Method to compute the stress-energy tensor for the massless spin $\frac{1}{2}$ field in a general static spherically symmetric spacetime

A method for computing the stress-energy tensor for the quantized, massless, spin $\frac{1}{2}$ field in a general static spherically symmetric spacetime is presented. The field can be in a zero temperature state or a nonzero temperature thermal state. An expression for the full renormalized stress-energy tensor is derived. It consists of a sum of two tensors both of which are conserved. One tensor is written in terms of the modes of the quantized field and has zero trace. In most cases it must be computed numerically. The other tensor does not explicitly depend on the modes and has a trace equal to the trace anomaly. It can be used as an analytic approximation for the stress-energy tensor and is equivalent to other approximations that have been made for the stress-energy tensor of the massless spin $\frac{1}{2}$ field in static spherically symmetric spacetimes.