Can One Model Gravitational Nonlinear Memory with Fractional Derivative Operators?

TL;DR

Problem: can fractional calculus model the nonlinear gravitational-wave memory predicted by GR? Approach: study two toy constructs—(i) a sequential Caputo time-fractional modification of the linearized Einstein field equations and (ii) a fractionalized quadrupole moment—revealing their history-dependent responses. Findings: both approaches produce memory-like offsets but all solutions decay to zero at late times, failing to reproduce the GR permanent displacement; this constitutes a no-go for naive fractionalization. Implications: viable fractional models must embed fractional kernels directly into GR's hereditary flux-balance integral while preserving gauge invariance and dimensional consistency, with potential links to modified gravity and memory behavior in higher dimensions ($D>4$).

Abstract

We investigate whether fractional calculus, with its intrinsic long-tailed memory and nonlocal features, can provide a viable model for gravitational-wave memory effects. We consider two toy constructions: ($i$) a fractional modification of the linearized Einstein field equations using a sequential Caputo operator, and ($ii$) a fractionalized quadrupole formula where the source moment is acted upon by the same operator. Both constructions yield history-dependent responses with small memory-like offsets. However, in all cases, the signal decays to zero at late times, failing to reproduce the permanent displacement predicted by GR. Our results, therefore, constitute a no-go demonstration: naive fractionalization is insufficient to model nonlinear gravitational memory. We argue that any successful model must incorporate fractional kernels directly into the hereditary flux-balance integral of General Relativity, while preserving gauge invariance and dimensional consistency. We also discuss possible connections to modified gravity and the absence of memory in the spacetime with dimensions $D>4$.

Figures (6)

• Figure 1: Time evolution of the field $u(t,x)$ at two observation points (a) near the source $x\approx0$ and (b) at distance $x\approx2.0$ for the source frequency $\omega=-6\pi$. The curves show the results for several fractional orders $\alpha$.
• Figure 2: Time evolution of the field $u(t,x)$ at two observation points (a) near the source $x\approx0$ and (b) at distance $x\approx2.0$ for the source frequency $\omega=-18\pi$ where the top-right panel shows the zoomed-in view of the region enclosed by the red dashed rectangle. The curves show the results for several fractional orders $\alpha$.
• Figure 3: Time evolution of the field $u(t,x)$ at two observation points (a) near the source $x\approx0$ and (b) at distance $x\approx2.0$ for the source frequency $\omega=-24\pi$ where the top-right panel shows the zoomed-in view of the region enclosed by the red dashed rectangle. The curves show the results for several fractional orders $\alpha$.
• Figure 4: Time evolution of the transverse-traceless gravitational waveform component $h_{11}^{\mathrm{TT}}$ for an equal mass circular binary with orbital frequency $f_0=0.2~\mathrm{Hz}$, observed along the $z$-axis. The two panels compare the fractional model for (a) $\alpha=0.8$ and (b) $\alpha=0.9$. In figures (a) and (b), bottom-left panel shows the chosen A and B values in \ref{['eq: Linear combination of Quadrupole moment classical/fractional']}, and top-right panel shows the zoomed-in view of the region enclosed by the red point.
• Figure 5: Time evolution of the transverse-traceless gravitational waveform component $h_{11}^{\mathrm{TT}}$ for an equal mass circular binary with orbital frequency $f_0=0.4~\mathrm{Hz}$, observed along the $z$-axis. The two panels compare the fractional model for (a) $\alpha=0.8$ and (b) $\alpha=0.9$. In figures (a) and (b), bottom-left panel shows the chosen A and B values in \ref{['eq: Linear combination of Quadrupole moment classical/fractional']}, and top-right panel shows the zoomed-in view of the region enclosed by the red point.