Interpreting spacetimes of any dimension using geodesic deviation

0.35 MB·English

by  Jiri Podolsky

Interpreting spacetimes of any dimension using geodesic deviation
Jiˇr´ı Podolsky´ and Robert Sˇvarc
∗ †
Institute of Theoretical Physics, Faculty of Mathematics and Physics,
Charles University in Prague, V Holeˇsoviˇck´ach 2, 180 00 Prague 8, Czech Republic
(Dated: January 30, 2012)
Wepresentageneralmethodwhichcanbeusedforgeometrical andphysicalinterpretationofan
arbitraryspacetimeinfouroranyhighernumberofdimensions. Itisbasedonthesystematicanalysis
ofrelativemotionoffreetestparticles. Wedemonstratethatlocaleffectofthegravitationalfieldon
particles,asdescribedbyequationofgeodesicdeviationwithrespecttoanaturalorthonormalframe,
can always be decomposed into a canonical set of transverse, longitudinal and Newton–Coulomb-
type components, isotropic influence of a cosmological constant, and contributions arising from
specificmattercontentoftheuniverse. Inparticular,exactgravitational wavesinEinstein’stheory
always exhibit themselves via purely transverse effects with D(D−3)/2 independent polarization
2
states. To illustrate the utility of this approach we study the family of pp-wave spacetimes in
1
higher dimensions and discuss specific measurable effects on a detector located in four spacetime
0
dimensions. For example, the corresponding deformations caused by a generic higher-dimensional
2
gravitational waves observed in such physical subspace, need not be tracefree.

n
a PACSnumbers: 04.50.-h,04.20.Jb,04.30.-w,04.30.Nk,04.40.Nr,98.80.Jk
J

7
2 I. INTRODUCTION 80]. This paved the way for a systematic study of wide
classes of algebraically special spacetimes in higher di-

c] In the last decade there has been a growing inter- mensions [81–84]. Investigation of asymptotic behaviour
q est in exact spacetimes within the context of higher- of the corresponding fields and their global structure, in
- dimensional General Relativity, primarily motivated by particular properties of gravitational radiation, has also
r
g finding particular models for string theories, AdS/CFT been initiated [85–99].
[ correspondence and brane-world cosmology. Such in- Nevertheless, in spite of the considerable effort de-

2 vestigations thus concentrated mainly on various types voted to this topic, there are still important aspects
v of black holes and black rings, see [1–8] for reviews concerning the nature of gravitational fields in higher-
0 and further references. More general static or station- dimensional gravity that remain open. Any sufficiently
9 ary axisymmetric [9–15], multi-black hole Majumdar– general method which could be used to probe geometri-
7 Papapetrou-type[16–23],andstaticsolutionswithcylin- calandphysicalpropertiesofagivenspacetimewouldbe
4
drical/toroidal symmetry [24–28] were also considered, useful. In the present work we suggest and develop such
.
1 including uniform and non-uniform black strings [29–36] an approach which is based on investigation and classi-
0 withthe aimto elucidate theirinstability [37–39]. Other fication of specific effects of gravity encoded in relative
2 important classes of higher-dimensional exact solutions motion of nearby test particles.
1
of Einstein’s equations have also been studied recently,
: In fact, in standard four-dimensional General Relativ-
v forexampleRobinson–TrautmanandKerr–Schildspace-
i times[40–45],extensionsoftheBertotti–Robinson,(anti- ity, this has long been used as an important tool for
X
studies of spacetimes. Relative motion of close free par-
)Nariai and Pleban´ski–Hacyan universes [46], higher-
r ticles helps us to clarify the structure of a gravitational
a dimensional FLRW-type [47–51] and multidimensional
field in which the test particles move. When they have
cosmologicalmodels[20,52](seealsoreferencestherein),
no charge and spin, this is mathematically described
specific solitons [24, 53, 54], or various exact gravita-
by the equation of geodesic deviation (sometimes also
tional waves — in particular those which belong to non-
calledtheJacobiequation)whichwasfirstderivedinthe
expandingKundtfamily[55,56],namelygeneralizedpp-
n-dimensional (pseudo-)Riemannian geometry by Levi-
waves[57–63](forastudyoftheircollisionssee[64]),VSI
Civita and Synge [100–103], see [104] for the historical
[62, 63] and CSI [65] spacetimes, or relativistic gyratons
account. Shortly after its application to Einstein’s grav-
[66–71].
itytheory[105–114]ithelped,forinstance,tounderstand
Fundamentalgeneralquestions concerningthe classifi-
the behaviour of test bodies influenced by gravitational
cation of higher-dimensional manifolds based on the al-
waves or the physical fate of observers falling into black
gebraic structure of the curvature tensor have been clar-
holes. Textbook descriptions of this equation, which is
ified [72–75], including generalizations of the Newman–
linear with respect to the separation vector connecting
Penrose and the Geroch–Held–Penrose formalisms [76–
thetestparticles,aregiven,e.g.,in[115–118]. Letusalso
mention that generalizations of the equation of geodesic
deviationtoadmitarbitraryrelativevelocitiesofthepar-
∗ [email protected]ff.cuni.cz ticles were obtained in the works [119–128]. Further ex-
† robert.svarc@mff.cuni.cz tensions, higher-order corrections to the geodesic devia-
2
tionequation,theirparticularapplicationsandreferences II. EQUATION OF GEODESIC DEVIATION
can be found in the recent papers [128–135] and in the
monograph [118].
