A study of the Riemann curvature tensor and Petrov classification in general relativity Текст научной статьи по специальности «Физика»

Том 153, кн. 3

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА

Физико-математические пауки

2011

UDK 514.763

A STUDY OF THE RIEMANN CURVATURE TENSOR AND PETROV CLASSIFICATION IN GENERAL RELATIVITY

Z. Ahsan

Abstract

In this article, we present, and support by examples a criterion for the existence of gravitational radiation in terms of the invariants of the Riemann curvature tensor. We give a classification of spacetimes in terms of electric and magnetic parts of the Weyl tensor and discuss some examples of spacetimes having purely magnetic and purely electric Weyl tensors. The Lanczos potential is studied using the method of general observers and tetrad formalisms. We obtain the Lanczos potentials for perfect fluid spacetimes, Godel cosmological model, and Kerr black hole. The work also considers the space-matter tensor, introduced by Pet.rov, and the perfect-fluid spacetimes with the divergence-free space-matter tensor.

Key words: Weyl tensor. Lanczos potential, tetrad formalisms, Godel model, Kerr black

Introduction

The general theory of relativity is a theory of gravitation in which gravitation emerges as the property of the space-time structure through the metric tensor gj. The metric tensor determines another object (of tensorial nature) known as Riemann curvature tensor. At any given event, this tensorial object provides all information about the gravitational field in the neighbourhood of the event. It may. in real sense, be interpreted as describing the enrvatnre of the spacetime. The Riemann enrvatnre tensor is the simplest non-trivial object one can build at a point: its vanishing is the criterion for the absence of genuine gravitational fields and its structure determines the relative motion of the neighbouring test particles via the equation of geodesic deviation. These discussions clearly illustrate the importance of the Riemann enrvatnre tensor in general relativity and it is for these reasons, a study of this enrvatnre tensor has been made here.

1. The invariants of Riemann tensor

The Riemann curvature tensor Rj is defined, for a covariant vector field Ak, through the Ricci identity fl]

Ai;ji — Ai;ij = RjAk, (1)

where = ^-V:, - ¿r^+r^-r^,

The Riemann enrvatnre tensor can be decomposed as fl]

Rijki = Cijki + Eijki + Gijki, (2)

where Cijki is the Weyl tensor, Eijki = ~\{gikSji + gjiSik - guSjk - gjkSu) is the Einstein enrvatnre tensor, with S'y = Rjj — -^gi.jR being the traceless tensor

arid Gijki = - — {gik9ji ~ 9a9jk)• The Ricci tensor /1':; is defined by //.-; =

and R = Rij is the Ricci scalar. These equations lead to a more convenient decomposition of the Riemann tensor as fl]

1 R

Rijki = Cijki + —(guRjk + 9jkRu — gikRji — gjiRik) — — (gugjk — gikgji)• (3)

R

There are four invariants of the Weyl tensor Cijkl. There are three invariants of the Einstein curvature tensor Eijkl and six invariants of the combined Weyl and Einstein curvature tensors. In empty spacetimes, there are four invariants of Riemann tensor which are given by

Ai = RijkiRljkl, A2 = Rjki Rijkl, 4 ..4

1 j _ _ i> T^mnrs j i ij 1 j _ _ />1 T^mnrs i> ij

D\ fLijmnfL £lrs , 132 g «J""1 rs '

These invariants have been calculated for the classifications of Riemann tensor according to Sharnia and Husain [2] and Petrov [3]. It has been found that A1, A2, B1, and B2 are all equal to zero for cases 111(a) and 111(b) or Petrov type III. This has led to the following criterion for the existence of gravitational radiation:

If Rabcd = 0 and A1 = A2 = B1 = B2 = 0, then the gravitational radiation is present; otherwise, there is no gravitational radiation.

