A NNALES DE L ’I. H. P., SECTION A
W. G. A NDERSON
R. G. M C L ENAGHAN F. D. S ASSE
Huygens’ principle for the non-self-adjoint scalar wave equation on Petrov type III space-times
Annales de l’I. H. P., section A, tome 70, n
o3 (1999), p. 259-276
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259
Huygens’ principle for the non-self-adjoint scalar
waveequation
onPetrov type III space-times
W. G. ANDERSON
R. G. McLENAGHAN
F. D. SASSE
Department of Physics, University of Wisconsin, Milwaukee, WI, U.S.A.
E-mail: warren@ricci.phys.uwm.edu
Department of Applied Mathematics,
University of Waterloo Waterloo, Ontario N2L 3G1, Canada E-mail: rgmclena@daisy.uwaterloo.ca
Department of Mathematics, Centre for Technological
Sciences-UDESC Joinville 89223-100, Santa Catarina, Brazil.
Ann. Inst. Henri Poincaré,
Vol. 70, n° 3, 1999, Physique theorique
ABSTRACT. - We prove that if the
non-self-adjoint
scalar waveequation
satisfies
Huygens’ principle
on Petrovtype
IIIspace-times,
then it isequivalent
to theconformally
invariant scalar waveequation.
@Elsevier,
Paris
Key words : Huygens’
principle,
scalar wave equation, Petrov type III, space-times.RESUME. - On demontre que la validite du
principe
deHuygens
pour Fequation
des ondes scalairesnon-auto-adjointe
sur unespace-temps general
de
type
III de Petrovimplique
que1’ equation
estequivalente
a1’ equation
invariante conforme des ondes scalaires. @
Elsevier,
ParisAnnales de l’Institut Henri Poincaré - Physique theorique - 0246-0211
Vol. 70/99/03/@ Elsevier, Paris
260 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
1.
INTRODUCTION
In this paper we consider the
general
linear second-orderhyperbolic equation
on a four-dimensional curvedspace-time 114,
with coefficients:where
gab
is a contravariantpseudo-Riemannian
metric withsignature (+ - --)
onY4, u
is the unknown scalarfunction,
denotes the covariant derivative withrespect
to the Levi-Civitaconnection,
Aa is avector field and C is a scalar field in
114.
All considerations are restrictedto a
geodesically
convex domain. Theequation ( 1 )
is also called thenon-self-adjoint
scalar waveequation.
Huygens’ principle
is said to be valid for anequation
of the form( 1 )
if andonly
if for everyCauchy
initialproblem,
and for eachpoint
xo E
V4,
the solutiondepends only
on theCauchy
data in anarbitrarily
small
neighbourhood
of S nC-(xo),
where ,5 denotes the initial surface andC- (xo )
thepast
null conoid of Suchequations
are calledHuygens’
equations.
Necessary
conditions for thevalidity
ofHuygens’ principle
for( 1 )
aregiven by
Annales de l’Institut Henri Poincaré - Physique theorique
261
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES
where
In the
expressions
aboveRabcd
denotes the Riemann tensor,Cabcd
theWeyl
tensor,Rab
:==gcdRcabd,
the Ricci tensor, and R :=gabRab
theRicci scalar associated to the metric and
Aa
:==gabAb.
Conditions I-IVwere
obtained by
Gunther[7],
condition V was derivedby
Wunsch[ 18]
for the
self-adjoint
case andMcLenaghan [8]
for thenon-self-adjoint
case.Condition VI was obtained
by
Anderson andMcLenaghan [8].
We shalluse here the
two-component spinor
formalism of Penrose[ 14]
and thespin-
coefficient formalism of Newman and Penrose
[ 13], [ 15],
whose conventionswe follow. The
spinor equivalents
of the tensors(8), (9), ( 10), ( 11 )
aregiven by:
where
03A8ABCD
= is theWeyl spinor,
A ==(1/24)R,
is the Maxwell
spinor,
and03A6ABAB = 03A6(AB)(AB) = 03A6ABAB is
the trace-freeRicci
spin or.
We can set up, at each
point
ofspace-time,
adyad
basissatisfying
the relationoAlA
= 1. TheNP-components
of anyspinor
aredefined
by projecting
thespinor
into the the local basis{oA,
For thecurvature
spinors
and Maxwellspinor
we havewhere GABCD == etc.
Vol. 70, n° 3-1999.