Themainobjectiveofthepresentworkistoinvestigate
andcharacterizethe curvature of an arbitraryspacetime
In 1965Szekeres[114] presentedanelegantanalysisof
of dimension D 4 by its local effects on freely falling
the behaviour of nearby test particles in a generic four- ≥
test particles (observers). The gravitational field mani-
dimensional spacetime. He demonstrated that the over-
fests itself, in Newtonian terminology, as specific “tidal
all effect consists of specific transverse, longitudinal and
forces”whichcausethenearbyparticlestoacceleraterel-
Newton–Coulomb-type components. This was achieved
ative to each other. This leads to a deviation of corre-
by decomposing the Riemann curvature tensor into the
sponding geodesics whose separation thus changes with
WeyltensorandthetermsinvolvingtheRiccitensor(and
time: in variousspatialdirections the particles approach
Ricciscalar). Whiletheformerrepresentsthe“freegrav-
or recede from themselves, exhibiting thus the specific
itationalfield”thelattercanbeexplicitlyexpressed,em-
characterofthespacetimeinthevicinityofagivenevent.
ploying Einstein’s field equations, in terms of the cor-
In standard and also higher-dimensional General Rel-
responding components of the energy-momentum tensor
ativity, such a behaviour of free test particles (without
which describes the matter content. In order to further
charge and spin) is described by the geodesic deviation
analyze the Weyl tensor contribution, Szekeres used the
equation [100–118]
formalism of self-dual bivectors [136, 137] constructed
fromnull frames. Thisenabledhimto deducethe effects
D2Zµ
of gravitational fields on nearby test particles in space- =Rµ uαuβZν , (1)
dτ2 αβν
times of various Petrov types. When these results are
re-expressedin a more convenient Newman–Penrosefor- where Rµ are components of the Riemann curva-
αβν
malism [138, 139], explicit physical interpretation of the ture tensor, uα are components of the velocity vector
corresponding complex scalars ΨA are obtained. In par- u=uα∂ of the reference (fiducial) particle moving
α
ticular, the Weyl scalar Ψ4 (the only non-trivial compo- along a timelike geodesic γ(τ) x0(τ),...,xD 1(τ) ,
−
nent in type N spacetimes) represents purely transverse uα = dxα, the parameter τ is i≡ts{proper time (so tha}t
effectofexactgravitationalwaves,thescalarΨ (present, dτ
3 u u g uαuβ = 1), and Zµ are components of the
e.g., in type III spacetimes) is responsible for longitudi- sep·ara≡tionαβvector Z−=Zµ∂ which connects the refer-
nal effects, and Ψ (typical for spacetimes of type D) µ
2 ence particle with another nearby test particle moving
gives rise to Newton-like deformations of the family of
along a timelike geodesic γ¯(τ). The situation is visual-
test particles (see [140–142] for more details; inclusion
ized in figure 1.
ofanonvanishingcosmologicalconstantwasdescribedin
[129]).
It is the purpose of the present work to extend these
results to arbitrary spacetimes in any dimension D 4.
≥
The paper is organized as follows. In section 2 we recall
the equation of geodesic deviation, including its invari-
ant form with respect to the interpretation orthonormal
frame adapted to an observer. In section 3 we perform
the canonical decomposition of the curvature tensor us-
ingEinstein’sequationsandtherealWeyltensorcompo-
nents Ψ with respectto an associatednull frame. We
A...
thus derive an explicit and general form of the equation
of geodesic deviation. Section 4 analyses the character
of all canonical components of a gravitationalfield. Sec-
tion 5 is devoted to the discussion of uniqueness of the
FIG. 1. In a curved D-dimensional spacetime, nearby test
interpretation frame, and derivation of explicit relations
particles moving along geodesics accelerate toward or away
whichgivethedependenceofthefieldcomponentsonthe
fromeachother,asgivenbytheequationofgeodesicdeviation
observer’s velocity. In section 6 we describe the effect of (1). Here u is the velocity vector of a reference particle, and
pure radiation,perfectfluid andelectromagneticfield on Z is the separation vector which represents actual relative
test particles. Final section 7 illustrates the method on position of thesecond test particle at a given propertime τ.
the family of pp-waves in higher dimensions. There are
also3appendices: InappendixAwegiverelationstothe Equation (1) explicitly expresses the relative accelera-
standardcomplexformalismofD =4GeneralRelativity, tion of two nearby particles by the second absolute (co-
and in appendix B we summarize alternative notations variant) derivative of the vector field Z along γ(τ),
commonly used in literature on D 4 spacetimes. Fi-
≥
nally, in appendix C the Lorentz transformations of the D2Zµ
ΨA... scalars are presented. dτ2 = Zµ;γuγ ;δuδ =Zµ;γδuγuδ , (2)
(cid:0) (cid:1)
3
intermsofthelocalcurvaturetensorandtheactualrela- Due to the fact that u is parallelly transported, for the
tivepositionofthe particles,describedbytheseparation zerothframe-componentZ(0) e(0) Z = u Z weim-
vector Z(τ) at the time τ. mediately obtain ≡ · − ·
To be geometrically more precise, the two geodesics
d2Z(0) D2Zµ
should be understood as specific representatives of a = u = R uµuαuβZν =0, (4)
congruence γ(τ,z), i.e. smooth one-parameter fam- dτ2 − µ dτ2 − µαβν
ily of geodesics, such that γ(τ) γ(τ,z =0) and
≡ usingthe skew-symmetryofthe Riemanntensor. There-
γ¯(τ) γ(τ,z =const.). The proper time τ and the pa- fore, Z(0)(τ) must be at most a linear function of the
≡
rameter z which labels the geodesics can be chosen as
proper time. By a natural choice of initial conditions,
coordinates on the submanifold spanned by the congru-
consistent with the above construction of the geodesic
ence. Thus u=∂τ and Z =∂z, and the deviation vec- congruence γ(τ,z), we set Z(0) =0. The temporal com-
tor field Z is Lie-transported along the geodesics gener- ponent of Z thus vanishes and the test particles always
ated by u. Consider now the positions of two test par-
stay in the same spacelike hypersurfaces synchronized
ticles at a given time, for example (located at z =0)
by τ.
P
and (for which z =1, say) at τ =0, as shown in fig-
Q Physicalinformation about relative motion of the test
ure 1. Their coordinates are related by the exponen-
particlesisthuscompletelycontainedinthespatialframe
wtiaelsmetazp=xµQ1=toexlopc(aztZe)x.µPIgfetnheerahtiegdhebry-oZrdearttPer,mwshaerree componentsZ(i)(τ)≡e(i)·Z oftheseparationvectorZ.