The validity of this assertion has been checked by considering the following metrics «•I'.. [4]):

(i) Takeno's plane wave, solution

ds2 = —A dx2 — 2D dxdy — B dy2 — dz2 + dt2;

(ii) Einstein-Rosen metric.

ds2 = e2Y-2^ (dt2 — dr2) — r2e-2^ d<j>2 — e2^ dz2,

where 7 Mid ^ are functions of r and t only, ^ = 0 and 7 = 7(r — t);

(iii) The Peres metric

ds2 = —dx1 — dx2 — dx^ — 2f (dx^ + dx^)2 + dx^;

(iv) The Schwarzchild exterior solution

1 _ ^ dr2 _ r2d()2 _ r2 gin2

rr

It is found that for the metrics (i) (iii), all the four invariants of the Riemann tensor vanish and thus correspond to the state of gravitational radiation. While for the

A1 = 0, B1 = 0, A2 = 0, B2 = 0;

D

2. The electric and magnetic spacetimes

It is known that a physical field is always produced by a source, which is termed as its charge. Manifestation of fields when charges are at rest is called electric and magnetic when they are in motion. This general feature is exemplified by the Maxwell's theory of electromagnetism from which the terms of electric and magnetic are derived. This

decomposition can be adapted in general relativity and the Weyl tensor can be decomposed into electric and magnetic parts. Based on this decomposition, a classification of spacetimes is given here which is supported by a number of examples.

An observer with time like 4-velocity vector u is said [5] to measure the electric and magnetic components, Eac and Hac respectively, of the Weyl tensor Cabcd by

where the dual is defined to be *Cabcd = 7^eabef C^j- It is possible to choose a null tetrad in NP formalism such that the invariants of enrvatnre tensor can be expressed in terms of the electric and magnetic parts of the Weyl tensor. Electric and magnetic Weyl tensors of types I and D have been considered by Mcintosh et al. [6]. For the remaining Petrov types, we have

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

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

It is seen [7] that plane-fronted gravitational waves and Robinson Trantman types N

Trantman type II metric, the Schwarzchild and the Reissner Nordstrom solutions are purely electric.

It is known that an electromagnetic field can be generated by a potential, the question then arises that whether it is possible to generate the gravitational field through a potential. The answer is affirmative: this indeed can be done through the covariant differentiation of a tensor field Ljk [8]. This tensor field is now known as Lanczos potential and the Weyl tensor Chijk is generated by Ljk through the equation [9, 10]

Chijk ~~ LhiJ-b - Lhob-'i Lnbh--i - L-ib-i-h — (L P. „ "I- L F'„•. „ 1 Qhb

This equation is known as Weyl Lanczos equation.

For a gravitational field with perfect fluid source, the basic covariant variables are: the fluid scalars 9 (expansion), p (energy density), p (pressure); the fluid spatial vectors ui (4-acceleration), wi (vorticity); the spatial trace-free symmetric tensors aij (fluid shear), the electric (Eij) and the magnetic (Hij) parts of the Weyl tensor; the projection tensor hij which projects orthogonal to the fluid 4-velocity vector ui. We have expressed these quantities and the equations satisfied by them in terms of the Newman-Penrose formalism and in the process have obtained the Lanczos potential for perfect fluid spacetimes. In fact we have proved the following [11]:

ui

irrotational and expanson-free, then the Lanczos potential is given by

(4)

3. Lanczos spin tensor

where u% = —,-{V" + n?) and the Lanczos scalars Li{i = 0,1,...,7) in this case are

11 1

L0 = — -K, ¿2 = — ^L0, Ls = ^L0, = ~~ Li = L'Î = LA = Le = 0.

Theorem 4. If in a given spacetime there is a field of observers ui which is geodetic, shear-free, expansion-free and the vorticity vector is covariantly constant (i.e., ai = d = aij = 0, = 0), then the Lanczos potential is given by

J2

Lijk = p{2('niinij - 'nij'nij)v,k + (nii'nik - 'nii'nik)uj - (m/mt - mjmk)ui}, (7)

where u% = + n%) and the Lanczos scalars Li{i = 0,1,... ,7) are L\ = Lq = j-p,

V 2 9

L0 = L2 = L3 = L4 = L5 = L7 = 0.

It may bo noted here that the hypothesis of Theorem 4 are in fact the conditions of the Godol solution and thus, through Eq. (7), a Lanczos potential for the Godol solution is obtained.