262 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
The covariant derivatives of the
dyad
basisspinors
aregiven
in termsof the
NP-spin
coefficientsby
where
The necessary conditions I-VI can be
expressed
in terms ofdyad
components, containing only
the Newman-Penrose scalars. This conversionprocedure
consists in twosteps. Firstly,
we have to convert the tensorialexpressions
intospinor form, using ( 12)-( 15). Secondly,
thespinor equations
must be
expressed
in terms ofdyad components, using ( 16), ( 17), ( 18),
and
( 19).
Weperform
theselengthy
calculationsautomatically
with theNP sp i no r package [4], [6],
available in theMaple computer algebra system.
Using
the fact that ==0,
it can be shown that thespinor
formof condition II is
given by:
For condition III a direct
application
of thecorrespondence
relationsyields
Instead of
(22)
we shall use astronger
form of thiscondition,
obtainedby
Wünsch
[ 19]
andMcLenaghan
and Williams[ 11 ] :
While the
original
necessary condition(22)
isHermitian, (23)
iscomplex.
The conversion of the
remaining
conditions to therespective spinor
form and the determination of the
dyad components
is doneautomatically by defining templates
in theNPspinor package.
Thespinor
form of the trace-freesymmetric part
of a tensor is obtainedby taking
thecorrespondent spinor equivalent
of that tensor andsymmetrizing
withrespect
to all dotted and undotted indices[17].
McLenaghan
and Walton[ 10]
have shown that anynon-self-adjoint equation ( 1 )
on any Petrovtype
Nspace-time
satisfiesHuygens’ principle
Annales de Henri Poincaré - Physique theorique
263
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES
if and
only
if it isequivalent
to a scalarequation
withAa
= 0 and C =0,
on an
space-time corresponding
to the exactplane-wave
metric.In this paper we prove the
correspondent
result for Petrovtype
IIIspace-times.
In this case theWeyl spinor
has the form:where 0152A and
~3A
are theprincipal spinors.
If we choose thespin
basissuch that aA is
proportional
to thedyad
basisspinor
oA and,~ proportional
to ~, we obtain from
(16):
The
dyad
transformationwhere
w and q
arecomplex functions,
induces thefollowing
transformationon W 3 :
Thus,
we can choose the tetrad such thatW3
==-1,
so thatIn a recent paper,
Anderson, McLenaghan
and Walton[3]
haveproved
the
following
theorems:THEOREM 1. - The
validity of Huygens’ principle for
anynon-self-adjoint
scalar wave
equation ( 1 )
in any Petrovtype
IIIspace-time implies
that thespace-time
isconformally
related to one in which everyrepeated
null vectorfield of
theWeyl
tensorla,
is recurrent,i.e.,
THEOREM 2. - There exist no
non-self-adjoint Huygens’ equations ( 1 )
onany Petrov type Ill
space-time
’for
which thefollowing
$ conditions holdIn the next section we show that the restrictions
imposed
in Theorem 2 may be removed.Vol. 70, n° 3-1999.
264 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
2. MAIN THEOREM
The main result of this paper is
expressed by
the theorem:THEOREM 3 . -
If
anon-self-adjoint
scalar waveequation of
theform ( 1 ) satisfies Huygens’ principle
on any Petrovtype
IIIspace-time,
then it mustbe
equivalent
to aconformally
invariant scalar waveequation
Proof. -
We shall first prove,using
the necessary conditions II to VIgiven by (3)-(7),
that theassumption
~ 0 leads to a contradiction.In terms of the
dyad components
of the Maxwell tensor this is thesame as
proving
that the necessary conditionsimply ~o
=~1
=~2
= O*Finally
we invoke a lemmaby
Gunther[7]
that states that everyequation of the form (1) for
which - 0 is relatedby a
trivialtransformation
to one
for
which 0. It then follows from the necessary condition I(eq. (2))
that C =.R/6.
We use a notation for thedyad components
of the necessary conditions in the form where X is the Roman numeralcorresponding
to the necessarycondition,
a denotes the number of indicescorresponding
to thedyad spinor c
and b the number of dotted indicescorresponding
to thedyad spinor
6. We shall refer to the Newman-Penrose fieldequations using
the notationNP1, NP2,
etc. as listed in theAppendix.
The methods
employed
in thisproof
are similar to those used in[ 12] .
