Derivation of the vacuum field equations of general relativity: method of geodesic deviation
Derivation of the vacuum field equations of general relativity: method of geodesic deviation.
Derivation of the vacuum field equations of general relativity: method of geodesic deviation.
This document provides the derivation of the solutions to Einstein's equation for a non-rotating black hole (the Schwarzschild solution discovered in 1915), the internal Schwarzschild solution (dicovered by Schwarzchild in 1916), a charged, non-rotationg black hole (the Reissner-Nordstrom solution discovered 1916-1918), and a rotating black hole (the Kerr solution). Although the Schwarzschild solution and the Reissner-Nordstrom solution were discovered soon after general relativity was invented, the solution for a rotating black hole was found by Kerr only in 1963. This document includes a derivation of the Einstein's equations in the presence of a perfect fluid and Einstein-Maxell equations from an action principle. The full field equations are obtained in a precisey analogous way to the way a source term for Maxwell's equations, a current Ja, arise from a variation of the coupled matter-Maxwell action with respect to the gauge potential.
A congruence is a system of nonintersecting geodesics. We will consider separately the case of timelike geodesics and null geodesics. We introduce the expansion scalar, the shear and twist tensors, as a means of describing the congruences's behaviour. The evolution equations for the expansion scalar, the shear and twist tensors are called the Raychaudhuri equations. This document provides the derivation of the Raychaudhuri equations for timelike and null geodesics.
Proof of the Hawking-Penrose Singularity Theorems.
We present a document detailing the spinor formalism of General Relativity (draft version).
Chapter 1: The first chapter introduces kinematic concepts of fluid mechanics. Including general deformation and an infinitesimal fluid elemnt, the rate of strain tensor and the rotation tensor, conservation of mass, continuity equation. The ideas of pathlines and streamlines. The velocity potential. Vortex lines and circulation. Chapter 2: We then move on to Euler's dynamical equations which apply to flows with high Reynolds number, where viscous effecs are small, most of the flow can be considered invisid. We consider potential flow (irrotational flow). This is described by Laplace's equation. Since the Laplace equation is linear we can superpose soutions to form more complicated flows of interest. Having constructed potential flows, the Euler equation can be used to find the pressure. Chapter 3: As mentioned in chapter 2 we can take simple potential flows and superpose them to form more complicated flows of interest. Exact solution are easily found and decribed in spherical or cylindrical coordinates. We introduce the mathematics for spherical and cylindrical coordinates with more details and derivations in appedix D and E. Chapter 4: We introduce viscous fluid flow described by the Navier-Stokes equaions. Includes alternative form of the NSE's and the vorticity equation. Chapter 5: Streamfunction is defined for steady, incompressible (divergence-free) flows in 2-dimensions. The flow velocity components can be expressed as the derivatives of the scalar stream function. Chapter 6: Plane potential flow. It is proven that any complex analytic function represents a 2-dimensional flow. Chapter 7: Boundary conditions. Chapter 8: Similarity, scaling, Reynold's number. Scaling the NSE's. Chapter 9: Solutions of the Navier-Stoke's equations. Chapter 10: Mathematical theorems. Chapter 11: Stoke's streamfunction. The Stokes stream function is used to describe the streamlines and flow velocity in a three-dimensional incompressible flow with axisymmetry. As an application, we calculate drag on a sphere. Appendix A: Glossary. Appendix B: Vector caculus, Divergence theorem, and Stoke's theorem. Appedix C: Orthogonal curvilinear coordinates. We develop the gradiant opertor, the divergence, the Laplacian of a scalar, and the curl in orthogonal curvilinear coordinates. Append D: Calculating the rate of strain tensor in cylindrical coordinates. Appendix E: Calculating the rate of strain tensor in shperical polar coordinates.
This is the heuritic approach based on the propagaor formulism developed by Feynman. We begin by developing the relativistic wave equations for spin-0 and spin-1/2, that is, the Klein-Gordon equation and the Dirac equation respectively. We apply perturbation theory and the propagator method to a large number of scattering and radiation processes involving electrons, positrons, and photons. From these general rues for calculating scattering and radiation processes are deduced known as Feynman's rules. Higher order proceses, regulariation techniques, and renormaisation are discused. At some point I will be writing up Ward's (incomplete) proof of renormalisability.
We begin the heuritic approach based on the propagaor formulism developed by Feynman. We begin by developing the relativistic wave equations for spin-0 and spin-1/2, that is, the Klein-Gordon equation and the Dirac equation respectively. We apply perturbation theory and the propagator method to a large number of scattering and radiation processes involving electrons, positrons, and photons. From these general rues for calculating scattering and radiation processes are deduced known as Feynman's rules. Higher order proceses, regulariation techniques, and renormaisation are discused.
We then move on to functional integral techniques. We derive the Feynman rules for a scalar quantum field theory in order to illustrate how to perform perturbation theory within the context of the functional integral approach.
We then move onto defining Grassmann functional integral, necessary in order to introduce fermions into the formulism. We rederive the Feynman rules for QED (write-up incomplete).
We then move onto non-Abelian gauge theories, that is, Yang-Mills theories. Perform the quantisation within the functional integral approach via the Fadeev-Popov technique. As an example we apply this method to QED. Before moving onto non-abelian gauge theories and deriving the Feynman rules.
At some point we will write up the proof the renormalisability.
We give the derivation of the analytical solution of the 2D Ising model following Schult et al, and calculation of thermodynamic functions. We explain the low and high temperature expansions of the partition function, and prove the Kramers-Wannier duality.
Expanding on the paper: "A spin network primer" by Seth A. Major. Giving all details of calculations.
An accompaniment to the book "Covariant Loop Quantum Gravity" by Carlo Rovelli and Francesca Vidotto. Giving much details of calculations (draft version).
Irreducible Representations of SU(2) and Racah’s Coefficients (draft verion).
We derive the Dynkin formula corresponding to the most general case. We demonstrate that the Dynkin formula yields the correct few terms. In particular we demonstrate reduces to the Baker-Campbell-Hausdorf formula in the special case where [X,[X,Y]] = [Y,[X,Y]] = 0. We give a simple calculus proof for this simple case. We also give an algebraic proof involving mathematical induction for this simple case.
We derive the forumla for and integral after making a coordinate transformation. We prove that the infintesimal volume element dnx is modified by the absoute value of the determinant of the Jacobian under the coordinate transformation.
We derive the Frobenius determinant formula for the number of standard Young tableaux, and derive from this the hook length forumla.
Any matrix over C is similar to an upper triangular matrix, but not necessarily similar to a diagonal matrix. Despite this we can still demand that it be similar to a matrix which is as ’nice as possible’, which is the Jordan Normal Form.
Euler's proof of Fermat's two square theoerem, which states that any odd p can be written as the sum of two squares if only if and only if p ≡ 1 (mod 4).
Don Zagier’s one-sentence proof of Fermat's two square theoerem, which states that any odd p can be written as the sum of two squares if only if and only if p ≡ 1 (mod 4).
An account of Alexander Spivak's visual proof of Fermat's two square theoerem, which states that any odd p can be written as the sum of two squares if only if and only if p ≡ 1 (mod 4).
Infinite series summationn technique for sequences of constants and functions via a comples contour integration trick (more to add - pardon the pun).
Infinite series summationn technique for sequences of constants and functions via use of \(\frac{1}{n^k} = \frac{1}{(k-1)!} \int_0^\infty e^{-nx} x^{k-1} \, dx\). This identity allows you to represent the summation as an integral, which is then mostly evaluated via contour integration (draft version).
Accessible derivation of Murihead inequality.
Muirhead.pdf