- Email: [email protected]
A cosmological model in which “singularity” does not require a “matter singularity”
Volume 28A, number 10
PHYSJCS
set of parameters at both frequencies indicates that any anisotropy in the exchange potential [3] for this system is less than 0.05 cm-l. The uncertainty here arises from the large width and asymmetry of the resonances near the turning points in fig. 1, where the resonance position would be most dependent on such anisotropy. The dipolar field at a manganese site is 8500G. Assuming that this is unchanged at the impurity site it contributes a zero field splitting of 2.27 cm-l. As the net zero-field splitting A = = 0.823 cm-l, the effective field from exchange and other multipole interactions is of opposite sign to Hd, but may be either larger or smaller than Hd by A/g3s. An interesting feature of magnetic impurities in a magnetically ordered lattice is their interaction with the spin-wave modes, which may influence the relaxation rate of the impurity spin. AS the spin wave anisotropy gap in MnF2 is at 12.55OK, energy concerving relaxation can only occur via two or more magnon scattering processes as has been observed for the F- nuclei in MnF2. [4] We have measured 2’1 for MnF2:Er3+ by pulse saturation techniques in the region 1.5oK to 4.2oK and observed a rapid temperature variation of the relaxation rate between 1.80-K and 3oK, which is not present in ZnF2:Er3+. Howsame
LETTTERS
24 February 1969
ever, below this temperature region a direct phonon relaxation appears dominant and above
3oK an Orbach-type phonon relaxation predominates involving the Er3+ level observed in the fluorescence spectrum at 36 cm-l above the ground state. The phonon relaxation has prevented any quantitative confirmation of spin-wave scattering for this system although the anomalous relaxation rate between 1.8oK and 3oK is of reasonable magnitude in relation to the expected rate for two-magnon scattering estimated from the Er3+-Mn2+ coupling. We are grateful to S. Geschwind for helpful discussions and the use of the 24 GHz spectrometer and to J. L. Davis for technical assistance.
References V. V. Eremenko, E. V. Matyushlein and S. V. Petrov, Phys. Stat. Sol. 18 (1966) 683. R. W. Alexander Jr. and A. J. Sievers, Optical properties of ions in crystals, eds. H. M. Crosswhite and H. W. Moos. K. A. Wickersheim and R. L. White, Phys. Rev. Letters 8 (1962) 483. N. Kaplan, R. Loudon, V. Jaccarino, H. J. Guggenheim, D. Beeman and P. A. Pincus, Phys. Rev. Letters 17 (1966) 357.
A COSMOLOGICAL MODEL IN WHICH “SINGULARITY” DOES NOT REQUIRE A “MATTER SINGULARITY”* L. C. SHEPLEY University
of Texas, Austin,
Received
3 September
Texas 78712,
USA
1968
A type V-homogeneous, relativistic, dust-filled model is presented in which there is a singular region R which is not accompanied by a matter singularity. This example throws light on the physical significance of singularities in general relativity.
Although all spatially-homogeneous, matterfilled cosmological models are incomplete [l], it is not known whether all are singular. If the structure constants of the group obey CEi = .O
(1)
then the models cannot be infinitely extended [2], * This research
was supported and by NSF Grant GU-1598.
by NSF Grant GP-8868
and so are singular and require a “matter larity” in the terminology. below.
singu-
In general, a region is called “incomplete” if it contains an open geodesic segment which cannot be extend. We will modify this terminology slightly to say that a region R in a cosmological model is “incomplete” if (a) R contains an inextendible (in terms of finite affine parameter values) open geodesic segment G, and (b) there is no closed 695
PHYSICS
Volume 28A, number9
set S containing R in which G is extendible. If, further, (c) there is no closed set containing R in which G may be covered by a compact set, then R will be called “singular”. A region R containing finite matter path segments will be said to require a “matter singularity” if (d) either there is no closed set S containing R in which the particle path segments of R are extendible (in terms of the proper time parameter) or else R contains a finite geodesic segment G on which the energy density is unbounded $. R and S are presumed to have a well-defined, non-singular, smooth metric at all points. In this note an example is presented to demonstrate that a non-empty singular region R need not require a matter singularity. It is concluded that the definition of singularity given above, which is equal to or stronger than the ones generally accepted, is defective in not having a definitive physical interpretation. The example demonstrating that (a), (b), (c) do not imply (d) is a spatially-homogeneous manifold with metric ds2 = 2dtdx+g( t)d$ + ( b(t)@dy)2 + (b(t)exdz)2 (2) The invariance group is Bianchi type V3, which does not obey eq. (1). The t = const homogeneous hypersurfaces are spacelike and have constant negative curvature for g > 0. The Einstein field equations for a dust-filled universe will be used. The density p is positive, and the unit matter velocity has components uo, ul, u2 = u3 = 0 obeying -guo2+2uou1
=- 1
(3)
in the dt, dx, @eddy), (bexdz) basis. Let p = b/b; with use of eq. (3), the field equations become /j = -/32-++,a; and
g = 4~-2@+~,2
~(1 -g2u04)/uo2
= 8g@- 8 - 4&
(4)
(5) ~(1 +guo2) = 2gp2 + Q-P - 8P Eqs. (4) and (5) show that if g, g, and p are given at t = to, then g and p may be determined for all t near to by integration of ordinary differential equations. (We will not explicitly do this integration here. ) Let to correspond tog = 0. $ I wish to thank R. Geroch for his comments on this definition. Definition (d) is appropriate only for fluid-filled models; a good, general definition of a physical singularity does not exist.
