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Paul R. Anderson

Publications and source records attributed to Paul R. Anderson.

At least 19 recordsLinked to original sources
arXiv2025

Vacuum polarization and stress-energy of a quantum field inside of two-dimensional black holes

Quantum effects are studied in both Schwarzschild spacetime and a spacetime in which a null shell collapses to form a black hole via the vacuum polarization $\langle ϕ^2 \rangle$ and stress-energy tensor $\langle T_{ab} \rangle$ for a massless minimally-coupled scalar field in two dimensions. For Schwarzschild spacetime, the Boulware, Unruh, and Hartle-Hawking states are considered. For the collapsing null shell spacetime, the \textit{in} vacuum state is used. Instabilities of the Unruh, Hartle-Hawking, and \textit{in} states resulting from the behavior of $\langle ϕ^2 \rangle$ in the regions inside and outside of the horizon are found. The question of how well the Unruh state for the eternal black hole approximates quantum effects in the interior of a black hole that forms from collapse is addressed.

gr-qc↗
arXiv2025

Linear Response Analysis of the Semiclassical Approximation to Spin 1/2 Quantum Electrodynamics in 1+1 Dimensions

An investigation of the validity of the semiclassical approximation to quantum electrodynamics in 1+1 dimensions is given. The criterion for validity used here involves the impact of quantum fluctuations introduced through a two-point function which emerges naturally when considering the stability of the backreaction equation to linear order perturbations, resulting in the linear response equation. Consideration is given to the case of a spatially homogeneous electric field generated by a classical source, coupled to a quantized massive spin 1/2 field. Solutions to the linear response equation as well as the impact of quantum fluctuations introduced through the current density two-point correlation function are presented for two relevant electric field-to-mass parameter values $qE/m^2$, indicative of the strength of the backreaction process. Previous efforts utilized approximate solutions to the linear response equation that were expected to be valid for early times. A comparative analysis is given between the exact and approximate solutions in order to validate this conjecture.

gr-qc↗
arXiv2025

Looking for traces of Hawking radiation in correlation functions of BEC acoustic black holes

Renaud Parentani was one of the leading figures in Quantum Field Theory in curved spacetime, in particular concerning its applications to Hawking-like radiation in analogue models. In this paper dedicated to him, we discuss the characteristic features appearing in the correlation functions in an acoustic black hole formed by a Bose-Einstein condensate, considered as signature of the presence of Hawking radiation in this system.

gr-qc↗
arXiv2025

Late-time behaviors of scalar field modes for a collapsing null shell spacetime and for the Unruh state in Schwarzschild spacetime

The behaviors of the modes for a massless minimally coupled scalar field are investigated for the Unruh state for Schwarzschild spacetime and the "in" vacuum state for a spacetime in which a spherically symmetric null shell collapses to form a nonrotating black hole. In both cases there are two different sets of solutions to the mode equation that make up the state. For both spacetimes, one set of modes oscillates forever with no damping of the oscillations and the other set approaches zero at late times. The difference between a mode that oscillates forever in the null-shell spacetime and the corresponding mode for the Unruh state vanishes as a power law in time. The modes that approach zero at late times also vanish at late times as a power law in time. In all cases the power-law damping is preceded by a period of oscillations that appear to be due to quasi-normal modes.

gr-qc↗
arXiv2025

Quantum Effects in 3+1 Schwarzschild-de Sitter Spacetime: Properties of the Hadamard Function

In a four-dimensional Schwarzschild-de Sitter background, the spherically symmetric $(\ell=0)$ contribution to the Hadamard two-point correlation function is computed for a massless minimally-coupled scalar field in the Unruh state. Consideration is given to spacetime points located between the black hole and cosmological horizons. Previously it was found in two dimensions at late times for spatially separated points that the Hadamard function exhibits unbounded linear growth in time, with a rate of growth proportional to the sum of the black hole and cosmological surface gravities. Here it is shown numerically that this instability persists in four dimensions, but with a modification of the two-dimensional result due to scattering effects associated with the scalar field modes. An analytic approximation is derived for the growth rate in four dimensions and, in the limit that the black hole vanishes, is found to be equivalent to the rate of growth for the Hadamard function found previously for de Sitter space in cosmological coordinates.

gr-qc↗
arXiv2023

Backreaction and order reduction in initially contracting models of the universe

The semiclassical backreaction equations are solved in closed Robertson-Walker spacetimes containing a positive cosmological constant and a conformally coupled massive scalar field. Renormalization of the stress-energy tensor results in higher derivative terms that can lead to solutions that vary on much shorter time scales than the solutions that would occur if the higher derivative terms were not present. These extra solutions can be eliminated through the use of order reduction. Four different methods of order reduction are investigated. These are first applied to the case when only conformally invariant fields, with and without classical radiation, are present. Then they are applied to the massive conformally coupled scalar field. The effects of different adiabatic vacuum states for the massive field are considered. It is found that if enough particles are produced, then the Universe collapses to a final singularity. Otherwise it undergoes a bounce, but at a smaller value of the scale factor (for the models considered) than occurs for the classical de Sitter solution. The stress-energy tensor incorporates both particle production and vacuum polarization effects. An analysis of the energy density of the massive field is done to determine when the contribution from the particles dominates.

