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The global uniqueness and $C^1$-regularity of geodesics in expanding impulsive gravitational waves

Jiri Podolsky, Clemens Sämann, Roland Steinbauer, Robert Svarc

TL;DR

This work establishes the existence and global uniqueness of $C^1$-geodesics for expanding impulsive gravitational waves in backgrounds of constant curvature by employing the continuous metric form and Filippov solution theory. It proves that geodesics crossing the impulse are $C^1$-curves, thereby rigorously validating the $C^1$-matching procedure used to derive their explicit behavior, including refraction across the wave surface. The authors derive explicit matching formulas that connect coordinates and velocities across the impulse via the generating function $h(Z)$, showing these refraction relations are independent of the curvature parameter $\epsilon$. The results provide a rigorous foundation for prior approaches and pave the way for linking continuous and distributional formulations and for deeper causality analyses in low-regularity Lorentzian geometries.

Abstract

We study geodesics in the complete family of expanding impulsive gravitational waves propagating in spaces of constant curvature, that is Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous form of the metric we rigorously prove existence and global uniqueness of continuously differentiable geodesics (in the sense of Filippov) and study their interaction with the impulsive wave. Thereby we justify the "$C^1$-matching procedure" used in the literature to derive their explicit form.

The global uniqueness and $C^1$-regularity of geodesics in expanding impulsive gravitational waves

TL;DR

This work establishes the existence and global uniqueness of -geodesics for expanding impulsive gravitational waves in backgrounds of constant curvature by employing the continuous metric form and Filippov solution theory. It proves that geodesics crossing the impulse are -curves, thereby rigorously validating the -matching procedure used to derive their explicit behavior, including refraction across the wave surface. The authors derive explicit matching formulas that connect coordinates and velocities across the impulse via the generating function , showing these refraction relations are independent of the curvature parameter . The results provide a rigorous foundation for prior approaches and pave the way for linking continuous and distributional formulations and for deeper causality analyses in low-regularity Lorentzian geometries.

Abstract

We study geodesics in the complete family of expanding impulsive gravitational waves propagating in spaces of constant curvature, that is Minkowski, de Sitter and anti-de Sitter universes. Employing the continuous form of the metric we rigorously prove existence and global uniqueness of continuously differentiable geodesics (in the sense of Filippov) and study their interaction with the impulsive wave. Thereby we justify the "-matching procedure" used in the literature to derive their explicit form.

Paper Structure

This paper contains 12 sections, 6 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Exact expanding impulsive gravitational waves in spacetimes of constant curvature
  3. The Penrose "cut and paste" method
  4. Continuous coordinates
  5. Limits of sandwich waves
  6. Impulses generated by infinitely accelerating sources
  7. Geodesics in expanding impulsive waves
  8. The geodesic equations
  9. Existence of $C^1$-geodesics
  10. Uniqueness of geodesics and crossing of the impulse
  11. Explicit $C^1$-matching of geodesics crossing the impulse
  12. Conclusion

Key Result

Theorem 3.1

Let $(M,g)$ be a smooth manifold with a $C^{0,1}$-semi-Rie-mann-ian metric $g$. Then there exist Filippov solutions of the geodesic equations which are $C^1$-curves.

Figures (2)

  • Figure 1: Minkowski or (anti-)de Sitter space is cut into two parts ${\cal M}^-$ and ${\cal M}^+$ along a future null cone ${\cal N}$. These parts are then re-attached with an arbitrary "warp" in which points on both sides of ${\cal N}$ are identified. Such a construction generates spherical impulsive gravitational waves expanding in these constant-curvature backgrounds.
  • Figure 2: Upper part: The family of future null cones $C_\tau$ with vertices along a timelike (${\epsilon=+1}$), null (${\epsilon=0}$), or a spacelike (${\epsilon=-1}$) line foliate in three different ways Minkowski space in the Robinson--Trautman form. Analogous foliations apply to (anti-)de Sitter space, see GP:09. Lower part: sandwich gravitational waves at a fixed time for different values of $\epsilon$ are indicated by the shaded regions. All wavefronts ${u =\>}$const. are spherical (hemispherical for ${\epsilon=-1}$) and expand with the speed of light. The inner region ${u<0}$ is free of topological defects, while the external region ${u>u_1>0}$ contains a cosmic string (thick line) which "disintegrates" within the sandwich (zigzag line), generating the wave.

Theorems & Definitions (9)

  • Theorem 3.1: Theorem 2 in S:14
  • Corollary 3.2: Existence
  • Remark 3.3: Local existence for non-smooth $H$
  • Lemma 3.4: Sufficient conditions for uniqueness, Lemma 2.10.2 in F:88
  • Theorem 3.5: Uniqueness
  • Corollary 3.6: Preservation of causal character
  • Corollary 3.7: Crossing the expanding impulse
  • Remark 3.8: Uniqueness for non-smooth $H$
  • Remark 3.9: Completeness