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Prevention of gravitational collapse
Volume 99B, number 5
PHYSICS LETTERS
5 March 1981
PREVENTION OF GRAVITATIONAL COLLAPSE J.W. M O F F A T
Department of Physics, University of Toronto, Toronto, Canada and J.G. T A Y L O R
Department of MathematT"cs,King's College, University of London, London, UK Received 3 November 1980 Revised manuscript received 8 January 1981
We apply a new theory of gravitation to the question of gravitational collapse to show that collapse is prevented in this theory under very reasonable conditions. This result also extends to prevent ultimate collapse of the Universe.
A new theory o f gravitation, based on a nonsymmetric bermitian field structure [ 1 - 3 ] has recently been analysed for its particle content [4]. This was shown to be the usual spin-2 graviton together with a massless scalar "skewon" field • , arising from the antisymmetric part o f the hermitian metric field guy" In the linearised theory the resulting second-order lagrangian for the skewon terms reduced to [4]
we wish to show, under certain assumptions, that they produce a contact interaction, which prevents gravitational collapse to black holes for any stellar or greater than stellar object. We first consider the form that T[u~] can take. In the rigorous newtonian expansion o t the theory [6], it was found that in the newtonian order of approximation the equations of motion led to the restriction
L(2) = (~,~)2 _LilY] [~-l L[uv I _ 8rrL [~v] ,vgq-l NU
T[uv] = ibN[u, v] ,
. . . . ta + Y6 4~ 2 :vu:v -2L[uoI'°D-2L[UVl,u
,
(1)
where
L[uv] = ~-*rN[u ' ~] - 87riTluv] .
(2)
T[uvl is an antisymmetric matter source which can be specified in the presence of strong sources [5]. N u is the fermion number current density, Nu = - a 2 i ~ e i ~ i V ~ ~i, I
(3)
with metric ( + - - - ) , where ~i and e i are the wave functions and charges of the various fermions; a is a constant of the dimensions of length. We wish to discuss here the physical implications of the additional terms in the r.h.s, of (1) beyond the first. In particular 396
(4)
where b is a constant expected to be of the order of unity. There may also be an additional intrinsic spin term proportional to euvooJ°'°. Since this intrinsic spin will average to zero for large systems such as a star, we may neglect the effects of this latter term and so concentrate on (4). We thus obtain the contact terms in (1) to be of value ~?rr2(l - 3b)NuNU.
(5)
This will be repulsive if b > ~ and attractive otherwise. Clearly the attractive case will not change collapse criteria in any qualitative fashion, b u t only cause collapse to occur more readily. Thus we will turn only to the repulsive case. We might expect b -- l for T[uvl arising from the skew part of the canonical e n e r g y - m o mentum tensor of the Dirac theory. We consider the equilibrium of a stellar object follow-
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
Volume 99B, number 5
PHYSICS LETTERS
ingthe simple arguments o f Landau [7] ; a more detailed analysis may be necessary at certain points, but should not change the conclusions to within an order of magnitude. Therefore we evaluate the energy per particle in a system of N similar particles, each of mass m and particle radius r as (6)
E = A/r + B/r 3 ,
where A = 1 - 4 G N 2 / 3 m 2,
B = 3Xa4N/47rG ,
3, = ?~ 7rZ(3b - 1),
(7) (8)
:rod the first term of (6) is the usual combination of kinetic and gravitational contributions (in units o f h = c = 1), as written by Landau. The second term arises from our contact interaction (5) by the following argument. We write the potential U due to the contact interaction term as
v=
v/a ,
(9)
where P i s the volume, V = 45rrr3N, we have dropped the spatial parts o f N u and reinserted G - I in (9) to give the correct dimensions. We take for N 4 the density a2(4rrr3) - 1 , so that U h a s the value given by tire last term in (6). For N < Ncrit = ( 4 G i n 2 ) 3/2 (for the neutron) we have that A > 0 so that the total energy E is monotonically increasing as r ~ 0 in (6); no collapse will result, as usual. For N > Ncrit, E develops a minimum as a function of r at r0 = (-3B/A)I/2 -- [ ( 9 X a 4 N / 4 n G ) / ( 4 G N Z / 3 m 2
- 1)] 1/2
(10)
This has a mininmm as N increases from Ncrit to 0% uith value rmin = (9Xa4/8~G ~1/2 ~3/4 ~ ~A/-1/2 "crit •
(11 )
In order that the model make sense we require that rmi n ~ rtz 1 (so that the neutrons are not collapsed inside themselves), so giving the condition on a that a > o~mll4G 518 ,
(12)
where ct = 2(4rr/X) 1 / 4 3 - 7 / 8 . For X of order unity this gives the limit a ~> 10 . 3 8 c m .
