Rigidity of geodesic completeness in the Brinkmann class of gravitational wave spacetimes

Abstract

We consider restrictions placed by geodesic completeness on spacetimes possessing a null parallel vector field, the so-called Brinkmann spacetimes. This class of spacetimes includes important idealized gravitational wave models in General Relativity, namely the plane-fronted waves with parallel rays, or pp-waves, which in turn have been intensely and fruitfully studied in the mathematical and physical literatures for over half a century. More concretely, we prove a restricted version of a conjectural analogue for Brinkmann spacetimes of a rigidity result obtained by M.T. Anderson for stationary spacetimes. We also highlight its relation with a longstanding 1962 conjecture by Ehlers and Kundt. Indeed, it turns out that the subclass of Brinkmann spacetimes we consider in our main theorem is enough to settle an important special case of the Ehlers-Kundt conjecture in terms of the well known class of Cahen-Wallach spaces.

The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

“…Indeed, the progress along these decades has focused on other aspects of the conjecture rather than in the crude geodesic equation (or the dynamical system (4)), concretely: (a) All plane waves (gravitational or not) are geodesically complete, as the equation (4) reduces to a second order linear system of differential equations [17], [ [26] gave a local characterization of pp-waves, argued that a natural extension of the conjecture follows when a locally pp-wave metric is taken on a compact M , and proved that this extension becomes equivalent to the standard one on R 4 . In this same line, the quoted article [16] also showed that plane waves are the universal coverings of certain spacetimes with non-trivial topology under some natural hypotheses for EK conjecture.…”

“…However, Leistner and Schliebner [26] gave a local characterization of pp-waves, argued that a natural extension of the conjecture follows when a locally pp-wave metric is taken on a compact M , and proved that this extension becomes equivalent to the standard one on R 4 . In this same line, the quoted article [16] also showed that plane waves are the universal coverings of certain spacetimes with non-trivial topology under some natural hypotheses for EK conjecture.…”

“…Choose θ + ∈ (θ 0 , π/(2n)) and take ρ 0 as in Proposition 3.7. Then γ remains in D[ρ 0 , θ + ] and, moreover, the bound (16) in Lemma 3.3 holds for some δ 0 > 0. So, from the first equation in (15) we have (as in (31)):…”

“…As cos(nθ) > cos(nθ + ) > 0 and ρ m−n ≤ 1/ρ in the large, some big ρ 0 > 1 so that δ 0 ≤ cos(nθ + ) − C/ρ 0 (for, say, nC = m m|λ m |) yield (16).…”

“…Therefore, EK conjecture becomes true if H is quadratically polynomially bounded, as such a bound implies directly that the pp-wave is a plane wave. There are some physical reasons supporting such a bound [19,20]; remarkably among them, the fact that otherwise the pp-wave would not be strongly causal in the autonomous case [16,Corollary 1.3]. (c) As quoted above, Ehlers and Kundt's original statement of the conjecture included the extra: "no matter which topology one chooses".…”