The two-parameter family of solutions which describe the spacetime around black holes is the Kerr family discovered by Roy Patrick Kerr in July 19C3. The two parameters are the mass and angular momentum of the black hole. Using GHP formalism (a tetrad formalism), we have obtained Lanczos potential for Kerr spacetime as [11]

_ 1 /v2V/3 r _ ~A v/3

3 \M) ' L2-—{M) '

which shows that Lanczos potential of Kerr spacetime is related to the mass parameter of the Kerr black hole and the Coulomb component of the gravitational field. Here A is a constant.

4. Space-matter tensor

Potrov [12] introduced a fourth rank tensor which satisfies all the algebraic properties of the Riemann cnrvatnro tensor and is more general than the Woyl conformal cnrvatnro tensor. This tensor is defined as

Pabcd = Rabcd — Aabcd + &(9ac gbd — gad 9bc), (8)

where Aabcd = —(gac Tbd + gbd Tac — gad Tbc — gbc Tad) and Tab is given by the

Einstein's field equations Rab — ^R gab = A Tab. Here A is a constant and Tab is the

energy-momentum tensor. The tensor Pabcd is known as space-matter tensor. The first part of this tensor represents the cnrvatnro of the space and the second part represents the distribution and motion of the matter. From the equations of Section 1, we have

Pabcd = Cabcd + (gad Rbc + gbc Rad — gac Rbd — gbd Rac) +

+ ( + & ) [gac gbd - gad gbc), (9)

which can also be expressed as

Pbcd = Ctd + (Shd Rbc - 5hc Rbd + gbc Rhd + gbd Rhc ) + ( ^R + a ) (<5cft gbd - Shd gbc). (10)

The algebraic properties (including spinor equivalent) and the classification of the space-matter tensor have been studied by Ahsan [13 15]. The concept of matter collineation, defined in terms of the space-matter tensor, has also been introduced by Ahsan [16]. who obtained the necessary and sufficient conditions under which a spacetime, including electromagnetic fields, may admit such collineation. In this section, the divergence of the space-matter tensor has been expressed in terms of the energy-momentnm and Ricci tensors. Pcrfcct-fhiid spacetimes with divergence-free space-matter tensor have also been considered.

The space matter tensor can also be written as

PL. = Rid + \(Shc Tbd - shd Tbc + gbd T* - gbc Thd ) + a(Shc gbd - Shd gbc) (11)

so that the divergence of Phcd is given by

Pbcd;h = Rbcd;h + 9 - Tbc;d) + -(gbd - gbc Td-h) + (T,c gbd. - v,d gbc• (12) which on using the contracted Bianchi identities and T.ab = 0 reduces to

Pbcd,h = \{Tbr-d. - Tbd,e) + \ l(T + 2a),c gbd — (T + 2a),d gbc\ . (13)

We thus have

Theorem 5. If Tab is a Codazzi tensor and T = -2a, then the space-matter tensor is diwe 1 yence- fi-ee.

While in terms of Ricci tensor, the divergence of the space-matter tensor takes the form 1

Pbcd;h = Rbc;d. ~ Rbd.;c ~ -^(R,c gbd. ~ R ,d. gbc) ■ (14)

We thus have

a=0

vanishes.

For a pcrfcct-fhiid distribution, the energy-momentnm tensor is given by-

Tab = (M + p) ua Ub + pgab, (15)

where m is the energy density, p is the isotropic pressure and ua is the fluid four velocity-vector. Consider now the fluid spacetime with divergence-free space-matter tensor and we have

Theorem 7. If a = —7T and the fluid spacetime has divergence-free space-matter

tensor, then either (m + p) = 0 (that is, the perfect fluid spacetime satisfies the vacuumlike equation of state) or the spacetime represents a Friedmann Robertson Walker cosmological model with (m — 3p) as constant.

For the proofs of these theorems and other related results, see [17].

