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On the Riemann Curvature

Tensor in General Relativity

Zafar Ahsan

Department of MathematicsAligarh Muslim University

Aligarh-202 002(India)e-mail:[email protected]

Dedicated to Prof. Hideki Yukawa

(First Noble Laureate in Physics from Asia)



LECTURE PLAN

1. Introduction

2. The Riemann curvature tensor

3. Electric and magnetic space-times

4. Lanczos spin tensor

5. Space-matter tensor

2



1. Introduction

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.

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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

Ai; jl − Ai;lj = Rkijl Ak

where

Rkijl =∂

∂x j Γkil −∂

∂xl Γkij + Γmil Γkmj − Γmij Γkml (1)

∗ The covariant form of the curvature tensor is

Rhijl =1

2(

∂ 2ghl

∂x j∂xi+

∂ 2gij

∂xl∂xh− ∂ 2gil

∂x j∂xh− ∂ 2ghj

∂xl∂xi)+gkm(Γk

ijΓmhl−Γk

ilΓmhj)

4



Properties:

(i) Rhijl = −Rhilj, Rhijl = −Rh

ilj

(ii) Rhijl = −Rihjl, Rhijl = R jlhi

(iii) Rkijl + Rk

jli + Rklij = 0, Rhijl + Rhjli + Rhlij = 0

(iv) Bianchi identities:

Rhijk;l + Rhikl; j + Rhilj;k = 0, Rhijk ;l + Rh

ikl; j + Rhilj;k = 0

∗ 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

5



E ijkl = −12

(gikS jl + g jl S ik − gilS jk − g jk S il), (3)

with S ij being the traceless tensor

S ij ≡ Rij − 1

4gij R, (4)

and where Gijkl is defined by

Gijkl ≡ − R12(gikg jl − gilg jk ) (5)

The Ricci tensor Rij is defined by

Rij ≡ Rkijk and R ≡ gij Rij (6)

is the Ricci scalar. Expression (2) may be regardedas the definition of the Weyl tensor, all of the other

quantities may be computed directly if the metric tensoris given.

∗ From eqns. (3)-(5), eqn (2) can be written as

Rijkl = C ijkl+1

2(gilR jk +g jk Ril−gikR jl−g jl Rik)−R

6(gilg jk−gikg jl )

(7)

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∗Geometric properties: Ruse, Geheniau and Debever

∗ 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.

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∗Type I (Algebraically general)

∗ Type II, D, III, N (Algebraically special)

∗ Types III, N : Gravitational Radiation

Remark:

∗ 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

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Rijkl = C ijkl+1

2(gilR jk +g jk Ril−gikR jl−g jl Rik)−R

6(gilg jk−gikg jl )

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

9



Weyl tensor C ijkl :

A1 = C ijklC ijkl, A2 = C ∗ijklC ijkl

B1 =4

3C ijmnC mnrsC ij

rs , B2 =4

3C ∗ijmnC mnrsC ij

rs

Einstein curvature tensor E ijkl:

E = 2E ijklE ijkl

F = 4E ijrsE rs

mn E mnpq

E ij pq −

1

2E 2

3G = 32E ijrsE rsmn E mnpq E uvpq E uvxyE ij

xy − 3EF − E 3

Combined Weyl and Einstein curvature tensors:

K 1 =1

4C ijrsE rs

mnE ijmn, K 2 =1

4C ∗ijrsE rs

mnE ijmn

2L1 = 32C ijrsE rsmnE pqmnC pqxyE xyuvE ij

uv − K 21 + K 22

2L2 = 32C ∗ijrsE rsmnE pqmnC pqxyE xyuvE ij

uv − 2K 1K 2

3M 1 = 256C ijrsE rsmnE ijmnC ijklE klpq E uvpq C uvwxE wxyz

E ijyz − K 31 + 3K 1K 22 − 18K 1L1 + 18K 2L2

3M 2 = 256C ∗ijrsE rsmn

E ijmnC ijkl

E klpq E uvpq

C uvwx

E wxyzE ijyz − 18K 1L2 − 18K 2L1 − 3K 21K 2 + K 32

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∗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).

