Session N08: Classical and Quantum Aspects of Black Holes

The Entropy of Dynamical Black Holes

We study the entropy of black holes in arbitrary classical diffeomorphism-invariant Lagrangian theories of gravity in $n$ dimensions. We propose a new formula for the entropy of a dynamical black hole, valid to second order in perturbation theory off of a stationary black hole background. In stationary eras, this formula agrees with the Noether charge formula proposed by one of us, but in nonstationary eras this formula introduces a nontrivial correction term to the Noether charge formula. In particular, in general relativity, our formula gives the entropy of a dynamical black hole as the area minus the integral of the expansion of the null generators of the horizon. Using our formula for the entropy of a black hole, we prove the ``physical process version'' of the first law of black hole thermodynamics for arbitrary perturbations of a stationary black hole. We also show that this entropy formula obeys a ``second law'' of black hole thermodynamics at leading order. We conclude with a derivation of a relation between our formula with those proposed recently by Wall and Dong.   

Late time behaviors of quantized scalar fields in the Unruh state in two dimensional spacetimes

It is shown that the late time behaviors for some of the modes of a scalar field in the Unruh state in two-dimensions depend on whether these modes diverge in the zero frequency or infrared limit.  There is an infrared divergence for the massless minimally coupled scalar field.  In this case the modes that are positive frequency with respect to the Kruskal time coordinate on the past horizon approach constant values at late times and, as has been shown previously, the two-point function grows linearly in time if the points are split in the spatial direction.  Two cases are considered where there are no infrared divergences in the mode functions. One is the case of a delta function potential and the other a massive scalar field in Schwarzschild-de Sitter spacetime.  In both cases the mode functions that are positive frequency with respect to the Kruskal time on the past horizon(s) vanish in the late time limit.  For the delta function potential it is also shown that the two-point function does not grow linearly in time.

    

Spherically symmetric contribution to the stress-energy tensor for a quantum field in a collapsing null shell background

The renormalized expectation value of the stress-energy tensor corresponding to a massless minimally-coupled scalar field in the ”in” vacuum state is numerically computed outside the event horizon of a four-dimensional black hole that forms from the collapse of a null shell. The computations are performed in the s-wave sector which is expected to make the largest contribution to the stress-energy tensor. The divergence in the unrenormalized stress-energy tensor is removed by subtracting the unrenormalized expression for the stress-energy

tensor when the field is in the Unruh state. Comparison is made between the components of the renormalized stress-energy tensor in four dimensions and the known results in the two dimensions. The convergence of the stress-energy tensor for the ”in” vacuum state to the stress-energy tensor for the Unruh state at late times is discussed.   

Quantum Effects in 4D Schwarzschild-de Sitter Spacetime: Properties of the Hadamard Two-Point Function

Schwarzschild-de Sitter spacetime offers a classical background structure in which quantum fields and their correlations can be studied in a case where there is both a black hole and cosmological horizon. To investigate these correlations, the Hadamard two-point function which will be computed for a massless minimally coupled scalar field for the Unruh state in four dimensions. It was previously found in two dimensions that the Hadamard function grows linearly in time if the points are split in a spatial direction for coordinates that are natural near the black hole horizon. The growth is proportional to the sum of the black hole and cosmological horizon surface gravities. A main goal in extending the analysis to four dimensions is to determine if this growth persists, or is modified by scattering effects due to the existence of an effective potential in the mode equation. If this growth in the two-point function is indeed present in four dimensions, this could be indicative of an instability which might have interesting physical implications, for example modification of the power spectrum if primordial black holes existed during inflation.   

Negative-temperature Hawking radiation seen near the inner horizon, the outer horizon, and beyond

The near-horizon geometry of an astrophysical black hole can be drastically altered when quantum effects like Hawking radiation are included on top of the classical vacuum metric. Here we study particle production effects through two different means: first, an effective temperature can be defined that in the adiabatic limit exactly reproduces the Hawking effect but generalizes to any inertial observer at an arbitrary position inside or outside the black hole, looking in any direction. Second, in the near-horizon regimes, the exact Bogoliubov coefficients governing the vacuum expectation value of the number operator can be computed to find the full Hawking greybody spectrum. When applied to a charged black hole with a quantum scalar field, these two calculations can help answer several questions about the semiclassical nature of astrophysical black holes: does Hawking radiation contribute to mass inflation at the inner horizon, and is it confined to a single point in the radial direction? Can the Hawking temperature ever be negative, and how should that be properly interpreted?   

On the loop quantization of an interior black hole model

In this talk we present some recent investigations on the quantum theory underlying a recent extension of Kruskal spacetime, proposed by Ashtekar, Olmedo and Singh in the context of effective loop quantum cosmology. To this aim, we first introduce a Hamiltonian description of the interior region that is consistent with the effective dynamics. In such description, the spacetime geometry is described in terms of the typical holonomy-flux algebra employed in loop quantum cosmology, thus providing the starting point for the quantization of the black hole interior. We then develop the first steps to construct this quantum theory. In particular, we propose specific operators for the constraints of the system and, under reasonable assumptions, we deduce the formal expression of the physical states as functions of the black hole mass. These results pave the road for future investigations that aim to include genuine loop quantum effects on black hole spacetimes and their perturbations.   

Author not Attending

We aim to construct a model of dilaton gravity in 1+1 spacetime dimensions for a nonsingular Reissner-Nordstrom (RNS) anti-de-Sitter (AdS) black hole. To construct a nonsingular black hole in 2D dilaton gravity, we start with the traditional RNS AdS spacetime solution. We use the maximal  Limiting Curvature Hypothesis (LCH) and the minimal curvature conjecture (MCC) to eliminate the singularity. By limiting the curvature at both extrema, i.e., bound curvatures at both UV and IR scales, we obtain a new static geodesically complete spacetime in two-dimensional dilaton gravity. We find the exact solution corresponding to a nonsingular RNS anti-de Sitter (AdS) black hole.

We use this solution to find the curvature tensor R and investigate the evaporation process of the nonsingular RNS AdS black hole.

Keywords: Black Hole Thermodynamics; 2D gravity; anti-de Sitter black holes; Quantum Gravity.

 

    

The true bijection between quantum entropy and the mass associated with every gravitationally closed horizon.

A black hole, (BH), is a closed space time horizon. It is the edge of space time. We can measure the amount of quantum information that has fallen behind it. Every bit of quantum information is the same. It has an entropy. In 1915 Karl Schwarzschild gave us the mass and radius of this mathematical object. It took 59 more years before Stephen Hawking realized this horizon has a temperature. It has taken another 48 years to prove that every bit of quantum information that falls in, has a quantum of entropy. Every quantum of entropy must have the same temperature. It cannot be zero and cannot be lower than the black hole temperature for the mass of our universe. Every gravitationally closed space-time horizon knows this temperature, (T_u). All closed space-time horizons are quantized by this temperature (T_u). I know this temperature and can prove it. (T_u) is the temperature of our universe as a BH. Every BH has a mass, (M), radius, (R), and temperature, (T).

(2M)/(R²)(T) = (4ϖk)/(c²)(L_p²) = 1/(T_u)