On the Riemann Curvature Tensor in General Relativity Zafar Ahsan Department of Mathematics Aligarh Muslim University Aligarh-202 002(India) e-mail:zafar.ahsan@rediffmail.com Dedicated to Prof. Hideki Y uk aw a (First Noble Laureate in Physics from Asia)
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Through his general theory of relativity, Einstein rede-fined gravity. From the classical point of view, grav-ity is the attractive force between massive objects inthree dimensional space. In general relativity, gravitymanifests as curvature of four dimensional space-time.
Conversely curved space and time generates effects thatare equivalent to gravitational effects. J.A. Wheelar hasdescribed the results by saying Matter tells space-time how to bend and space-time returns the complement by telling matter how to move.
The general theory of relativity is thus a theory of grav-itation in which gravitation emerges as the property of
the space-time structure through the metric tensor gij.The metric tensor determines another object (of tensor-ial nature) known as Riemann curvature tensor. At anygiven event this tensorial object provides all informa-tion about the gravitational field in the neighbourhoodof the event. It may, in real sense, be interpreted as de-scribing the curvature of the space-time. The Riemanncurvature tensor is the simplest non-trivial object onecan build at a point; its vanishing is the criterion for theabsence of genuine gravitational fields and its structuredetermines the relative motion of the neighbouring testparticles via the equation of geodesic deviation.
The above discussion clearly illustrates the importanceof the Riemann curvature tensor in general relativityand it is for these reasons, a study of this curvaturetensor has been made here.
2. The Riemann curvature tensor
∗ The Riemann curvature tensor Rkijl is defined, for a
covariant vector field Ak, through the Ricci identity
∗ A tensor of rank four has n4 components in an n-dimensional space and for n = 4, it is 256. For Rhijl, dueto symmetry properties, the number of algebraically in-dependent components, in an n-dimensional manifold,is
1
12n
2
(n
2
− 1)so that in 4-dimension, this no. is 20.
∗ Geheniau and Debever, have shown that the Riemanncurvature tensor can be decomposed as
Rijkl = C ijkl + E ijkl + Gijkl (2)
where C ijkl is the Weyl tensor, E ijkl is the Einstein cur-vature tensor, defined by
∗ Algebraic classification of the vacuum Riemann ten-sor: Petrov and Pirani.
∗ Witten and Penrose have redeveloped both algebraicand geometric properties (spinor calculus).
∗ The relevance of Petrov classification to the gravita-tional theory was suggested by Pirani.
∗ Petrov classes were further developed: Lichnerowicz,Bel, Ehlers, Sachs and Sharma and Husain
∗ In empty space-time, Riemann tensor reduces to Weyltensor. Thus, in order to have a classification of vacuumRiemann tensor, it is sufficient to classify the Weyl ten-
sor.
∗ Three main approaches for the classification of Weyltensor-namely: the matrix method (Synge, Petrov), thespinor method (Penrose, Pirani) and the tensor method(Sachs). The connection between these approaches hasbeen found out by Ludwig. The tensor method is equiv-alent to the other two.
∗ We have seen that the number of independent compo-nents of Riemann tensor in n-dimension is 112
n2(n2 − 1).The symmetries of the Riemann tensor thus leads to
(i) if n=1, Rhijk = 0;
(ii) if n=2, Rhijk has only one independent component,namely R1212 = 1
2gR;
(iii) if n=3, Rhijk has six independent components. TheRicci tensor has also six independent components andthus Rhijk can be expressed in terms of Rij as
Rhijk = ghj Rik + gikRhj − ghkRij − gij Rhk − 1
2(ghjgik − ghkgij)R
(iv) if n=4, Rhijk has twenty independent components-ten of which are given by Ricci tensor and the Remain-ing ten by the Weyl tensor C hijk.
It may be noted that Weyl tensor C hijk makes its ap-pearance only in a four dimensional space-time throughequation
and in empty space-time (i.e., Rij = 0) the gravitationalfield is characterized by the Weyl tensor.
∗ Thus according to general relativity, if we lived in a three di-mensional Universe, gravity could not exist in a vacuum region.
∗ Maxwell eqns (vector): · E = 0, · H = 0
× E = −1
c
∂H
∂t, × H = −1
c
∂E
∂t
∗ Invariants: E · H = 0, E 2 = H 2
∗ Tensor: F ij;k + F jk ;i + F ki; j = 0, F ij; j = 0
∗ Invariants: F ij F ij
= 0, F
ijF ij
= 0 - (em radiation)
∗ The Riemann curvature tensor has fourteen invari-ants. There is the Ricci scalar R. There are 04 invari-ants of the Weyl tensor C ijkl. There are 03 invariantsof the Einstein curvature tensor E ijkl and 06 invariantsof the combined Weyl and Einstein curvature tensors.The component form of these invariants are
∗Importance : For example, the behaviour of the scalar
RijklRijkl is studied in connection with the existence of
any geometrical singularity.
