Tag Archives: theoretical physics
From the Equivalence Principle to Curved Geometry of Spacetime
Newtonian mechanics contains two masses with no obvious reason to be related. One appears in the law of gravitation, $$\mathbf{F}=-\frac{GMm_g}{r^{2}}\hat{\mathbf r},$$ where $m_g$ measures how strongly a body couples to a gravitational field. The other appears in the second law, … Continue reading
Posted in Expository, Notes
Tagged Classical Gravity, curvature, differential geometry, Einstein equations, equivalence principle, general relativity, geodesic deviation, Geodesics, gravity, Lorentzian Geometry, manifolds, Mathematical Physics, spacetime, Tensors, theoretical physics
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Why $R_{\mu\nu}=0$ does not imply $R_{\mu\nu\rho\sigma}=0$
Almost everyone makes this mistake once, and it’s a good one to make. You learn the Einstein field equations. You learn that in vacuum, with no matter and no energy and no cosmological constant, they collapse down to something very … Continue reading
Derivation of the Cosmic-Ray Diffusion Coefficient from Pitch-Angle Scattering
Cosmic rays arrive at Earth from every direction almost perfectly evenly, to about one part in a thousand. That’s a disaster if you wanted to know where they came from, because they’re charged and the Galaxy is magnetised, so whatever … Continue reading
Posted in Expository, Projects, Research
Tagged astronomy, Astrophysics, Mathematical Physics, physics, science, theoretical physics
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The rubber sheet gets the geometry right and the mechanism wrong!
You have seen this demonstration. Somebody stretches a rubber sheet over a frame, drops a bowling ball in the middle, and the sheet sags. Then they roll a marble across it and the marble curves toward the bowling ball, maybe … Continue reading
Posted in Diary, Expository, Notes, Scratch essays
Tagged Astrophysics, debunking, equivalence principle, general relativity, Geodesics, gravitational waves, gravity, manifolds, Mathematical Physics, philosophy of science, physics, Schwarzschild metric, science, science writing, spacetime, spacetime curvature, theoretical physics
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Deriving the Euler–Lagrange Equation for Particles, Fields, and Curved Spacetime
A good way to understand the laws of motion is not to memorize a separate equation for each situation, but to notice there is really only one trick, and every equation of motion is what falls out when you apply … Continue reading
The mathematics behind a “wormhole”
Let me tell you something that will sound ridiculous at first. Take a piece of paper. Draw a dot on the left and a dot on the right. The shortest path between them, if you are a little ant walking … Continue reading
How many solutions of Einstein’s equations are there?
How many solutions does the most beautiful equation in physics have? More than you’d think… probably infinitely many, and most of them will never have names. Continue reading
Posted in Expository, Notes
Tagged Astrophysics, Black holes, differential geometry, Einstein equation, general relativity, Kerr spacetime, Lorentzian Geometry, manifolds, Mathematical Physics, riemannian geometry, Schwarzschild spacetime, Tensor Calculus, theoretical physics, Wormholes
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On the Consistency of Published M87* Mass Measurements
A useful way to test a black hole spacetime is not only to ask whether one observational method agrees with Kerr, but to ask whether several independent methods agree with each other. In the case of M87*, this question is … Continue reading
Some remarks on quasinormal modes for Euler–Heisenberg black holes in a PFDM background
One of the recurring themes in black hole perturbation theory is that many apparently complicated dynamical questions eventually reduce to a rather geometric spectral problem. One begins with a black hole spacetime, perturbs it slightly, separates variables, and discovers that … Continue reading
Posted in Expository
Tagged Black holes, general relativity, Mathematical Physics, theoretical physics
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Recent notes on covariance-weighted consistency tests for Kerr parameter estimates
A recurring issue in strong-field tests of General Relativity is the question of how one should compare parameter estimates inferred from genuinely independent observational sectors. In the case of stationary black hole spacetimes, the Kerr hypothesis predicts that all sufficiently … Continue reading