On Effective Potential in Tortoise Coordinate

TL;DR

This paper analyzes the effective potential in Tortoise coordinates for scalar and spinor fields in various three-dimensional black hole backgrounds, revealing distinct asymptotic behaviors and potential barriers.

Contribution

It derives a general form of the effective potential in Tortoise coordinates for different fields and black hole types, highlighting differences in asymptotic behavior.

Findings

Potential in BTZ and new type black holes diverges at infinity

Vanishing horizon potential approaches a fixed value at infinity

Distinct asymptotic behaviors between black hole types

Abstract

In this paper, we study the field dynamics in Tortoise coordinate where the equation of motion of a scalar can be written as Schrodinger-like form. We obtain a general form for effective potential by finding the Schrodinger equation for scalar and spinor fields and study its global behavior in some black hole backgrounds in three dimension such as BTZ black holes, new type black holes and black holes with no horizon. Especially, we study the asymptotic behavior of potential at infinity, horizons and origin and find that its asymptotic in BTZ and new type solution is completely different from that of vanishing horizon solution. In fact, potential for vanishing horizon goes to a fixed quantity at infinity, while in BTZ and new type black hole we have an infinite barrier.