Bouncing Geodesics in Geometry & Cosmology

• Bouncing geodesics are curves that follow geodesic paths but are redirected by boundaries, metric degeneracies, or cosmological scale factor reversals.
• They manifest in diverse settings such as Riemannian billiards, bouncing FLRW cosmologies, black-hole interiors, and singular obstacle problems, each with unique reflection laws.
• Analytical methods—including quasigeodesic approximations, the Raychaudhuri equation, and PDE frameworks—provide insights into geodesic completeness and singularity behavior.

“Bouncing geodesics” denotes a family of related but non-identical notions across differential geometry, mathematical billiards, cosmology, and black-hole physics. In billiard theory, they are curves that are geodesic in the interior of a table and satisfy a reflection law at the boundary; in non-singular cosmology, they are timelike or null geodesics that extend through a classical bounce where the scale factor reaches a nonzero minimum; in black-hole and holographic settings, they are null limits of spacelike or timelike geodesics that approach the curvature singularity and return; and near conical or cuspidal singularities they are geodesics that reach a smallest distance $\delta$ and then move away while winding around the singularity (Giannetto, 3 Feb 2026, Güngör et al., 2020, Grozdanov et al., 16 Mar 2026, Grieser et al., 2023). In Euclidean obstacle problems, the same language applies to shortest paths that alternate between interior line segments and boundary geodesics, with finitely many switch points (Yang, 2023).

1. Terminological scope

The term has no single universal definition. Its meaning is controlled by the ambient category: smooth manifold with boundary, singular metric space, FLRW spacetime, or black-hole geometry. The following usages are all standard within their respective subliteratures.

Setting	Meaning of “bouncing geodesic”	Characteristic relation
Riemannian billiard table	Interior geodesic segments with boundary reflection	$g_{x_0}(u+v,w)=0$ for all $w\in T_{x_0}\partial K$
Closed bouncing cosmology	Causal geodesics extend through a non-singular minimum of $a(t)$	$H(t_b)=0,\ \dot H(t_b)>0$
Static black hole	Null limit of a geodesic that approaches $r=0$ and returns	$E^2=\dot r^2-\epsilon f(r)$
Conical or cuspidal singularity	Geodesic reaches minimal radius $\delta$ and winds away	$\ell(y_\delta)\sim C_f/f'(\delta)$
Analytic obstacle in $\mathbb R^3$	Shortest path alternating line segments and boundary geodesics	finitely many switch points

This suggests a unifying theme: the geodesic is not “free” in the Euclidean sense, but is redirected by boundary reflection, by the reversal of cosmological contraction, by an effective potential near a curvature singularity, or by the degeneration of the metric near a singular point. The redirection mechanism, however, is context-dependent and should not be conflated across these settings.

2. Billiards, quasigeodesics, and fold limits

In the geometric theory of billiards, the slogan “billiard trajectories are geodesics that bounce” is made precise by a quasigeodesic formalism (Giannetto, 3 Feb 2026). A Riemannian billiard table is a compact manifold with boundary $g_{x_0}(u+v,w)=0$0, where $g_{x_0}(u+v,w)=0$1 is the closure of an open orientable connected subset of $g_{x_0}(u+v,w)=0$2 with $g_{x_0}(u+v,w)=0$3 boundary and $g_{x_0}(u+v,w)=0$4 is induced from an ambient complete smooth Riemannian metric. At a boundary point $g_{x_0}(u+v,w)=0$5, two unit vectors $g_{x_0}(u+v,w)=0$6 satisfy the billiard reflection law when

$g_{x_0}(u+v,w)=0$7

equivalently,

$g_{x_0}(u+v,w)=0$8

with $g_{x_0}(u+v,w)=0$9 the inward unit normal. A billiard trajectory is an arclength-parameterized Lipschitz regular curve that is locally length-minimizing away from bounce times and satisfies this reflection law at every boundary hit.

The same paper reformulates this in Alexandrov-comparison language through $w\in T_{x_0}\partial K$0-quasigeodesics. For a space of curvature bounded below by $w\in T_{x_0}\partial K$1, a Lipschitz curve $w\in T_{x_0}\partial K$2 is a $w\in T_{x_0}\partial K$3-quasigeodesic when, for every point $w\in T_{x_0}\partial K$4,

$w\in T_{x_0}\partial K$5

satisfies

$w\in T_{x_0}\partial K$6

in the barrier sense. When the manifold has no boundary, $w\in T_{x_0}\partial K$7-quasigeodesics are exactly arclength geodesic segments. With boundary, quasigeodesics may have corners at the boundary; the one-sided tangent vectors are then polar, and in billiard tables the polar condition is equivalent to the billiard reflection law. In this sense, a billiard trajectory is a geodesic in the interior and a quasigeodesic globally.

