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Singularities in homogeneous world models
PHYSICS
Volume 17, number 3
SINGULARITIES
IN
LETTERS
HOMOGENEOUS
15 July 1965
WORLD
MODELS
S. HAWKINS and G. F. R. ELLIS Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England
Received 5 June 1965
The question of singularities in cosmological models in General Relativity has received considerable attention [e.g. l-61. If a world model is spatially homogeneous and isotropic, and contains matter with ordinary properties, then there is a singularity [7,8]. If we assume homogeneity but not isotropy of the universe filled with ordinary matter this does not exclude rotation or acceleration *, and so there is a possibility of nonsingular world models. It has been shown by Shepley [2], for certain particular homogeneous world models, that there must be a singularity. We will prove a more general result for homogeneous world models, using methods similar to those of Penrose [9]. Consider a general relativistic model of the universe, filled with continuous matter; with field equations Gab = R.ab - igab R = Stab. Tab will represent the stress energy tensor of matter. The character of the space-time will be specified as follows: a) the space-time manifold M4 is a Riemanian manifold, signature (+---), in which all null and time-like geodesics can be extended to arbitrary null and affine parameter distances. Any models in which this were not the case would not seem to be reasonable models of the universe. b) the tensor Tab represents a perfect fluid, i.e. Tab = (p-p) UaUb + #gab, where ~1> 0, p > p > -5~; ua is the velocity vector of matter, and is uniquely defined geometrically as the time-like eigenvector of l&b. If the flow lines ua intersect, the density will be infinite (because of the continuity equation Tab; b = 0) and hence there will be a physical singularity. Also b) implies R,& VaVb> 0 for any time-like Or ndl VeCtor va. The condition of homogeneity will be assumed in a strong form: c) a group of motions Gr exists in Mq, r > 3, which is transitive on at least one spacelike hypersurface; but spacetime is not stationary. This implies that a spacelike hypersurface of transitivity exists, defined uniquely by the group of motions. 246
We will show that a), b), c) imply there is a physical singularity in the universe. Consider a spacelike hypersurface H of transitivity of the group. This must be a surface of constant T; therefore T, a will be in the direction of the timelike normal to the surface. Let T, .T’ a = f z 0. Then f is constant on H, as f is a differential COnCOmitaId of Rabcd. Let va = e(TT a)T, a/Jf, where e is an indicator: 9 = + 1 if Tt a is in the positive time direction, e = - 1 if T, a is in the negative time direction. Then = 0. Thus vais a tional geodesic unit vecrors. If fl = va; a and uab = v(a; bj +
- ;(gaJ, - vavb) 8, then 8, bvb = -+ e2 - oabsb
$
- Rabvavb (Raychaudhuri’ s equation [I, lo]) gives the time development of the expansion 0 of the world lines. But oaboab 3 0, and RabvaVb > 0 by (b), and so e bvb < +2. If e c 0, then e will become infinite ‘in finite proper time. If e > 0, then let via = -vat and 81, = - 8. Then @bvb < - 48’2, so 8’ will become infinite in finite proper time.
1) If we assume that the surfaces T = constant remain spacelike, this means that the convergence of their unit normals becomes infinite, which implies that they must degenerate into, at most, a 2-surface T = To: call this C. Suppose M4 is non-singular everywhere. Then a flow line through C must have passed through H. Let F be the set of all flow lines through C. Then M, the intersection of F and H, is not empty. Since C is uniquely defined by the geometry, F is also uniquely defined, and so M is a uniquely defined subset of H. Since there is a group of motions transitive on H, M must be H itself. Thus al2 the flow lines intersect in C. Hence, there is a * It has been sometimes assumed that rotation being zero implies that flow lines are orthogonalto the surfaces of transitivity, and that acceleration ii therefore zero. This is not necessarily SO.
Volume 1’7, number 3
PHYSICS
physical singularity at C, where n is infinite. 2) If the surfaces of transitivity are not all spacelike, there must be at least one lightlike surface T = const, say S. Then on S, f = 0; T9 a f 0 *. Suppose IQ is non-singular everywhere. TYa is then tangent to a null geodesic on 5. Introduce a null geodesic congruence La throughout SpZi.Ce, SCC3; ,, Lb = 0 and LaL, = 0; andtakeL,aT ,a Define p = -$La; a, the convergence of the
LETTERS
15 July 1965
one homogeneous space section; and add a) there are equations of state such that the Cauchy development of M is determinate; then it can be shown that succeeding space like surfaces of constant T are homogeneous; and much the same proof can be given that there is a physical singularity if a) b) c) c’) d) hold. Work is proceeding on the generalization of these results to the case when there is no exact group of motions; full details of the extended results wilI be published shortly.
