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

TL;DR

This paper introduces a numerical method to compute the stress-energy tensor of a quantized scalar field outside a black hole formed by null shell collapse, extending previous 2D results to 4D with scattering considerations.

Contribution

It presents a new numerical approach for calculating the stress-energy tensor in 4D black hole spacetimes formed from collapse, including mode construction with scattering effects.

Findings

Numerical results agree with known 2D analytic solutions.

Mode construction in 4D accounts for scattering effects.

Method enables analysis of quantum fields in dynamic black hole spacetimes.

Abstract

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.