Archives for General Relativity and Cosmology
Schrödinger’s Space-Time Structure – Chapter 1
Let’s work through Erwin Schrödinger’s Space-Time Structure one chapter at a time, explaining both the argument and the mathematics. The book’s overall approach Schrödinger builds spacetime geometry in three stages:…
write this out in expanded form
The repeated index \(\nu\) means sum over all four coordinates, while \(\mu\) labels which transformed coordinate we mean:\ Writing out all four equations:\ dx'^1 &= \frac{\partial x'^1}{\partial x^0}dx^0 +\frac{\partial x'^1}{\partial…
Non Commutavity of the Minkowski Group
The Minkowski transformation group (more commonly the Lorentz group and, when translations are included, the Poincaré group) has several important algebraic properties. One of the most important is that it…
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…