Thesedeterminetheactualrelativespatialposition ofthe
negligiblethisexpressioQnreducestoxµ xµ (Zxµ) ,
demonstratingthattheseparationvecQto−rZPde≈scribesthPe ttwioonneqeaurabtyionpa(r1t)icolenst.oBey(i)p=roejectiwnegotbhteagineodesic devia-
relativepositionofthe twotestparticles,andZ(τ) gives (i)
itsevolutionthatisobtainedbysolvingtheequation(1). Z¨(i) =R(i) Z(j) , (5)
Suchlinearapproximationimproveswhenthesecondtest (0)(0)(j)
particle moves very close to the reference one, i.e. along
where i,j=1,2,...,D 1, and we denote the physical
thegeodesicz =const. 1,inwhichcasetheseparation −
is described by the vect≪or field zZ(τ). relative acceleration as
Itshouldalsoberecalledthattheequationofgeodesic Z¨(i) e(i) D2Z =e(i) D2Zµ . (6)
deviation (1) is linear with respect to the components ≡ · dτ2 µ dτ2
of the separation vector, neglecting higher-order terms
The frame components of the Riemann tensor are
in the Taylor expansion of exact expression for relative R R eµ uαuβeν . Let us note thatPirani
acceleration of free test particles. It can thus be used (i)(0)(0)(j) ≡ µαβν (i) (j)
[105,106]labeled,inD =4,theframecomponentsofthe
when the relative velocities of the particles are negligi-
“tidal stress tensor” that occurs in equation (5) (with
ble, i.e. their geodesics are almost parallel. Generaliza-
an opposite sign) as Ka Ra =Ra ucud. They
tionsofequation(1)toadmitarbitraryrelativevelocities b 0b0 cbd
≡
are equivalent to the electric part of the Riemann tensor
were obtained and applied in the works [119–124, 126–
R =R ucud, see [118].
128]. Further extensions, higher-order corrections to the ab a0b0 acbd
E ≡
FollowingPiraniitisalsousuallyassumedthattheor-
geodesic deviation equation and their specific applica-
thonormal frame e is parallelly propagatedalong the
tions can be found in [130–135] (for reviews and other a
{ }
referencegeodesic. However,inourworkwedonotmake
referencessee[118,128,131,135]). Ouraim,however,in
suchanassumption. Infact,asakeyideaoftheproposed
thispaperistoinvestigatelocalrelativemotionofnearby
interpretation method, we align the orthonormal frame
free test particles which are initially at rest with respect
with the algebraic structure of a given spacetime instead
toeachother. Forsuchananalysis,theclassicalgeodesic
(see also section V). This makes the investigation of its
deviation equation (1) will be fully sufficient.
physical properties much easier.
Now, in order to obtain invariant results independent
of the choice of coordinates, it is natural to adopt the
Pirani approach [105, 106] based on the use of compo-
III. CANONICAL DECOMPOSITION OF THE
nents of the above quantities with respect to a suitable
CURVATURE TENSOR
orthonormal frame e . At any point of the reference
a
{ }
geodesic this defines an observer’s framework in which
Next step is to express the frame components of the
physicalmeasurementsaremadeandinterpreted. Inpar-
ticular, the separationvector is expressed as Z =Zae . Riemann tensor R(i)(0)(0)(j). Using the standard decom-
a
position of the curvature tensor into the traceless Weyl
The timelike vector of the frame is identified with the
velocity vector of the observer, e =u, and e , where tensorCabcd andspecificcombinationsoftheRiccitensor
(0) (i) R and Ricci scalar R,
i=1,2,...,D 1, are perpendicular spacelike unit vec- ab
−
torswhichformitslocalCartesianbasisinthe hypersur- 2
face orthogonal to u (see also figure 2), Rabcd =Cabcd+ D 2 ga[cRd]b−gb[cRd]a ,
−
2 (cid:0) (cid:1)
ea·eb ≡gαβeαaeβb =ηab ≡diag(−1,1,...,1). (3) −(D−1)(D−2)Rga[cgd]b (7)
4
we immediately obtain Usingthenotationof[93],thecomponentsoftheWeyl
tensorinsucha nullframearedeterminedbythe follow-
1 ing scalars (grouped by their boost weight):
R =C + R δ R
(i)(0)(0)(j) (i)(0)(0)(j) D 2 (i)(j)− ij (0)(0)
Rδ − (cid:16) (cid:17) Ψ0ij=Cabcd kambikcmdj ,
ij
−(D 1)(D 2) . (8) Ψ1ijk=Cabcd kambimcjmdk , Ψ1Ti =Cabcd kalbkcmdi ,
− −
Ψ =C mambmc md , Ψ =C kalblckd ,
2ijkl abcd i j k l 2S abcd
Beforesubstitutingthisintothegeodesicdeviationequa-
Ψ =C kalbmcmd , Ψ =C kamblcmd ,
tion(5)wealsoemploythe Einsteinfieldequations,gen- 2ij abcd i j 2Tij abcd i j
eralized to any dimension D 4, Ψ =C lambmcmd , Ψ =C lakblcmd ,
≥ 3ijk abcd i j k 3Ti abcd i
Ψ = C lamblcmd , (14)
R 1Rg +Λg =8πT , (9) 4ij abcd i j
ab− 2 ab ab ab
where i,j,k,l=2,...,D 1. All other frame compo-
−
whereΛisacosmological constant andTab istheenergy- nents can be obtained using the symmetries of the Weyl
momentum tensor of the matter field. Using (9) and its tensor. The scalars in the left column are independent,
trace R= 2 (8πT DΛ), we rewrite (8) as up to the obvious constraints
2 D −
−
Ψ =0, Ψ k =0,
2Λδ 0[ij] 0k
ij
R(i)(0)(0)(j) = (D 1)(D 2) +C(i)(0)(0)(j) (10) Ψ1i(jk)=0, Ψ1[ijk] =0,
− −
Ψ =Ψ , Ψ =0,
8π 2T 2ijkl 2klij 2(ij)
+ T δ T + .
