---
title: Finite-Distance Jacobi-Metric Gauss-Bonnet Framework
url: https://www.emergentmind.com/topics/finite-distance-jacobi-metric-gauss-bonnet-framework
type: topic
---

# Finite-Distance Jacobi-Metric Gauss-Bonnet Framework

Searching arXiv for the cited finite-distance Jacobi/optical Gauss–Bonnet literature and recent extensions.
The finite-distance Jacobi-metric Gauss-Bonnet framework is a geometric formulation of weak gravitational deflection in which the trajectory problem is recast on a two-dimensional optical or Jacobi manifold and the deflection angle is extracted from the Gauss-Bonnet theorem. In the review “Deriving Weak Deflection Angle by Black Holes or Wormholes using Gauss-Bonnet Theorem” [2111.02805], this framework appears through the Gibbons-Werner construction for asymptotically flat spacetimes, Ishihara et al.’s finite-distance formulation for source and receiver at finite radii, and the Jacobi metric as an alternative to the optical metric for massive particles and for finite-distance geometry. Later work sharpened different components of this picture: the role of the Jacobi-Maupertuis metric near degenerate Hill boundaries [1407.7163], finite-distance optical-metric constructions [1604.08308], finite-distance Jacobi-metric treatments for massive particles in stationary spacetimes [1912.05194], charged massive particles [2012.14226], spinning massive particles [2603.00650], quadrupolar spinning bodies [2603.20657], radial-integral reformulations [2601.17059], and reference-renormalized curvature-primitive normalizations [2604.16807].

## 1. Geometric core of the framework

The central idea is to recast the light-deflection problem into a purely 2D Riemannian (or Finsler-Randers) geometry problem and then compute the deflection angle via the Gauss-Bonnet theorem (GBT) [2111.02805]. For a 2D oriented Riemannian manifold \(\mathcal{D}\) with piecewise smooth boundary \(\partial\mathcal{D}\), the theorem is written as
\[
\iint_{\mathcal{M}} \mathcal{K}\, dS + \int_{\partial\mathcal{M}} \kappa\, dt + \sum_i \alpha_i = 2\pi \chi(\mathcal{M}),
\]
where \(\mathcal{K}\) is the Gaussian curvature, \(dS\) is the induced area element, \(\kappa\) is the geodesic curvature of boundary curves, \(\alpha_i\) are the exterior angles, and \(\chi(\mathcal{M})\) is the Euler characteristic [2111.02805]. In gravitational lensing, the relevant 2D surface is the optical geometry or the Jacobi surface, and the projected light or particle trajectories are geodesics of that effective metric.

For static, spherically symmetric, asymptotically flat spacetimes, Gibbons and Werner choose a domain bounded by the light ray and an auxiliary circular arc \(r=R\), with \(R\to\infty\), and obtain
\[
\hat{\alpha} = -\iint_{\mathcal{M}} \mathcal{K}\, dS.
\]
With the straight-line approximation \(r(\varphi)=u/\sin\varphi\), this becomes
\[
\hat{\alpha} = -\int_0^\pi \int_{u/\sin\varphi}^\infty \mathcal{K}(r,\varphi)\,\sqrt{\det \bar g}\,dr\,d\varphi,
\]
which is the central Gibbons-Werner result in the weak-field regime [2111.02805]. In Schwarzschild, where \(f(r)=g(r)=1-2M/r\), the optical Gaussian curvature is \(\mathcal{K}=-2M/r^3\), yielding \(\hat{\alpha}=4M/u\) at leading order [2111.02805].

This suggests a bulk-boundary decomposition of deflection: the curvature integral carries the accumulated effect of the effective spatial geometry, while the boundary terms encode how the chosen lensing domain is closed. That interpretation becomes essential once source and receiver are not sent to infinity.

## 2. Optical metric, Jacobi metric, and their relation

For a static spherically symmetric spacetime
\[
ds^2 = -f(r)\,dt^2 + \frac{1}{g(r)}\,dr^2 + r^2(d\theta^2+\sin^2\theta\,d\varphi^2),
\]
null geodesics restricted to the equatorial plane define the optical metric
\[
dt^2 = \bar g_{rr}\,dr^2+\bar g_{\varphi\varphi}\,d\varphi^2,
\qquad
\bar g_{rr}=\frac{1}{f(r)g(r)},\quad
\bar g_{\varphi\varphi}=\frac{r^2}{f(r)},
\]
with area element
\[
dS=\sqrt{\frac{r^2}{f(r)^2 g(r)}}\,dr\,d\varphi
\]
[2111.02805]. Ishihara et al. explicitly describe this optical metric as the geometric arena for finite-distance bending [1604.08308].