11



(iii) The Peres metric

ds2 = −dx21 − dx2

2 − dx23 − 2f (dx4 + dx3)2 + dx2

4

(iv) The Schwarzchild exterior solution

ds2 = −(1 − 2m

r)−1dr2 − r2dθ2 − r2sin2θdφ2 + (1 − 2m

r)dt2

∗ 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.

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∗An observer moving with time-like 4-velocity ua in a

relativistic electromagnetic field measures the electricand magnetic components of the field, respectively, by

E a = F ab ub, H a = F ∗ab ub

Here F ab = −F ba is the Maxwell stress tensor with dualF ∗ab = 1

2abcd F cd, and the 4-velocity of the observer is

normalized so that uc uc = −1

∗ An observer with time like 4-velocity vector u is saidto measure the electric and magnetic components, E ac

and H ac respectively, of the Weyl tensor C abcd by

E ac = C abcd ub ud, H ac = ∗C abcd ub ud

where the dual is defined to be ∗C abcd = 12

abef C ef

cd.

AlsoE ab + iH ab = Qab = C abcd ub ud

whereQab = Qba, Qa

a = Qab ub = 0,

C abcd = C abcd + i ∗C abcd

andE ab = E ba, E ab ub = 0, E ab gab = 0

H ab = H ba, H ab ub = 0, H ab gab = 0

∗ The Weyl tensor is said to be purely electric if H ac = 0and purely magnetic if E ac = 0. The Weyl tensor in termsof E and H can be decomposed as

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C cdab = 2u[aE b]

[cud] + δ [c[aE

d]b] − ηabef ue H f [cud] − ηcdef ue H f [aub]

which equivalently can be written as

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

14



ψ0 = ψ4 = 12

(λ2 − λ1), ψ1 = ψ3 = 0, ψ2 = −12

λ3

Clearly, if in this frame the Newman-Penrose compo-nents ψABCD are real (imaginary) the Weyl tensor ispurely electric (magnetic).

∗ The two independent invariants I and J of Weyl tensorin terms of Q are

I = Qab Qb

a, J = Qab Qb

c Qca,

(or, I = ψCDAB ψAB

CD , J = ψCDAB ψEF

CD ψABEF )

which can be written as

I = (E ab E ba−H ab H ba)+2i E ab H ba =1

8(C abcd C abcd +iC abcd

∗C abcd)

and

J = (E ab E bc E ca − 3E ab H bc H ca) − i(H ab H bc H ca − 3E ab E bc H ca)

=1

16(C abcd C cdef C ab

ef + i ∗C abcd C cdef C abef )

From above eqns.

I = 12(λ21 + λ22 + λ23), J = 16(λ31 + λ32 + λ33)

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∗In terms of the components of the Weyl tensor

I = ψ0 ψ4 − 4ψ1 ψ3 + 3ψ22

J =

ψ4 ψ3 ψ2

ψ3 ψ2 ψ1

ψ2 ψ1 ψ0

∗ Types I and D electric and magnetic Weyl tensorshave been considered by Mc Intosh et al.

∗ For the remaining Petrov types we proceed as follows:

∗ For Petrov type II, the Newman-Penrose componentsof Weyl tensors are

ψ0 = ψ1 = ψ3 = 0, ψ2 = −λ

2, ψ4 = −2

where

λ1 = λ2 = −λ2

, λ3 = λ,

which, from above eqns. take the form

I =3

4λ2, J =

λ3

8

andI = 3ψ2

2, J =

−ψ3

2

alsoI 3 = 27J 2

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∗We thus have

Theorem 1. For type II, the Weyl tensor is purely electric (magnetic) if and only if ψ2 (or λ) is real (imaginary).

∗ Further, for type N , we have

λ1 = λ2 = λ3 = 0 = λ and ψ0 = ψ1 = ψ2 = ψ3 = 0, ψ4 = −2

which gives I = J = 0.

∗ While, for type III, we have

λ1 = λ2 = λ3 = 0 and ψ0 = ψ1 = ψ2 = ψ4 = 0, ψ3 = −i

which gives I = J = 0. We thus have

Theorem 2. Types III and N Weyl tensors are neither purely electric nor purely magnetic .

∗ Examples:

(a) Plane Fronted Gravitational Waves . The metric is

ds2 = 2H (u,x,y) du2 + 2 du dr − dx2 − dy2.

This is a space of Petrov type N .

