Embedding nonrelativistic physics inside a gravitational wave

TL;DR

<3-5 sentence high-level summary>The paper develops a geometric framework to embed nonrelativistic physics inside relativistic gravitational waves by extending the Eisenhart lift to a broad class of spacetimes called Platonic gravitational waves. It shows how nonrelativistic trajectories arise as shadows of ambient geodesics, derives the Schrödinger equation on curved space via null reduction of Klein-Gordon, and provides a comprehensive geometric classification of Platonic waves (including Bargmann-Eisenhart, Julia-Nicolai, and Kundt relations). The work also analyzes global properties such as geodesic completeness and causal structure, and classifies curvature invariants (CSI/VSI) within this framework, highlighting the physical relevance of AdS-gyratons, Siklos spacetimes, and Schrödinger backgrounds. Overall, the ambient approach clarifies how nonrelativistic dynamics are encoded in the geometry of gravitational waves and offers a robust platform for further holographic and quantum-gravity explorations.

Abstract

Gravitational waves with parallel rays are known to have remarkable properties: Their orbit space of null rays possesses the structure of a non-relativistic spacetime of codimension-one. Their geodesics are in one-to-one correspondence with dynamical trajectories of a non-relativistic system. Similarly, the null dimensional reduction of Klein-Gordon's equation on this class of gravitational waves leads to a Schroedinger equation on curved space. These properties are generalized to the class of gravitational waves with a null Killing vector field, of which we propose a new geometric definition, as conformally equivalent to the previous class and such that the Killing vector field is preserved. This definition is instrumental for performing this generalization, as well as various applications. In particular, results on geodesic completeness are extended in a similar way. Moreover, the classification of the subclass with constant scalar invariants is investigated.

Figures (9)

• Figure 1: The screen worldvolume is transverse to the congruence of null rays. (In all figures, we will follow the standard spacetime diagram convention, i.e. time flows from bottom to top and null directions are at 45$^\circ$.)
• Figure 2: 1. Screen worldvolume; 2. Screen at $t=t_1$; 3. Congruence of null geodesics generating the wavefront worldvolume $t=t_1$; 4. Wavefont worldvolume $t=t_1$; 5. Screen at $t=t_0$; 6. Congruence of null geodesics generating the wavefront worldvolume $t=t_0$; 7. Wavefont worldvolume $t=t_0$
• Figure 3: The Eisenhart lift, 1. Geodesic of ambient spacetime; 2. Shadow of the ambient geodesic on the screen worldvolume; 3. Emission of a graviton by the geodesic; 4. Detection on the screen at $t=t_1$; 5. Emission of a graviton by the geodesic; 6. Detection on the screen at $t=t_0$;
• Figure 4: Two-dimensional Minkowski spacetime as a gravitational wave. The wavefront worldvolumes are the lines $x^-=\hbox{const}$.
• Figure 5: Several choices of screen worldvolumes are possible e.g. the timelike screen worldvolume axis $x^0$, so that leaves of the foliation are labeled by the retarded time $t$, or the lightlike screen worldvolume axis $x^-$ which labels leaves with light-cone time $x^-$. The event $E_1$ is encoded on the timelike screen worldvolume by its position $x_1$ and the time of emission ($E_2$) of the graviton intersecting it: $t_2=t_1-x_1$. Alternatively, on the lightlike screen worldvolume, the moment of emission ($E_3$) of the graviton intersecting $E_1$ has for light-cone time $x^-=\frac{t_1-x_1}{\sqrt{2}}$.

Theorems & Definitions (30)

• Theorem 2.1: Eisenhart-Lichnerowicz Eisenhart1928Lichnerowicz
• Definition 4.1
• Lemma 4.2
• Lemma 4.3
• Definition 4.4
• Lemma 4.5
• Definition 4.6
• Lemma 4.7
• Definition 4.8
• Lemma 4.9