The Jacobi metric is introduced as an alternative framework, especially suited for massive particles with fixed energy and static spacetimes [2111.02805]. For
\[
ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)\,d\Omega^2,
\]
the equatorial Jacobi metric is given in the review as
\[
dl^2 = m^2\frac{1}{1-v^2-A(r)}
\left(
\frac{B(r)}{A(r)}\,dr^2+\frac{C(r)}{A(r)}\,d\varphi^2
\right),
\]
with \(m\) the rest mass and \(v\) the speed in units \(c=1\) [2111.02805]. The associated orbit equation is written as
\[
\left(\frac{d\tilde r}{d\varphi}\right)^2
=
\frac{C^4\tilde r^4}{AB}
\left(\frac{1}{u^2}-\frac{A}{C}\right),
\]
with \(\tilde r=1/r\) and \(u\) the impact parameter [2111.02805].

In the null case, the Jacobi metric reduces, up to a conformal factor, to the optical metric, and the review states that the deflection angle obtained from GBT is invariant under positive conformal rescalings of the 2D metric, so both metrics yield the same bending angle [2111.02805]. For massive particles, the Jacobi metric encodes non-null geodesics and extends the same geometric machinery to timelike probes. That extension is developed explicitly for stationary spacetimes in “The finite-distance gravitational deflection of massive particles in stationary spacetime: a Jacobi metric approach” [1912.05194] and for charged massive particles in [2012.14226].

A plausible implication is that the phrase “Jacobi-metric Gauss-Bonnet framework” is best understood not as a distinct theorem, but as a unifying effective-geometry prescription: select the appropriate optical/Jacobi metric, compute \(\mathcal{K}\) and relevant geodesic curvatures, and then use Gauss-Bonnet on a lensing domain adapted to the physical endpoints.

## 3. Finite-distance formulation

Ishihara et al. showed that, for a static spacetime, the total deflection angle for source at \(S\) and observer at \(R\) can be written geometrically as
\[
\hat{\alpha}=-\iint \mathcal{K}\,dS \equiv \Psi_R-\Psi_S+\Phi_{RS},
\]
where \(\Psi_{R,S}\) are the angles between the ray and the radial direction at receiver and source, and \(\Phi_{RS}=\Phi_R-\Phi_S\) is the coordinate angular separation [2111.02805]. Finite-distance corrections are then defined by
\[
\delta\hat{\alpha}=\hat{\alpha}-\hat{\alpha}_\infty,
\]
with \(\hat{\alpha}_\infty\) the limiting deflection angle as \(S,R\to\infty\) [2111.02805].

An equivalent orbit-based form uses
\[
F(\tilde r)\equiv \left(\frac{d\tilde r}{d\varphi}\right)^2
\]
and gives
\[
\hat{\alpha}
=
\Psi_R-\Psi_S
+
\left[
\int_{\tilde r_R}^{\tilde r_L}\frac{d\tilde r}{\sqrt{F(\tilde r)}}
+
\int_{\tilde r_S}^{\tilde r_L}\frac{d\tilde r}{\sqrt{F(\tilde r)}}
\right],
\]
where \(\tilde r_L\) is the turning point [2111.02805]. The 2016 paper “Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem” [1604.08308] presents the same finite-distance observable as
\[
\alpha = \Psi_R-\Psi_S+\phi_{RS},
\]
and emphasizes that this formulation remains meaningful even when the spacetime is not asymptotically flat.

The asymptotically flat and non-asymptotically flat cases differ in the treatment of the outer boundary. In asymptotically flat geometries, the outer circular arc can be pushed to infinity and its geodesic curvature tends to unity, yielding the standard Gibbons-Werner simplification [2111.02805]. In non-asymptotically flat spacetimes such as Kottler and Weyl gravity, that construction is no longer available; one instead uses finite-radius boundaries or a background reference metric and defines relative deflection angles [2111.02805; 1604.08308].

