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Geometric Kinematics in Lorentzian Geometry: Relativistic Motions and Their Characterizations (eBook)

Type: e-book
Genre: Mathematics, Science & Technology
Language: English
Price: ₹180
(Immediate Access on Full Payment)
Available Formats: PDF


In this work, I've focused on the study of three notable motions in General Relativity, Uniformly Accelerated Motion (UAM), Unchanged Direction Motion (UDM), and Uniform Circular Motion (UCM) which are well-established concepts in Classical Mechanics. UAM was explored in Lorentz-Minkowski spacetime L^4 and referred to as hyperbolic motion. However, I aim to extend the analysis of these motions to arbitrary spacetimes using modern Lorentzian Geometry techniques.

In Chapter 1, I have explained the essential tools of (differential) Lorentzian Geometry, including the notion of time orientation for a Lorentzian manifold, which is fundamental in defining spacetime.
The main focus of Chapter 2 is to introduce the Fermi-Walker covariant derivative of an observer and the associated Fermi-Walker parallel transport. This enables the analysis of an observer that follows a UAM, where their 4-acceleration remains constant in a general spacetime. Such an observer can be geometrically characterized as a Lorentzian circle. I also explored how trajectories of uniformly accelerated observers can be represented as the projection of integral curves of a specific vector field defined on a particular fiber bundle over the spacetime. Moreover, I discussed assumptions that ensure the absence of singularities for inextensible UAM observers within a finite proper time. Chapter 2 also delved into the study of UDM and UCM in arbitrary spacetimes. I analyzed these three motions from the perspective of the corresponding ordinary differential equations that define each motion. Throughout the work, I emphasized the significance of accurate techniques from Lorentzian Geometry, particularly the Fermi-Walker connection, which provides each observer with a moving gyroscope in their relative space at each instant of their proper time. These advanced tools allowed me to gain a deeper understanding of the motions observed in General Relativity.

About the Author

Riddhiman Bhattacharya FRSA, a High School Student and Fellow of the Royal Society of Arts, London, is a passionate enthusiast of Physics and Mathematics. Delving into the depths of his being, he finds solace in intertwining the realms of these two profound sciences into every aspect of his life.

In the quantum analogy, Riddhiman's enthusiasm can be described mathematically by a wave function, where the square of its absolute value determines the likelihood of finding him in a particular state. This enthusiasm is deeply connected to his inherent creativity and thinking, represented by matrix A, which contributes to shaping his state of mind.

To Riddhiman, life presents a mysterious challenge, waiting to be unraveled. He firmly believes that the profound tools of Mathematics (M) and Physics (P) hold the key to unlocking hidden truths and navigating the unknown.

Book Details

Number of Pages: 48
Availability: Available for Download (e-book)

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