Transport equation approach to calculations of Hadamard Green functions and non-coincident DeWitt coefficients

Building on an insight due to Avramidi, we provide a system of transport equations for determining key fundamental bitensors, including derivatives of the world function, $\sigma (x,x^{\prime })$, the square root of the Van Vleck determinant, ${\Delta }^{1/2}(x,x^{\prime })$, and the tail term, $V(x,x^{\prime })$, appearing in the Hadamard form of the Green function. These bitensors are central to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity. Their transport equations may be used either in a semi-recursive approach to determining their covariant Taylor series expansions, or as the basis of numerical calculations. To illustrate the power of the semi-recursive approach, we present an implementation in Mathematica, which computes very high order covariant series expansions of these objects. Using this code, a moderate laptop can, for example, calculate the coincidence limit $[a_7(x,x)]$ and $V(x,x^{\prime })$ to order $({\sigma }^a{)}^{20}$ in a matter of minutes. Results may be output in either a compact notation or in xTensor form. In a second application of the approach, we present a scheme for numerically integrating the transport equations as a system of coupled ordinary differential equations. As an example application of the scheme, we integrate along null geodesics to solve for $V(x,x^{\prime })$ in Nariai and Schwarzschild spacetimes.

Figure 1

Comparison of numerical and exact analytic calculations of ${\Delta }^{1/2}$ as a function of the angle, $\varphi$, along an orbiting null geodesic in Nariai spacetime. Left: The numerical calculation (blue dots) is a close match with the analytic expression (red line). Right: With parameters so that the code completes in 1 minute, the relative error is within 0.0001% up to the boundary of the normal neighborhood (at $\varphi =\pi$).

Figure 2

Comparison of numerical and exact analytic calculations of $V_0$ for a massless, minimally coupled scalar field as a function of the angle, $\varphi$, along an orbiting null geodesic in Nariai spacetime. Left: The numerical calculation (blue dots) is a close match with the analytic expression (red line). The coincidence value is $V_0(x,x)=\frac{1}{2}(\xi -\frac{1}{6})R=-\frac{1}{3}$, as expected. Right: With parameters so that the code completes in 1 min, the relative error in the numerical calculation is within 1% up to the boundary of the normal neighborhood (at $\varphi =\pi$).

Figure 3

${\Delta }^{1/2}$ along the light cone in Schwarzschild spacetime. The point $x$ at the vertex of the cone is fixed at $r=10M$. ${\Delta }^{1/2}$ increases along a geodesic up to the caustic where it is singular.

Figure 4

$V(x,x^{\prime })$ for a massless, scalar field along the light cone in Schwarzschild spacetime. The point $x$ (the vertex of the cone) is fixed at $r=6M$. $V(x,x^{\prime })$ is 0 initially, then, travelling along a geodesic, it goes negative for a period before turning positive and eventually becoming singular at the caustic. (Note that $V$ coincides with $V_0$ on the light cone.)

Figure 5

$V_0(x,x^{\prime })$ (solid blue line) and ${\Delta }^{1/2}$ (dashed purple line) for a massless, scalar field along the timelike circular orbit at $r=10M$ in Schwarzschild spacetime as a function of the angle, $\varphi$ through which the geodesic has passed. In the logarithmic plot, the absolute value of $V_0(x,x^{\prime })$ is plotted to illustrate the divergence close to the caustic. [Since $V_0(x,x^{\prime })$ passes through 0 at $\varphi \approx 0.6$ and $\varphi \approx 2.6$ there are corresponding features in the logarithmic plot.]