NO C H A N C E FOR B L A C K H O L E S ?
(Letter to the Editor) H. D E H N E N
andF.
GHABOUSSI
Universitiit Konstanz, Fakultiit J~r Physik, F.R.G.
(Received 7 October, revised 4 November, 1987) Abstract, The influence of the results of the grand unified theory of interactions on the formation of black holes is discussed and compared with the observations of the supernova 1987a.
Until now there is no unique evidence for the reality of black holes in spite of intensive observational search. However, in the last time there exist considerations, which bring together the observation of the second neutrino pulse connected with the supernova 1987a (Hirata et al., 1987) with the fact, that the neutron star born in the supernova has been collapsed subsequently - after approximately 5 hours - to a black hole (Hillebrandt et al., 1987b). In this connection we want to point to alternative possibilities for explaining the observations from SN 1987a, which have their origin in the grand unified theory of electro weak and strong interactions (GUT). As well known, in Schwarzschild's interior solution of Einstein's field equations for a non-rotating sphere of incompressible matter the pressure diverges in the center of the sphere before the Schwarzschild-radius is reached; this happens when the quotient between radius R of the sphere and Schwarzschild-radius 2M takes the value
R/ZM = 9
(1)
corresponding to a redshift z = 2 for light emitted from the surface of the object. In all other cases of compressible matter the divergence of the central pressure and herewith of the central density appears even for larger values of the quotient (1) and for smaller values of the redshift than z = 2 (see, e.g., Stephani, 1977). This divergent pressure and density mean, that the kinetic energy of the baryons of the object, which was a neutron star in case of SN 1987a, would exceed all limits in the central regions of the body, when it is collapsing. This is not the case, if the collapse is a free fall; but this model neglects all non-gravitating interaction forces and is not realistic. We emphasize that we are persuing here the idea that the grand unified theory has its practical importance not only in the big bang of the Universe but more directly in the collapse of a massive star as estimated above. Following this line one has to expect, that with reaching of the unification energy (~_ 1015 GeV) in the star the generation of X- and Y-bosons takes place very intensively, in consequence of which all quarks of the baryons can decay into leptons and Astrophysics and Space Science 140 (1988) 209-211, Q 1988 by D. Reidel Publishing Company
210
H. DEHNEN AND F. GHABOUSSI
quark/antiquark-pairs (mesons) described by the balance formula n--.rc
+e + .
(2)
Subsequently, we get by the weak decay of ~ - - i.e., 7t-~#
+v,,
#-~e-
+~e+V ".
(3)
The time-scale of these reactions is very short in comparison with the collapse time. Consequently, a rapid decay of the baryons into ~e, v,, ~ and e +, e - happens. The neutrinos leave the object immediately, whereas the e + / e - - p l a s m a annihilates to photons. In this way all matter in the central region of the object is converted into outgoing neutrino-radiation and photon emission. Considering the neutrino emission times we find the following result. The very dense inner core of the object will be surrounded by normal nuclear matter. The neutrino diffusion time is then given by (4)
t = R2/4D,
where D = cl/3 is the diffusion coefficient and I = (o-n)- 1 the mean-free-path length of the neutrinos with o-their cross-section and n the mean nuclear particle number density. According to Schwarzschild's interior solution the energy density is given in the case of the extreme situation (1) by (5)
pc 2 = 16c4/2437rM2G
(G, Newtonian gravitational constant). The neutron stars become unstable with respect to the gravitational collapse, when the nucleons go over in the state of a degenerate relativistic Fermi gas, which is the case, after cooling, for neutron star masses beyond the Oppenheimer-Volkoff limit. Therefore, the energy density (5) consists mainly of the zero-point energy density according to the relativistic Fermi statistics given by pc 2 -
(91Z)2/3
(6)
hcn 4/3 .
4
From (5) and (6) the mean nuclear particle number density results in 32 //3"~ 1/4
C9/4
n = 2437r 1-~1\4~/ M 3 / 2 G 3 / 4 h 3/4
(7)
If we insert (7) into D and calculating (4) with the use of (1) we obtain immediately =
t
1 (L~l/4
2rr\4~/
C5/4M 1/2
o- G3/4h 3/4
(8)
For electron neutrinos o- is of the order of 10-43 (E~/MeV)2 cm2; in case of a typical neutrino energy of 10 MeV, as observed from SN 1987a, and for a neutron star of 2 solar
NO CHANCE FOR BLACK HOLES?
211
masses the emission time (8) results in t = 0.3 s. For muon neutrinos the cross-section is smaller, so that this time can be regarded as an upper limit for the neutrino emission time, which may be herewith shorter than the collapse time under consideration of the processes discussed above. Concerning the photon emission we note that the radiation coming out of the inner parts of the star undergoes a very strong redshift of the order of z > 2 with respect to (1). Therefore, in view of the relation between observed and emitted luminosity L o = L / ( 1 + z)2 in the case of the Schwarzschild solution, a very rapid increase of the luminosity of the object is not to be expected, although a significant fraction of the star's restmass is converted into photons. A main part of the photon energy is used for dissolving the bound state of the star. Instead of the collapse into the very bound state of a black hole, an expanding non-bounded radiation field results at last. In this connection we note that non-bounded stationary solution of Einstein's field equation exists for an isolated object consisting of pure radiation (Klein, 1947). In this situation no relic is to be expected at the position of the supernova. Our results are in qualitative agreement with the observations: a main contribution of the luminosity of SN 1987a comes from the e+/e--annihilation processes; the second neutrino pulse consists of~e, v,, and ~, according to (3). This is in accordance with the analysis of Hillebrandt et al. (1987b), according to which the second pulse consists of two neutrino kinds (probably with different masses; see also Hillebrandt et al., 1987a). However, we can give here only a qualitative description of the scenario; a detailed quantitative calculation of the dynamic processes may not be easily. It was our aim to point to the possibility, that the consequences of modern elementary particle physics influence the black hole formation drastically. Finally, we note that the considered processes could be realized on a larger scale in the quasars. In this way the enormous energy output of these objects would find a plausible explanation. References
Hitlebrandt, W. et aL: 1987a,Astron. Astrophys. 177, L4[. Hillebrandt, W. et al. : 1987b,Astron. Astrophys. 180, L20. Hirata, K. et al. : 1987,Phys. Rev. Letters 58, 1490. Klein, O.: 1947,Arkiv Math., Astron. Fys. 34A, No. 19. Stephani, H.: 1977,Algemeine Relativitiitstheorie, VEB DeutseherVerlag der Wissenschaften,Berlin,p. 215.
