VoLUME 39,NUMBER 22
PHYSICAL
REVIEW
LETTERS
28NovEMBER 1977 the perturbation from thefirst
solvent layer.These calculations then suggest that the
vibration-al dephasing
is
mainly determined by the averageforce
field that the excited molecules experience from the fluctuating number ofnearest neighbors.Also in agreement with this picture
is
theincrese
in the correlation time (Table I)with increasingmolecular interaction.
"
The investigations were supported in part by the Netherlands Foundation for Chemical
Re-search
(SON) with financial aid from the Nether-lands Organization for the Advancement of Pure Research (ZWO).~D. von der Linde, A. Laubereau, and W.Kaiser, Phys. Bev. Lett. 26, 954 (1971).
S.
F.
Fisher and A. Laubereau, Chem. Phys. Lett. 85, 6 (1975).3A.Laubereau, Chem. Phys. Lett. 27, 600(1974). A. Laubereau, G. Wochner, and W.Kaiser, Phys.
Bev.A 18, 2212 (1976).
'D.
J.
Diestler, Chem. Phys. Lett. 99, 99(1976).6D. W. Oxtoby and
S.
A.Rice, Chem. Phys. Lett. 42, 1(1976).VW. G.Rothschild,
J.
Chem. Phys. 65, 2958(1976). "A.Rahman, Phys. Rev. 186,405 (1964).9J.A.Giordmaine and W. Kaiser, Phys. Rev. 144, 676(1966).
OW.
F.
Calaway and G.E.
Ewing, Chem. Phys. Lett. 80, 485(1975).Y.le Duff,
J.
Chem. Phys. 59, 1984(1978).~ M. Scotto,
J.
Chem. Phys. 49, 5862 (1968).
i3
B.
Kobo, in Fluctuations, Relaxation and Resonance in Magnetic Systems, edited by D. Ter Haar (Plenum, New York, 1962), p. 27.
S.Bratos and
E.
Marechal, Phys. Bev.A4,
1078(1971).
'P.
S.
Y.Cheung andJ.
G.Powless, Mol. Phys. 80, 921 (1975).B.
Quentrec and C. Brot, Phys. Rev. A 12, 272(1975).
K.Toukubo and K.Nakanishi,
J.
Chem. Phys. 65, 19S7(1976).Colhding Impulsive Gravitational Waves
Y.Nutku
Joseph IIenry Laboratories, Princeton University, Princeton, New Jersey 08540, and' Department ofPhysics, Middle East Technical Unioersity, Ankara, Turkey
and M. Halil
Department of Physics, Middle East Technical University, Ankara, Turkey (Received 18July 1977)
We formulate the problem ofcolliding plane gravitational waves with two polarizations asthe harmonic mappings of Riemannian manifolds and construct an exact solution of the vacuum Einstein field equations describing the interaction ofcolliding impulsive gravita-tional waves which inthe limit of collinear polarization reduces to the solution ofKhan
and. Penrose.
In this
Letter
we adopt an approach to theEin-stein field equations of gravitation which
is
basedon the theory of
Eells
and Sampson of harmonic mappings ofRiemannian manifolds.'
We hadearlier
pointed out the connection bebveen theseproblems"
andit
seems worthwile toremarkthat the theory of harmonic mappings of Riemann-ian manifolds
is
also applicable toa
wide varietyof problems in other branches ofphysics. In
particular, we can readily recognize the Nambu
string, solitons, nonlinear 0 model, and the Heisenberg ferromanget in the expression of
Eells
and Sampsonfor
their invariant functionals of the mapping. The formulation ofa
problem interms ofharmonic mappings provides us with
a
powerful formalismfor
the discussion ofa
num-ber
of questions. ranging from the constructionof exact solutions to considerations of topology
related to the index of the mapping, and
its
ad-vantage
lies
in thedirect
geometrical insightit
brings into the problem. We shall now present
a
new exact solution of the vacuum Einstein
equa-tions which
is
of physical interest and which wasobtained by the use of these techniques.
