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

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Superstring theoryをめぐって(放談室)

We present a holographic dual of four-dimensional, large N_c QCD with massless flavors. This model is constructed by placing N_f probe D8-branes into a D4 background, where supersymmetry is completely broken. The chiral symmetry breaking in QCD is manifested as a smooth interpolation of D8 - anti-D8 pairs in the supergravity background. The meson spectrum is examined by analyzing a five-dimensional Yang-Mills theory that originates from the non-Abelian DBI action of the probe D8-brane. It is found that our model yields massless pions, which are identified with Nambu-Goldstone bosons associated with the chiral symmetry breaking. We obtain the low-energy effective action of the pion field and show that it contains the usual kinetic term of the chiral Lagrangian and the Skyrme term. A brane configuration that defines a dynamical baryon is identified with the Skyrmion. We also derive the effective action including the lightest vector meson. Our model is closely related to that in the hidden local symmetry approach, and we obtain a Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin-type relation among the couplings. Furthermore, we investigate the Chern-Simons term on the probe brane and show that it leads to the Wess-Zumino-Witten term. The mass of the \eta' meson is also considered, and we formulate a simple derivation of the \eta' mass term satisfying the Witten-Veneziano formula from supergravity.

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Low energy hadron physics in holographic QCD

We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

https://arxiv.org/pdf/cond-mat/0102119.pdf

Quantum Phase Transitions

We derive a general relation between the ground state entanglement Hamiltonian and the physical stress tensor within the path integral formalism. For spherical entangling surfaces in a CFT, we reproduce the local ground state entanglement Hamiltonian derived by Casini, Huerta and Myers. The resulting reduced density matrix can be characterized by a spatially varying “entanglement temperature”. Using the entanglement Hamiltonian, we calculate the first order change in the entanglement entropy due to changes in conserved charges of the ground state, and find a local first law-like relation for the entanglement entropy. Our approach provides a field theory derivation and generalization of recent results obtained by holographic techniques. However, we note a discrepancy between our field theoretically derived results for the entanglement entropy of excited states with a non-uniform energy density and current holographic results in the literature. Finally, we give a CFT derivation of a set of constraint equations obeyed by the entanglement entropy of excited states in any dimension. Previously, these equations were derived in the context of holography.

Entanglement temperature and entanglement entropy of excited states

The type-IIB supergravity solution describing a collection of regular and fractional D3-branes on the conifold (hep-th/0002159) was recently generalized to the case of the deformed conifold (hep-th/0007191). Here we present another generalization - when the conifold is replaced by the resolved conifold. This solution can be found in two different ways: (i) by first explicitly constructing the Ricci-flat Kahler metric on resolved conifold and then solving the supergravity equations for the D3-brane ansatz with constant dilaton and (self-dual) 3-form fluxes; (ii) by generalizing the ``conifold" ansatz of hep-th/0002159 in a natural ``asymmetric" way so that the 1-d action describing radial evolution still admits a superpotential and then solving the resulting 1-st order system. The superpotentials corresponding to the ``resolved" and ``deformed" cases turn out to have essentially the same simple structure. The solution for the resolved conifold case has the same asymptotic UV behaviour as in the conifold case, but unlike the deformed conifold case is still singular in the IR. The naked singularity is of repulson type and may have a brane resolution.

https://iopscience.iop.org/article/10.1088/1126-6708/2000/11/028/pdf

3-branes on resolved conifold

We discuss various Penrose limits of conformal and nonconformal backgrounds. InAdS5£T 1;1 , for a particular choice of the angular coordinate inT 1;1 the resulting Penrose limit coincides with the similar limit forAdS5£S 5 . In this case an identiflcation of a subset of fleld theory operators with the string zero mode creation operators is possible. For another limit we obtain a light-cone string action that resembles a particle in a magnetic fleld. We also consider three difierent types of backgrounds that are dual to nonconformal fleld theories: The Schwarzschild black hole in AdS5, D3-branes on the small resolution of the conifold and the Klebanov-Tseytlin background. We flnd that in all three cases the introduction of nonconformality renders a theory that is no longer exactly solvable and that the form of the deformation is universal. The corresponding world sheet theory in the light-cone gauge has a ? = x + dependent mass term.

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On Penrose limits and gauge Theories

Motivated by the duality with thermalization in field theory, we study gravitational collapse of a minimally coupled massless scalar field in Einstein gravity with a negative cosmological constant. We investigate the system numerically and establish that for small values of the initial amplitude of the scalar field there is no black hole formation, rather, the scalar field performs an oscillatory motion typical of geodesics in AdS. For large enough values of the amplitude of the scalar field we find black hole formation which we detect numerically as the emergence of an apparent horizon. Using the time of formation as an estimate for thermalization in the field theory we conclude that thermalization occurs very rapidly, close to the causal bound for a very wide range of black hole masses. We further study the thermalization time in more detail as a function of the amplitude and the width of the initial Gaussian scalar field profile and detect a rather mild structure.

Rapid Thermalization in Field Theory from Gravitational Collapse

We construct backreacted D3/D7 supergravity backgrounds which are dual to four-dimensional N = 1 and N = 2 supersymmetric Yang-Mills at large Nc with ∞avor quarks in the fundamental representation of SU(Nc). We take into account the backreaction of D7-branes on either AdS5£S 5 or AdS5£T 1;1 , or more generically on backgrounds where the space transverse to the D3-branes is Kahler. The construction of the backreacted geometry splits into two stages. First we determine the modiflcation of the six-dimensional space transverse to the D3 due to the D7, and then we compute the warp factor due to the D3. TheN = 2 background corresponds to placing a single stack of Nf D7-branes in AdS5£S 5 . Here the Kahler potential is known exactly, while the warp factor is obtained in certain limits as a perturbative expansion. By placing another D7 0 probe in the backreacted D3/D7 background, we derive the efiect of the D7-branes on the spectrum of the scalar ∞uctuations to flrst order in Nf. The two systems with N = 1 supersymmetry that we discuss are D3/D7/D7 0 and D3/D7 on the conifold. In both cases, the Kahler potential is obtained perturbatively in the number of D7-branes. We provide all the ingredients necessary for the computation of each term in the expansion, and in each case give the flrst few terms explicitly. Finally, we comment on some aspects of the dual gauge theories.

http://iopscience.iop.org/article/10.1088/1126-6708/2005/02/022/pdf

Leopoldo A. Pando Zayas

Papers

Entanglement temperature and entanglement entropy of excited states

3-branes on resolved conifold

On Penrose limits and gauge Theories

Rapid Thermalization in Field Theory from Gravitational Collapse

Holographic duals of flavored Script N = 1 super Yang-Mills: beyond the probe approximation