Archives for General Relativity and Cosmology
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…
Gödel’s Solution to Einstein’s Field Equations
Gödel's Solution to Einstein's Field Equations Kurt Gödel's solution to Einstein's field equations is a fascinating example of how general relativity (GR) can predict exotic spacetime geometries. Gödel introduced a…
Richard Gott’s Three Universes at the Big Bang
Gott's Theory on the Big Bang Singularity J. Richard Gott, a Princeton physicist, proposed an intriguing theory concerning the Big Bang Singularity. His idea explores what happens if we consider…
Particle Trajectory in Curved Space Using Jacobi Equation
To derive the trajectory of a particle in a curved gravitational field using the Hamilton-Jacobi equation, we will follow these steps: 1. **Hamilton-Jacobi Equation**: \ Here, \( S \) is…
Curved Space does not need to be embedded in a flat space time
One of the conceptual problems we humans had was visualizing curved Space as something that exists in the foreground of a FLAT background. Riemann showed that there does not need…
Schwarzschild Solution for a Black Hole
Schwarzschild Solution for a Black Hole 1. Schwarzschild Metric: The Schwarzschild solution is the simplest solution to Einstein's field equations in general relativity. It describes the spacetime geometry surrounding a…