Archives for General Relativity and Cosmology - Page 2
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…
Scalar Wave Equation in a Schwarzschild Background
Scalar Wave Equation in Schwarzschild Background: For the Schwarzschild metric, the scalar wave equation becomes: (1−2GMc2r)−1∂2Φ∂t2−1r2∂∂r(r2(1−2GMc2r)∂Φ∂r)−1r2sinθ∂∂θ(sinθ∂Φ∂θ)−1r2sin2θ∂2Φ∂ϕ2=0.\left(1 - \frac{2GM}{c^2r}\right)^{-1} \frac{\partial^2 \Phi}{\partial t^2} - \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \left(1 - \frac{2GM}{c^2r}\right)…
Compactified versus non compactified Kaluza-Klein Theories
Kaluza-Klein Gravity - by J. M. Overduin and P. S. Wesson Compactified Kaluza-Klein Theories Definition and Mechanism: In compactified Kaluza-Klein theories, the extra dimensions are curled up into very small…
Kaluza-Klein theory and general relativity
The relationship between Kaluza-Klein theory and general relativity is a foundational aspect of the former. Kaluza-Klein theory extends general relativity by incorporating additional dimensions beyond the familiar four-dimensional spacetime. Here…
Hawking Radiation and Black Hole Thermodynamics
The document titled "Hawking Radiation and Black Hole Thermodynamics" by Don N. Page provides a comprehensive review of the theoretical developments in the understanding of black holes, focusing on their…
Creating black holes and baby Universes in a lab
Take a solid sphere and compress it. The Internal High Density will cause a negative pressure (dense on the inside, pulling the outside in) This will lead to an implosion.…
The Surface of the Balloon – Galaxies Receding from Each Other
Galaxies are receding from each other Isn't this true of ANY explosive event (say a large firecracker in a closed room) - things will appear to recede from each other?…
The cosmological constant
Einstein did not have the benefit of knowing that a) There were other galaxies besides our own (Hubble found Andromeda in 1923) b) These 'other galaxies' were receding from us…