A second major result is an approximation theorem in the opposite direction. For a billiard table $w\in T_{x_0}\partial K$8, one constructs fold-type hypersurfaces

$w\in T_{x_0}\partial K$9

which flatten onto $a(t)$0 as $a(t)$1. Under a uniform lower curvature bound on these folds, any sequence of arclength geodesic segments on $a(t)$2 with $a(t)$3 has a subsequence converging uniformly to a billiard trajectory in $a(t)$4. In convex Euclidean tables the required curvature bound is automatic, because the sectional curvatures of the folds are nonnegative. The resulting correspondence is two-sided: boundary geodesics can be approximated by billiard trajectories, and billiard trajectories can be approximated by honest geodesics on smooth fold surfaces.

3. Cosmological bounces and geodesic completeness

In classical bouncing cosmology, “bouncing geodesics” usually means causal geodesics that pass through a non-singular minimum of the scale factor rather than terminating at $a(t)$5. A concrete realization is a closed FLRW universe in General Relativity coupled to a real scalar field with canonical kinetic term, a renormalizable quartic Jordan-frame potential, and a non-minimal $a(t)$6 coupling (Güngör et al., 2020). The metric is

$a(t)$7

and the bounce occurs at $a(t)$8 when

$a(t)$9

Positive spatial curvature supplies the crucial term allowing a bounce without violating the null energy condition. The universe is vacuum-energy dominated and contracting at $H(t_b)=0,\ \dot H(t_b)>0$0, the Ricci scalar and other invariants remain finite, and the scale factor never vanishes.

For comoving timelike geodesics, the four-velocity is $H(t_b)=0,\ \dot H(t_b)>0$1, so proper time equals cosmic time,

$H(t_b)=0,\ \dot H(t_b)>0$2

Since the metric remains regular and $H(t_b)=0,\ \dot H(t_b)>0$3 extends to $H(t_b)=0,\ \dot H(t_b)>0$4, these geodesics are past complete. The same model argues that null geodesics are extendible as well: there is no curvature blow-up, the geodesic equations remain regular for all $H(t_b)=0,\ \dot H(t_b)>0$5, and the spacetime is described as geodesically complete. The usual past-incompleteness logic of Borde–Guth–Vilenkin does not apply because the spacetime has an eternal contracting phase, compact positively curved spatial slices, and no strictly positive average Hubble parameter along past-directed geodesics.

A broader effective-cosmology perspective emphasizes that in homogeneous bouncing FLRW models the geodesics themselves do not literally reverse direction (1803.01961). Comoving timelike geodesics remain $H(t_b)=0,\ \dot H(t_b)>0$6, radial null geodesics remain monotone in conformal time, and what “bounces” is the scale factor or, equivalently, the physical separation $H(t_b)=0,\ \dot H(t_b)>0$7 between neighboring geodesics. In that usage, “bouncing geodesics” means geodesically complete congruences whose physical distances decrease during contraction, attain a finite minimum, and then increase again during expansion.

4. Raychaudhuri, focusing, and dynamical horizons

The Raychaudhuri equation supplies the congruence-level criterion for whether a contracting family of timelike geodesics can cross a bounce instead of focusing to a singularity. For a timelike geodesic, irrotational, shear-free FLRW congruence, the expansion scalar is

$H(t_b)=0,\ \dot H(t_b)>0$8

and the Raychaudhuri equation reduces to

$H(t_b)=0,\ \dot H(t_b)>0$9

or equivalently

$r=0$0

(Chakraborty et al., 2023). For a local-minimum bounce $r=0$1, one has $r=0$2 before the bounce, $r=0$3, $r=0$4 after it, and $r=0$5 near $r=0$6. This implies

$r=0$7

so the convergence condition is violated and the focusing theorem does not force $r=0$8 in finite proper time. In perfect-fluid language this is $r=0$9, and in the flat case $E^2=\dot r^2-\epsilon f(r)$0 also gives $E^2=\dot r^2-\epsilon f(r)$1, so NEC and SEC are violated near the bounce. For a local-maximum bounce $E^2=\dot r^2-\epsilon f(r)$2, $E^2=\dot r^2-\epsilon f(r)$3, and the paper distinguishes a usual-matter case with $E^2=\dot r^2-\epsilon f(r)$4 from an exotic-matter case with $E^2=\dot r^2-\epsilon f(r)$5 but $E^2=\dot r^2-\epsilon f(r)$6.

