Soft synchronous gauge: principal value prescription

Abstract

The synchronous gauge in gravity ($g_{0\hat{I}»}=-\hat{I}{´}_{0\hat{I}»}$) is ill-defined due to the singularity at $p_0=0$ in the graviton propagator. Previously we studied "softening" this gauge by considering instead the gauge $n^{\hat{I}}»g_{\hat{I}»\hat{I}¼}=0$, $n^{\hat{I}}»=(1,-\epsilon ({\partial }^j{\partial }_j{)}^{-1}{\partial }^k)$ in the limit $\epsilon \to 0$. We now explore the possibility of using a principal value prescription (not in the standard Cauchy sense), which amounts, roughly speaking, to replacing singularities $p_0^{-j}\Rightarrow [(p_0+i\epsilon {)}^{-j}+(p_0-i\epsilon {)}^{-j}]/2$, which then behave like distributions. We show that such a propagator follows upon adding to the action a gauge-violating term of a general form, which reduces to $ \sim \int f_λΛ^{λμ} f_μ\d^4 x $ with a constant operator $\hat{I}{›}^{\hat{I}»\hat{I}¼}$ depending on $\partial$ and a metric functional $f_{\hat{I}}»$. The contribution of the ghost fields to the effective action is analysed. For the required intermediate regularization, the discrete structure of the theory at small distances is implied. It is shown that the ghost contribution can be disregarded in the limit $\epsilon \to 0$.

19 pages, our previous paper arXiv:2312.17119 discusses a gravitational analogue of the {\it Landshoff} prescription; v2 adds consideration of some choice of free parameters that makes the required gauge fixing term non-singular (Eqs. (52-54))