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Abstract 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. 

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. 

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 {\it 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 the construction of the in modes more complicated. The numerical computation of the in modes in this case is given. 

A bstract 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.