A distinct spacetime realization is McVittie geometry on a bouncing background with

$E^2=\dot r^2-\epsilon f(r)$7

(Pérez et al., 2021). In areal radius $E^2=\dot r^2-\epsilon f(r)$8, the metric contains the factor

$E^2=\dot r^2-\epsilon f(r)$9

and radial null geodesics satisfy

$\delta$0

Trapping horizons are determined by

$\delta$1

In the contracting phase a dynamical black hole is present; as the universe approaches the bounce, the trapping horizons merge and disappear, so the black hole ceases to exist. Immediately after the bounce the central weak singularity is naked, whereas at large positive time the null-geodesic behavior again indicates a black hole. Some radial null geodesics traverse the bounce without hitting the singularity, while others asymptote to the late-time event horizon. The paper’s conclusion is that neither the contracting nor the expanding epoch can accommodate a black hole at all times.

5. Black-hole interiors, bouncing singularities, and holography

In static black-hole geometries, bouncing geodesics are defined directly in terms of radial motion in

$\delta$2

with conserved quantity

$\delta$3

and radial equation

$\delta$4

(Grozdanov et al., 16 Mar 2026). A bouncing geodesic is the null limit of a spacelike or timelike geodesic that approaches the curvature singularity from finite radius, comes arbitrarily close to $\delta$5, and then returns to finite radius. A sufficient condition is

$\delta$6

which yields turning points $\delta$7 as $\delta$8. The same paper places this in a rigorous PDE framework: by the local Hadamard form and global propagation of singularities, the bulk retarded Green’s function is singular whenever two points are connected by a null geodesic, and the same conclusion survives the holographic boundary limit for boundary retarded propagators. Bouncing geodesics therefore correspond to “bouncing singularities” in retarded correlators.

For planar AdS black holes, these singularities are explicit. Retarded two-point functions of local scalar operators, and of displacement operators on a Wilson line, develop a singularity at

$\delta$9

interpreted as the complex time of a null geodesic that leaves the boundary, probes the black-hole interior, and bounces back (Giombi et al., 11 Mar 2026). The high-frequency retarded correlator contains an exponentially suppressed factor

$\ell(y_\delta)\sim C_f/f'(\delta)$0

and the same structure is recovered both from a WKB analysis with infalling horizon boundary conditions and from an asymptotic OPE analysis based only on near-boundary data. The paper argues that this reflects a universal high-frequency structure controlled by multi-stress-tensor OPE data.

The same geodesic logic extends beyond AdS. In asymptotically flat Schwarzschild and Schwarzschild–de Sitter black holes, bouncing geodesics produce singularities in the bulk retarded Green’s function at computable critical times, with anchorings at finite timelike walls, null infinity, static spheres, or cosmological horizons (Grozdanov et al., 9 Jun 2026). Placing the black hole in a reflecting cavity yields a cavity thermal product formula and the asymptotic relation

$\ell(y_\delta)\sim C_f/f'(\delta)$1

linking the bouncing time $\ell(y_\delta)\sim C_f/f'(\delta)$2 to the asymptotic cavity quasinormal-mode spectrum. The paper verifies this relation for scalar, electromagnetic, and gravitational perturbations.

A central limitation is that curvature singularity does not imply bouncing geodesics. An explicit counterexample is a black hole in the self-dual linear axion model: the curvature scalar diverges at $\ell(y_\delta)\sim C_f/f'(\delta)$3, but the blackening factor remains of BTZ type and there are no bouncing geodesics (Grozdanov et al., 16 Mar 2026). A different limitation emerges in black-to-white-hole bounce models: in a thin-shell approximation, crossing the transition surface changes the conserved energy according to

$\ell(y_\delta)\sim C_f/f'(\delta)$4

and repeated mass-decreasing cycles squeeze bounded radial timelike geodesics into a stretched-horizon layer, producing Planck-scale blueshifts relative to regular infalling trajectories (Hong et al., 2022). In that setting, “bouncing geodesics” are not benign probes but dynamically problematic degrees of freedom.