The 2026 “Radial Integral Reformulation of the Gauss-Bonnet Weak Deflection Angle at Finite Distance” [2601.17059] re-expresses the finite-distance deflection as a sum of two radial integrals over \([r_0,r_S]\) and \([r_0,r_R]\) plus the finite-distance angular bookkeeping term, by converting Li’s curvature-primitive line integral into a purely radial form. The 2026 “Reference-renormalized curvature-primitive Gauss-Bonnet formalism” [2604.16807] then interprets the curvature primitive as defined only modulo an additive constant and fixes that gauge by matching to a reference optical geometry, such as Minkowski in asymptotically flat settings or de Sitter in Kottler-type backgrounds.

This suggests that finite-distance lensing is not a minor correction to the infinite-distance formula but a different geometric problem, in which the endpoint angles, closure prescription, and reference geometry all become part of the definition of the observable.

## 4. Representative spacetimes and explicit formulas

The review compiles a range of black-hole and wormhole examples built from the same workflow: write the optical or Jacobi metric, compute \(\mathcal{K}\), and insert it into the Gauss-Bonnet integral or its finite-distance variant [2111.02805].

| Spacetime | Key formula stated in the data | Finite-distance or weak-field outcome |
|---|---|---|
| Schwarzschild | \(\mathcal{K}=-2M/r^3\) | \(\hat{\alpha}=4M/u\) [2111.02805] |
| Kottler | \(F(\tilde r)=\frac{1}{u^2}-\tilde r^2+r_g\tilde r^3+\frac{\Lambda}{3}\) | finite-distance \(\Lambda\)-dependent terms plus mixed \(r_g\Lambda\) term [2111.02805] |
| Weyl conformal gravity | finite-distance formula with parameter \(\gamma\) | reduces to \(4M/u\) plus Weyl corrections [2111.02805] |
| Reissner-Nordström | leading orders include charge correction | \(\hat{\alpha}=\frac{4M}{u}-\frac{3\pi Q^2}{4u^2}+\cdots\) [2111.02805] |
| Einstein-Maxwell-dilaton wormhole | \(\mathcal{K}\approx -16PQ/r^4+\Sigma^2/r^4+\mathcal{O}(r^{-6})\) | \(\hat{\alpha}=\frac{3\pi PQ}{2u^2}-\frac{\pi\Sigma^2}{4u^2}+\cdots\) [2111.02805] |
| Black-bounce traversable wormhole | \(\mathcal{K}\approx 3M^2/r^4-2M/r^3-a^2/r^4+\cdots\) | \(\hat{\alpha}=\frac{4M}{u}+\frac{\pi a^2}{4u^2}+\cdots\) [2111.02805] |

For Kottler,
\[
\hat{\alpha}
=
\frac{r_g}{u}
\left[
\sqrt{1-u^2\tilde r_R^2}
+
\sqrt{1-u^2\tilde r_S^2}
\right]
-
\frac{\Lambda u}{6}
\left[
\frac{\sqrt{1-u^2\tilde r_R^2}}{\tilde r_R}
+
\frac{\sqrt{1-u^2\tilde r_S^2}}{\tilde r_S}
\right]
+
\frac{r_g\Lambda u}{12}
\left[
\frac{1}{\sqrt{1-u^2\tilde r_R^2}}
+
\frac{1}{\sqrt{1-u^2\tilde r_S^2}}
\right],
\]
and in the far-source/observer limit
\[
\hat{\alpha}
=
\frac{2r_g}{u}
-\frac{\Lambda u}{6}\left(\frac{1}{\tilde r_R}+\frac{1}{\tilde r_S}\right)
+\frac{r_g\Lambda u}{6},
\]
which reduces to the standard Schwarzschild result when \(\Lambda\to0\) and \(r_{R,S}\to\infty\) [2111.02805]. The same mixed \(r_g\Lambda\) term is reproduced in the reference-renormalized curvature-primitive formalism [2604.16807].

For Weyl gravity,
\[
\hat{\alpha}
=
\frac{2M}{u}
\left(
\sqrt{1-u^2\tilde r_R^2}+\sqrt{1-u^2\tilde r_S^2}
\right)
-\gamma M u
\left(
\frac{\tilde r_R}{\sqrt{1-u^2\tilde r_R^2}}
+
\frac{\tilde r_S}{\sqrt{1-u^2\tilde r_S^2}}
\right),
\]
showing explicitly how finite-distance endpoint factors enter non-asymptotically flat lensing [2111.02805].