(b) The Schwarzchild solution . The Schwarzchild solutionin null coordinates is

ds2 = (1 − 2m

r) du2 + 2 du dr − r2(dθ2 + sin2θ dφ2)

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(c) The Reisner-Nordstrom Solution . This is just the Schwarzchildspace-time with a static non-singular electromagneticfield and is given by

ds2 = 2(1 − m

r+

2e2

r2) du2 + 2 du dr − r2

2 p2dξ dξ.

(d) The Robinson-Trautman space-time . This is a family of

metrics which generalize the Schwarzchild solution toPetrov types II, III and N and is given by

ds2 = (−2U 0 + 4γ 0 r + 2ψ0

2

r) du2 + 2 du dr − r2

2 p2dξ dξ.

∗ Plane fronted gravitational waves and Robinson-Trautmantypes III and N metrics: neither purely electric norpurely magnetic.

∗ Radiative space-times : neither purely electric norpurely magnetic.

∗ The Robinson-Trautman type II metric, the Schwarzchildand the Reissner-Nordstrom solutions are purely elec-tric.

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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:

(40 conditions ) Lijk = −L jik

(4 conditions ) L tit = 0 (or, gkl Lkil = 0)

(4 conditions ) Lijk + L jki + Lkij = 0 (or, ∗L tit = 0)

L kij;k = 0 or (Lk

ij + Ljk

ij ); k = 0

The Weyl tensor C hijk is generated by Lijk through theequation (Weyl-Lanczos equation)

C hijk = Lhij;k − Lhik; j + L jkh;i − L jki;h + L(hk) gij + L(ij) ghk

−L(hj) gik − L(ik) ghj +2

3 L pq p;q (ghj gik − ghk gij)

where

19



Lij = L ki j;k − L k

i k; j

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

20



wave equation obtained by Illge

∗ Edgar and Hoglund (1997): identities

∗ Novello and Velloso (1988): An algorithm for calculat-ing the Lanczos potential for perfect fluid space times,under certain conditions

∗ Lopez-Bonilla, Ares deParge and coworkers (1989, 93,

97,98,2000): using the Newmann-Penrose formalism,obtained the Lanczos potential for various algebraicallyspecial space times

∗ Bergqvist (1997): Lanczos potential in Kerr geometry

∗ Dolan and Muratori (1998): relationship between theLanczos potential for vacuum space times and the Ernstpotential

∗ Edgar and Hoglund (2002): Lanczos potential forWeyl tensor does not exist for all spaces of dim n ≥ 7

∗ Cartin (2003): Lpt act as a tensor potential for thelinear spin theory

∗ Ahsan, Barrera-Figueroa and Lopez-Bonilla (2006):

Using the methods of local and isometric embedding,Lovelock’s theorem and method of wave equation, ob-tained a potential for Godel cosmological model

21



∗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

22



as follows:

(i) The projection tensor hij :

hij = gij − ui u j, hij = h ji , h ki hkj = hij , hi

i = 3, hij u j = 0

(ii) The expansion scalar θ: θ = ui;i

(iii) The acceleration vector ai:

ai = .ui = ui; j u j, ai ui = 0

(iv) The symmetric shear tensor σij:

σij = hikh j

lu(k;l) − 1

3θ hij , σij u j = 0, σ i

i = 0

(v) The anti-symmetric vorticity or rotation tensor ωij:

ωij = hi

k

h j

l

u[k;l], ωij u

i

= ωij u

j

= 0and is equivalent to a vorticity vector

ωi =1

2ηijkl ω jk ulso that ωij = ηijkl ωk ul

where ηijkl is completely anti-symmetric Levi-Civita ten-sor. It may be noted that ai, σij and ωij are space like.

