Jerk is the time derivative of an acceleration vector and, hence, the third time derivative of the position vector. In this paper, we consider a particle moving in the three dimensional Euclidean space and resolve its jerk vector along the tangential direction, radial direction in the osculating plane and the other radial direction in the rectifying plane. Also, the case for planar motion in space is given as a corollary. Furthermore, motion of an electron under a constant magnetic field and motion of a particle along a logarithmic spiral curve are given as illustrative examples. The aforementioned decomposition is a new contribution to the field and it may be useful in some specific applications that may be considered in the future.
In this MSc. thesis, we have attempted to give an overview of the firewall paradox and various approaches towards its resolution. After an introductory chapter on some basic concepts in quantum field theory in curved spacetimes such as Hawking radiation, we introduce the paradox. It arises out of application of principles each of which is thought or assumed to be correct: 1) unitary black hole evaporation, 2) validity of quantum field theory in curved spacetime, 3) a measure of the number of black hole quantum states, 4) Einstein's equivalence principle. Then, we present various approaches that exist in the literature towards the resolution of the paradox. ÖzetBu yüksek lisans tezinde, ateşten set paradoksu ve onun çözümüne yönelik çeşitli yaklaşımları anlatma çabası içinde bulunuldu. Egri uzay-zamanlarda kuantum alanlar kuramındaki, Hawking ışınımı gibi, bazı temel kavramlar üzerine olan bir giriş kısmından sonra ateşten set paradoksu anlatılıyor. Paradoks ayrı ayrı dogru oldugu düşünülen ya da sanılan ilkelerin hep beraberce uygulanması sonucunda doguyor: 1) kara deliklerin kuantum mekanigi ile uyumlu biçimde buharlaşmaları, 2) egri uzay-zamanlarda kuantum alanlar teorisinin geçerliligi, 3) kara deliklerin kuantum hallerini saymak için bir ölçü, 4) Einstein'ın eşdegerlilik ilkesi. Sonrasında paradoksun çözümüne yönelik literatürde yer alan çeşitli yaklaşımlara yer veriliyor. vii
In this study, we have considered the elliptical harmonic motion which is the superposition of two simple harmonic motions in perpendicular directions with the same angular frequency and phase difference of π/2 . It is commonly recognized that a convenient formulation for problems in planar kinematics is obtained by using number systems. Here the elliptical numbers are used to derive the Bobillier formula with two different methods for aforesaid motion; the first method depends on the Euler-Savary formula and the second one uses the usual relations of the velocities and accelerations.
No abstract
Considering the role of black hole singularity in quantum evolution, a resolution to the firewall paradox is presented. It is emphasized that if an observer has the singularity as a part of his spacetime, then the semi-classical evolution would be nonunitary as viewed by him. Specifically, a free-falling observer inside the black hole would have a Hilbert space with non-unitary evolution; a quantum jump for particles encountering the singularity to outside of the horizon as late Hawking radiations. The non-unitarity in the jump resembles the one in collapse of wave function, but preserves entanglements. Accordingly, we elaborate the first postulate of black hole complementarity: freely falling observers who pass through the event horizon would have non-unitary evolution, while it does not have physically measurable effects for them. Besides, no information would be lost in the singularity. Taking the modified picture into account, the firewall paradox can be resolved, respecting No Drama. A by-product of our modification is that roughly half of the entropy of the black hole is released close to the end of evaporation in the shape of very hot Hawking radiation.
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