We start with the result obtained
by
Anderson andMcLenaghan [3], expressed
infollowing
lemma:LEMMA 1. - For the
non-self-adjoint
scalarequation of the form (1),
inPetrov type III
space-times,
the necessary conditionsII, III, IV,
V and VItogether
with theassumption
that the Maxwellspinor 03C6AB is
nonzero,imply
that there exists a
spinor dyad
and aconformal trans.f’ormation
such that
In what follows we shall use the relations
(32)
where necessary. It was shown in[3]
that~o
=~i
= O.Thus,
what remains to beproved
is thatthe
assumption ~2 7~ 0
leads to a contradiction.Let us assume
initially
thato;/?7r ~
0.Annales de l’Institut Henri Poincaré - Physique theorique
265
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES
From
H01, II00, III10, (NP6)
and(NP25 )
wehave, respectively,
By adding (NP22)
to thecomplex conjugate
of(NP23),
andsolving
forD$i2
weget
and
Subtracting (NP24)
from(NP29)
andsolving
for8P12
we obtainAdding (NP24)
to two times(NP29)
andsolving
forÐq>22
weget
By substituting (34)
intoIV10
we findUsing (41)
and(42), V20
andVI03
can be writtenrespectively
asEliminating 6(a
+27r)
from(43)
and(44)
we findThe denominator
-~22 + 4 in
theexpression
above must be nonzero, since~2
cannot be constant.Otherwise,
from(34),
we would have/3
= 0.Vol. 70, n° 3-1999.
266 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
In order
to determine
further side relations we still need to find the Pfaffians~/3,
and in terms From(NP6), (NP7), (NP8), (NP9)
and
(NP 12)
we haveUsing (38)
and(41),
we now evaluate the NP commutator[03B4,D]03A622 - [0394,D]03A612
to obtainFrom we
get
From
(NP26),
Substituting (52), (51 ), (36), (46), (47)
and(48)
into(50)
we haveNow, using (33), (34), (42), (47), (49)
and(53)
in the commutator~~, b~ ~2,
and
solving
forD,u
we obtainAnnales de l’Institut Henri Poincaré - Physique théorique
267
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES
The
explicit
form of Da can be determined from(43), (44)
and(46),
andis
given by:
Finally,
fromYl2
weget
We have now determined all the Pfaffians needed for
finding
new siderelations
using integrability
conditions. Beforeproceeding
we go back to(45)
and introduce a furthersimplification by expressing ~2
in terms ofSubstituting (57)
into the numerator of thecomplex conjugate
of(45),
we find
Another side relation can be determined from the NP commutator
[~](~+27r):
(-820~$n
+957r~n
+2680a3a - 5448o~ii - 102367r~n
+
5360~7f - 9648~/? - 38016~/3~ - 13926~~ - 2508~~
-
3816/3o~ - 720~~
+61628a~2~
+40128~r3~r
+200647r~
+
308147r~
+157527r~a - 38376~/3
+315047r~7f
-
14592~7r~a - 26887rc~7? - 15087ra$n)/(l97r
+10a)
= 0.(59)
It follows from
(57)
that the numerator 19 Tr+10 a in thepreceding equation
must be non-zero.
Solving
thisequation
for~n
we obtainVol. 70, nO 3-1999.
268 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
where,
for now, we assume that the denominator in theexpression above,
is non-zero.
Evaluating 03B403C62
+ 203C6203B2 = 0(cf. Eq. (38)), using (57),
andsolving for 03A611
we findwhere we assume, for now, that the denominator of
(62), given by
is nonzero.
Evaluating
the commutators[8, 8]:8
and[8, 8]4>2’
andsolving
each one for bc~ we
get, respectively,
where we assume, for the moment, that the denominators of
(64)
and(65), given by
are nonzero.
By subtracting (64)
from(65)
andsolving
for$n
we haveAnnales de Henri Poincaré - Physique theorique
269
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES
where the denominator of
(68),
is assumed to be nonzero for the moment.
Subtracting (60)
from(62)
andtaking
the numerator we findBy subtracting (62)
from(68)
andtaking
the numerator, we obtain a third side relation:We can eliminate 7r in the above
equations by defining
new variables xland x2
by
Vol. 70, n° 3-1999.
270 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
The side
relations
now assume the form(modulo
non-zerofactors)
In order to
study
thepossible
solutions of thepolynomial systems
that appear from the siderelations,
weapply
theprocedure gs o lve
which ispart
of theMaple package grobner [5].
Thisprocedure computes
acollection of reduced
lexicographic
Grobner basescorresponding
to a setof
polynomials.
