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From the Cover…

In this modern and distinctive textbook, Helliwell and Sahakian present classical mechanics as a thriving and contemporary field with strong connections to cutting-edge research topics in physics. Each part of the book concludes with a capstone chapter describing various key topics in quantum mechanics, general relativity, and other areas of modern physics, clearly demonstrating how they relate to advanced classical mechanics, and enabling students to appreciate the central importance of classical mechanics within contemporary fields of research. Numerous and detailed examples are interleaved with theoretical content, illustrating abstract concepts more concretely. Extensive problem sets at the end of each chapter further reinforce students' understanding of key concepts, and provide opportunities for assessment or self-testing. A detailed online solutions manual and lecture slides accompany the text for instructors. Often a flexible approach is required when teaching advanced classical mechanics, and, to facilitate this, the authors have outlined several paths instructors and students can follow through the book, depending on background knowledge and the length of their course.


In the context of the Bank-Fishler-Shenker-Susskind Matrix theory, we analyze a spherical membrane in light-cone M theory along with two asymptotically distant probes. In the appropriate energy regime, we find that the membrane behaves like a smeared Matrix black hole; and the spacetime geometry seen by the probes can become non-commutative even far away from regions of Planckian curvature. This arises from nonlinear Matrix interactions where fast matrix modes lift a flat direction in the potential — akin to the Paul trap phenomenon in atomic physics. In the regime where we do have a notion of emergent spacetime, we show that there is non-zero entanglement entropy between supergravity modes on the membrane and the probes. The computation can easily be generalized to other settings, and this can help develop a dictionary between entanglement entropy and local geometry — similar to Ryu-Takayanagi but instead for asymptotically flat backgrounds.


When matter falls past the horizon of a large black hole, the expectation from string theory is that the configuration thermalizes and the information in the probe is rather quickly scrambled away. The traditional view of a classical unique spacetime near a black hole horizon conflicts with this picture. The question then arises as to what spacetime does the probe actually see as it crosses a horizon, and how does the background geometry imprint its signature onto the thermal properties of the probe. In this work, we explore these questions through an extensive series of numerical simulations of D0 branes. We determine that the D0 branes quickly settle into an incompressible symmetric state—thermalized within a few oscillations through a process driven entirely by internal nonlinear dynamics. Surprisingly, thermal background fluctuations play no role in this mechanism. Signatures of the background fields in this thermal state arise either through fluxes, i.e. black hole hair; or if the probe expands to the size of the horizon—which we see evidence of. We determine simple scaling relations for the D0 branes’ equilibrium size, time to thermalize, lifetime, and temperature in terms of their number, initial energy, and the background fields. Our results are consistent with the conjecture that black holes are the fastest scramblers as seen by matrix theory.

Recent Publications by

Vatche Sahakian

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Thomas McCaffree Helliwell and Vatche Sahakian

Modern Classical Mechanics

Cambridge University Press, 2021.
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Vatche Sahakian, Yossathorn Tawabutr, and Xinrui Yan

Emergent Spacetime & Quantum Entanglement in Matrix Theory

Journal of High Energy Physics 08 (2017) 140.

Paul L. Riggins and Vatche Sahakian

Black hole thermalization, D0 brane dynamics, and emergent spacetime

Physical Review D 86 (2012) 046005.
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Gregory Minton and Vatche Sahakian

New mechanism for nonlocality from string theory: UV-IR quantum entanglement and its imprints on the CMB

Physical Review D 77 (2008) 026008.

A. Murugan and Vatche Sahakian

Emergence of the fuzzy horizon through gravitational collapse

Physical Review D 74 (2006) 106010.

Vatche Sahakian

Hairy strings

Physical Review D 73 (2006) 026002.

Vatche Sahakian

Closed strings in Ramond-Ramond backgrounds

Journal of High Energy Physics 0404 (2004) 026.