Are Padé approximants suitable for modelling shape functions of traversable wormholes?

TL;DR

The paper investigates whether Padé approximants can systematically generate shape functions for traversable wormholes that satisfy essential geometric constraints, such as $b(r_0)=r_0$, the flare-out condition, throat regularity, and asymptotic flatness. By analyzing low-order approximants $[1/0]$, $[0/1]$, and $[1/1]$, it demonstrates how to transform generic or unsuitable shape profiles into analytically tractable, physically viable candidates, while outlining parameter regimes that ensure feasibility. It also shows that high-order approximants frequently introduce spurious poles and overly constrained parameterizations, which can jeopardize the wormhole geometry, thus cautioning against indiscriminate use of large orders. Overall, Padé approximants emerge as a useful tool for shaping and simplifying wormhole geometry at low order, but their applicability at higher orders requires careful, case-by-case analysis and remains limited without further constraints such as energy conditions.

Abstract

This study investigates the applicability of Padé approximants in constructing suitable shape functions for traversable wormholes, emphasizing their ability to satisfy essential geometric constraints. By analyzing low-order Padé approximants, we demonstrate their effectiveness in transforming inadequate shape functions into physically consistent candidates, while inherently fulfilling critical criteria such as asymptotic flatness, flare-out conditions, and throat regularity. Specific parameter restrictions are established to ensure compliance with these constraints; for instance, low-order rational approximations help to avoid artificial singularities and maintain asymptotic behavior when derivative conditions at the throat are controlled. In contrast, high-order Padé approximants introduce challenges, including spurious poles within the physical domain, which disrupt geometric requirements. Our findings highlight that low-order Padé approximants provide a robust framework for simplifying complex shape functions into analytically tractable forms, balancing mathematical flexibility with physical feasibility. This work underscores their potential as a systematic tool in traversable wormhole modeling, while cautioning against unphysical artifacts in higher-order approximations.

Figures (11)

• Figure 1: The "'flare-out" condition for (\ref{['original']}) as a function of $r$ with $r_0 = 0.6$ and $d = 0.1, 0.3$ and $0.4$.
• Figure 2: (a) The "flare-out" condition and (b) the fourth condition for $[1/0]$-order Padé approximant of (\ref{['original']}), with $b(r)$ as the approximated function, as a function of $r$ for $d \le 0.43$ ($d=0.1$ and $0.3$) and $d>0.43$ ($d=0.5$) and $r_0 = 0.6$.
• Figure 3: (a) The "flare-out" condition and (b) the fourth condition for $[1/0]$-order Padé approximant of a generic shape function $b(r)$, in the format of \ref{['firsttaylor']}, as a function of $r$, with $r_0 = 1.1$ and $b'(r_0) = 1/nr_0$, with $n=1,2,3$.
• Figure 4: (a) The "flare-out" condition and (b) the fourth condition for $[0/1]$-order Padé approximant of a generic shape function $b(r)$, in the format of \ref{['pade01_3']}, as a function of $r$, with $r_0 = 1.1$ and $b'(r_0) = 1/nr_0 > 0$, with $n=1,1.5,2$.
• Figure 5: (a) The "flare-out" condition and (b) the fourth condition for $[0/1]$-order Padé approximant of a generic shape function $b(r)$, in the format of \ref{['pade01_3']}, as a function of $r$, with $r_0 = 1.1$ and $b'(r_0) = n<0$, with $n=-4,-2,-\frac{4}{3}$.