The same framework extends beyond photons. Charged massive particles in a four-dimensional charged Einstein-Gauss-Bonnet black hole are treated by constructing a Jacobi metric with electrostatic potential and applying Gauss-Bonnet in Jacobi space, with finite-distance effects included through local endpoint angles [2012.14226]. Massive particles in stationary spacetimes, including Kerr and Teo wormholes, are handled by a generalized Jacobi metric of Randers-Finsler type, yielding finite-distance corrections in \(r_S\), \(r_R\), \(v\), and spin parameters of the background [1912.05194].

## 5. Spin, non-geodesic rays, and higher-order structure

The spinless framework assumes that the physical ray is a geodesic of the Jacobi manifold. That assumption fails for spinning particles governed by Mathisson-Papapetrou-Dixon dynamics. “Gauss-Bonnet lensing of spinning massive particles in static spherically symmetric spacetimes” [2603.00650] states that the spatial ray is generically driven away from Jacobi geodesics, so the Gauss-Bonnet construction must be reformulated to accommodate a non-geodesic particle boundary. The resulting spin-generalized identity adds a single boundary functional: the geodesic-curvature integral of the physical ray in the Jacobi manifold [2603.00650].

A more general extension appears in “A Rigorous Jacobi-Metric Approach to the Gauss-Bonnet Lensing of Spinning Particles: Extension to Quadrupole Order” [2603.20657]. There, the background spacetime is static and spherically symmetric,
\[
ds^2=-A(r)\,dt^2+B(r)\,dr^2+C(r)(d\theta^2+\sin^2\theta\,d\phi^2),
\]
and the Jacobi metric for a massive test particle with rest mass \(m\) and conserved energy \(E\) is
\[
dl^2=\mathcal{G}_{ij}\,dx^i dx^j
=
\frac{E^2-m^2A(r)}{A(r)}\left[B(r)\,dr^2+C(r)\,d\phi^2\right]
\]
[2603.20657]. The Gauss-Bonnet theorem on the 2D Jacobi manifold gives
\[
\alpha=\iint_D K\,dA+\int_{\gamma_p}\kappa_g\,dl,
\]
where the boundary term \(\int_{\gamma_p}\kappa_g\,dl\) is now central because the physical trajectory \(\gamma_p\) is non-geodesic [2603.20657].

At quadrupole order, the MPD force law includes
\[
\frac{DP^\mu}{d\lambda}
=
-\frac{1}{2}R^\mu{}_{\nu\alpha\beta}u^\nu S^{\alpha\beta}
-
\frac{1}{6}J^{\alpha\beta\gamma\delta}\nabla^\mu R_{\alpha\beta\gamma\delta},
\]
with a spin-induced quadrupole
\[
J^{\alpha\beta\gamma\delta}
=
\frac{C_Q}{m}
\left(
S^{\alpha\gamma}S^{\beta\delta}
-
S^{\alpha\delta}S^{\beta\gamma}
\right)
\]
[2603.20657]. The quadrupole piece generates a non-geodesic force in Jacobi space, and the corresponding geodesic-curvature contribution produces the deflection correction
\[
\delta\alpha_{\text{quad}}
=
\frac{\pi C_Q s^2 M}{2m^2 b^3}
\left[\frac{3(1+v^2)}{4v^4}\right]
=
\frac{3\pi}{8}\,\frac{C_Q s^2 M}{m^2 b^3}\,\frac{1+v^2}{v^4}
\]
in Schwarzschild [2603.20657]. The same paper emphasizes that the formulation is valid at finite distance even though the explicit displayed quadrupole formula is evaluated in the large-distance regime.

Charged signals in a dipole magnetic field provide a related non-Riemannian extension. There the Jacobi metric is a Finsler metric of Randers type, and the difference between deflection angles for opposite rotation directions is argued to be a Finslerian effect of non-reversibility [2208.11458]. In that setting, the generalized Jacobi method is particularly natural for finite-distance constructions because the Riemannian part of the Jacobi metric remains explicit while the one-form contributes through geodesic curvature [2208.11458].

