Thus the first order of business is the construction of the stress tensor and then to derive the symmetric stress-energy tensor two point function, the noise kernel, in terms of the Wightman Green function. In this section we will use the traditional notation for index tensors in the point-separation context.
An object like the Green function is an example of a bi-scalar: It transforms as scalar at both points and . We can also define a bi-tensor : Upon a coordinate transformation, this transforms as a rank tensor at and a rank tensor at . We will extend this up to a quad-tensor which has support at four points , transforming as rank tensors at each of the four points. This also sets the notation we will use: unprimed indices referring to the tangent space constructed above , single primed indices to , double primed to and triple primed to . For each point, there is the covariant derivative at that point. Covariant derivatives at different points commute, and the covariant derivative at, say, point does not act on a bi-tensor defined at, say, and :
Having objects defined at different points, the coincident limit is defined as evaluation “on the diagonal”, in the sense of the spacetime support of the function or tensor, and the usual shorthand is used. This extends to -tensors asSynge’s theorem in this context; we follow Fulling’s discussion .
The bi-tensor of parallel transport is defined such that when it acts on a vector at , it parallel transports the vector along the geodesics connecting and . This allows us to add vectors and tensors defined at different points. We cannot directly add a vector at and vector at . But by using , we can construct the sum . We will also need the obvious property .
The main bi-scalar we need is the world function . This is defined as a half of the square of the geodesic distance between the points and . It satisfies the equationend point expansion of a bi-scalar is of the form
The last object we need is the VanVleck–Morette determinant , defined as . The related bi-scalar
Further details on these objects and discussions of the definitions and properties are contained in [67, 68] and . There it is shown how the defining equations for and are used to determine the coincident limit expression for the various covariant derivatives of the world function (, , etc.) and how the defining differential equation for can be used to determine the series expansion of . We show how the expansion tensors are determined in terms of the coincident limits of covariant derivatives of the bi-scalar . ( details how point separation can be implemented on the computer to provide easy access to a wider range of applications involving higher derivatives of the curvature tensors.)
© Max Planck Society and the author(s)