92

Z. AHSAN

Резюме

3. Axcau. Исследование римаиова тензора кривизны и классификации Петрова в общей теории относительности.

На основе инвариантов римаиова тензора кривизны представлен и проиллюстрирован различными примерами критерий существования гравитационного излучения. Дана классификация пространств-времен исходя из электрической и магнитной составляющих тензора Вейля. Рассмотрены некоторые примеры прострапств-времеп, у которых тензор Вейля содержит только магнитную или только электрическую часть. Проведено исследование потенциала Лапцоша с помощью метода общих наблюдателей и тетрадного формализма, получен потенциал Лапцоша для прострапств-времеп идеальной жидкости. Кроме того, получен потенциал Лапцоша для космологической модели Гёделя и черной дыры Керра. Рассмотрены тензор пространства-материи, введенный Петровым, и пространства-времена идеальной жидкости с тензором пространства-материи, не содержащим дивергентной части.

Ключевые слова: тензор Вейля. потенциал Лапцоша, тетрадный формализм, модель Гёделя. черпая дыра Керра.

References

1. Ahsan Z. Tensor Analysis with Applications. - New Delhi: Anamaya Publ.; Tunbridge Wells, UK: Anshan. - 2008. - 260 p.

2. Sharma D.N., Husain S.I. Algebraic Classification of the Curvature Tensor in General Theory of Relativity // Proc. Nat. Acad. Sci. India A. - 1969. - V. 39, No 3. - P. 405-413.

3. Petrov A.Z. Klassifikacya prostranstv, opredelyayushchikh polya tyagoteniya // Uchenye Zapiski Kazanskogo Universiteta. - 1954. - V. 114, bk. 8. - P. 55-69.

4. Ahsan Z., Husain S.I. Invariants of curvature tensor and gravitational radiation in general relativity // Indian J. Pure Appl. Maths. - 1977. - V. 8. - P. 656-662.

5. Ellis G.F.R. Relativistic Cosmology // General Relativity and Cosmology / Ed. by R.K. Sachs. - N. Y.; London: Acad. Press, 1971. - P. 104-182.

6. McIntosh C.B.G., Arianrhod R., Wade S.T., Hoenselaers C. Electric and magnetic Weyl tensors: classification and analysis // Class. Quantum. Grav. - 1994. - V. 11, No 6. -P. 1555-1564.

7. Ahsan Z. Electric and magnetic Weyl tensors // Indian J. Pure Appl. Maths. - 1999. -V. 30. - P. 863-869.

8. Lanczos C. The splitting of the Riemann tensor // Rev. Mod. Phys. - 1962. - V. 34, No 3. - P. 379-389.

9. Dolan P., Kim C. W. Some Solutions of the Lanczos Vacuum Wave Equation // Proc. R. Soc. London A. - 1994. - V. 447. - P. 577-585.

10. Dolan P., Muratori B.D. The Lanczos potential for vacuum space-times with an Ernst potential // J. Math. Phys. - 1998. - V. 39, No 10. - P. 5406-5420.

11. Ahsan Z., Ahsan N., Ali S. Lanczos potential and tetrad formalisms // Bull. Cal. Math. Soc. - 2001. - V. 93, No 5. - P. 407-422.

12. Petrov A.Z. Einstein Spaces. - Oxford: Pergamon Press, 1969. - 411 p.

13. Ahsan Z. . Algebraic classification of space-matter tensor in general relativity // Indian J. Pure Appl. Maths. - 1977. - V. 8. - P. 231-237.

14. Ahsan Z. Algebra of space-matter tensor in general relativity // Indian J. Pure Appl. Maths. - 1977. - V. 8. - P. 1055-1061.

15. Ahsan Z. A note on the space-matter spinor in general relativity // Indian J. Pure Appl. Maths. - 1978. - V. 9. - P. 1154-1157.

16. Ahsan Z. A Symmetry Property of the Space-Time of General Relativity in Terms of the Space-Matter Tensor // Braz. J. Phys. - 1996. - V. 26, No 3. - P. 572-576.

17. Ahsan Z., Siddiqui S.A. On the Divergence of the Space-matter Tensor in General Relativity // Adv. Studies Theor. Phys. - 2010. - V. 4, No 11. - P. 543-556.

Поступила в редакцию 19.10.10

Ahsan, Zafar PliD. Professor. Department of Mathematics, Aligarli Muslim University, Aligarli, India.

Ахсан, Зафар доктор паук, профессор отделения математики Алигархского мусульманского университета, г. Алигарх, Индия. E-mail: zafar.ahsanOndiffmail.com