(13)
5 March 1981
Since this is five orders of magnitude below the Planck length x/G = I/6 × 10 -33 cln, we feel safe in neglecting the possibility that a violates the bound (13). From strong source theory we expect that a ~ 10 -23 cm [5], so that in this case inequality (13) is clearly well satisfied. Our conclusion is that provided a satisfies (13) (and X ~ 1), then no gravitational collapse to black holes can ensue. This appears to be against the established view that black holes are (a) inevitable consequences of gravitational collapse for massive enough stars [7,8], and that (b) observational evidence for the existence o f black holes is available [9] from Cygnus X-I, and other variable X-ray stars, globur clusters like M15 and as the energy sources in quasars. We have presented a model above in which (a) is violated by the fact that geodesics are not maximal in the static spherically symmetric background metric [1,101. On the other hand, the above-mentioned observations cannot be used to distinguish a collapsed object in this theory from a black hole. The only distinguishing feature is the specific fluctuation time of the X-ray emission from near the event horizon o f a black hole; this has not yet been observed [9]. We remark that our model with a satisfying (13) should also prevent the ultimate collapse o f the Universe, producing in its place a "bounce". It would be necessary to give a more detailed analysis to follow through such a process, but this is not expected to change the general features we have presented here. It should also be pointed out that in General Relativity apparently repulsive terms can increase the tendency to collapse rather than decrease it because of the phenomenon of pressure regeneration [7]. This would not show up in the newtonian analysis presented here; we hope to present a discussion of this elsewhere. We note finally that the particle spectrum (0, 2) is a subset of that arising naturally in extended supergravity models [1 l ] . Thus we expect that such models will display a similar singularity-avoidance mechanism, provided that the constant b in (4) satisfies b 1 > 5- We expect that such a theory has b = 1, as mentioned earlier, so that extended supergravity lrray be expected to be singularity avoiding. It may also be renormalisable [12] so that we may have a physically reasonable and sensible theory. 397
Volume 99B, number 5
PHYSICS LETTERS
References [1] [2] [3} [4]
J.W. Moffat, Phys. Rev. D19 (1979) 3554. J.W. Moffat, Phys. Rev. D19 (1979) 3562. J.W. Moffat, J. Math. Phys. 21 (1980) 1798. R.B. Mann, J.W. Moffat and J.G. Taylor, Phys. Lett. 97B (1980) 73. [5] J.W. Moffat, Gauge invariance and strong interactions in a generalized theory of gravitation, Univ. of Toronto preprint (August 1980). [6] R.B. Mann and J.W. Moffat, Post-newtonian approximation of a new theory of gravity, Univ. of Toronto preprint (Jan. 1980)
398
5 March 1981
[7] B.K. ttarrison, K.S. Thorne, M. Wakano and J. Wheeler, Gravitation theory and gravitational collapse (Chicago U.P., 1965). [8] S. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge U.P., London, 1973). [91 R.D. Blandford and K.S. Thorne, in: General relativity: an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U.P., 1979). [10] R. Mann and J.W. Moffat, Equations of motions in a new theory of gravitation, Univ. of Toronto preprint (March 1980). [111 P. Fayet and S. Ferrara, Phys. Rep. 32 (1977) 249. [12] J.G. Taylor, The ultra-violet divergences of superfield supergravity, Proc. EPS Conf. (Geneva, 1979) (CERN, Geneva).