6. Singular Riemannian spaces and analytic obstacles

Near an isolated conical or cuspidal singularity, geodesic motion is governed by a warped metric

$\ell(y_\delta)\sim C_f/f'(\delta)$5

on $\ell(y_\delta)\sim C_f/f'(\delta)$6, where $\ell(y_\delta)\sim C_f/f'(\delta)$7 is convex and $\ell(y_\delta)\sim C_f/f'(\delta)$8 is a closed Riemannian manifold (Grieser et al., 2023). On the unit-energy shell,

$\ell(y_\delta)\sim C_f/f'(\delta)$9

and with $\mathbb R^3$0 one has

$\mathbb R^3$1

Every maximal non-radial geodesic has a unique minimum radius $\mathbb R^3$2, enters from $\mathbb R^3$3, reaches $\mathbb R^3$4, and exits again to $\mathbb R^3$5. As $\mathbb R^3$6, its radial part satisfies

$\mathbb R^3$7

Its angular length obeys

$\mathbb R^3$8

In the conical case $\mathbb R^3$9, $g_{x_0}(u+v,w)=0$00 stays bounded; in the cuspidal case $g_{x_0}(u+v,w)=0$01, $g_{x_0}(u+v,w)=0$02, so the number of windings diverges as the geodesic approaches the singularity. For $g_{x_0}(u+v,w)=0$03, $g_{x_0}(u+v,w)=0$04, this gives power-law divergence of the winding number.

In Euclidean obstacle geometry, “bouncing geodesics” are shortest paths in $g_{x_0}(u+v,w)=0$05 constrained by analytic hypersurfaces (Yang, 2023). With one analytic obstacle, a geodesic is a $g_{x_0}(u+v,w)=0$06 curve consisting of finitely many interior line segments alternating with boundary hypersurface geodesics. The paper proves that, near any boundary point $g_{x_0}(u+v,w)=0$07, the number of line segments in a sufficiently small ball is uniformly bounded independently of the initial velocity; more precisely, there are at most two complete or partial line segments in that ball. With two transversally intersecting analytic hypersurfaces $g_{x_0}(u+v,w)=0$08, the paper proves that if $g_{x_0}(u+v,w)=0$09, then there exists $g_{x_0}(u+v,w)=0$10 such that $g_{x_0}(u+v,w)=0$11 has no switch point for $g_{x_0}(u+v,w)=0$12, whether the intersection angle is less than $g_{x_0}(u+v,w)=0$13 or at least $g_{x_0}(u+v,w)=0$14. Infinite local oscillation between the two surfaces is therefore excluded.

These results show that outside smooth manifolds without boundary, “bouncing” often means either finite switching between strata or repeated winding near a singular metric degeneration. Reflection is only one special case.

7. Limits of the concept

Several common misconceptions are corrected by the current literature. First, a bounce in cosmology does not require that individual geodesics reverse parameter direction; in homogeneous FLRW models, what reverses is the sign of $g_{x_0}(u+v,w)=0$15 and the physical separation within a congruence, while timelike and null geodesics can remain smoothly extendible through the bounce (1803.01961, Chakraborty et al., 2023). Second, a black-hole curvature singularity does not by itself guarantee bouncing geodesics or bouncing singularities in correlators; the near-singularity behavior of the blackening factor is decisive, and explicit counterexamples exist (Grozdanov et al., 16 Mar 2026). Third, in billiard geometry the slogan “geodesics that bounce” is exact only under structural hypotheses such as curvature lower bounds on the billiard table or on the fold approximants; for non-convex examples these hypotheses can fail, and the fold curvature can blow up negatively (Giannetto, 3 Feb 2026).

The term is therefore best read contextually. In billiards it denotes a precise reflection law and quasigeodesic structure; in cosmology it denotes congruence reversal and geodesic completeness across a non-singular minimum of $g_{x_0}(u+v,w)=0$16; in black-hole physics it denotes null limits of singularity-probing geodesics and the corresponding singularities of retarded Green’s functions; and in singular or stratified geometry it denotes turning and winding behavior enforced by metric degeneration or obstacle boundaries. The shared motif is continuation through or against an apparent obstruction, but the obstruction itself—boundary, horizon, singularity, or stratified interface—varies from one theory to another.