(vi) The electric and magnetic parts of the Weyl tensor

as measured by an observer with a time like 4-velocityvector ui are:

E ik = C ijkl u j ul, H ik = ∗C ijkl u j ul =1

2ηij

mnC mnkl u j ul

23



AlsoE ik = E ki, E ik uk = 0, E ik gik = E tt = 0

H ik = H ki, H ik uk = 0, H ik gik = H tt = 0

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

T ij; j = 0

24



∗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

found to be as follows:

L◦ = −1

2κ, L2 = −1

3L◦, L5 =

1

3L◦, L7 = −L◦

L1 = L3 = L4 = L6 = 0

25



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

26



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:

27



Fields F ij C hijk

Potential Ai Lijk

Field relations F ij = Ai; j − A j;i C hijk = W (L)hijk

Gauge invariance Ai = Ai + χ,i L

ijk = Lijk + χijk

W ij(A) = W ij(A) W hijk(L) = W hijk(L

Gauge conditions Ai;i = 0 L t

ij;t = 0

Field equations F ij; j = J i C t

ijk ;t = J ijk

Potential wave Ai + R ki Ak = J i Lijk + 2R t

k Lijt

equation in −R ti L jkt − R t

j Lk

matter −gik R pt L pjt + g jk R pt

−1

2R L

ijk =J

ijk

Potential wave Ai + R ki Ak = 0 Lijk = 0

equation in vacuo

Field wave F ij + Rti F tj − Rt

j F ti Rhijk + 4Rhpq [ jRk]q

equation in matter −2Rrisj F rs − F ti ;t; j −Rhipq R

pq jk

+F t j ;t;i = 0 +2Rq [kR j]qhi

+2R j[i;h];k + 2Rk[h;i]; j =

Field wave F ij + Rti F tj C hijk + 4C hpq [ jC k]

q

equation in vacuo −Rt j F ti − 2Rrisj F rs = 0 −C

pq hi C pqjk = 0

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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.

Introduce a fourth order tensor

Aabcd =λ

2(gac T bd + gbd T ac − gad T bc − gbc T ad)

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From the definition this tensor has properties:

Aabcd = −Abacd = −Aabdc = Acdab, Aabcd + Aacbd + Aadbc = 0

which leads to

Aac = λ T ac +λ

2T gac = λ T ac − R

2gac

Define a new fourth order tensor

P abcd = Rabcd − Aabcd + σ(gacgbd − gadgbc)

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.

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(v) If gab, the metric tensor, σ, the scalar and P abcd areknown, then T ab can be determined uniquely.

In section 2, we have seen that the Riemann curva-ture tensor can be decomposed as

Rabcd = C abcd + E abcd + Gabcd

where C abcd is the Weyl tensor, E abcd is the Einstein cur-vature tensor, defined by

E abcd = −1

2(gacS bd + gbdS ac − gadS bc − gbcS ad)

where

S ab = Rab − 1

4gab R

being the trace-less Ricci tensor, and Gabcd is defined by

Gabcd = − R

12(gacgbd − gadgbc)

These eqns lead to

Rabcd = C abcd +1

2(gadRbc + gbcRad − gacRbd − gbdRac)

−R

6(gadgbc − gacgbd)

Above eqns. also lead to

Aabcd = 12

(gacRbd + gbdRac − gadRbc − gbcRad)

−R

2(gacgbd − gadgbc)

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∗Thus space-matter tensor may be decomposed as

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.

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∗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.

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∗Now consider a V n for which Rij = 0 = σ and denote

this space as V 0n , then

Theorem 6. In a V 0n every motion is a matter collineation .

∗ The energy momentum tensor T ab of an electromag-netic field is defined by

T ab = F akF kb − 1

4gabF ij F ij

∗ Representation of space-matter tensor in case of anon-null electromagnetic field

P abcd = C abcd − gacF bkF kd − gbdF apF pc + gadF btF tc + gbcF axF xd

+(σ +1

2F ijF ij)(gacgbd − gadgbc)

from which we have

P hbcd = C hbcd − δ hc F bkF kd − gbdF h p F pc + δ hd F btF tc + gbcF hx F xd

+(σ +1

2F ijF ij)(δ hc gbd − δ hd gbc)

∗ Since for non-null electromagnetic fields, that Lξ gij =0 ⇒ Lξ F ij = 0. Thus taking the Lie derivative of theabove eqn.

Theorem 7. A non-null electromagnetic field admits matter collineation if and only if it admits motion and σ = 0.

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∗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

Lξ T ab = 0, where T ab is defined

as above.

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∗Since a null electromagnetic field with T ab = φ2kakb is

known to be a pure radiation field, taking the Lie deriv-ative of this T ab, we have

∗ Theorem 9. Pure radiation fields admit total radiation collineation if and only if the tangent vector ka defines a null geodesic.

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tensors-analysis of spaces

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