# Vaidya Spacetime in the Diagonal Coordinates

## TL;DR

This paper derives an exact transformation from Vaidya to diagonal coordinates for accreting black holes, revealing regions with coordinate singularities that are not physical, and introduces a new coordinate system avoiding divergences.

## Contribution

It provides an exact solution for the Vaidya metric in diagonal coordinates with a linear mass function, identifying coordinate singularities and proposing a new coordinate system without divergences.

## Key findings

- Identified a narrow region near the horizon with qualitative metric differences from Schwarzschild.
- Discovered that multiple diagonal coordinate sets are needed to cover the spacetime outside the horizon.
- Found that certain coordinate singularities are false firewalls with finite tidal forces.

## Abstract

We have analyzed the transformation from initial coordinates $(v,r)$ of the Vaidya metric with light coordinate $v$ to the most physical diagonal coordinates $(t,r)$. An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate $v$. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates $(t,r)$ is insufficient to cover the entire range of initial coordinates $(v,r)$ outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either $g_{00}=0$ or $g_{00}=\infty$.). The energy-momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false firewalls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates $(\eta,y)$, in which the solution is obtained in explicit form, and there is no energy-momentum tensor divergence.