congruence and aa = $(L% b)La; b - 2p2), representing the shear of the congruence [ll]. Then o; bLb = o2 + 05 + $HabLaLb *. By a Similar argument to the previous, pbecomes infinite within a finite affine parameter distance; we can reach this finite parameter value, by (b). The only freedom we have in the choice of La on the surface S is L’, = PL,, where B bLb = 0, i.e. p is constant along the null ray defin;d by L, a. Thus L’a. a = PL”. and so L’a.,.willbe infinite when La’ is i&&e. Thus the 2-‘s&face (which can be shown to be a l-surface) on S on which La. a is infinite, is uniquely defined; which implids that the group has a set of fixed points on S. The argument used previously shows there must be a physical singularity there since all the flow lines again pass through this l-surface. It it then not meaningful to call the singular surface S null; in fact, case (2) turns out to be the same as case (1). The assumption of a group of motions throughout space-time is rather strong. If we replace c) by c’); c’) there is a spacelike hypersurface II in which there are at least 3 independent vectors Ea such that d%;Lagbc= 0, &?xaI&de = 0 On H, but M is not stationary **. I.e. the universe has
The authors would like to thank D. W. Sciama, C. W. Misner and L. Shepley for helpful criticism.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
A.Raychaudhuri, Phys.Rev.98 (1955) 1123. L. Shepley, Proo. Nat. Acad.Soi. 52 (1964) 1403. C.Behr, Z.Ap.60 (1965) 286. E.M.LifshitzandI.M.KhaIainikov,Adv.Phys.12 (1963) 185. O.IIeclanamr and E.SohUckingz in Gravitation, ed. L.Witten (Wiley. N.Y., 1962). A.Komar, Phye.Rev.104 (1956) 544. H.P.Robertson, Rev.Mod.Phys.5 (1933) 62. II. Bondi, Cosmology (Cambri&Ie University Press, 1961). R.Penrose, Phycl.Rev.Lettera 14 (1965) 57. J. EhIers, Abh. Akad. Wise .Mainz Mat. -Nat. KI. No. 11 (1961). E.Newman and R.Penrose, J.Math.Phys.3 (1962) 566.
*IfT*‘=O, wetakeanyotherecaIarpolynomiaIin the curvature tensor and ita covariant derivatives instead of T. Theee cannot alI be can&ant if the universe ia non-stationary. ** 2 x8 denotes the Lie derivative in the direction of Xa.
*****
PLASMA
ACOUSTIC
WAVES
ON AN INHOMOGENEOUS
COLUMN
+
F. W. CRAWFORD and J. A. TATAHONIS Institute
for Plasma
Research,
Stanford University,
Stanford,
California
Received 9 June 1965
In the last year or two careful experiment and rather sophisticated computation have established t This work was eupported by the U.S. Atomic Energy Commission.
clearly the mechanism of the well-known TonksDattner resonances exhibited by a plasma column irradiated by a transverse r-f. electric field [l-4]. Their origin has been shown to be due to radial propagation of plasma acoustic waves. At 241
Volume 17, number 3
SINGULARITIES
IN
LETTERS
HOMOGENEOUS
15 July 1965
WORLD
MODELS
S. HAWKINS and G. F. R. ELLIS Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England
Received 5 June 1965
The question of singularities in cosmological models in General Relativity has received considerable attention [e.g. l-61. If a world model is spatially homogeneous and isotropic, and contains matter with ordinary properties, then there is a singularity [7,8]. If we assume homogeneity but not isotropy of the universe filled with ordinary matter this does not exclude rotation or acceleration *, and so there is a possibility of nonsingular world models. It has been shown by Shepley [2], for certain particular homogeneous world models, that there must be a singularity. We will prove a more general result for homogeneous world models, using methods similar to those of Penrose [9]. Consider a general relativistic model of the universe, filled with continuous matter; with field equations Gab = R.ab - igab R = Stab. Tab will represent the stress energy tensor of matter. The character of the space-time will be specified as follows: a) the space-time manifold M4 is a Riemanian manifold, signature (+---), in which all null and time-like geodesics can be extended to arbitrary null and affine parameter distances. Any models in which this were not the case would not seem to be reasonable models of the universe. b) the tensor Tab represents a perfect fluid, i.e. Tab = (p-p) UaUb + #gab, where ~1> 0, p > p > -5~; ua is the velocity vector of matter, and is uniquely defined geometrically as the time-like eigenvector of l&b. If the flow lines ua intersect, the density will be infinite (because of the continuity equation Tab; b = 0) and hence there will be a physical singularity. Also b) implies R,& VaVb> 0 for any time-like Or ndl VeCtor va. The condition of homogeneity will be assumed in a strong form: c) a group of motions Gr exists in Mq, r > 3, which is transitive on at least one spacelike hypersurface; but spacetime is not stationary. This implies that a spacelike hypersurface of transitivity exists, defined uniquely by the group of motions. 246
We will show that a), b), c) imply there is a physical singularity in the universe. Consider a spacelike hypersurface H of transitivity of the group. This must be a surface of constant T; therefore T, a will be in the direction of the timelike normal to the surface. Let T, .T’ a = f z 0. Then f is constant on H, as f is a differential COnCOmitaId of Rabcd. Let va = e(TT a)T, a/Jf, where e is an indicator: 9 = + 1 if Tt a is in the positive time direction, e = - 1 if T, a is in the negative time direction. Then = 0. Thus vais a tional geodesic unit vecrors. If fl = va; a and uab = v(a; bj +
- ;(gaJ, - vavb) 8, then 8, bvb = -+ e2 - oabsb
$
- Rabvavb (Raychaudhuri’ s equation [I, lo]) gives the time development of the expansion 0 of the world lines. But oaboab 3 0, and RabvaVb > 0 by (b), and so e bvb < +2. If e c 0, then e will become infinite ‘in finite proper time. If e > 0, then let via = -vat and 81, = - 8. Then @bvb < - 48’2, so 8’ will become infinite in finite proper time.
