We initiate an investigation into 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 in which the same operator acts on the source moment. Both constructions yield history-dependent responses with small memory-like offsets. However, in all cases we studied, the signal decays to zero at late times, failing to reproduce the permanent displacement predicted by General Relativity. We showed that, under asymptotic and spatial flatness of spacetime, the solutions of the proposed models decay to zero at late times when the time derivatives of the perturbed metric are temporally localized and bounded at each spatial point. Therefore, our results constitute a no-go demonstration: naive fractionalization is insufficient to model the permanent offset in the metric without explicitly building in flux-balance laws or asymptotic symmetry structure. 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 spacetime with $D>4$ dimensions.
"With fractional time operators of the form above, a gravitational waveform generated by a finite-duration burst must decay to zero at late times" (§4.1, Eq. 45): under Assumption A the auxiliary ḡ^α_µν is uniformly bounded, t^(α−1)ḡ^α_µν → 0, the equation reduces to Δh̄_µν = 0, and asymptotic flatness plus the maximum principle force h̄_µν → 0.
Assumption A (§4.1): for every k ∈ ℕ⁺ and every spatial point x, ∂_t^k h̄_µν(t,x) is bounded by some M_k AND has compact support in time (vanishes for t > τ'(x)). Compact temporal support of all derivatives at each point is essentially "the field is eventually time-independent at every x," logically close to the no-memory conclusion the theorem derives. Whether this is an independent physical hypothesis or a near-tautology is unclear.
1/ GW "memory" is a permanent shift in detector arms after a burst — in GR tied to flux-balance laws at null infinity and BMS supertranslations. The authors ask: can fractional calculus, with built-in long-tailed memory kernels, reproduce it? 2/ They try two toys: a sequential Caputo-fractionalized linearized Einstein equation, and a fractionalized quadrupole formula. Both produce small hereditary offsets, but in every simulation the signal decays back to zero. 3/ They prove a late-time decay theorem: under boundedness and temporal localization of all time derivatives of h̄, the t^(α−1) prefactor forces Δh̄=0 asymptotically, hence h̄→0. Conclusion: naive fractionalization can't substitute for flux-balance / BMS structure.
Gravity waves leave a permanent stretch behind. The authors show that a popular "memory math" trick cannot make that permanent stretch.
Full text reviewed. Modest, well-scoped negative result: two toy fractionalizations (sequential Caputo wave eq. and fractionalized quadrupole) shown numerically and analytically not to produce a permanent metric offset. Numerical sections are illustrative; the load-bearing content is the late-time decay theorem in §4: integration by parts on the weakly singular Caputo kernel, bounding ḡ^α via dominated estimates, then using t^(α−1)→0 to reduce to Δh̄=0 and applying max principle with asymptotic flatness. The weakest link is Assumption A, which requires every ∂_t^k h̄ to be bounded AND compactly supported in time at each x. Compact temporal support of all derivatives pointwise is essentially "the field eventually stops evolving," which is close to assuming the conclusion. The authors flag it as physically reasonable for burst-like systems, but the logical tightness as a no-go deserves more care. The choice of sequential Caputo with t^{1−α} prefactor is one of several inequivalent dimensionally consistent fractionalizations; the no-go does not obviously cover Erdélyi–Kober or tempered kernels, despite abstract phrasing. Result is plausible and consistent with the broader observation that GR memory is an asymptotic-symmetry/flux-balance phenomenon, not a hereditary-kernel one. Confidence MODERATE: I followed the analytic argument under its assumptions; have not independently re-derived all integration-by-parts steps or verified numerical convergence. Novelty is limited but real.