D 2 (i)(j)− ij (0)(0) D 1 Ψ2(ij)kl=Ψ2ij(kl) =Ψ2i[jkl] =0, (15)
− (cid:20) (cid:16) − (cid:17)(cid:21) Ψ =0, Ψ =0,
3i(jk) 3[ijk]
The equation of geodesic deviation (5) thus takes the Ψ =0, Ψ k =0,
4[ij] 4k
following invariant form
while those in the right column of (14) are not indepen-
2Λ dent because they can be expressed as the contractions
Z¨(i) = Z(i)+C Z(j) (11)
(D 1)(D 2) (i)(0)(0)(j) (hence the symbol “T” which indicates “tracing”)
− −
+ 8π T Z(j) T + 2 T Z(i) . Ψ1Ti=Ψ1kki ,
D−2(cid:20) (i)(j) −(cid:16) (0)(0) D−1 (cid:17) (cid:21) Ψ2S=Ψ2Tkk = 21Ψ2klkl , (16)
The first term represents the isotropic influence of the Ψ2Tij= 21(Ψ2ikjk +Ψ2ij)
cosmologicalconstantΛonfreetestparticles,thesecond where Ψ2T(ij) = 12Ψ2ikjk , Ψ2T[ij] = 21Ψ2ij ,
term describes the effect of a “free” gravitational field Ψ3Ti=Ψ3kki .
encodedinthe Weyltensor,whilethe secondline in(11)
gives a direct effect of specific matter present in a given In the case D =4, these Weyl tensor components in the
spacetime. nulltetradreducetothestandardNewman–Penrose[138,
The terms proportional to the coefficients C 139] complex scalars ΨA. Explicit expressions are given
(i)(0)(0)(j)
can further be conveniently expressed using the in appendix A.
Newman–Penrose-type scalars, which are the compo- Using relations e(0) = √12(k+l), e(1) = √12(k−l)
nents of the Weyl tensor with respect to an associated and the definition (14), a straightforward calculation
(real) null frame k,l,m . This frame is introduced by then leads to the following expressions for the compo-
i
{ }
the relations nents C of the Weyl tensor which appear in
(i)(0)(0)(j)
equation (11):
k= 1 (u+e ), l= 1 (u e ),
√2 (1) √2 − (1) C(1)(0)(0)(1) =Ψ2S ,
m =e for i=2,...,D 1, (12)
i (i) 1
−
C = (Ψ Ψ ),
(1)(0)(0)(j) √2 1Tj − 3Tj
where u e is the velocity vector of the observer.
aTrheuDs k a2≡nsdpa(l0t)iaarleCfuarttuerseiaonriveenctteodrsnourllthvoegcotonrasl,taontdhemmi, C(i)(0)(0)(1) = √12(Ψ1Ti −Ψ3Ti), (17)
satisfy−ing 1
C = (Ψ +Ψ ) Ψ ,
(i)(0)(0)(j) −2 0ij 4ij − 2T(ij)
k l= 1, mi mj =δij, where i,j =2,...,D 1 label the spatial directions or-
k·k=−0=l l, · k m =0=l m . (13) thogonal to the privile−ged spatial direction of e .
i i (1)
· · · ·
5
Putting this into (11) we obtain the final invariant and fully general form of the equation of geodesic deviation:
2Λ 1
Z¨(1) = Z(1)+Ψ Z(1)+ (Ψ Ψ )Z(j)
(D 1)(D 2) 2S √2 1Tj − 3Tj
− −
8π 2
+ T Z(1)+T Z(j) T + T Z(1) , (18)
D 2 (1)(1) (1)(j) − (0)(0) D 1
− (cid:20) (cid:16) − (cid:17) (cid:21)
2Λ 1 1
Z¨(i) = Z(i) Ψ Z(j)+ (Ψ Ψ )Z(1) (Ψ +Ψ )Z(j)
(D 1)(D 2) − 2T(ij) √2 1Ti − 3Ti − 2 0ij 4ij
− −
8π 2
+ T Z(1)+T Z(j) T + T Z(i) . (19)
D 2 (i)(1) (i)(j) − (0)(0) D 1
− (cid:20) (cid:16) − (cid:17) (cid:21)
This completely describes relative motion of nearby free test particles in any spacetime of an arbitrary dimension D.
In the next section we will discuss the specific effects given by particular scalars which represent the contributions
from various components of the gravitationaland matter fields.
Finally, we remark that our notation which uses Ψ interaction simplifies considerably to
A...
in any dimension is simply related to the notations em- 2Λ
ployed, e.g., in [72, 73], in [76, 81] and recently in [79]. Z¨(1) = Z(1)+Ψ2SZ(1) (20)
(D 1)(D 2)
The identifications for the components presentin the in- − −
1
variant form of the equation of geodesic deviation are + (Ψ Ψ )Z(j) ,
summarized in table I. More details are given in ap- √2 1Tj − 3Tj
pendix B, in particular see expressions (B8), (B11) and Z¨(i) = 2Λ Z(i) Ψ Z(j) (21)
(B13). (D 1)(D 2) − 2T(ij)
− −
1 1
+ (Ψ Ψ )Z(1) (Ψ +Ψ )Z(j) .
√2 1Ti − 3Ti − 2 0ij 4ij
TABLEI.Differentequivalentnotationsusedintheliterature
The overall effect of the gravitationalfield on test parti-
for theWeylscalars which occur in theequations of geodesic
cles is thus naturally decomposed into clearly identified
deviation (18), (19).
components proportionalto the cosmologicalconstantΛ
refs. [72, 73] refs. [76, 81] ref. [79] and the Weyl scalars ΨA.... Of course, for algebraically
Ψ2S −C0101 −Φ −Φ special spacetimes some (or many) of these coefficients
vanish completely,andeveninalgebraicallygeneralcases
ΨΨ21TTijj −−CC001i10jj −Φij −−ΦΨijj sgpueischifitchneudmomeriincaanltvateluremssoffrotmhetshcoasleartshaΨtAa..r.ecnaengldigisitbilne-.
Ψ3Tj C101j Ψj Ψ′j Let us now briefly describe the character of each term
Ψ0ij C0i0j Ωij separately, including its physical interpretation.
Ψ4ij C1i1j 2Ψij Ω′ij Λ: isotropic influence of the cosmological back-
•
ground
The presence of the cosmologicalconstant Λ is en-
coded in the term
Z¨(1) 2Λ 1 0 Z(1)
IV. EFFECT OF CANONICAL COMPONENTS = , (22)
OF A GRAVITATIONAL FIELD ON TEST Z¨(i) ! (D−1)(D−2) 0 δij ! Z(j) !
