Archives for General Relativity and Cosmology
The unconnected manifold versus the affine connected manifold – describe what these are. Densities, Derivatives, Integrals, Invariant Derivatives in particular
This is a deep and beautiful distinction—one that sits right at the boundary between pure geometry and physics-ready geometry. Let’s build it cleanly and intuitively, then connect it to densities,…
Using the expansion as a Universal Clock
The Universe Has a Clock — And It’s Hidden in Its Expansion We often say the Big Bang has a “past.” But that statement quietly assumes something profound: there exists…
Gravitons emerging from String theory
<!doctype html> How the Graviton Emerges from String Theory In perturbative string theory, the graviton is not added by hand; it appears automatically as one of the vibration modes of…
Why are there Gravitational Quadrupoles?
Why Are There Gravitational Quadrupoles? 1. No Monopole Radiation A gravitational monopole would require changes in the total mass of a system. However, total mass is conserved in isolated systems,…
Gravitational Monopoles and Dipoles
Why There Are No Gravitational Monopoles or Dipoles Why Are There No Gravitational Monopoles? Gravitational monopole would mean a single, isolated “negative mass” (similar to a negative electric charge), which…
Projection OPerators and Christoffel Symbols
Are Christoffel Symbols Related to Projection Operators? This is a deep and fascinating question — and it's insightful to sense a connection. While Christoffel symbols and projection operators arise in…
Understanding Rindler Space
Understanding Rindler Space 1. What is Rindler Space? Rindler space describes the spacetime experienced by an observer undergoing constant acceleration in special relativity. It is useful for understanding: Uniformly accelerated…
Understanding the Event Horizon
Understanding the Event Horizon 1. The Event Horizon as a Global Concept The event horizon is not a physical surface but a mathematical boundary beyond which nothing can escape. Its…
Tangent Vectors, Affine Parametrization of Curves
Tangent Vectors, Affine Parameterization, and Tangent Spaces Tangent Vectors Definition: A tangent vector at a point on a manifold (a space that locally resembles Euclidean space) represents the "direction" and…
Poincare Recurrence Theorem – Mathematical
Poincaré Recurrence Theorem The Poincaré Recurrence Theorem is a foundational result in dynamical systems and ergodic theory, which asserts that a system in a finite phase space will, after sufficient…