gr-qc↗
arXiv2023

Infrared Effects and the Unruh State

Detailed behaviors of the modes of quantized scalar fields in the Unruh state for various eternal black holes in two dimensions are investigated. It is shown that the late-time behaviors of some of the modes of the quantum fields and of the symmetric two-point function are determined by infrared effects. The nature of these effects depends upon whether there is an effective potential in the mode equation and what form this potential takes. Here, three cases are considered, one with no potential and two with potentials that are nonnegative everywhere and are zero on the event horizon of the black hole and zero at either infinity or the cosmological horizon. Specifically, the potentials are a delta function potential and the potential that occurs for a massive scalar field in Schwarzschild-de Sitter spacetime. In both cases, scattering effects remove infrared divergences in the mode functions that would otherwise arise from the normalization process. When such infrared divergences are removed, it is found that the modes that are positive frequency with respect to the Kruskal time on the past black hole horizon approach zero in the limit that the radial coordinate is fixed and the time coordinate goes to infinity. In contrast, when there is no potential and thus infrared divergences occur, the same modes approach nonzero constant values in the late-time limit when the radial coordinate is held fixed. The behavior of the symmetric two-point function when the field is in the Unruh state is investigated for the case of a delta function potential in certain asymptotically flat black hole spacetimes in two dimensions. The removal of the infrared divergences in the mode functions results in the elimination of terms that grow linearly in time.

gr-qc↗
arXiv2022

Horizons and Correlation Functions in 2D Schwarzschild-de Sitter Spacetime

Two-dimensional Schwarzschild-de Sitter is a convenient spacetime in which to study the effects of horizons on quantum fields since the spacetime contains two horizons, and the wave equation for a massless minimally coupled scalar field can be solved exactly. The two-point correlation function of a massless scalar is computed in the Unruh state. It is found that the field correlations grow linearly in terms of a particular time coordinate that is good in the future development of the past horizons, and that the rate of growth is equal to the sum of the black hole plus cosmological surface gravities. This time dependence results from additive contributions of each horizon component of the past Cauchy surface that is used to define the state. The state becomes the Bunch-Davies vacuum in the cosmological far field limit. The two point function for the field velocities is also analyzed and a peak is found when one point is between the black hole and cosmological horizons and one point is outside the future cosmological horizon.

hep-th↗
arXiv2022

Linear growth of the two-point function for the Unruh state in $1+1$ dimensional black holes

The symmetric two-point function for a massless, minimally coupled scalar field in the Unruh state is examined for Schwarzschild-de Sitter spacetime in two dimensions. This function grows linearly in terms of a time coordinate that is well-defined on the future black hole and cosmological horizons, when the points are split in the space direction. This type of behavior also occurs in two dimensions for other static black hole spacetimes when the field is in the Unruh state, and at late times it occurs in spacetimes where a black hole forms from the collapse of a null shell. The generalization to the case of the symmetric two-point function in two dimensions for a massive scalar field in Schwarzschild-de Sitter spacetime is discussed.

gr-qc↗
arXiv2021

Stress-energy Tensor for a Quantized Scalar Field in a Four-Dimensional Black Hole Spacetime that Forms From the Collapse of a Null Shell

A method is presented which allows for the numerical computation of the stress-energy tensor for a quantized massless minimally coupled scalar field in the region outside the event horizon of a 4D Schwarzschild black hole that forms from the collapse of a null shell. This method involves taking the difference between the stress-energy tensor for the in state in the collapsing null shell spacetime and that for the Unruh state in Schwarzschild spacetime. The construction of the modes for the in vacuum state and the Unruh state is discussed. Applying the method, the renormalized stress-energy tensor in the 2D case has been computed numerically and shown to be in agreement with the known analytic solution. In 4D, the presence of an effective potential in the mode equation causes scattering effects that make the construction of the in modes more complicated. The numerical computation of the in modes in this case is given.

gr-qc↗
arXiv2021

Semiclassical predictions regarding a pre-inflationary era and its effects on the power spectrum

An investigation is undertaken into the properties and effects of a preinflationary era during at least part of which semiclassical gravity was valid. It is argued that if the Universe (or our part of it) was approximately homogeneous and isotropic during that era, then the Universe was likely to have been radiation dominated. A simple model in which the Universe contains classical radiation and a cosmological constant is used to investigate potential effects of such a preinflationary era on the cosmic microwave background. The power spectrum is computed using the mode functions of a quantized massless minimally coupled scalar field. Various choices of state for this field are considered, including adiabatic vacuum states of various orders and the vacuum state that would naturally occur if the Universe made a sudden transition from being radiation dominated to de Sitter space. In all cases investigated, there is a suppression of the power spectrum at large angles, and, when plotted as a function of the momentum parameter, there are always oscillations with state-dependent amplitudes.

