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• 1
Electronic Resource
Springer
Communications in mathematical physics 25 (1972), S. 167-171
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract It is shown that a stationary space containing a black hole is a solution of the Brans-Dicke field equations if and only if it is a solution of the Einstein field equations. This implies that when the star collapses to form a black hole, it loses that fraction (about 7%) of its measured gravitational mass that arises from the scalar interaction. This mass loss is in addition to that caused by emission of scalar or tensor gravitational radiation. Another consequence is that there will not be any scalar gravitational radiation emitted when two black holes collide.
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• 2
Electronic Resource
Springer
Communications in mathematical physics 25 (1972), S. 152-166
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract It is assumed that the singularities which occur in gravitational collapse are not visible from outside but are hidden behind an event horizon. This means that one can still predict the future outside the event horizon. A black hole on a spacelike surface is defined to be a connected component of the region of the surface bounded by the event horizon. As time increase, black holes may merge together but can never bifurcate. A black hole would be expected to settle down to a stationary state. It is shown that a stationary black hole must have topologically spherical boundary and must be axisymmetric if it is rotating. These results together with those of Israel and Carter go most of the way towards establishing the conjecture that any stationary black hole is a Kerr solution. Using this conjecture and the result that the surface area of black holes can never decrease, one can place certain limits on the amount of energy that can be extracted from black holes.
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• 3
Electronic Resource
Springer
Communications in mathematical physics 55 (1977), S. 133-148
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.
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• 4
Electronic Resource
Springer
Communications in mathematical physics 35 (1974), S. 287-296
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract Causally continuous general relativistic spacetimes are defined and analyzed. In a causally continuous spacetime, the past and future of a local observer behave continuously under small perturbations of the metric or small changes in his location. Causally simple spacetimes are causally continuous; causally continuous spacetimes are causally stable. Possible reasons for taking causal continuity as a basic postulate in macrophysics are briefly discussed.
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• 5
Electronic Resource
Springer
Communications in mathematical physics 43 (1975), S. 199-220
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature $$\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K$$ where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.
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• 6
Electronic Resource
Springer
Communications in mathematical physics 88 (1983), S. 295-308
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface Σ which is regular outside an apparent horizonH. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface Σ which is regular outside an apparent horizon.
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• 7
Electronic Resource
Springer
Communications in mathematical physics 87 (1982), S. 395-415
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract Quantum gravity seems to introduce a new level of unpredictability into physics over and above that normally associated with the uncertainty principle. This is because the metric of spacetime can fluctuate from being globally hyperbolic. In other words, the evolution is not completely determined by Cauchy data at past or future infinity. I present a number of axioms that the asymptotic Green functions should obey in any reasonable theory of quantum gravity. These axioms are the same as for ordinary quantum field theory in flat spacetime, except that one axiom, that of asymptotic completeness, is omitted. This allows pure quantum states to decay into mixed states. Calculations with simple models of topologically non-trivial spacetime indicate that such loss of quantum coherence will occur but that the effect will be very small except for fundamental scalar particles, if any such exist.
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• 8
Electronic Resource
Springer
Communications in mathematical physics 87 (1983), S. 577-588
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract The Einstein equations with a negative cosmological constant admit black hole solutions which are asymptotic to anti-de Sitter space. Like black holes in asymptotically flat space, these solutions have thermodynamic properties including a characteristic temperature and an intrinsic entropy equal to one quarter of the area of the event horizon in Planck units. There are however some important differences from the asymptotically flat case. A black hole in anti-de Sitter space has a minimum temperature which occurs when its size is of the order of the characteristic radius of the anti-de Sitter space. For larger black holes the red-shifted temperature measured at infinity is greater. This means that such black holes have positive specific heat and can be in stable equilibrium with thermal radiation at a fixed temperature. It also implies that the canonical ensemble exists for asymptotically anti-de Sitter space, unlike the case for asymptotically flat space. One can also consider the microcanonical ensemble. One can avoid the problem that arises in asymptotically flat space of having to put the system in a box with unphysical perfectly reflecting walls because the gravitational potential of anti-de Sitter space acts as a box of finite volume.
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• 9
Electronic Resource
Springer
Communications in mathematical physics 27 (1972), S. 283-290
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
Notes: Abstract The area of the event horizon round a rotating black hole will increase in the presence of a non-axisymmetric or time dependent perturbation. If the perturbation is a matter field, the area increase is related to the fluxes of energy and of angular momentum into the black hole in such a way as to maintain the formula for the area in the Kerr solution. For purely gravitational perturbations one cannot define angular momentum locally but one can use the area increase and the expression for area in terms of mass and angular momentum to calculate the slowing down of a black hole caused by a non-axisymmetric distribution of matter at a distance. It seems that the coupling between the rotation of a black hole and the orbit of a particle going round it can be significant if the angular momentum of the black hole is close to its maximum possible value and if the angular velocity of the particle is nearly equal to that of the black hole.
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• 10
Electronic Resource
Springer
Communications in mathematical physics 46 (1976), S. 206-206
ISSN: 1432-0916
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics , Physics
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