PARTICLES
which can be written in a unified way
Z¨(i) = 2Λ Z(i) for all spatial compo-
(D 1)(D 2)
Let us consider a set of freely falling test particles, nents i=−1,2,.−..,D 1. In parallelly propagated
−
initially at rest relative to each other, which form, e.g., frames, this yields the following explicit solutions
a small (hyper-)sphere. In any curved spacetime, such a
Λ=0: Z(i) =A τ +B ,
configurationundergoestidaldeformationswhichcanbe i i
deduced from the accelerations measured by the fiducial Λ>0: Z(i) =A cosh 2Λ τ
i (D 1)(D 2)
observer attached to the reference test particle in the − −
hq i
center. Theresultingrelativemotionrepresentstheeffect +B sinh 2Λ τ ,
i (D 1)(D 2)
of a given gravitational field, whose specific structure is − −
hq i
explicitly characterizedby the system (18), (19). Λ<0: Z(i) =Aicos (D 21)|(ΛD| 2)τ
theFisryststceomncoefnterqautaintigonosndtehsecrviabcinugumpucraesley,gi.rea.viTtaabti=on0a,l +Bisinhq(D−21)|(ΛD|−2)τi,
− −
hq i
6
where A,B are constants of integration. These namely those in the vicinity of a spherically sym-
i i
are characteristic relative motions of test parti- metric static source. Recall that Ψ2S =Ψ2Tkk
cles in spacetimes of constant curvature, namely (see (16) and (15) for further relations), so that
Minkowskispace,deSitterspaceandanti-deSitter the (D 1) (D 1)-dimensional matrix in (25)
− × −
space, respectively, as derived by Synge [102, 103]. is symmetric and traceless. These terms are typ-
ically present in type D spacetimes, for which the
• Ψth4eijd:irtercatniosnve+rsee gravitational wave propagating in notation Ψ2S ≡−Φ and −Ψ2T(ij) ≡ΦSij is com-
(1) monly used [76, 78–82, 99], see (B11). As shown
Thispartofagravitationalfieldinfluences the test in (A6), the only nonvanishing coefficients of this
particles as type in four dimensions are the diagonal elements
1Ψ =Ψ =Ψ ReΨ .
2 2S 2T(22) 2T(33) ≡− 2
Z¨(1) 1 0 0 Z(1)
= . (23) Ψ : longitudinal component of a gravitational
Z¨(i) ! −2 0 Ψ4ij ! Z(j) ! • fie1lTdiwith respect to e
(1)
−
The corresponding effect on test particles is
Obviously,thisisapurelytransverseeffectbecause
there is no accelerationin the privilegedspatialdi-
mreecttrioicn(eΨ(14)ij. =ThΨe4jsie)taonfdsctaralacresleΨss4i(jΨf4okrkm=s a0)symma-- ZZ¨¨((1i)) != √12 Ψ10Ti Ψ10Tj ! ZZ((1j)) !, (26)
trix of dimension (D 2) (D 2), cf. the last
line in (15), so that it−has 1×D(D− 3) independent which is very similar to the acceleration caused
components corresponding2to po−larization modes by the longitudinal component Ψ3Ti, as de-
scribed by (24). In fact, it is its counter-
(see also [72, 78, 79]). In direct analogy with a
part: it follows from the definition (14) that
linearized Einstein gravity in four [115, 116] and
higherdimensions[86,93],Ψ4ij representsthegrav- athned sΨcalars+ΨΨ1Ti ≡+ΨΨ1kki =(w0h)eraereΨe1qijukiv=al−enΨt1itkoj
itational wave which propagates along the null di- 1ijk 1jki 1kij
okrefcrteiolanktkio,neis.e.(1=in2)0thtfoehresrpiea=tiis2a,lk.d(.1i.r),e≡Dctikon·1e+)(.1e)S(1>p)a(0cientwvimhieielwes Ψksc(31aT)lai≡rskuΨn·1edT(ei1r)re>pthr0eesewninthtitelhrecehlalo(nn1g)gei≡tuldk·in↔ea(1ll)c.<om0p,oSnintehnceet
(i) ≡ · (i) − of the field associated with the spatial direction
ofalgebraictypeN(forwhichonlythecomponents
e .
Ψ4ij C1i1j arenonvanishing[72,73])canthusbe − (1)
≡
interpreted as exact gravitational waves in any di- Ψ : transverse gravitational wave propagating in
mension D 4. • th0eijdirection e
≥ − (1)
Ψ : longitudinal component of a gravitational This component of a gravitational field is charac-
• fie3lTdiwith respect to +e terized by
(1)
Suchtermscauselongitudinaldeformationsofaset Z¨(1) 1 0 0 Z(1)
of test particles given by = , (27)
Z¨(i) ! −2 0 Ψ0ij ! Z(j) !
Z¨(1) = 1 0 Ψ3Tj Z(1) . (24) which is fully equivalent to (23) under k l. The
Z¨(i) ! −√2 Ψ3Ti 0 ! Z(j) ! scalarsΨ0ij (whichformasymmetricand↔traceless
These(D 2)scalarsΨ3Ti,whichcombinemotion 0(D) t−hu2s)d×es(cDri−be2t)hemtartarnixs:verΨse0ijgr=avΨit0ajtii,onΨa0lkwkav=e
in the pri−vileged spatial direction e(1) with mo- propagating along the null direction l, i.e. in the
tion in the transverse directions e(i), are also ob- spatial direction e(1). Superposition of gravita-
tained using Ψ3Ti ≡Ψ3kki, where Ψ3ijk =−Ψ3ikj tional waves whic−h would propagate in both direc-
andΨ3ijk +Ψ3jki +Ψ3kij =0. Longitudinaleffects tions simultaneously (that is an “outgoing” wave
of this type occur in spacetimes of type III and in givenbyΨ andan“ingoing”wavegivenbyΨ )
4ij 0ij
algebraically more general cases. can only be present in spacetimes which are of al-
gebraically general type.
Ψ ,Ψ : Newton–Coulomb components of a
• 2S 2T(ij)
gravitational field
The terms V. UNIQUENESS OF THE INTERPRETATION
FRAME AND DEPENDENCE OF THE FIELD
Z¨(1) Ψ 0 Z(1) COMPONENTS ON THE OBSERVER
= 2S (25)
Z¨(i) ! 0 −Ψ2T(ij) ! Z(j) !