A plausible implication is that the finite-distance Jacobi-metric Gauss-Bonnet framework is now better viewed as a hierarchy. At the lowest level, the ray is a geodesic of the effective metric and only the curvature-area term matters. At higher multipole order, the effective manifold remains useful, but the physical trajectory acquires nonzero geodesic curvature that must be integrated as an additional boundary term.

## 6. Global geometry, normalization issues, and Hill-boundary considerations

Two distinct global issues appear in the literature. The first concerns normalization of the curvature primitive. Li-type reductions treat the curvature-area contribution as a line integral of a primitive defined only up to an additive constant. The 2026 reference-renormalized formalism fixes this gauge by reference subtraction, defining a unique renormalized discrepancy primitive \(\mathcal{P}_e(r)\) by matching to a physical reference optical geometry in an outer regime [2604.16807]. In asymptotically flat settings the canonical reference is Minkowski, while in Kottler-type backgrounds the canonical reference is de Sitter within the static patch [2604.16807]. The paper states that this yields the Ishihara-Li finite-distance deflection angle without invoking any circular null orbit and remains fully compatible with orbit-normalized prescriptions whenever a suitable photon sphere exists [2604.16807].

The second concerns the Jacobi-Maupertuis metric near degenerate boundaries. “Who’s Afraid of the Hill Boundary?” [1407.7163] studies the Jacobi-Maupertuis metric
\[
ds^2_{\mathrm{JM}}=2(E-V(q))\,g_{ij}(q)\,dq^i dq^j
\]
for Newtonian systems and shows that it degenerates at the Hill boundary \(E=V(q)\). Near a regular boundary point, Seifert’s coordinates put the metric into the local form
\[
ds^2_{\mathrm{JM}} = y\,dy^2 + y\,f(x,y)\left(\sum_{i=1}^{n-1}dx_i^2+\sum_{i,j=1}^{n-1}h_{ij}(x,y)\,dx_i dx_j\right),
\]
with leading behavior
\[
ds^2_{\mathrm{JM}}\approx y(dy^2+dx^2)
\]
[1407.7163]. The paper proves that any JM geodesic sufficiently close to a regular point of the Hill boundary contains a pair of conjugate points close to the boundary and thus fails to minimize JM length [1407.7163]. It also shows that the first conjugate locus is a smooth hypersurface tangent to the boundary and of fold type [1407.7163].

These results are not lensing formulas, but they are directly relevant to any finite-distance Jacobi-metric Gauss-Bonnet framework whenever the effective Jacobi geometry approaches a degeneracy surface. The paper notes that sectional curvatures of planes containing the normal direction blow up as \(K\sim 1/y^3\), while the JM distance to the boundary behaves like \(y^{3/2}\) [1407.7163]. This suggests that finite-distance curvature concentration can force conjugate points on the same scale as the distance to the boundary.

A common misconception is that Jacobi-metric Gauss-Bonnet methods are purely asymptotic or purely topological. The cited literature indicates otherwise. The finite-distance formulations are explicitly local in \(r_S\) and \(r_R\) [1604.08308; 2111.02805; 2601.17059], and the Hill-boundary analysis shows that local degeneracy structure, conjugate loci, and curvature blow-up can matter even at finite Jacobi distance [1407.7163]. Another misconception is that photon-sphere normalization is intrinsic; the reference-renormalized formulation makes clear that it is a gauge choice rather than a necessity [2604.16807].

The framework therefore comprises a family of closely related constructions rather than a single canonical algorithm. In its most compact static spherical form, it consists of selecting an effective 2D optical or Jacobi metric, computing its Gaussian curvature and any needed boundary geodesic curvatures, choosing a finite lensing domain bounded by the physical ray and auxiliary geodesics, and applying Gauss-Bonnet to obtain the finite-distance deflection angle. In the asymptotically flat null limit, this recovers the Gibbons-Werner surface-integral formula. In non-asymptotically flat spacetimes, it requires finite-distance bookkeeping or a reference geometry. For massive, charged, spinning, or quadrupolar probes, the same geometric language survives, but the physical ray is generically non-geodesic in the Jacobi manifold, and the boundary functional \(\int \kappa_g\,dl\) becomes indispensable [2111.02805; 1912.05194; 2012.14226; 2603.00650; 2603.20657].

Source: https://www.emergentmind.com/topics/finite-distance-jacobi-metric-gauss-bonnet-framework