PHYSICS LETTERS
5 March 1981
PREVENTION OF GRAVITATIONAL COLLAPSE J.W. M O F F A T
Department of Physics, University of Toronto, Toronto, Canada and J.G. T A Y L O R
Department of MathematT"cs,King's College, University of London, London, UK Received 3 November 1980 Revised manuscript received 8 January 1981
We apply a new theory of gravitation to the question of gravitational collapse to show that collapse is prevented in this theory under very reasonable conditions. This result also extends to prevent ultimate collapse of the Universe.
A new theory o f gravitation, based on a nonsymmetric bermitian field structure [ 1 - 3 ] has recently been analysed for its particle content [4]. This was shown to be the usual spin-2 graviton together with a massless scalar "skewon" field • , arising from the antisymmetric part o f the hermitian metric field guy" In the linearised theory the resulting second-order lagrangian for the skewon terms reduced to [4]
we wish to show, under certain assumptions, that they produce a contact interaction, which prevents gravitational collapse to black holes for any stellar or greater than stellar object. We first consider the form that T[u~] can take. In the rigorous newtonian expansion o t the theory [6], it was found that in the newtonian order of approximation the equations of motion led to the restriction
L(2) = (~,~)2 _LilY] [~-l L[uv I _ 8rrL [~v] ,vgq-l NU
T[uv] = ibN[u, v] ,
. . . . ta + Y6 4~ 2 :vu:v -2L[uoI'°D-2L[UVl,u
,
(1)
where
L[uv] = ~-*rN[u ' ~] - 87riTluv] .
(2)
T[uvl is an antisymmetric matter source which can be specified in the presence of strong sources [5]. N u is the fermion number current density, Nu = - a 2 i ~ e i ~ i V ~ ~i, I
(3)
with metric ( + - - - ) , where ~i and e i are the wave functions and charges of the various fermions; a is a constant of the dimensions of length. We wish to discuss here the physical implications of the additional terms in the r.h.s, of (1) beyond the first. In particular 396
(4)
where b is a constant expected to be of the order of unity. There may also be an additional intrinsic spin term proportional to euvooJ°'°. Since this intrinsic spin will average to zero for large systems such as a star, we may neglect the effects of this latter term and so concentrate on (4). We thus obtain the contact terms in (1) to be of value ~?rr2(l - 3b)NuNU.
(5)
This will be repulsive if b > ~ and attractive otherwise. Clearly the attractive case will not change collapse criteria in any qualitative fashion, b u t only cause collapse to occur more readily. Thus we will turn only to the repulsive case. We might expect b -- l for T[uvl arising from the skew part of the canonical e n e r g y - m o mentum tensor of the Dirac theory. We consider the equilibrium of a stellar object follow-
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
Volume 99B, number 5
PHYSICS LETTERS
ingthe simple arguments o f Landau [7] ; a more detailed analysis may be necessary at certain points, but should not change the conclusions to within an order of magnitude. Therefore we evaluate the energy per particle in a system of N similar particles, each of mass m and particle radius r as (6)
E = A/r + B/r 3 ,
where A = 1 - 4 G N 2 / 3 m 2,
B = 3Xa4N/47rG ,
3, = ?~ 7rZ(3b - 1),
(7) (8)
:rod the first term of (6) is the usual combination of kinetic and gravitational contributions (in units o f h = c = 1), as written by Landau. The second term arises from our contact interaction (5) by the following argument. We write the potential U due to the contact interaction term as
v=
v/a ,
(9)
where P i s the volume, V = 45rrr3N, we have dropped the spatial parts o f N u and reinserted G - I in (9) to give the correct dimensions. We take for N 4 the density a2(4rrr3) - 1 , so that U h a s the value given by tire last term in (6). For N < Ncrit = ( 4 G i n 2 ) 3/2 (for the neutron) we have that A > 0 so that the total energy E is monotonically increasing as r ~ 0 in (6); no collapse will result, as usual. For N > Ncrit, E develops a minimum as a function of r at r0 = (-3B/A)I/2 -- [ ( 9 X a 4 N / 4 n G ) / ( 4 G N Z / 3 m 2
- 1)] 1/2
(10)
This has a mininmm as N increases from Ncrit to 0% uith value rmin = (9Xa4/8~G ~1/2 ~3/4 ~ ~A/-1/2 "crit •
(11 )
In order that the model make sense we require that rmi n ~ rtz 1 (so that the neutrons are not collapsed inside themselves), so giving the condition on a that a > o~mll4G 518 ,
(12)
where ct = 2(4rr/X) 1 / 4 3 - 7 / 8 . For X of order unity this gives the limit a ~> 10 . 3 8 c m .
