arXiv subjects

Karol Urbański

Publications and source records attributed to Karol Urbański.

3 recordsLinked to original sources

arXiv2026

A visual introduction to curved geometry for physicists

This article provides a gentle, visual introduction to the basic concepts of differential geometry appropriate for students familiar with special relativity. Visual methods are used to explain basics of differential geometry and build intuition for all types of Riemannian and Lorentzian manifolds of constant curvature. A visual derivation of the Thomas precession is given, showcasing the utility of differential geometry while also pointing a spotlight at certain intricacies of Minkowski space crucial from a pedagogical perspective. In addition, a straightforward method to generate some Carter-Penrose diagrams -- suitable for students with no differential geometry knowledge -- is presented, and a new method of indicating distortion on spacetime diagrams is shown.

physics.ed-ph↗

arXiv2025

Visual relativistic mechanics

This article shows how to express relativistic concepts in a visual manner using the full power of hyperbolic trigonometric functions. Minkowski diagrams in energy-momentum space are used in conjunction with hyperbolic triangles. Elegant new derivations of the relativistic rocket equation and the relativistic Doppler effect are presented that use this visual approach.

gr-qc↗

arXiv2024

Prolonging The Inevitable: Maximising survival time of an engine-equipped spacecraft between spatial hypersurfaces, as applied to the Schwarzschild spacetime

The fate of an astronaut unfortunate -- or foolish -- enough to find themselves hurtling towards spaghettification after passing the event horizon of a black hole is a common anecdote told by scientists to the regular population. However, despite the fact the Schwarzschild spacetime has been discovered over a century ago, the simple question of how long can such a space traveller live has not been fully elaborated on since. In fact, a few textbooks even give a mistaken or easily misread description of what happens. We address those inconsistencies. We calculate the proper time a space traveller equipped with means of propulsion can expect to live in these circumstances, giving analytical expressions (as elliptic integrals) wherever possible. We prove a principle that explains the best strategy to extend their life, and show its' generalisation for other spacetimes. Finally, we give quantitative answers to what gains due to optimal control can be expected in typical and somewhat `realistic' circumstances.

gr-qc↗