LETTERS
1969
Eqs. (5) yield, at t = to zlo2 = Q(4 -&b/(2 If
g< -2,
+&);
p =
p
2(&-4)#I (8)
then g, &, and /3 yield positive p, uo2, andg and P may be found for all tin an open set P about to. Let R be a region in this model whose t values lie strictly within P, and include to. R contains the nullg = 0 hypersurface L, which contains an incomplete open null geodesic segment G. (The proof that G can be extended in only one direction to infinite values of its affine parameter was given by Hawking and Ellis Cl].) G cannot be extended nor can it be covered by a compact set in the closure of R, yet the closure of R has a well-defined stress-energy tensor, with finite p and well-defined matter paths. The proof of Hawking and Ellis concerning the onset of singularities in spatially-homogeneous models consists of showing that a null invariant hypersurface is a necessary feature and that incomplete regions R thereby result. Similar discussions [4] show that more general cosmological models contain incomplete regions (which may or may not be singular regions). There are three interpretations of singularities in general relativity: (1) The necessity of singular regions is a defect of the theory itself. (2) The theory is all right; the onset of a singularity is the development of a shock front. (3) If smoothness is required, the set of space-time events must be enlarged (to include non-Hausdorff sets [5], for example) until all geodesics are extendible. In the present example, because L appears simultaneously over all space, (2) is inappropriate. Because there is no matter singularity, (1) seems a bit farfetched. Hence extension, as in (3), should be considered more carefully than it has been.
References 1. S. Hawking and G. F. R. Ellis, Phys. Letters 17 (1965) 246. 2. L.C. Shepley (unpublished). See L.C. Shepley, Proc. Nat. Acad. Sci. (USA) 52 (1964) 1403. J. Math. Phys. 8 (1967) 2315. 3. D. L. Farnsworth, 4. For example, S. W.Hawking, Proc. Roy. Sot. A294 (1966) 511. 5. R. Geroch, J. Math. Phys. 9 (1968) 450: E. H. Kronheimer and R. Penrose. Proc. Camb. Phil. Sot. 63 (1967) 481.
* * * *
696
24 February
PHYSJCS
set of parameters at both frequencies indicates that any anisotropy in the exchange potential [3] for this system is less than 0.05 cm-l. The uncertainty here arises from the large width and asymmetry of the resonances near the turning points in fig. 1, where the resonance position would be most dependent on such anisotropy. The dipolar field at a manganese site is 8500G. Assuming that this is unchanged at the impurity site it contributes a zero field splitting of 2.27 cm-l. As the net zero-field splitting A = = 0.823 cm-l, the effective field from exchange and other multipole interactions is of opposite sign to Hd, but may be either larger or smaller than Hd by A/g3s. An interesting feature of magnetic impurities in a magnetically ordered lattice is their interaction with the spin-wave modes, which may influence the relaxation rate of the impurity spin. AS the spin wave anisotropy gap in MnF2 is at 12.55OK, energy concerving relaxation can only occur via two or more magnon scattering processes as has been observed for the F- nuclei in MnF2. [4] We have measured 2’1 for MnF2:Er3+ by pulse saturation techniques in the region 1.5oK to 4.2oK and observed a rapid temperature variation of the relaxation rate between 1.80-K and 3oK, which is not present in ZnF2:Er3+. Howsame
LETTTERS
24 February 1969
ever, below this temperature region a direct phonon relaxation appears dominant and above
3oK an Orbach-type phonon relaxation predominates involving the Er3+ level observed in the fluorescence spectrum at 36 cm-l above the ground state. The phonon relaxation has prevented any quantitative confirmation of spin-wave scattering for this system although the anomalous relaxation rate between 1.8oK and 3oK is of reasonable magnitude in relation to the expected rate for two-magnon scattering estimated from the Er3+-Mn2+ coupling. We are grateful to S. Geschwind for helpful discussions and the use of the 24 GHz spectrometer and to J. L. Davis for technical assistance.
References V. V. Eremenko, E. V. Matyushlein and S. V. Petrov, Phys. Stat. Sol. 18 (1966) 683. R. W. Alexander Jr. and A. J. Sievers, Optical properties of ions in crystals, eds. H. M. Crosswhite and H. W. Moos. K. A. Wickersheim and R. L. White, Phys. Rev. Letters 8 (1962) 483. N. Kaplan, R. Loudon, V. Jaccarino, H. J. Guggenheim, D. Beeman and P. A. Pincus, Phys. Rev. Letters 17 (1966) 357.