The canonical components of a gravitational field de-
giverise to deformations which generalizethe clas- scribedinprevioussectionarerepresentedbytherealco-
sical Newton–Coulomb-type tidal effects in D =4, efficients Ψ . These are projections of the Weyl tensor
A...
7
ontoparticularcombinationsofthenullframe k,l,m , by m , see the explicit relation (C4) presented in ap-
i i
{ }
asdefinedin(14). Theyarespacetimescalarsandinthis pendix C.
sense the above physical interpretation is invariant. On Forany spacetime oftype N(in which the WAND has
the other hand, the values of ΨA... depend on the choice maximal alignment order) the null vector k is unique.
of the basis vectors of the frame. In this section we will In spacetimes of other algebraic types (namely III , II ,
i i
arguethatsuchadependencecorrespondstosimplelocal I and D) different WANDs exist. These can alterna-
i
Lorentz transformations related to the choice of specific tively be used as the vector k of the interpretation null
observer in a given event, and that the natural interpre- frame k,l,m . Because the distinct WANDs can al-
i
tation null frame is essentially unique. ways b{e related}using the null rotation with fixed l, as
Let us consider an observer attached to the reference given explicitly by equation (C2) in appendix C, it is
(fiducial) testparticle movingthroughsome eventin the straightforwardtoevaluatethe“new”valuesoftheWeyl
spacetime, such as the point in figure 1, whose ve- scalars Ψ using the expressions (??). Notice that the
A...
locity vector is u. This timeliPke vector (normalized as coefficients Ψ , which are the amplitudes of transverse
4ij
u u= 1)definesanorthogonalspatialhypersurfaceof gravitational waves propagating along k, are invariant
dim· ensio−n D 1 spanned by the Cartesian vectors e , under such a change.
(i)
−
where i=1,2,...,D 1. Assuming the spacetime is of Let us now consider another observer moving through
an algebraic type I or−more special, it is most natural to the same event with a different velocity u˜. Locally,
P
associate the corresponding Weyl aligned null direction this transition is just the Lorentz transformation from
(WAND) withthe nullvectork ofthe interpretationref- the original reference frame e to e˜ for which
a a
{ } { }
erence frame, see figure 2.
u˜ = u+ iD=−11vie(i) , (28)
1P D−1v2
− i=1 i
q
P
where v ,...,v are components of the spatial veloc-
1 D 1
ityofthenewobs−erverwithrespecttotheoriginalCarte-
sian basis e . This can be obtained as the combination
(i)
of a boost in the k l plane followed by a null rotation
withfixedk,seeequ−ations(C3)and(C1)inappendixC,
if we take the specific parameters
B = 1− iD=−11vi2 , L = vi , (29)
FIG. 2. Natural choice of the interpretation null frame and q 1P−v1 i 1 D−1v2
the related orthonormal frame (12), (13). Up to spatial ro- − i=1 i
tations of mi =e(i), they are uniquely given by the velocity where i=2,...,D 1: q P
vectoru of the observerand theWAND k at any eventP of −
thespacetime.
k˜ = 1− iD=−11vi2 k,
The privileged unit vector e , defining the longitudinal q 1 v
(1) P 1
−
spatialdirection,isthenuniquelyobtainedbyprojecting D 1
k onto the spatial subspace orthogonal to u. This also ˜l= 1 (1 v )l+√2 − v m
1 i i
fixes the normalization of k (to satisfy the first relation 1 D−1v2(cid:20) − i=2
in (12) we require k u= 1 ). The complementary − i=1 i X
null vector l of the fra·me is−th√e2n also uniquely given via q P + iD=−21vi2 k , (30)
ttrhaenrsevleartsieonspla=tia√lv2euct−orsk.e(I2t),o.n.l.y,ere(Dma1i)n,si.teo. cmhoio=seet(hi)e. m˜ =m +√2 vi kP. 1−v1 (cid:21)
As shown in figure 2, these must li−e in the (D 2)- i i 1 v
1
dimensional subspace orthogonal both to u and e −, so −
that k m =0=l m as required by (13). Ne(g1)lect- Indeed, u˜ 1 (k˜+˜l) gives exactly the relation (28).
· i · i ≡ √2
ing possible inversions, the only remaining freedom are The corresponding change of the Weyl scalars ΨA... can
thus standard spatial rotations represented by the rota- thus be obtained by combining (C7) with (C5), which
tion group SO(D 2) which acts on the space spanned yields
−
8
1
Ψ˜ =Ψ ,
B2 0ij 0ij
1
Ψ˜ =Ψ 2√2Ψ X ,
B 1ijk 1ijk − 0i[j k]
1
Ψ˜ =Ψ +√2Ψ Xj ,
B 1Ti 1Ti 0ij
Ψ˜ =Ψ 2√2 X Ψ X Ψ +4 Ψ X X +Ψ X X ,
2ijkl 2ijkl − [l 1k]ij − [i 1j]kl 0i[k l] j 0j[l k] i
Ψ˜2S =Ψ2S −2√2Ψ(cid:0)1TiXi−2Ψ0ijXiXj ,(cid:1) (cid:0) (cid:1)
Ψ˜ =Ψ +√2Ψ Xk 2√2Ψ X 4Ψ X Xk ,
2ij 2ij 1kij − 1T[i j]− 0k[i j]
Ψ˜ =Ψ +√2Ψ Xk √2Ψ X 2Ψ XkX +Ψ X 2 , (31)
2Tij 2Tij 1ikj − 1Ti j − 0ik j 0ij| |
BΨ˜ =Ψ +√2 Ψ Xl Ψ X +2X Ψ
