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’s function. In this section we will use the traditional notation for index tensors in the point-separation context.
An object like the Green’s function is an example of a bi-scalar; it transforms as a scalar at both
points
and
. We can also define a bitensor
: 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
and
, transforming
as rank
and
tensors at each of the four points. This also sets the notation we
will use: unprimed indices refer 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 bitensor 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 as
The bitensor 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 equation
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 [83, 84] and [90, 301]. 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
. ([301] 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.)
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