Penrose has introduced the notion of impulsive gravitational waves where space-time
is
flateverywhere except along
a
hypersurface with the Riemann tensor sufferinga
&-function discontinui-tyat
thissurface.
Here we shall be concernedwith the
case
where the discontinuity surfaceis
a
null plane and the impulsive wave
is
thena
famili-ar
ppwave.
sFor
p-urely impulsive gravitationalVOLUME 39,NUMBER 22
PHYSICAL
RE
VIE%'
LETTERS
28NOVEMBER 1977 waves the surface of discontinuityis
shear-free;
however, when two impulsive waves collide they produce shear and the mutual focusing and back-scattering of these waves
results
ina
space-timesingularity. This picture emerges from the exact
solution of Khan and Penrose' which describes
colliding impulsive plane gravitational waves with
collinear polarization.
It
illustrates the necessityof working with
exact
solutions of the Einsteinfield equations since,
as
Szekeres"'
has shown,arbitrarily weak incoming gravitational waves inevitably produce
a
space-time singularity. Inthis
Letter
we shall present an exact solution of the vacuum Einstein field equations which enablesus to study the geometry of space-time resulting
from the collision of bvo linearly polarized impul-sive waves with arbitrary relative polarization.
We find that in the interaction region the solution
is
no longer linearly polarized and the focusingproperties' of the waves
result inthe
development!
of an angular-momentum
as
wellas
a
massas-H(x',
y') =—,(x"
-y")
cosn +x'y'sina,
(2)where &
is
a
constant which measures the angle ofpolarization. InEq.
(2) H has been scaled bya
constant factor to set the amplitude of the wave equal to unity. This can be accomplished without.loss
of generality since such a constant can al-ways berestored
bya
coordinate transformation.The coordinate patch used inwriting the metric (1)has the advantage of manifestly displaying the Minkowski form of the metric on both sides of the discontinuity surface, but for reasons which
are
discussed in detail by Szekeresit
is
inap-propriate for the colliding waves problem.
For
this purpose we transform to the Bosen form
pect,
but once again the collision producesa
spacelike curvature singularity onthe space-time
manif old.
Impulsive gravitational plane waves
are
des-cribed by the metric for P-P wavesds =2du'dv'-dx'2 —dy'2-2H(x', y')5(u')du', (1)
ds'=2e"
dudv-e
u(e coshWdx'—
2sinhWdxdy+e"coshWdy'),
which has two mutually nonorthogonal Killing
vectors
$„and
$,
and where the metric coefficientsare
C'-differentiable functions ofthe null coordinatesu,
n. The transformation which accomplishes this taskis
given byx = [1.+ue(u) coso.]x+[uS(u)sino.
]y,
y' =[1-u8(u)cosn]y
+[u8(u)sinn]x,M =Q7
v' =v +[~z(cosa)(x'
-y')
+(sinn)xy] 0 (u)+a(x'+y')u&(u),(4)
where 0
is
the Heaviside unit step function. Nowit
can be readily verified that the metric (1)is
in the form of Eq. (3)witheU1p2
2v (1 +2p cosQ+p )
(1 —2pcoso.+p')
'
sinhW= —2p sinn/(1 —
p'),
M=0,
with the conventional definition p=u6(u).
In the region IIthe metric
for
the incominggravi-tational wave will be given by
Eqs.
(3)and(5).
There will be
a
similar expression for the metricin region III ifwe replace all the
p's
inEq.
(5)with
q =v8(v),
and the polarization angle & by another angle which will be called
P.
In this situation the waves in regions IIand IIIare
linearly polarized buttheir polarizations differ by the angle u —
P.
Thecollision problem
is
then formulatedas
follows:We consider two gravitational waves traveling in
+z and
-~
directions. Before the collision, the region between them (I)is
aMinkowski spaceand for impulsive waves region IIju&0, v&Oj and region III
(u&0, v&0j are
flatas
well. The junction conditions between regions IandII,
and between I andIII, are
the usual impulsive wave conditions. In region IV the two waves collidehead-on and their interaction
is
determined by the Einstein field equations for the metric (3)VoLUME )9,NUMBER 22
PHYSICAL
REVIEW LETTERS
28NovEMBER 1977 with the requirement that the solution reduces tothe known solutions in regions II and III
[cf.
