Exact Stress Tensor Flows in Causal Electrodynamics: Courant-Hilbert Approach

Abstract

It has been shown that for theories whose Courant-Hilbert function depends on a single dimensionful coupling, the energy-momentum tensor is controlled by the two combinations $τ\dot{\ell}$ and $\ell-τ\dot{\ell}$, so that the quadratic invariants $X=T_{μν}T^{μν}$ and $Y=(T^μ_μ)^2$ depend on a single auxiliary variable. Eliminating this variable in favour of $Y/X$ yields the exact autonomous flow $\partial_λ\mathcal L=X\,\mathcal F(Y/X)$, with no explicit dependence on the coupling, the electromagnetic invariants or the CH parameter, on every branch where the map is invertible. The expansion of $\mathcal F$ in powers of $\sqrt{Y/X}$ shows which flows are built from integer powers of $X$ and $Y$ and which require half-integer powers. In this paper, we work out four examples in closed form, each with its analytic structure and physical interpretation: the generalized Born-Infeld theory, whose flow is quadratic in the stress tensor; the $q$-deformed family, in which the odd coefficients of the expansion vanish at $q=1/2$ and are nonzero otherwise; the no-$τ$-maximum theory; and the causal Russo-Townsend logarithmic theory, whose flow involves the lower real branch of the Lambert $W$ function and reproduces the known perturbative result. The singular points of $\mathcal{F}$ are related to the critical fields, energy conditions, and strong-field behavior of each theory.