A COSMOLOGICAL MODEL IN WHICH “SINGULARITY” DOES NOT REQUIRE A “MATTER SINGULARITY”* L. C. SHEPLEY University
of Texas, Austin,
Received
3 September
Texas 78712,
USA
1968
A type V-homogeneous, relativistic, dust-filled model is presented in which there is a singular region R which is not accompanied by a matter singularity. This example throws light on the physical significance of singularities in general relativity.
Although all spatially-homogeneous, matterfilled cosmological models are incomplete [l], it is not known whether all are singular. If the structure constants of the group obey CEi = .O
(1)
then the models cannot be infinitely extended [2], * This research
was supported and by NSF Grant GU-1598.
by NSF Grant GP-8868
and so are singular and require a “matter larity” in the terminology. below.
singu-
In general, a region is called “incomplete” if it contains an open geodesic segment which cannot be extend. We will modify this terminology slightly to say that a region R in a cosmological model is “incomplete” if (a) R contains an inextendible (in terms of finite affine parameter values) open geodesic segment G, and (b) there is no closed 695
PHYSICS
Volume 28A, number9
set S containing R in which G is extendible. If, further, (c) there is no closed set containing R in which G may be covered by a compact set, then R will be called “singular”. A region R containing finite matter path segments will be said to require a “matter singularity” if (d) either there is no closed set S containing R in which the particle path segments of R are extendible (in terms of the proper time parameter) or else R contains a finite geodesic segment G on which the energy density is unbounded $. R and S are presumed to have a well-defined, non-singular, smooth metric at all points. In this note an example is presented to demonstrate that a non-empty singular region R need not require a matter singularity. It is concluded that the definition of singularity given above, which is equal to or stronger than the ones generally accepted, is defective in not having a definitive physical interpretation. The example demonstrating that (a), (b), (c) do not imply (d) is a spatially-homogeneous manifold with metric ds2 = 2dtdx+g( t)d$ + ( b(t)@dy)2 + (b(t)exdz)2 (2) The invariance group is Bianchi type V3, which does not obey eq. (1). The t = const homogeneous hypersurfaces are spacelike and have constant negative curvature for g > 0. The Einstein field equations for a dust-filled universe will be used. The density p is positive, and the unit matter velocity has components uo, ul, u2 = u3 = 0 obeying -guo2+2uou1
=- 1
(3)
in the dt, dx, @eddy), (bexdz) basis. Let p = b/b; with use of eq. (3), the field equations become /j = -/32-++,a; and
g = 4~-2@+~,2
~(1 -g2u04)/uo2
= 8g@- 8 - 4&
(4)
(5) ~(1 +guo2) = 2gp2 + Q-P - 8P Eqs. (4) and (5) show that if g, g, and p are given at t = to, then g and p may be determined for all t near to by integration of ordinary differential equations. (We will not explicitly do this integration here. ) Let to correspond tog = 0. $ I wish to thank R. Geroch for his comments on this definition. Definition (d) is appropriate only for fluid-filled models; a good, general definition of a physical singularity does not exist.
LETTERS
1969
Eqs. (5) yield, at t = to zlo2 = Q(4 -&b/(2 If
g< -2,
+&);
p =
p
2(&-4)#I (8)
then g, &, and /3 yield positive p, uo2, andg and P may be found for all tin an open set P about to. Let R be a region in this model whose t values lie strictly within P, and include to. R contains the nullg = 0 hypersurface L, which contains an incomplete open null geodesic segment G. (The proof that G can be extended in only one direction to infinite values of its affine parameter was given by Hawking and Ellis Cl].) G cannot be extended nor can it be covered by a compact set in the closure of R, yet the closure of R has a well-defined stress-energy tensor, with finite p and well-defined matter paths. The proof of Hawking and Ellis concerning the onset of singularities in spatially-homogeneous models consists of showing that a null invariant hypersurface is a necessary feature and that incomplete regions R thereby result. Similar discussions [4] show that more general cosmological models contain incomplete regions (which may or may not be singular regions). There are three interpretations of singularities in general relativity: (1) The necessity of singular regions is a defect of the theory itself. (2) The theory is all right; the onset of a singularity is the development of a shock front. (3) If smoothness is required, the set of space-time events must be enlarged (to include non-Hausdorff sets [5], for example) until all geodesics are extendible. In the present example, because L appears simultaneously over all space, (2) is inappropriate. Because there is no matter singularity, (1) seems a bit farfetched. Hence extension, as in (3), should be considered more carefully than it has been.
References 1. S. Hawking and G. F. R. Ellis, Phys. Letters 17 (1965) 246. 2. L.C. Shepley (unpublished). See L.C. Shepley, Proc. Nat. Acad. Sci. (USA) 52 (1964) 1403. J. Math. Phys. 8 (1967) 2315. 3. D. L. Farnsworth, 4. For example, S. W.Hawking, Proc. Roy. Sot. A294 (1966) 511. 5. R. Geroch, J. Math. Phys. 9 (1968) 450: E. H. Kronheimer and R. Penrose. Proc. Camb. Phil. Sot. 63 (1967) 481.
* * * *
696
24 February