Thesystem corresponding
to the set is first subdividedby
factorization. Then a variant ofBuchberger’s algorithm
which factors allintermediate
results isapplied
to eachsubsystem.
The result is a list of reducedsubsystems
whose roots are those of theoriginal system,
but whose variables have beensuccessively eliminated
andseparated
as far aspossible.
This means that instead oftrying
to find a Grobnerbasis,
thepackage attempts
to factor thepolynomials
that form thesystem
after eachstep
of the reductionalgorithm.
In thealgorithm,
the variables ~l, x2 and theircomplex conjugates, ~
and ~2, are treated asindependent
variables.In the
subsequent analysis
we use the fact thatthey
arecomplex conjugates
of each other.
Annales de l’Institut Henri Poincaré - Physique theorique
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES 271
Applying g s o lve
to the set ofequations
formedby 52
=0, 53
= 0(cf.
(75)
and(76)),
theircomplex conjugates,
and51
= 0(cf. (74)),
we find thethe
only possible
solution for which~1 ~
0 andx2 ~
0 isgiven by
By substituting (77)
into any of theprevious expressions
forI> 11
we findthat 1>11
== 0.Using
this and 1f = - a6 / 11
inV11
onegets
It is easy to
verify
that(78)
and the firstequation
in(77),
nowgiven
in theform
1089~ -
Q0152 ==0, imply
that a== ;3 ==
0.Let us consider first the cases in which each one of the denominators
dl, d2, d3, d4
andd5, given respectively by (61), (63), (66), (67)
and(69),
is zero.
(i) d 1
= 0From
(61)
wehave,
in terms of the variables xl and ~2:Since the numerator of
(60)
must alsovanish,
we haveApplying g s o lve
to the set ofequations consisting
of(79), (80),
theircomplex conjugates,
and81
= 0(cf.
eq.(74))
we find that allpossible
solutions
require
that x2 == 0.(ii) d2
= 0In this case, from
(63),
we haveThis
implies
that the numerator of(62)
mustvanish,
so we haveVol. 70, n° 3-1999.
272 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
Applying gs o lve
to the set ofequations consisting
of(81 ), (82),
theircomplex conjugates,
and51
= 0 we findagain
that all solutionsrequire
that ~2 = 0.
(ill) d3
= 0From
(66)
we now haveBy subtracting (83)
from itscomplex conjugate
we obtainSubtracting (83)
from(60)
andtaking
the numerator, weget
An additional side relation is obtained
by subtracting
thecomplex conjugate
of
(83)
from(62)
andtaking
the numerator of theresulting expression:
Applying grobner
to the setconsisting
ofE2,
itscomplex conjugate, E1, E3
and81
we find that thissystem
admits no solution.(iv) d4
= 0When the denominator of
(65), given by (67),
is zero, its numeratormust be zero,
implying
that a =/3(443/3). This,
on the other handimplies immediately,
from81
= 0(cf. (58)),
that/?
= 0.(v) ds
= 0When the denominator of
(68)
is zero, its numerator must be zero too.Thus,
weget
Applying gs o lve
to thepolynomial system
definedby
thesystem
ofpolynomials
definedby (87), (88),
theircomplex conjugates,
and81
=0,
we find that there are no
possible
solutions.Annales de Poincaré - Physique theorique
273
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES
The case a =
{3
== 7r =0,
which leads to Theorem2,
was consideredin
[3]
and results in a contradiction to thehypothesis ~2 ~
0 . The moregeneral
case,c~7r/?
==0,
also leads to a contradiction. Theproof,
found in[ 16],
is tedious butstraightforward,
and will not bepresented
here.Thus,
the necessary conditions I-VI for thevalidity
ofHuygens’ principle imply
thatwe must have
Hab A~a,b~ -
0. Thiscompletes
theproof
of Theorem 3.APPENDIX
In this
Appendix
wegive
the Newman-Penrose fieldequations
andcommutation relations referred in the paper. We note that many lists in the literature contain
typographic mistakes,
or use different conventionsBianchi identities
Vol. 70, nO 3-1999.
274 W. G. ANDERSON, R. G. McLENAGHAN AND F. D. SASSE
Ricci identities
Annales de Henri Poincare - Physique theorique
275
HUYGENS’ PRINCIPLE ON PETROV TYPE III SPACE-TIMES
NP commutation relations
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(Manuscript received on August 25th, 1997. )
Annales de Henri Poincaré - Physique theorique