Modified Gravity and Cosmology

QUANTUM gravitational effects are usually ignored in calculations of the formation and evolution of black holes. The justification for this is that the radius of curvature of space-time outside the event horizon is very large compared to the Planck length (Għ/c3)1/2 ≈ 10−33 cm, the length scale on which quantum fluctuations of the metric are expected to be of order unity. This means that the energy density of particles created by the gravitational field is small compared to the space-time curvature. Even though quantum effects may be small locally, they may still, however, add up to produce a significant effect over the lifetime of the Universe ≈ 1017 s which is very long compared to the Planck time ≈ 10−43 s. The purpose of this letter is to show that this indeed may be the case: it seems that any black hole will create and emit particles such as neutrinos or photons at just the rate that one would expect if the black hole was a body with a temperature of (κ/2π) (ħ/2k) ≈ 10−6 (M/M)K where κ is the surface gravity of the black hole1. As a black hole emits this thermal radiation one would expect it to lose mass. This in turn would increase the surface gravity and so increase the rate of emission. The black hole would therefore have a finite life of the order of 1071 (M/M)−3 s. For a black hole of solar mass this is much longer than the age of the Universe. There might, however, be much smaller black holes which were formed by fluctuations in the early Universe2. Any such black hole of mass less than 1015 g would have evaporated by now. Near the end of its life the rate of emission would be very high and about 1030 erg would be released in the last 0.1 s. This is a fairly small explosion by astronomical standards but it is equivalent to about 1 million 1 Mton hydrogen bombs. It is often said that nothing can escape from a black hole. But in 1974, Stephen Hawking realized that, owing to quantum effects, black holes should emit particles with a thermal distribution of energies — as if the black hole had a temperature inversely proportional to its mass. In addition to putting black-hole thermodynamics on a firmer footing, this discovery led Hawking to postulate 'black hole explosions', as primordial black holes end their lives in an accelerating release of energy.

Black Hole Explosions

We present a candidate quantum field theory of gravity with dynamical critical exponent equal to $z=3$ in the UV. (As in condensed-matter systems, $z$ measures the degree of anisotropy between space and time.) This theory, which at short distances describes interacting nonrelativistic gravitons, is power-counting renormalizable in $3+1$ dimensions. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory flows naturally to the relativistic value $z=1$, and could therefore serve as a possible candidate for a UV completion of Einstein's general relativity or an infrared modification thereof. The effective speed of light, the Newton constant and the cosmological constant all emerge from relevant deformations of the deeply nonrelativistic $z=3$ theory at short distances.

Quantum Gravity at a Lifshitz Point

We present in a manifestly gauge-invariant form the theory of classical linear gravitational perturbations in part I, and a quantum theory of cosmological perturbations in part II. Part I includes applications to several important examples arising in cosmology: a univese dominated by hydrodynamical matter, a universe filled with scalar-field matter, and higher-derivative theories of gravity. The growth rates of perturbations are calculated analytically in most interesting cases. The analysis is applied to study the evolution of fluctuations in inflationary universe models. Part II includes a unified description of the quantum generation and evolution of inhomogeneities about a classial Friedmann background. The method is based on standard canonical quantization of the action for cosmological perturbations which has been reduced to an expression in terms of a single gauge-invariant variable. The spectrum of density perturbations originating in quantum fluctuations is calculated in universe with hydrodynamical matter, in inflationary universe models with scalar-field matter, and in higher-derivative theories of gravity.

The gauge-invariant theory of classical and quantized cosmological perturbations developed in parts I and II is applied in part III to several interesting physical problems. It allows a simple derivation of the relation between temperature anistropes in the cosmic microwave background. radiation and the gauge-invariant potential for metric perturbations. The generation and evolution of gravitational waves is studied. As another example, a simple analysis of entropy perturbations and non-scale-invariant spectra in inflationary universe models is presented. The gauge-invariant theory of cosmological perturbations also allows a consistent and gauge-invariant definition of statistical fluctuations.

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Theory of Cosmological Perturbations

Extended Theories of Gravity

(1970). Oscillatory approach to a singular point in the relativistic cosmology. Advances in Physics: Vol. 19, No. 80, pp. 525-573.

Oscillatory approach to a singular point in the relativistic cosmology

(by translator) A detailed report is given here of the general investigations carried out by the authors in the field of relativistic cosmology during the past years. The paper consists of two parts. The first part is devoted to a study of the singularities of the cosmological solutions of the gravitational equations. An attempt is made to provide an answer to one of the principal questions of modern cosmology: ‘does the general solution of the gravitational equations have a singularity?’ The authors give a negative answer to this question. The study carried out leads, in fact, to the general conclusion that the presence of a singularity with respect to time is not a necessary property of cosmological models of the general theory of relativity, and that the general case of an arbitrary distribution of matter and gravitational field does not lead to the appearance of a singularity. This result, however, does not exclude the possibility of the existence of more restricted classes of cosmological so...

Investigations in relativistic cosmology

This paper is a concluding review exposition of the investigations aimed at the construction of a general cosmological solution of the Einstein equations with a singularity in time; thus it is a direct continuation of the previous (1970) paper by the authors in this Journal. A detailed description is given of the analysis which leads to the construction of such a solution, and of its properties.

A General Solution of the Einstein Equations with a Time Singularity

Inflationary stages in cosmological models with a scalar field

Contents 1. Introduction 745 2. Kasner's Generalized Solution 746 3. Oscillatory Approach to the Singular Point 748 4. Evolution of Model Under the Influence of Two Perturbations 751 5. Evolution of Model in the Asymptotic Region of Arbitrarily Short Times 752 6. Statistical Analysis of the Evolution of the Model on Approaching the Singular Point 54 7. Construction of General Solution for a Long Era with Small Oscillations 756 8. Concluding Remarks 759 Appendices A. Some Information from the Theory of Continuous Fractions 760 B. Refinement of the Calculations in Section 4 761 C. Homogeneous Spaces 762 D. Homogeneous Spaces of Type VIII and IX , 764 Cited Literature

http://iopscience.iop.org/article/10.1070/PU1971v013n06ABEH004279/pdf

Isaak M. Khalatnikov

Papers

Oscillatory approach to a singular point in the relativistic cosmology

Investigations in relativistic cosmology

A General Solution of the Einstein Equations with a Time Singularity

Inflationary stages in cosmological models with a scalar field

Oscillatory approach to the singular point in relativistic cosmology