We investigate all geodesics in the entire class of nonexpanding impulsive gravitational waves propagating in an (anti-)de Sitter universe using the distributional metric. We extend the regularization approach of part I [Sämann, C. et al., Classical Quantum Gravity 33(11), 115002 (2016)] to a full nonlinear distributional analysis within the geometric theory of generalized functions. We prove global existence and uniqueness of geodesics that cross the impulsive wave and hence geodesic completeness in full generality for this class of low regularity spacetimes. This, in particular, prepares the ground for a mathematically rigorous account on the “physical equivalence” of the continuous form with the distributional “form” of the metric.

1.
Aichelburg
,
P. C.
and
Balasin
,
H.
, “
Generalized symmetries of impulsive gravitational waves
,”
Classical Quantum Gravity
14
,
A31
A41
(
1997
).
2.
Bainov
,
D.
and
Simeonov
,
P. S.
,
Integral Inequalities and Applications
(
Springer
,
1992
), Vol. 57.
3.
Barrabès
,
C.
and
Hogan
,
P. A.
,
Singular Null Hypersurfaces in General Relativity: Light-Like Signals From Violent Astrophysical Events
(
World Scientific Publishing Co., Inc.
,
River Edge, NJ
,
2003
).
4.
Burtscher
,
A.
and
Kunzinger
,
M.
, “
Algebras of generalized functions with smooth parameter dependence
,”
Proc. Edinburgh Math. Soc.
55
(
1
),
105
124
(
2012
).
5.
Candela
,
A. M.
,
Flores
,
J. L.
, and
Sánchez
,
M.
, “
On general plane fronted waves. Geodesics
,”
Gen. Relativ. Gravitation
35
,
631
649
(
2003
).
6.
Candela
,
A. M.
,
Flores
,
J. L.
, and
Sánchez
,
M.
, “
Geodesic connectedness in plane wave type spacetimes. A variational approach
,” in
Dynamic Systems and Applications
(
Dynamic
,
Atlanta, GA
,
2004
), Vol. 4, pp.
458
464
.
7.
Candela
,
A. M.
and
Sánchez
,
M.
, “
Geodesics in semi-Riemannian manifolds: Geometric properties and variational tools
,” in
Recent Developments in Pseudo-Riemannian Geometry
, ESI Lectures in Mathematics Physics (
European Mathematical Society
,
Zürich
,
2008
), pp.
359
418
.
8.
Colombeau
,
J.-F.
,
Elementary Introduction to New Generalized Functions
, Volume 113 of North-Holland Mathematics Studies, Notes on Pure Mathematics, 103 (
North-Holland Publishing Co.
,
Amsterdam
,
1985
).
9.
De Rham
,
G.
, “
Differentiable manifolds
,” in
Grundlehren der Mathematischen Wissenschaften
(
Springer
,
Berlin
,
1984
).
10.
De Roever
,
J. W.
and
Damsma
,
M.
, “
Colombeau algebras on a C -manifold
,”
Indag. Math.
2
(
3
),
341
(
1991
).
11.
Erlacher
,
E.
and
Grosser
,
M.
, “
Inversion of a ‘discontinuous coordinate transformation’ in general relativity
,”
Appl. Anal.
90
(
11
),
1707
1728
(
2011
).
12.
Erlacher
,
E.
and
Grosser
,
M.
, “
Ordinary differential equations in algebras of generalized functions
,” in
Pseudo-Differential Operators, Generalized Functions and Asymptotics
, Volume 231 of Operator Theory: Advances and Applications (
Birkhäuser/Springer Basel AG
,
Basel
,
2013
), pp.
253
270
.
13.
Flores
,
J. L.
and
Sánchez
,
M.
, “
Causality and conjugate points in general plane waves
,”
Classical Quantum Gravity
20
,
2275
2291
(
2003
).
14.
Flores
,
J. L.
and
Sánchez
,
M.
, “
On the geometry of pp-wave type spacetimes
,” in
Analytical and Numerical Approaches to Mathematical Relativity
, Volume 692 of Lecture Notes in Physics (
Springer
,
Berlin
,
2006
), pp.
