t'Hooft Meets a String: A Proposal on Shockwave Production by Massive Particles

t’Hooft Meets a String: A Proposal on Shockwave Production by Massive Particles Aman Chawla November 3, 2025 Abstract In this note, the authors propose to extend shockwave production by photons to the string theory domain and more massive particles. In a paper with Dray, Gerard t’Hooft gives the analysis for spacetime shockwaves1(gravitational) generated as a result of the massless photon [2]. In [3] (see Chapter 4.5, page 125), Becker, Becker and Schwarz mention how in the negative sector, mass is created by raising operators of the oscillator - much like high energy states of a quantum simple harmonic oscillator are created by the raising operators [4]. In the present note we propose to integrate these two results. That is, we propose the following investigation: what happens if the massless particle of Dray-t’Hooft is subjected to the raising operator? What is the shockwave or gravitational signature generated by it as the mass moves near an event horizon? This proposal would correctly be framed in the string description of black holes [5] and their event horizons, masses, and gravitational waves [6]. Acknowledgments This work was prepared with the assistance of LLMs. 1We note an important terminological distinction in Penrose’s work on impulsive gravitational waves. According to Penrose [1], a gravitational shock wave refers to a metric with a curvature discontinuity (i.e., a C1metric where first derivatives are discontinuous), while an impulsive wave refers to a metric with a curvature delta-function (i.e., a C0metric where the metric itself is only continuous). The Dray--’tHooft analysis produces metrics of the form given in equation (B.4), which contain explicit delta-function terms δ(ˆu)in the metric coefficients. This results in delta-function singularities in the Ricci tensor (equations (B.7)--(B.8)) and a stress-energy tensor Tuu ∝δ(p)δ(u)containing delta-functions. By Penrose’s stricter terminology, these are therefore impulsive waves rather than shock waves. Nevertheless, following Dray and ’tHooft, we employ the term “shockwave” throughout this work to describe these delta-function singularities in spacetime curvature, recognizing that this represents a somewhat more relaxed usage than Penrose’s original definition. The physical distinction between discontinuities and delta-functions represents different orders of singularity in the gravitational field and should be kept in mind when comparing results across different frameworks. 1 References [1] Roger Penrose. The geometry of impulsive gravitational waves. General Relativity, Papers in Honour of JL Synge, pages 101–115, 1972. [2] Tevian Dray and Gerard’t Hooft. The gravitational shock wave of a massless particle. Nuclear physics B, 253:173–188, 1985. [3] Katrin Becker, Melanie Becker, and John H Schwarz. String theory and M-theory: A modern introduction. Cambridge university press, 2006. [4] David J Griffiths and Darrell F Schroeter. Introduction to quantum mechanics. Cambridge university press, 2018. [5] Stephen W Hawking, Malcolm J Perry, and Andrew Strominger. Soft hair on black holes. Physical Review Letters, 116(23):231301, 2016. [6] Ignatios Antoniadis and NA Obers. Plane gravitational waves in string theory. Nuclear Physics B, 423(2-3):639–660, 1994. 2