gr-qc↗
arXiv2021

Correlations between a Hawking particle and its partner in a 1+1D Bose-Einstein condensate analog black hole

The Fourier transform of the density-density correlation function in a Bose-Einstein condensate (BEC) analog black hole is a useful tool to investigate correlations between the Hawking particles and their partners. It can be expressed in terms of $\langle ^{\text{out}}\hat{a}_{\text{up}}^{\text{ext}}\ ^{\text{out}}\hat{a}_{\text{up}}^{\text{int}} \rangle$, where $^{\text{out}}\hat{a}_{\text{up}}^{\text{ext}}$ is the annihilation operator for the Hawking particle and $^{\text{out}}\hat{a}_{\text{up}}^{\text{int}} $ is the corresponding one for the partner. This basic quantity is calculated for three different models for the BEC flow. It is shown that in each model the inclusion of the effective potential in the mode equations makes a significant difference. Furthermore, particle production induced by this effective potential in the interior of the black hole is studied for each model and shown to be nonthermal. An interesting peak that is related to the particle production and is present in some models is discussed.

gr-qc↗
arXiv2021

Pair production due to an electric field in 1+1 dimensions and the validity of the semiclassical approximation

Solutions to the backreaction equation in 1+1-dimensional semiclassical electrodynamics are obtained and analyzed when considering a time-varying homogeneous electric field initially generated by a classical electric current, coupled to either a quantized scalar field or a quantized spin-$\frac{1}{2}$ field. Particle production by way of the Schwinger effect leads to backreaction effects that modulate the electric field strength. Details of the particle production process are investigated along with the transfer of energy between the electric field and the particles. The validity of the semiclassical approximation is also investigated using a criterion previously implemented for chaotic inflation and, in an earlier form, semiclassical gravity. The criterion states that the semiclassical approximation will break down if any linearized gauge-invariant quantity constructed from solutions to the linear response equation, with finite nonsingular data, grows rapidly for some period of time. Approximations to homogeneous solutions of the linear response equation are computed and it is found that the criterion is violated when the maximum value, $E_{\rm max}$, obtained by the electric field is of the order of the critical scale for the Schwinger effect, $E_{\rm max} \sim E_{\rm crit}\equiv m^2/q$, where $m$ is the mass of the quantized field and $q$ is its electric charge. For these approximate solutions the criterion appears to be satisfied in the extreme limits $\frac{qE_{\rm max}}{m^2} \ll 1$ and $\frac{qE_{\rm max}}{m^2} \gg 1$.

gr-qc↗
arXiv2020

Method to compute the stress-energy tensor for a quantized scalar field when a black hole forms from the collapse of a null shell

A method is given to compute the stress-energy tensor for a massless minimally coupled scalar field in a spacetime where a black hole forms from the collapse of a spherically symmetric null shell in four dimensions. Part of the method involves matching the modes for the in vacuum state to a complete set of modes in Schwarzschild spacetime. The other part involves subtracting from the unrenormalized expression for the stress-energy tensor when the field is in the in vacuum state, the corresponding expression when the field is in the Unruh state and adding to this the renormalized stress-energy tensor for the field in the Unruh state. The method is shown to work in the two-dimensional case where the results are known.

gr-qc↗
arXiv2019

Particle Production in the Interiors of Acoustic Black Holes

Phonon creation inside the horizons of acoustic black holes is investigated using two simple toy models. It is shown that, unlike what occurs in the exterior regions, the spectrum is not thermal. This non-thermality is due to the anomalous scattering that occurs in the interior regions.

gr-qc↗
arXiv2019

Late time approach to Hawking radiation: terms beyond leading order

Black hole evaporation is studied using wave packets for the modes. These allow for approximate frequency and time resolution. The leading order late time behavior gives the well known Hawking radiation that is independent of how the black hole formed. The focus here is on the higher order terms and the rate at which they damp at late times. Some of these terms carry information about how the black hole formed. A general argument is given which shows that the damping is significantly slower (power law) than what might be naively expected from a stationary phase approximation (exponential). This result is verified by numerical calculations in the cases of 2D and 4D black holes that form from the collapse of a null shell.

gr-qc↗
arXiv2019

Black hole remnants may exist if Starobinsky inflation occurred

Zero temperature black hole solutions to the semiclassical backreaction equations are investigated. Evidence is provided that certain components of the stress-energy tensors for free quantum fields at the horizon only depend on the local geometry near the horizon. This allows the semiclassical backreaction equations to be solved near the horizon. It is found that macroscopic uncharged zero temperature black hole solutions to the equations may exist if the coefficient of one of the higher derivative terms in the gravitational Lagrangian is large enough and of the right sign for Starobinsky inflation to have occurred in the early Universe.

gr-qc↗