Eq.
(5)],as
wellas
satisfying the necessary junction conditionsacross
the various regions. TheRicci
tensor will be zero throughout. The Einstein field equations for this problem
are
well known, but we shall now presenta
new formulation ofthese equations using the theory of harmonic map-pings of Riemannian manifolds which brings an element of simplicity into the problem.
We consider two Riemannian manifolds (M, g) and (M',
g')
with amappingf:M -M'.
Aninvari-ant functional of this mapping
is
thetrace
withre-spect tothe metric
g
of the induced metricf*g'
on
M.
The integral of this quantity over thevol-ume of
I
is
the Eells-Sampson energy functionalE(f).
In local coordinatesit is
given byAgB
E(f)
= "g (8)and those maps
for
which thefirst
variationvan-ishes,
5E(f)
=0,
are
called harmonic. The Einstein field equationsfor the metric in
Eq.
(3)are
obtainedas
harmon-ic
maps ifwe consider the following bvo Riemann-ian manifolds:Let
I
bea
flat two-dimensional manifold with the metricds2
=2'
dvcosh+ =
cosh'
coshW,cosv =(1+sinh'Vcoth'W)
'
',
(12)
and pass on to Klein's representation of the space
of constant curvature by defining
Riemannian manifolds
is
essentiallya
"mini-su-perspace"'
approach; indeed,it
grew out ofmany stimulating conversations with
Professor
C.
W.Misner.It
is
helpful in constructing exactsolutions because the requirement that the map-ping
f:
M-M'
be harmonicis
an invariant state-ment whichis
not affected by any choice of coordi-nates on MandI'.
Thus by performingcoordi-nate transformations onMand M' we can obtain
all possible forms of the Einstein field equations which
respect
the choices madeearlier
about the Killing directions in the space-timemani-fold. They will furthermore be consistent with the freedom inherited from general covariance
and the
arbitrariness
available in choosing any Ansatz about the form of the nonvanishing metriccoefficients in the original space-time
metric.
It
is
therefore natural to look for coordinatetrans-formations onM and M' such that the partial dif-ferential equations resulting from the
require-ment of harmonic mapping
are
simple. To thisend we
first
note that thelast
two terms inEq.
(1) describea
space of constant negative curvature but the line elementis
not in the canonical form.So we transform tonew coordinates v,
~
such thatand consider M' to be endowed with the metric g
=e"
tanh &&. (13)ds'
=e (2dMdU+dU-dW
—cosh2W dV).
(11)It
should be noted tha,t
themetrics
(10)and (11)are
given knownmetrics,
and even though wewrote them using particular coordinate patches,
it
is
evident that their significanceis
not tied to any coordinate system. Now, ifwe form theen-ergy functional (8) using the
metrics
(10)and (11),we obtain the Hilbert action principle specialized to the metric (3)and the condition for the map-ping tobe harmonic
is
equivalent to satisfactionof the Einstein field equations.
The formulation of the Einstein field equations
a,
s
the problem of the harmonic mappings of[
On
I,
it
is
convenient to choose new coordinatesby the transformation
T=g
(I
v2)~~2 yv(I g~)&~2v=8(l
—
v )v(l
Q )(14)
which up to
a
conformalfactor
brings the lineelement (10) to a form similar to the standard
expression in prolate spheroidal coordinates.
"
The conformal factor itself
is
irrelevant becauseit
washes out of the energy functional (8) when Mis
iwo dimensional. In the variables (13)and thecoordinates (14)the solution
is
very straight-forward. The space-time metricfor
two colliding impulsive wavesis
then given by=2
1-kk
dudv t —[(1——k)dx +i(1+k)dy] [(1-k)dx —i(1
+k)dy],t~so
1-kk
VoLUME 39,NUMBER 22
PH
YSI
CAL
RE
VIE%'
LETTER
S 28NovEMBER 1977 wherebars
denote complex conjugation and wehave used the definitions
t
=1-p
—
q2, r2=1,-pa,
to2=1—
q2, k=e'"pu+e'aqr.