(13)
5 March 1981
Since this is five orders of magnitude below the Planck length x/G = I/6 × 10 -33 cln, we feel safe in neglecting the possibility that a violates the bound (13). From strong source theory we expect that a ~ 10 -23 cm [5], so that in this case inequality (13) is clearly well satisfied. Our conclusion is that provided a satisfies (13) (and X ~ 1), then no gravitational collapse to black holes can ensue. This appears to be against the established view that black holes are (a) inevitable consequences of gravitational collapse for massive enough stars [7,8], and that (b) observational evidence for the existence o f black holes is available [9] from Cygnus X-I, and other variable X-ray stars, globur clusters like M15 and as the energy sources in quasars. We have presented a model above in which (a) is violated by the fact that geodesics are not maximal in the static spherically symmetric background metric [1,101. On the other hand, the above-mentioned observations cannot be used to distinguish a collapsed object in this theory from a black hole. The only distinguishing feature is the specific fluctuation time of the X-ray emission from near the event horizon o f a black hole; this has not yet been observed [9]. We remark that our model with a satisfying (13) should also prevent the ultimate collapse o f the Universe, producing in its place a "bounce". It would be necessary to give a more detailed analysis to follow through such a process, but this is not expected to change the general features we have presented here. It should also be pointed out that in General Relativity apparently repulsive terms can increase the tendency to collapse rather than decrease it because of the phenomenon of pressure regeneration [7]. This would not show up in the newtonian analysis presented here; we hope to present a discussion of this elsewhere. We note finally that the particle spectrum (0, 2) is a subset of that arising naturally in extended supergravity models [1 l ] . Thus we expect that such models will display a similar singularity-avoidance mechanism, provided that the constant b in (4) satisfies b 1 > 5- We expect that such a theory has b = 1, as mentioned earlier, so that extended supergravity lrray be expected to be singularity avoiding. It may also be renormalisable [12] so that we may have a physically reasonable and sensible theory. 397
Volume 99B, number 5
PHYSICS LETTERS
References [1] [2] [3} [4]
J.W. Moffat, Phys. Rev. D19 (1979) 3554. J.W. Moffat, Phys. Rev. D19 (1979) 3562. J.W. Moffat, J. Math. Phys. 21 (1980) 1798. R.B. Mann, J.W. Moffat and J.G. Taylor, Phys. Lett. 97B (1980) 73. [5] J.W. Moffat, Gauge invariance and strong interactions in a generalized theory of gravitation, Univ. of Toronto preprint (August 1980). [6] R.B. Mann and J.W. Moffat, Post-newtonian approximation of a new theory of gravity, Univ. of Toronto preprint (Jan. 1980)
398
5 March 1981
[7] B.K. ttarrison, K.S. Thorne, M. Wakano and J. Wheeler, Gravitation theory and gravitational collapse (Chicago U.P., 1965). [8] S. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge U.P., London, 1973). [91 R.D. Blandford and K.S. Thorne, in: General relativity: an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U.P., 1979). [10] R. Mann and J.W. Moffat, Equations of motions in a new theory of gravitation, Univ. of Toronto preprint (March 1980). [111 P. Fayet and S. Ferrara, Phys. Rep. 32 (1977) 249. [12] J.G. Taylor, The ultra-violet divergences of superfield supergravity, Proc. EPS Conf. (Geneva, 1979) (CERN, Geneva).