3ijk 3ijk 2lijk − 2jk i [j 2Tk]i
+4Ψ1T[j(cid:0)Xk]Xi−2(Ψ1jliXk+Ψ1ljkXi−Ψ(cid:1)1kliXj)Xl+Ψ1ijk|X|2
+4√2Ψ X X Xl 2√2Ψ X X 2 ,
0l[j k] i − 0i[j k]| |
BΨ˜ =Ψ +√2Ψ Xj √2 Ψ Xk+Ψ X
3Ti 3Ti 2ij − 2Tki 2S i
+2 2Ψ1TjXi−Ψ1kjiX(cid:0) k Xj −Ψ1Ti|X|2(cid:1)+2√2Ψ0jkXjXkXi−√2Ψ0ijXj|X|2 ,
B2Ψ˜4ij =Ψ4ij +(cid:0)2√2 Ψ3T(iXj)−Ψ3((cid:1)ij)kXk
+2Ψ2ikjl(cid:0)XkXl−4Ψ2Tk(iXj)Xk(cid:1)+2Ψ2T(ij)|X|2−2Ψ2SXiXj −4Ψ2k(iXj)Xk
2√2(2Ψ X XkXl+Ψ Xk X 2+Ψ X X 2 2Ψ XkX X )
− 1kl(i j) 1(ij)k | | 1T(i j)| | − 1Tk i j
+4Ψ XkXlX X 4Ψ X Xk X 2+Ψ X 4 ,
0kl i j − 0k(i j) | | 0ij| |
where we denoted
v
i
X BL = . (32)
i i
≡ 1 v
1
−
Inparticular,forspacetimesofalgebraictypeN,which by the invariant form of the equation of geodesic devia-
admitaWANDofthemaximalalignmentorder,theonly tion (18), (19). Setting the cosmological constant Λ and
nonvanishingcomponentofthegravitationalfieldisΨ all components of the Weyl tensor to zero, it reduces to
4ij
representing the transversegravitationalwave propagat-
ifnrogmin(3t1h)easnpdati(a2l9)ditrheacttiotnheet(1r)a.nsIittioimnmtoedaiantyeloythfoerlloowbs- Z¨(1) = D8π2 T(1)(1)Z(1)+T(1)(j)Z(j)
− (cid:20)
server results just in a simple rescaling of the gravita- 2
T + T Z(1) ,
tional wave amplitudes − (0)(0) D 1
(cid:16) − (cid:17) (cid:21)
Ψ˜4ij = 1−(1−iD=v−11)12vi2 Ψ4ij . (33) Z¨(i) = D8−π2(cid:20)T(i)(1)Z(1) +2T(i)(j)Z(j)
If the new observer movPes only in the spatial direc- − T(0)(0)+ D 1T Z(i) . (34)
tion in which the wave propagates, v1 >0 and vi =0 (cid:16) − (cid:17) (cid:21)
for i=2,...,D 1. Then Ψ˜ =(1 v )/(1+v )Ψ Itwillbeillustrativetoinvestigatesomeimportanttypes
− 4ij − 1 1 4ij
which is smaller than Ψ . If the observer’svelocity ap- of matter usually contained in the families of exact so-
4ij
proachesthespeedoflight,v 1,theamplitudesofthe lutions of Einstein’s equations, namely pure radiation,
1
gravitational wave disappear, →Ψ˜ 0. Contrary, when perfect fluids and electromagnetic fields.
4ij
→
theobservermovesagainstthewaveitsamplitudesgrow,
and for v 1 they diverge.
1
→−
VI. THE EFFECT OF MATTER ON TEST
PARTICLES
Letusnowconsiderthedirecteffectofspecificformsof
matter on relative motion of test particles, as described
9
pure radiation so that its trace is T = 1 (4 D)F Fab. The
• 16π − ab
framecomponentsofT whichoccurinexpressions
The energy-momentum tensor of a pure radiation ab
(34) are
field (or “null dust”) aligned along the null direc-
tion k is
1 1
T = F F c+ F Fab ,
T =ρk k , (35) (0)(0) 4π (0)c (0) 4 ab
ab a b
1 (cid:16) 1 (cid:17)
T = F F c F Fab ,
where ρ is a function representing the radiation (1)(1) 4π (1)c (1) − 4 ab
density. Its trace vanishes, T =0, and using (12) 1 (cid:16) (cid:17)
we derive that the only nonvanishing components T(1)(i) = 4πF(1)cF(i)c , (40)
of T in the equation of geodesic deviation are
ab 1 1
T(0)(0) =T(1)(1) = 12ρ. Equations (34) thus reduce T(i)(j) = 4π F(i)cF(j)c− 4δijFabFab .
considerably to (cid:16) (cid:17)
Inthiscasetheequationofgeodesicdeviationtakes
Z¨(1) 4πρ 0 0 Z(1) the following more complicated form:
= . (36)
Z¨(i) ! −D−2 0 δij ! Z(j) !
Z¨(1) Z(1)
cInelearnataiornbiitnratrhyedloimngeintusidoinnaDlspthaetriaelidsirtehcutsionnoeac-. Z¨(i) != TTi TTijj ! Z(j) !, (41)
(1)
The effects in the transversesubspaceare isotropic
where
and(since ρ>0) they cause the radialcontraction
which may eventually lead to an exact focusing. 2
= F F c F F c
perfect fluid T D 2 (1)c (1) − (0)c (0)
• − (cid:16) 3 (cid:17)
Foraperfect fluidofenergydensity ρ andpressure FabFab ,
−(D 1)(D 2)
p (which is assumed to be isotropic) the energy- − −
2
momentum tensor is = F F c , (42)
Ti D 2 (1)c (i)
−
T =(ρ+p)u u +pg . (37) 2
ab a b ab = F F c δ F F c
Tij D 2 (i)c (j) − ij (0)c (0)
Providedthe fluid is comoving,its velocity u coin- − (cid:16) 3 (cid:17)
cides with the observer’s velocity which is the vec- δijFabFab .
tor e(0) of the orthonormal frame. The trace is −(D−1)(D−2)
T =(D 1)p ρ, and the relevant nonvanishing
− − We observe that the clear distinction between the
frame components are T =ρ, T =p and
(0)(0) (1)(1) longitudinalandtransversespatialdirectionsisnot
T =pδ . The equation of geodesic deviation
(i)(j) ij present,exceptatveryspecialsituations. Someim-
thus takes the form
portantparticularsubcases canbe easily identified
Z¨(1) (D 3)ρ+(D 1)p 1 0 Z(1) and analyzed, for example a null electromagnetic
= 8π − − . field for which the invariant vanishes, F Fab =0,
Z¨(i) ! − (D−1)(D−2) 0 δij ! Z(j) ! or purely electric aligned field in the avbicinity of
(38)
static black holes.
