In this article, HH spaces of type [N] ⊗ [N] with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic symmetries. The general form of the homothetic vector fields is found. New coordinates are introduced, which enable us to reduce the HH system of partial differential equations to one ordinary differential equation (ODE) on one holomorphic function. In a special case, this is a second-order ODE and its general solution is explicitly given. In the generic case, one gets rather involved fifth-order ODE.

1.
A.
Trautman
, “
Radiation and boundary conditions in the theory of gravitation
,”
Bull. Acad. Polon. Sci.
6
(
6
),
407
(
1958
).
2.
F.
Pirani
, “
Invariant formulation of gravitational radiation theory
,”
Phys. Rev.
105
,
1089
(
1957
).
3.
A.
Lichnerowicz
, “
Sur les ondes et radiations gravitationnelles
,”
C. R. Acad. Sci. Colon.
246
,
893
(
1958
).
4.
H.
Bondi
,
M. G. J.
van der Burg
, and
A. W. K.
Metzner
, “
Gravitational waves in general relativity. VII. Waves from axisymmetric isolated systems
,”
Proc. R. Soc. A
269
,
21
(
1962
).
5.
R. K.
Sachs
, “
Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time
,”
Proc. R. Soc. A
270
,
103
(
1962
).
6.
E. T.
Newman
and
R.
Penrose
, “
An approach to gravitational radiation by a method of spin coefficients
,”
J. Math. Phys.
3
,
566
(
1962
);
Erratum
, 4,
998
(
1963
).
7.
F. A. E.
Pirani
,
Introduction to Gravitational Radiation Theory
, Lecture in General Relativity Brandeis Summer Institute (
Prentice Hall, Inc.
,
Englewood Cliffs, NJ
,
1965
), Vol. 1, p.
249
.
8.
V. D.
Zakharov
,
Gravitational Waves in Einstein’s Theory
(
Halsted Press, a Division of John Wiley & Sons, Inc.
,
New York
,
1973
).
9.
H.
Stephani
,
D.
Kramer
,
M. A. H.
MacCallum
,
C.
Hoenselaers
, and
E.
Hertl
,
Exact Solutions to Einstein’s Field Equations
, 2nd ed. (
Cambridge University Press
,
Cambridge
,
2003
).
10.
C. D.
Hill
and
P.
Nurowski
, “
How the green light was given for gravitational wave search
,”
Not. Am. Math. Soc.
64
(
7
),
686
692
(
2017
).
11.
I.
Robinson
and
A.
Trautman
, “
Some spherical gravitational waves in general relativity
,”
Proc. R. Soc. A
265
,
463
(
1962
).
12.
J. F.
Plebański
,
Type N Solutions of Gμν = −ϱkμkν With Null kμ
(
Cinvestav
,
Mexico City
,
1978
).
13.
C. B. G.
McIntosh
, “
Symmetries of vacuum type-N metrics
,”
Classical Quantum Gravity
2
,
87
(
1985
).
14.
I.
Hauser
, “
Type N gravitational field with twist
,”
Phys. Rev. Lett.
33
,
1112
(
1974
).
15.
I.
Hauser
, “
Type N gravitational field with twist. II
,”
J. Math. Phys.
19
,
661
(
1978
).
16.
W. D.
Halford
, “
Petrov type N vacuum metrics and homothetic motions
,”
J. Math. Phys.
20
,
1115
(
1979
).
17.
C. D.
Collinson
, “
Homothetic motions and the Hauser metric
,”
J. Math. Phys.
21
,
2601
(
1980
).
18.
H.
Stephani
and
E.
Herlt
, “
Twisting type-N vacuum solutions with two non-commuting Killing vectors do exist
,”
Classical Quantum Gravity
2
,
L63
(
1985
).
19.
E.
Herlt
, “
Reduktion der einsteinschen feldgleichungen für type-N-vacuum-Lösungen mit Killingvector und homothetischler gruppe auf eine gewöhnliche differentialgleichung 3 ordnung für eine reelle funktion Wiss
,”
Z. Friedrich Schiller Univ. Jena Natur. Reihe
35
,
735
(
1986
).
20.
J. D.
Finley
 III
, Toward Real-Valued HH Spaces: Twisting Type N Gravitation and Geometry, A Volume in Honor of I. Robinson, edited by
W.
Rindler
and
A.
Trautman
(
Bibliopolis
,
Naples
,
1987
), p.
131
.
21.
F. J.
Chinea
, “
Twisting type-N vacuum gravitational fields with symmetries
,”
Phys. Rev. D
37
,
3080
(
1988
).
22.
E.
Herlt
, “
Reduction of Einstein’s field equations for twisting type-N vacuum fields with an H2
,”
Gen. Relativ. Gravitation
23
,
477
(
1991
).