∗ If Rij = 0 (empty space-time) then Riemann tensorreduces to Weyl tensor; and in this case there are fourinvariants of Riemann tensor which are given by
A1 = RijklRijkl, A2 = R∗
ijklRijkl
B1 =4
3RijmnRmnrsR ijrs , B2 =4
3R∗ijmnRmnrsR ijrs
∗ Calculating these invariants for the classification of Riemann tensor due to Sharma and Husain and Petrov:
∗ If Rabcd = 0 and A1 = A2 = B1 = B2 = 0, then the gravi-tational radiation is present; otherwise there is no gravitational
radiation.
∗ Check for validity:
(i) Takeno’s plane wave solution
ds2 = −Adx2 − 2Ddxdy − Bdy2 − dz 2 + dt2
(ii) Einstein-Rosen metric
ds2 = e2γ −2ψ(dt2
−dr2)
−r2e−2ψdφ2
−e2ψdz 2
where γ and ψ are functions of r and t only, ψ = 0 andγ = γ (r − t).
∗ For the metrics (i)-(iii), all the four invariants of theRiemann tensor vanish and thus correspond to the state
of gravitational radiation. For the Schwarzchild exteriorsolution A1 = 0, B1 = 0, A2 = 0, B2 = 0; and Schwarzchildsolution, being a Petrov type D solution, is known tobe non-radiative.
3. Electric and magnetic space-times
∗ The correspondence between electromagnetism and
gravitation are very rich and detailed. Some of thesecorrespondence are still uncovered while some of themare further developed.
∗ A physical field is always produced by a source -charge. Manifestation of fields when charges are at restis called electric and magnetic when they are in motion.This general feature is exemplified by the Maxwell’s the-
ory of electromagnetism from which the terms of elec-tric and magnetic are derived. This decomposition canbe adapted in general relativity and the Weyl tensorcan be decomposed into electric and magnetic parts.
C abcd = (ηacef ηbdpq − gacef gbdpq ) ue u p E f q
+(ηacef gbdpq − gacef ηbdpq ) ue u p H f q
∗ Now assume Weyl tensor to be of Petrov type I. A co-ordinate frame can be chosen in which the componentsof u are (1,0,0,0). For this observer, a frame rotationcan be made such that components of Q are observedby that observer to be
Qab =
0 0 0 00 λ1 0 00 0 λ2 0
0 0 0 λ3
whereλ3 = −(λ1 + λ2)
∗ The Weyl tensor is purely electric if and only if Q
is real (i.e., λ1, λ2, λ3 are real) and the Weyl tensoris purely magnetic if and only if Q is imaginary (i.e.,λ1, λ2, λ3 are imaginary).
∗ A null tetrad can be chosen such that the Newman-Penrose components ψABCD of the Weyl tensor in thattetrad are
4. Lanczos spin tensor∗ Maxwell eqns (Tensor): F ij;k + F jk ;i + F ki; j = 0, F
ij; j = 0
∗ Generated through a potential: F ij = Ai; j − A j;i
∗ Is it possible to generate the gravitatinal field througha potental?
∗ YES. Through the covariant differentiation of a tensorfield Lijk . This tensor field is now known as Lanczos potential or Lanczos spin tensor and satisfies the followingsymmetries:
From above symmetric relations, the Weyl-Lanczos re-lations can also be expressed as
C hijk = Lhij;k − Lhik; j + L jkh;i − L jki;h
+1
2(L
pi j; p + L
p j i; p) ghk +
1
2(L
ph k; p + L
pk h; p) gij
−12
(Lp
h j; p + Lp
j h; p) gik − 12
(Lp
i k; p + Lp
k i; p) ghj
∗ Although the existence of a tensor Lijk as a potentialto the Weyl tensor C abcd was established by C. Lanczosin 1962, there was a little development in the subjectfor quite some time.
∗Zund (1975): spinor calculus
∗ Bampi and Caviglia (1983): proved the existence of Lanczos potential to a larger class of 4-tensors and to alarger class of 3-tensors
∗ Illge (1988): spinor formalism, proved the existenceof Lanczos potential in four dimension and obtained thewave equation for the Lanczos potential both in spinor
and tensor forms
∗ Dolan and Kim (1994): wave equations for the Lanc-zos potential and gave a correct tensor version of the
∗The physical meaning of Lanczos tensor is not yet
very clear, but the quest for studying this tensor is ONwith results of elegance-and the list of workers in thisparticular field of interest is very long, we have men-tioned here only a few of them.