The resulting motion is isotropic, the same in the
longitudinal and all transverse spatial directions.
Forpositiveρ andp,the fluidmatter causesacon-
VII. AN EXPLICIT EXAMPLE: PP-WAVES IN
traction, such as in the case of dust (p=0), inco- HIGHER DIMENSIONS
herent radiation (p= D 3ρ), or stiff fluid (p=ρ).
D−1
However, for matter wit−h a negative pressure, the
We conclude this paper by demonstrating the useful-
set of test particles may expand. In particular, if
ness of the above interpretation method on an impor-
the matter is described by the equation of state
tant family of exact spacetimes, namely the pp-waves.
p= ρ=const., it mimics the cosmological con-
− These are defined geometrically as admitting a covari-
stant Λ=8πρ since (38) is then completely equiv-
antly constant null vector field k. Such CCNV space-
alent to (22).
times thus form a special subclass of the Kundt space-
electromagnetic field times because the geodesic congruencegeneratedby k is
• twist-free, shear-free and non-expanding.
The energy-momentum tensor of an electromag-
In [55] we investigated general Kundt spacetimes in
netic field is given by
higher dimensions, admitting a cosmological constant Λ
1 1 and a Maxwell field aligned with k (which is necessarily
Tab = 4π FacFbc− 4gabFcdFcd , (39) a multiple WAND). In natural coordinates the metric of
(cid:16) (cid:17)
10
all suchpp-wavescan be written in the Brinkmannform of geodesic deviation (18) and (19):
[57]
1
ds2 =gijdxidxj +2eidxidu 2dudr+cdu2 , (43) Ψ2S = (D 1)(D 2)sR,
− − −
1 1
Ψ = sR mkml sRδ ,
wherek ∂ andg ,e ,carefunctionsofthetransverse 2Tij D 2 kl i j − (D 1)(D 2) ij
r ij i − − −
spatial c∝oordinates xk and the null coordinate u. The √2
Ψ = sR x˙m+R u˙ mk , (46)
explicitEinstein–Maxwellequationscanbefoundin[55], 3Ti −D 2 km ku i
namely equations (115)–(118). − (cid:0) 1 (cid:1)
Ψ =2 sR g sR x˙mx˙n
For the metric (43) the interpretation null frame 4ij kmln− D 2 kl mn
(cid:20)(cid:16) − (cid:17)
adapted to a general observer which has the velocity 1
u=r˙∂r +u˙∂u+x˙2∂x2 +...+x˙D−1∂xD−1 is +2 Rkmlu− D 2gklRmu x˙mu˙
(cid:16) − (cid:17)
1
+ R g R u˙2 mkml .
kulu− D 2 kl uu (i j)
k= 1 ∂ , (cid:16) − (cid:17) (cid:21)
r
√2u˙
l= √2r˙ 1 ∂ +√2u˙∂ This is a general result valid for any pp-wave spacetime
r u
− √2u˙ because no particular field equations have not yet been
(cid:16)+√2x˙2∂x2 +(cid:17)...+√2x˙D−1∂xD−1 , (44) imposed.
1 Notice that Ψ =Ψ . Moreover, in accordance
m = (e u˙ +g x˙j)mk∂ 2Tij 2T(ij)
i u˙ k jk i r with the relations (16) and (15), Ψ2S =Ψ2Tkk and
+m2i ∂x2 +...+miD−1∂xD−1 , Ψ4kk =0 so that any pp-wave is traceless.
The relative tidal motion of nearby test particles in
general pp-waves will thus be caused by the combina-
where g mkml =δ ,andnontrivialcomponentsofthe tionofthetransversegravitationalwave(23)propagating
kl i j ij
Weyl tensor are alongkwithamplitudeΨ4ij,thelongitudinalcomponent
(24) of the gravitational field with amplitude Ψ , and
3Ti
the Newton–Coulomb contribution (25) determined by
1 the scalars Ψ2S and Ψ2Tij:
C = sR,
ruru
−(D 1)(D 2)
− −
1 1
C = sR sRg , Z¨(1) = Ψ Z(1) 1 Ψ Z(j) , (47)
riuj D 2 ij − (D 1)(D 2) ij 2S − √2 3Tj
C = 1− R − 1 − sRe , Z¨(i) =−Ψ2T(ij)Z(j)− √12Ψ3TiZ(1)− 21Ψ4ijZ(j) .
ruui ui i
D 2 − (D 1)(D 2)
− − −
2
C = sR (g sR g sR )
ijkl ijkl − D 2 i[k l]j − j[k l]i There is also the isotropic background influence (22) if
− 2 thecosmologicalconstantΛispresent,ortheinteraction
+(D 1)(D 2) sRgi[kgl]j , (45) (41) with the electromagnetic field.
− −
2 The scalars (46) which enter (47) combine kinematics
Cuijk =Ruijk − D 2(e[jsRk]i−gi[jRk]u) (namely the velocity components x˙m, u˙ of the observer)
− 2 with the specific curvature of spacetime encoded in the
+(D 1)(D 2) sRe[jgk]i , onlynonvanishingcomponentsoftheRiemannandRicci
− − tensors, namely
1
C =R (csR 2e R +g R )
iuju iuju− D 2 ij − (i j)u ij uu
−
1
+ sR(cgij eiej). Rijkl = sRijkl ,
(D 1)(D 2) −
− − Ruijk = 21(ek,ij −ej,ik+gij,uk−gik,uj)
+sΓm 1g +e sΓm 1g +e ,
ij 2 km,u [m,k] − ik 2 jm,u [m,j]
Uthseinigntdeerfipnreittiaotnio(n14n)uwllefreavmaleua(t4e4t)h.eLWenegytlhtyencsaolrcu(4la5t)ioinn Riuju = 12(ei,uj(cid:0)+ej,ui−c,ij −(cid:1)gij,uu) (cid:0) (4(cid:1)8)
+gkl 1g +e 1g +e
(with some “miraculous” cancelations) gives the follow- 2 ik,u [k,i] 2 jl,u [l,j]
ing nonvanishing Weyl scalars which enter the equations −sΓki(cid:0)j ek,u− 21c,k (cid:1),(cid:0) (cid:1)
(cid:0) (cid:1)