1) If we assume that the surfaces T = constant remain spacelike, this means that the convergence of their unit normals becomes infinite, which implies that they must degenerate into, at most, a 2-surface T = To: call this C. Suppose M4 is non-singular everywhere. Then a flow line through C must have passed through H. Let F be the set of all flow lines through C. Then M, the intersection of F and H, is not empty. Since C is uniquely defined by the geometry, F is also uniquely defined, and so M is a uniquely defined subset of H. Since there is a group of motions transitive on H, M must be H itself. Thus al2 the flow lines intersect in C. Hence, there is a * It has been sometimes assumed that rotation being zero implies that flow lines are orthogonalto the surfaces of transitivity, and that acceleration ii therefore zero. This is not necessarily SO.
Volume 1’7, number 3
PHYSICS
physical singularity at C, where n is infinite. 2) If the surfaces of transitivity are not all spacelike, there must be at least one lightlike surface T = const, say S. Then on S, f = 0; T9 a f 0 *. Suppose IQ is non-singular everywhere. TYa is then tangent to a null geodesic on 5. Introduce a null geodesic congruence La throughout SpZi.Ce, SCC3; ,, Lb = 0 and LaL, = 0; andtakeL,aT ,a Define p = -$La; a, the convergence of the
LETTERS
15 July 1965
one homogeneous space section; and add a) there are equations of state such that the Cauchy development of M is determinate; then it can be shown that succeeding space like surfaces of constant T are homogeneous; and much the same proof can be given that there is a physical singularity if a) b) c) c’) d) hold. Work is proceeding on the generalization of these results to the case when there is no exact group of motions; full details of the extended results wilI be published shortly.
congruence and aa = $(L% b)La; b - 2p2), representing the shear of the congruence [ll]. Then o; bLb = o2 + 05 + $HabLaLb *. By a Similar argument to the previous, pbecomes infinite within a finite affine parameter distance; we can reach this finite parameter value, by (b). The only freedom we have in the choice of La on the surface S is L’, = PL,, where B bLb = 0, i.e. p is constant along the null ray defin;d by L, a. Thus L’a. a = PL”. and so L’a.,.willbe infinite when La’ is i&&e. Thus the 2-‘s&face (which can be shown to be a l-surface) on S on which La. a is infinite, is uniquely defined; which implids that the group has a set of fixed points on S. The argument used previously shows there must be a physical singularity there since all the flow lines again pass through this l-surface. It it then not meaningful to call the singular surface S null; in fact, case (2) turns out to be the same as case (1). The assumption of a group of motions throughout space-time is rather strong. If we replace c) by c’); c’) there is a spacelike hypersurface II in which there are at least 3 independent vectors Ea such that d%;Lagbc= 0, &?xaI&de = 0 On H, but M is not stationary **. I.e. the universe has
The authors would like to thank D. W. Sciama, C. W. Misner and L. Shepley for helpful criticism.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
A.Raychaudhuri, Phys.Rev.98 (1955) 1123. L. Shepley, Proo. Nat. Acad.Soi. 52 (1964) 1403. C.Behr, Z.Ap.60 (1965) 286. E.M.LifshitzandI.M.KhaIainikov,Adv.Phys.12 (1963) 185. O.IIeclanamr and E.SohUckingz in Gravitation, ed. L.Witten (Wiley. N.Y., 1962). A.Komar, Phye.Rev.104 (1956) 544. H.P.Robertson, Rev.Mod.Phys.5 (1933) 62. II. Bondi, Cosmology (Cambri&Ie University Press, 1961). R.Penrose, Phycl.Rev.Lettera 14 (1965) 57. J. EhIers, Abh. Akad. Wise .Mainz Mat. -Nat. KI. No. 11 (1961). E.Newman and R.Penrose, J.Math.Phys.3 (1962) 566.
*IfT*‘=O, wetakeanyotherecaIarpolynomiaIin the curvature tensor and ita covariant derivatives instead of T. Theee cannot alI be can&ant if the universe ia non-stationary. ** 2 x8 denotes the Lie derivative in the direction of Xa.
*****
PLASMA
ACOUSTIC
WAVES
ON AN INHOMOGENEOUS
COLUMN
+
F. W. CRAWFORD and J. A. TATAHONIS Institute
for Plasma
Research,
Stanford University,
Stanford,
California
Received 9 June 1965
In the last year or two careful experiment and rather sophisticated computation have established t This work was eupported by the U.S. Atomic Energy Commission.
clearly the mechanism of the well-known TonksDattner resonances exhibited by a plasma column irradiated by a transverse r-f. electric field [l-4]. Their origin has been shown to be due to radial propagation of plasma acoustic waves. At 241