∗ For a given geometry, the construction of Lijk is equiv-alent to solving Wely-Lanczos equation along with thesymmetry eqns of Lanczos tensor; and as seen from the
above discussion that there are several ways of solvingthis equation although none of them are as straightfor-ward as one would like them to be.
∗ Using the method of general observers, we have givenyet another method for finding the Lanczos potentialand found the Lanczos potential for the perfect fluidspace times in terms of the spin coefficients.
∗ For a gravitational field with perfect fluid source, thebasic covariant variables are: the fluid scalars θ (expan-
sion),∼ρ (energy density), p (pressure); the fluid spa-
tial vectors ui (4-acceleration), wi (vorticity); the spa-tial trace-free symmetric tensors σij (fluid shear), theelectric (E ij) and the magnetic (H ij) parts of the Weyltensor; and the projection tensor hij which projects or-thogonal to the fluid 4-velocity vector ui.
These quantities, for a unit time like vector field ui suchthat ui ui = 0 (physically, the time like vector field ui isoften taken to be the 4-velocity of the fluid), are defined
The Weyl tensor is said to be purely electric if H ik = 0and purely magnetic if E ik = 0 and in terms of E ik andH ik, the Weyl tensor can be decomposed as
C hijk = 2 uh u j E ik + 2 ui uk E hj − 2 uh uk E ij − 2 ui u j E hk
+ghk E ij + gij E hk − ghj E ik − gik E hj
+ηhi pq u p uk H qj−ηhi
pq u p u j H qk+η jk pq ui u p H hq −η jk
pq uh u p H iq
(vii) The covariant derivative of ui may be decomposedinto its irreducible parts
ui; j
= σij
+1
3θ h
ij+ ω
ij+ a
iu
j
where hij, θ, ai, σij and ωij are, respectively, define through(i)-(v).
(viii) The energy density∼ρ and the pressure p are given
by the energy momentum tensor T ij of the perfect fluid
T ij =∼ρ ui u j − p hij
The relativistic equations of the conservation of energyand momentum are
∗Translated the above kinematical quantities and the
equations satisfied by them into the language of spin-coefficient formalism due to Newman and Penrose andin the process have obtained the Lanczos potential forperfect fluid space-times. In fact we have proved thefollowings
Theorem 3. If in a given space-time there is a field of observers ui that is shear-free, irrotational and expanson-free, then the
Lanczos potential is given by
Lijk = −κ{m[i uj ] uk − 1
3m[i gj ]k} − κ{m[i uj ] uk − 1
3m[i gj ]k}
where
ui =1√
2(li + n j)
The Lanczos scalars Li(i = 0, 1, ....., 7) in this case are
Theorem 4. If in a given space-time there is a field of observers ui which is geodetic, shear-free, expansion-free and the vorticity vector is covariantly constant (i.e., ai = θ = σij = 0, ωi; j = 0)then the Lanczos potential is given by
Lijk =
√ 2
9ρ{2(mim j − mim j)uk
+(mimk − mimk)u j − (m jmk − m jmk)ui}where
ui =1
√ 2(li + ni)
The Lanczos scalars Li(i = 0, 1, ....., 7) in this case arefound to be as follows:
L1 = L6 =1
9ρ
L◦ = L2 = L3 = L4 = L5 = L7 = 0
∗Remarks:
1. There is some structural link between the spin coef-ficients and the Lanczos scalars
2. The Godel solution is characterized by
ai = θ = σij = 0, ωi; j = 0
ω =1
2
√ ωij ωij =
1
a√ 2= constant
The Godel solution is not a realistic model of the Uni-verse but it does possess a number of interesting prop-erties. The matter in this universe does not expand but
rotate. The solution also contains time-like lines, i.e.,an observer can influence his past. It may be notedhere that the hypothesis of Theorem 4 are infact theconditions of the Godel solution and thus obtained apotential for the Godel solution. Also it is shown thatGodel solution is of Petrov type D.
∗ The two parameter family of solutions which describethe space-time around black holes is the Kerr family
discovered by Roy Patrick Kerr in July 1963. The twoparameters are the mass and angular momentum of theblack hole. Kerr solution is just the Schwarzchild ex-terior solution with angular momentum. Using GHPformalism (a tetrad-formalism), we have obtained theLanczos potential for Kerr space-time as
L1 = (ψ2
M )1
3τ , L5 = −A(ψ2
M )1
3 ρ
which shows that Lanczos potential of Kerr space-timeis related to the mass parameter of the Kerr black holeand the Coulomb component of the gravitational field.