We note that in regions IIand III
Eg.
(15) reducestothe desired form
[cf.
Egs.
(5)j and for o.'=p
=0the
results
of Ref. 6are
obtained. The solutionhas
a
curvature singularity on the open interval$t'=0;u&0, o&0).
We have also found new exactsolutions
for
colliding gravitational waves which in the limit of collinear polarization reduce to theSzekeres family.
'
These will be published else-where,.
This work was done mostly while we were
visit-ing the University of
Texas
at Austin. Weare
grateful toProfessor
J.
A. %'heelerfor
advice,encouragement, and his generous hospitality. We
are
also grateful toProfessor
S.
Chandrasekharfor his kind interest in this work. We thank
Dr. R.
Guvenfor
many interesting conversations. This work was supported bya
NATO ResearchGrant No. 1002and
a
National Science Foundation Grant No. PHV-76-82662.~J.Eells,
Jr.
, andJ.
H.Sampson, Am.J.
Math. 86, 109(1964).Y.Nutku, Ann. Inst. Henri Poincare A21, 175{1974).
A.Eris and Y.Nutku,
J.
Math, Phys. (N.Y.) 16,141(1975);A.
Eris,
J.
Math. Phys. (N.Y.) 18,824 (1977).4
B.
Penrose, in General Relativity; I'aPet'sin Honor
of
J;
L.Synge, edited by0
Baifeartaigh (Oxford Vaiv Press, Oxford, 1972).5J. Ehlers and W.Kundt, in Gravitation, An Introduc-tion to Cun"ent Research, edited by
L.
Witten (Wiley, New York, 1962).6K.Khan and
B.
Penrose, Nature (London) 229, 185(1971).
P.
Szekeres, Nature (London) 228, 1188 (1970).P.
Szekeres,J.
Math. Phys. (N.Y.) 18,286 (1972).9
B.
Penrose,in I'exsPectives in Geometry and
Rela-tivity, edited by
B.
Hoffmann (Indiana Univ. Press, Bloomington, 1966).C.W.Misner, in Magic without Magic: John A~chi-bald iVbeeler; A Collection of Essays in Honor ofHis 60th Birthday, edited by
J.
B.
IQauder (Freeman, SanFrancisco, 1972).
Ageometrical picture oftransformations (14) emerg-es ifwe consider asquare which isto be covered by ellipses with principal axes lying along the diagonals and tangent tothe boundary ofthe square. The interac-tion region is one quarter ofthis square where the per-pendicular bisector ofthe sides are the u,v axes and the loci ofconstant v or oare portions ofthese ellipses. For g= 0and T= 0we obtain two orthogonal straight lines but in the limit of 0 1and
T-
1both sets of el-lipses degenerate into the circle inscribed by the square and this coincides with the eventual singularityofthe solution.
M. Halil, thesis, Middle East Technical University, 1977(unpublished).
Solitonlike Properties
of
Heat
PulsesT.
Schneider andE.
StollIBM Zurich Research Laboratory, 8808Ruschlikon, Switzerland
Y.
HiwatariSeminar fiir Tbeoretische Physi@, Eidgenossische Tecbniscbe Hocbscbule Honggerberg— ,
8098Zurich, Switzer Eand
(Beceived 21June 1977)
Using the molecular-dynamics technique, we establish solitonlike properties of heat pulses in a lattice-dynamic model when the ambient temperature is in the second-sound
regime.
Inthis
Letter
we report on molecular-dynamicsresults in which we have observed propagation
ofheat pulses under conditions in which soliton-like properties
are
established. Recently, con-siderable effort has been devoted to the study of strong anharmonicity, anda
variety ofinterest-ing effects
are
predicted,'
'
including optic-modesecond sound.
'
Our present study
is
performed ina
regimewhere the equilibrium system exhibits second sound, and we use the heat-pulse method. We
consider an incompressible model system, where
acoustic modes
are
absent. Itbelongs to thefam-ily ofmodels which have been used with
remark-able
success
to elucidate thecritical
properties associated with distortive phase transitions.'
In this