79
98
.
15.
Geroch
,
R.
and
Traschen
,
J.
, “
Strings and other distributional sources in general relativity
,”
Phys. Rev. D
36
,
1017
1031
(
1987
).
16.
Grosser
,
M.
,
Kunzinger
,
M.
,
Oberguggenberger
,
M.
, and
Steinbauer
,
R.
,
Geometric Theory of Generalized Functions with Applications to General Relativity
, Volume 537 of Mathematics and its Applications (
Kluwer Academic Publishers
,
Dordrecht
,
2001
).
17.
Griffiths
,
J. B.
and
Podolský
,
J.
,
Exact Space-Times in Einstein’s General Relativity
, Cambridge Monographs on Mathematical Physics (
Cambridge University Press
,
Cambridge
,
2009
).
18.
Hörmann
,
G.
,
Kunzinger
,
M.
, and
Steinbauer
,
R.
, “
Wave equations on non-smooth space-times
,” in
Evolution Equations of Hyperbolic and Schrödinger Type
, Volume 301 of Progress in Mathematics (
Birkhäuser/Springer Basel AG
,
Basel
,
2012
), pp.
163
186
.
19.
Hotta
,
M.
and
Tanaka
,
M.
, “
Shock-wave geometry with non-vanishing cosmological constant
,”
Classical Quantum Gravity
10
,
307
314
(
1993
).
20.
Kunzinger
,
M.
and
Steinbauer
,
R.
, “
A rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves
,”
J. Math. Phys.
40
,
1479
1489
(
1999
).
21.
Kunzinger
,
M.
and
Steinbauer
,
R.
, “
A note on the Penrose junction conditions
,”
Classical Quantum Gravity
16
(
4
),
1255
1264
(
1999
).
22.
Kunzinger
,
M.
and
Steinbauer
,
R.
, “
Foundations of a nonlinear distributional geometry
,”
Acta Appl. Math.
71
(
2
),
179
206
(
2002
).
23.
Kunzinger
,
M.
and
Steinbauer
,
R.
, “
Generalized pseudo-Riemannian geometry
,”
Trans. Am. Math. Soc.
354
(
10
),
4179
4199
(
2002
).
24.
Kunzinger
,
M.
,
Oberguggenberger
,
M.
,
Steinbauer
,
R.
, and
Vickers
,
J. A.
, “
Generalized flows and singular ODEs on differentiable manifolds
,”
Acta Appl. Math.
80
(
2
),
221
241
(
2004
).
25.
Kunzinger
,
M.
,
Steinbauer
,
R.
,
Stojković
,
M.
, and
Vickers
,
J. A.
, “
A regularisation approach to causality theory for C1,1-Lorentzian metrics
,”
Gen. Relativ. Gravitation
46
(
8
),
1738
(
2014
).
26.
Kunzinger
,
M.
,
Steinbauer
,
R.
,
Stojković
,
M.
, and
Vickers
,
J. A.
, “
Hawking’s singularity theorem for C1,1-metrics
,”
Classical Quantum Gravity
32
,
075012
(
2015
).
27.
Kunzinger
,
M.
,
Steinbauer
,
R.
, and
Vickers
,
J. A.
, “
The Penrose singularity theorem in regularity C1,1
,”
Classical Quantum Gravity
32
(
15
),
155010
(
2015
).
28.
LeFloch
,
P.
and
Madare
,
C.
, “
Definition and stability of Lorentzian manifolds with distributional curvature
,”
Port. Math.
64
,
535
573
(
2007
).
29.
Marsden
,
J. E.
, “
Generalized Hamiltonian mechanics
,”
Arch. Ration. Mech. Anal.
28
(
4
),
323
361
(
1968
).
30.
Oberguggenberger
,
M.
,
Multiplication of Distributions and Applications to Partial Differential Equations
, Volume 259 of Pitman Research Notes in Mathematics Series (
Longman Scientific & Technical
,
Harlow
,
1992
), copublished in the United States with John Wiley & Sons, Inc., New York.
31.
