An independent analysis of exact Schwarzschild–de Sitter geodesics finds that, at the comparable order considered there, cosmology enters through the unlensed distances and redshift factor rather than an additional local $\mathrm{\Lambda }$ term [18]. The mixed term above instead records how the chosen static de Sitter reference and endpoint identification act before that observable reduction.

Figure 4 uses $\mathrm{\Lambda }{R}^2/3=0.033$ and remains inside the static patch. The Minkowski subtraction is dominated by the de Sitter background. Subtracting that background exposes the lens coefficient, and endpoint matching adds a finite shift of the same type found in Schwarzschild.

5. Proper Time and Differential Lensing Observables

A static receiver at $r_R$ measures

$$\mathrm{d}{\tau }_R=\sqrt{A\left(r_R\right)}\mathrm{d}t.$$

(58)

For two rays a and b received at the same static worldline,

$$\Delta {\tau }_R=\sqrt{A\left(r_R\right)}[T_a-T_b].$$

(59)

The common lapse factors out, but it does not cancel. It may be set to unity only after normalizing the time coordinate to the receiver or when $A\left(r_R\right)\simeq 1$ at the required order.

A measured lensing delay requires each branch to satisfy the same emission event and the appropriate reception condition. Equations (39), (48), and (57) are therefore time-transfer building blocks. Evaluating them at arbitrary $r_0$ values without solving the endpoint condition does not produce an image-to-image observable. Once the branches are fixed, common coordinate- and reference-dependent constants drop from their difference. If the receiver moves, as in a cosmological Kottler setup, the two reception events need not have the same static radius and Equation (59) must be replaced by an integration along the receiver worldline [18].

5.1. Independent Numerical Validation

The primitives and the perturbation kernels were differentiated symbolically. The exact integrals were also evaluated after the change of variable

$$z=\sqrt{r^2-r_0^2},$$

(60)

which removes the square-root turning-point singularity. For the RN–Kottler family,

$$\begin{array}{cc}\hfill A\left(r_0\right)r^2-A\left(r\right)r_0^2=& z^2-2M\left({\displaystyle \frac{r^2}{r_0}}-{\displaystyle \frac{r_0^2}{r}}\right)\hfill \\ \hfill & +Q^2\left({\displaystyle \frac{r^2}{r_0^2}}-{\displaystyle \frac{r_0^2}{r^2}}\right),\hfill \end{array}$$

(61)

which is numerically stable near $z=0$ and displays the exact cancellation of $\mathrm{\Lambda }$ from the orbit radicand. A simple turning point requires

$$1-{\displaystyle \frac{3M}{r_0}}+{\displaystyle \frac{2Q^2}{r_0^2}}>0,$$

(62)

which is satisfied throughout the weak-branch tests.

For endpoint-matched tests, the exact physical angular sweep was evaluated first and Equation (23) was then solved for the reference turning radius. Figure 5 uses $r_S=r_R=10r_0$, expansion parameters from ${10}^{-5}$ to ${10}^{-2}$, and $\mathrm{\Lambda }{r}_e^2={10}^{-2}$ in the Kottler test. The plotted error is $|\delta T_{\mathrm{exact}}-\delta T^{\left(1\right)}|/|\delta T_{\mathrm{exact}}|$. Log–log fits give slopes between $0.98$ and $1.00$, consistent with an absolute remainder that starts at second order. The upper end of the range also satisfies Equation (34).

6. Discussion

The optical-length identity and the Gauss–Bonnet construction refer to the same two-dimensional optical manifold, but to different data on it. In an endpoint-fixing isothermal chart, the first variation of time samples the conformal perturbation on the reference ray; bending contains its normal derivative together with the boundary closure. Their canonical link is Equation (21), not a harmonic-conjugate relation.

A boundary primitive does not remove the geodesic problem. It performs the radial integration after a branch and turning point have been selected. Equation (25) then supplies the missing endpoint conversion. The distinction matters whenever a common-$r_0$ subtraction is reported as a Shapiro delay: that subtraction compares different angular events unless $\delta {\mathrm{\Phi }}_{r_0}=0$.

The kernels (30) and (31) separate two effects that are otherwise mixed in a direct expansion: propagation in the perturbed optical metric and displacement of the reference ray needed to keep the endpoints fixed. For a gravitational effective field theory, a, $\beta$, and c can be expanded in Wilson coefficients and the endpoint conversion applied to each coefficient. A bound on a Wilson coefficient still requires an image or ranging model with the same endpoint convention; an unmatched primitive is not itself an observable.

The present construction assumes a static spherical metric, one simple radial turning point, and no caustic crossing or winding. A static isotropic plasma can be incorporated through its frequency-dependent Riemannian optical metric. Timelike propagation admits a related Jacobi–Maupertuis description [20]. Rotation produces a Randers–Finsler optical geometry; the Hamilton–Jacobi endpoint relation survives, whereas the two-dimensional Riemannian conformal reduction used here does not.

7. Conclusions

Finite-distance time transfer requires both a propagation calculation and an endpoint prescription. A common-turning-radius subtraction provides the first but not, in general, the second. The Hamilton–Jacobi relation converts it to the same-event quantity through the exact ray-space integral (25); Equation (2) and the quadratic term in Equation (26) are its local expansions.

All Schwarzschild, RN, and Kottler primitives pass symbolic derivative checks and exact numerical tests. Endpoint matching changes the finite Schwarzschild contribution and reduces the leading far-field RN charge coefficient from $3\pi /2$ to $3\pi /4$. In Kottler spacetime, de Sitter subtraction separates background propagation from the static-coordinate lens coefficient, while the final interpretation of any $M\mathrm{\Lambda }$ term depends on the complete observable setup. The general perturbation kernels extend the same calculation to static spherical effective metrics without rederiving the endpoint shift for each model.