23.
J. F.
Plebański
and
M.
Przanowski
, “
Comments on twisting type N vacuum equations
,”
Phys. Lett. A
152
,
257
(
1991
).
24.
J. D.
Finley
 III
and
J. F.
Plebański
, “
Equations for twisting, type-N, vacuum Einstein spaces without a need for Killing vectors
,”
J. Geom. Phys.
8
,
173
(
1992
).
25.
G.
Ludwig
and
Y. B.
Yu
, “
Type N twisting vacuum gravitational fields
,”
Gen. Relativ. Gravitation
24
,
93
(
1992
).
26.
H.
Stephani
, “
A note on the solutions of the diverging twisting, type N, vacuum field equations
,”
Classical Quantum Gravity
10
,
2187
(
1993
).
27.
J. D.
Finley
 III
and
A.
Price
, “
The involutive prolongation of the (complex) twisting, type-N vacuum field equations
,” in
Aspects of General Relativity and Mathematical Physics Proceedings of a Conference in Honor of Jerzy Plabański
, edited by
N.
Bretón
,
R.
Capovilla
, and
T.
Matos
(
CINVESTAV
,
Mexico City
,
1993
), p.
1
.
28.
J. D.
Finley
 III
,
J. F.
Plebański
, and
M.
Przanowski
, “
Third-order ODES for twisting type-N vacuum solutions
,”
Classical Quantum Gravity
11
,
157
(
1994
).
29.
J. D.
Finley
 III
,
J. F.
Plebański
, and
M.
Przanowski
, “
An iterative approach to twisting and diverging, type-N, vacuum Einstein equations: A (third-order) resolution of Stephani’s ‘paradox
,’”
Classical Quantum Gravity
14
,
489
(
1997
).
30.
S. B.
Edgar
and
G.
Ludwig
, “
Integration in the GHP formalism. II: An operator approach for spacetimes with Killing vectors, with application to twisting type N spaces
,”
Gen. Relativ. Gravitation
29
,
19
(
1997
).
31.
J.
Bičák
and
V.
Pravda
, “
Curvature invariants in type N spacetime
,”
Classical Quantum Gravity
15
,
1539
(
1998
).
32.
L.
Palacios
and
J. F.
Plebański
, “
A note on the iterative approach to twisting type N solutions
,” e-print arXiv:gr-gc/9805021 (
1998
).
33.
J. F.
Plebański
,
M.
Przanowski
, and
S.
Formański
, “
Linear superposition of two type-N nonlinear gravitons
,”
Phys. Lett. A
246
,
25
(
1998
).
34.
P.
MacAlevey
, “
Approximate solutions of Einstein’s vacuum field equations in the type N, twisting and diverging case
,”
Classical Quantum Gravity
16
,
2259
(
1999
).
35.
F. J.
Chinea
and
F.
Navarro-Lérida
, “
Twisting type-N vacuum fields with a group H2
,”
Classical Quantum Gravity
17
,
4587
(
2000
).
36.
G.
Ludwig
and
S. B.
Edgar
, “
A generalized Lie derivative and homothetic or Killing vectors in the Geroch-Held-Penrose formalism
,”
Classical Quantum Gravity
17
,
1683
(
2000
).
37.
P.
Nurowski
and
J. F.
Plebański
, “
Non-vacuum twisting type N metrics
,”
Classical Quantum Gravity
18
,
341
(
2001
).
38.
D.
Finley
, “
Equations for complex-valued, twisting, type-N, vacuum solutions with one or two Killing/homothetic vectors
,” e-print arXiv:gr-gc/0108055v1 (
2001
).
39.
J. F.
Plebański
and
M.
Przanowski
,
On Vacuum Twisting Type-N Again Revisiting the Foundations of Relativistic Physics
, Festschrift in Honor of John Stachel, edited by
A.
Ashtekar
, et al
(
Kluwer Academic Publishers
,
Dordrecht
,
2003
), p.
361
.
40.
M.
Przanowski
and
M. A. R.
Segura
, “
Diverging and twisting type N solutions of vacuum Einstein equations: An approximative approach
,”
Classical Quantum Gravity
23
,
761
(
2006
).
41.
P.
Nurowski
, “
Twisting type N vacuums with cosmological constant
,”
J. Geom. Phys.
58
,
615
(
2008
).
42.
A.
Chudecki
and
M.
Przanowski
, “
A simple example of type-[N][N]HH-spaces admitting twisting null geodesic congruence
,”
Classical Quantum Gravity
25
,
055010
(
2008
).
43.
X.
Zhang
and
D.