Penrose
,
R.
, “
Gravitational collapse and space-time singularities
,”
Phys. Rev. Lett.
14
,
57
59
(
1965
).
32.
Penrose
,
R.
, “
The geometry of impulsive gravitational waves
,” in
General Relativity: Papers in Honour of J. L. Synge
(
Clarendon Press
,
Oxford
,
1972
), pp.
101
115
.
33.
Podolský
,
J.
, “
Non-expanding impulsive gravitational waves
,”
Classical Quantum Gravity
15
,
3229
3239
(
1998
).
34.
Podolský
,
J.
, “
Exact impulsive gravitational waves in spacetimes of constant curvature
,” in
Gravitation: Following the Prague Inspiration
(
World Scientific
,
Singapore
,
2002
), pp.
205
246
.
35.
Podolský
,
J.
and
Griffiths
,
J. B.
, “
Impulsive waves in de Sitter and anti-de Sitter space-times generated by null particles with an arbitrary multipole structure
,”
Classical Quantum Gravity
15
,
453
463
(
1998
).
36.
Podolský
,
J.
and
Griffiths
,
J. B.
, “
Nonexpanding impulsive gravitational waves with an arbitrary cosmological constant
,”
Phys. Lett. A
261
,
1
4
(
1999
).
37.
Podolský
,
J.
and
Ortaggio
,
M.
, “
Symmetries and geodesics in (anti-)de Sitter spacetimes with non-expanding impulsive gravitational waves
,”
Classical Quantum Gravity
18
,
2689
2706
(
2001
).
38.
Podolský
,
J.
,
Sämann
,
C.
,
Steinbauer
,
R.
, and
Švarc
,
R.
, “
The global existence, uniqueness and C1-regularity of geodesics in nonexpanding impulsive gravitational waves
,”
Classical Quantum Gravity
32
(
2
),
025003
(
2015
).
39.
Podolský
,
J.
and
Vesely
,
K.
, “
Continuous coordinates for all impulsive pp-waves
,”
Phys. Lett. A
241
,
145
147
(
1998
).
40.
Sämann
,
C.
and
Steinbauer
,
R.
, “
On the completeness of impulsive gravitational wave spacetimes
,”
Classical Quantum Gravity
29
(
24
),
245011, 11
(
2012
).
41.
Sämann
,
C.
and
Steinbauer
,
R.
, “
Geodesic completeness of generalized space-times
,” in
Pseudo-Differential Operators and Generalized Functions
, Volume 245 of Operator Theory Advances and Applications (
Birkhäuser/Springer
,
Cham
,
2015
), pp.
243
253
.
42.
Sämann
,
C.
,
Steinbauer
,
R.
,
Lecke
,
A.
, and
Podolský
,
J.
, “
Geodesics in nonexpanding impulsive gravitational waves with Λ. Part I
,”
Classical Quantum Gravity
33
(
11
),
115002
(
2016
).
43.
Sämann
,
C.
,
Steinbauer
,
R.
, and
Švarc
,
R.
, “
Completeness of general pp-wave spacetimes and their impulsive limit
,”
Classical Quantum Gravity
33
(
21
),
215006
(
2016
).
44.
Sfetsos
,
K.
, “
On gravitational shock waves in curved spacetimes
,”
Nucl. Phys. B
436
,
721
746
(
1995
).
45.
Steinbauer
,
R.
, “
Geodesics and geodesic deviation for impulsive gravitational waves
,”
J. Math. Phys.
39
(
4
),
2201
2212
(
1998
).
46.
Steinbauer
,
R.
, “
A note on distributional semi-Riemannian geometry
,”
Novi Sad J. Math.
38
(
3
),
189
199
(
2008
).
47.
Steinbauer
,
R.
and
Vickers
,
J. A.
, “
On the Geroch–Traschen class of metrics
,”
Classical Quantum Gravity
26
,
065001
(
2009
).
48.
Weissinger
,
J.
, “
Zur theorie und anwendung des iterationsverfahrens
,”
Math. Nachr.
8
,
193
212
(
1952
).
You do not currently have access to this content.