∗ A comparison between electromagnetism and gravita-tion is given in the Table on the next page:
5. Space-matter tensor∗ Through Sections 1-4, the importance of Riemann ten-sor is seen.
∗ Some other fourth rank tensors which involve Rie-mann tensor.
∗One such tensor, known as space-matter tensor, is
studied. A decomposition of this tensor is given in termsof Riemann tensor and an attempt has been made toexpress the space-matter in terms of electromagneticfield tensor. A symmetry of the space-time is definedin terms of the space-matter tensor and studied.
∗ Petrov (1969) introduced a fourth rank tensor whichsatisfies all the algebraic properties of the Riemann cur-vature tensor and is more general than the Weyl con-formal curvature tensor. This tensor is introduced asfollows:Let the Einstein’s field equations be
Rab − 1
2R gab = λ T ab
where λ is a constant and T ab is the energy-momentumtensor.
This tensor is known as space-matter tensor. The firstpart of this tensor represents the curvature of the spaceand the second part represents the distribution and mo-tion of the matter. This tensor has the following prop-erties:
(i) P abcd = −P bacd = −P abdc = P cdab, P abcd + P acdb + P adbc = 0
(ii) P ac = Rac − λ T ac + R2
gac + 3σgac = (R + 3σ)gac
(iii) If the distribution and the motion of the matter,i.e., T ab and the space-matter tensor, P abcd are given, thenRabcd, the curvature of the space is determined to withinthe scalar σ.
(iv) If T ab = 0 and σ = 0, then P abcd is the curvature of the empty space-time.
P abcd = C abcd + (gadRbc + gbcRad − gacRbd − gbdRac)
+(2
3R + σ)(gacgbd − gadgbc)
which can also be expressed as
P
h
bcd = C
h
bcd+(δ
h
d Rbc−δ
h
c Rbd+gbcR
h
d+gbdR
h
c )+(
2
3R+σ)(δ
h
c gbd−δ
h
d gbc)
∗ A classification of space-matter tensor, using matrixmethod, has been given by Ahsan and the different casesthat have been arrived at are compared with the Petrovclassification. It is found that case III(a) correspond toPetrov type III gravitational field. The algebraic prop-erties and the spinor equivalent of the space-matter ten-
sor have been obtained by Ahsan. Moreover, they havefound the covariant form of the invariants of the space-matter tensor and presented a criterion for the existenceof gravitational radiation, in terms of the invariants of the space-matter tensor.
∗ In general theory of relativity the curvature tensor de-scribing the gravitational field consists of two parts viz,the matter part and the free gravitational part. The in-
teraction between these two parts is described throughBianchi identities.
∗For a given distribution of matter, the construction of
gravitational potentials satisfying Einstein’s field equa-tions is the principal aim of all investigations in gravi-tation physics, and this has been often achieved by im-posing symmetries on the geometry compatible with thedynamics of the chosen distribution of matter. The geo-metrical symmetries of the space-time are expressiblethrough the vanishing of the Lie derivative of certaintensors with respect to a vector. This vector may be
time-like, space-like or null.
∗ Motivated by the role of symmetries in general rel-ativity, we have defined a symmetry in terms of thevanishing of the Lie derivative of the space-matter ten-sor and termed it as Matter Collineation
Definition 1. A matter collineation is defined to be apoint transformation xi
→xi + ξ idt leaving the form of
the space-matter tensor P hbcd invariant, that is
Lξ P hbcd = 0
where Lξ denotes the Lie derivatives along the vector ξ .
∗ Since every motion in a V n is a Weyl conformal collineation,we thus have
Theorem 5. A V n admits matter collineation if it admits mo-tion, Ricci collineation and σ = 0.
∗The energy-momentum tensor for a null electromag-
netic field is given by
T ab = F acF cb
where F ac = satc − tasc and sasa = sata = 0, tata = 1, vectorss and t are the propagation and polarization vectors,respectively.∗ The representation of space-matter tensor for a nullelectromagnetic field
P hbcd = C hbcd + 2(δ hd F bkF kc − δ hc F btF td + gbcF h p F pd − gbdF hf F f
c )
∗ Theorem 8. A null electromagnetic field admits matter collineation along the vector ξ (propagation/polarization) if ξ
is Killing and expansion-free.
∗ As we are working with the null electromagnetic field,it is therefore natural to expect that the Lichnerowiczcondition for total radiation are satisfied and we have
T ab = φ2kakb
where ka is the tangent vector. We now have
Definition 2. A null electromagnetic field admits a totalradiation collineation if