Finley
, “
Lower order ODEs to determine new twisting type N Einstein spaces via CR geometry
,”
Classical Quantum Gravity
29
,
065010
(
2012
).
44.
A.
Held
, “
On the type N twisting vacuum solution of the Einstein equations
,”
Gen. Relativ. Gravitation
49
,
127
(
2017
).
45.
J.
Leroy
, “
Un espace d’Einstein de type N à rayons non intégrables
,”
C. R. Acad. Sci. Paris A
270
,
1078
(
1970
).
46.
P.
Sommers
and
M.
Walker
, “
A note on Hauser’s type N gravitational field with twist
,”
J. Phys. A: Math. Gen.
9
,
357
(
1976
).
47.
J. F.
Plebański
and
I.
Robinson
, “
Left-degenerate vacuum metrics
,”
Phys. Rev. Lett.
37
,
493
(
1976
).
48.
J. D.
Finley
 III
and
J. F.
Plebański
, “
The intrinsic spinorial structure of hyperheavens
,”
J. Math. Phys.
17
,
2207
(
1976
).
49.
C. P.
Boyer
,
J. D.
Finley
 III
, and
J. F.
Plebański
, “
Complex general relativity, H and HH spaces: A survey of one approach
,” in
General Relativity and Gravitation
, Einstein Memorial Volume 2, edited by
A.
Held
(
Plenum
,
New York
,
1980
), p.
241
.
50.
K.
Rózga
, “
Real slices of complex spacetime in general relativity
,”
Rep. Math. Phys.
11
,
197
(
1977
).
51.
J. F.
Plebański
and
G. F.
Torres del Castillo
, “
.HH spaces with an algebraically degenerate right side
,”
J. Math. Phys.
23
,
1349
(
1982
).
52.
J. F.
Plebański
and
S.
Hacyan
, “
Null geodesic surfaces and Goldberg-Sachs theorem in complex Riemannian spaces
,”
J. Math. Phys.
16
,
2403
(
1975
).
53.
M.
Przanowski
and
J. F.
Plebański
, “
Generalized Goldberg-Sachs theorems in complex and real space-times. II
,”
Acta Phys. Polon. B
10
,
573
(
1979
).
54.
J. F.
Plebański
and
K.
Rózga
, “
The optics of null strings
,”
J. Math. Phys.
25
,
1930
(
1984
).
55.
A.
Chudecki
, “
On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces
,”
J. Math. Phys.
58
,
112502
(
2017
).
56.
R.
Sachs
, “
Gravitational waves in general relativity. VI. The outgoing radiation condition
,”
Proc. R. Soc. A
264
,
309
(
1961
).
57.
P.
Szekeres
, “
On the propagation of gravitational fields in matter
,”
J. Math. Phys.
7
,
751
(
1966
).
58.
J. F.
Plebański
,
Spinors, Tetrads and Forms Unpublished Monograph
(
Centro de IEA del IPN
,
Mexico City
,
1974
).
59.
W.
Ślebodziński
,
Exterior Forms and Their Applications
, Monografie Matematyczne Vol. 52 (
PWN-Polish Scientific Publishers
,
Warszawa
,
1970
).
60.
S.
Kobayashi
and
K.
Nomizu
, in
Foundations of Differential Geometry Volume II
, edited by
L.
Bers
,
R.
Courant
, and
J. J.
Stoker
(
John Wiley & Sons, Inc.
,
New York
,
1969
).
61.
D. B.
Fairlie
and
A. N.
Leznov
, “
General solutions of the Monge-Ampère equation in n-dimensional space
,”
J. Geom. Phys.
16
,
385
(
1995
).
62.
A.
Chudecki
, “
Homothetic Killing vectors in expanding HH-spaces with Λ
,”
Int. J. Geom. Methods Mod. Phys.
10
,
1250077
(
2013
).
63.
S. A.
Sonnleitner
and
J. D.
Finley
 III
, “
The form of Killing vectors in expanding HH spaces
,”
J. Math. Phys.
23
,
116
(
1982
).
64.
W. D.
Halford
and
R. P.
Kerr
, “
Einstein spaces and homothetic motions. I
,”
J. Math. Phys.
21
,
120
(
1980
).
65.
K.
Yano
, “
On groups of homothetic transformations in Riemannian spaces
,”
J. Indian Math. Soc.
15
,
105
(
1951
).
66.
A.
Chudecki
, “
Classification of the complex and real type [N] ⊗ [N] spaces
,” e-print arXiv:1804.02039 [gr-qc] (
2018
).
67.
G. W.
Gibbons
, “
Dark energy and Schwarzian derivative
,” e-print arXiv:1403.5431 [hep.th] (
2014
).
You do not currently have access to this content.