Category Archives: Notes
Why MHD Works in Weakly Collisional Astrophysical Plasmas
Here is a number that should bother you. In the solar wind near the Earth’s orbit the proton density is about $5$ per cubic centimetre and the temperature is around $10^{5}$ K. Work out how far a proton travels before … Continue reading
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
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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A Scientist Before “Scientist”
Mary Somerville is so difficult to put into words that it’s funny. To be honest, who was she? Although you would be correct in calling her a mathematician, you would then be required to clarify the astronomy. The word “astronomer” … 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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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
Inspiral-merger-ringdown consistency tests and the reconstruction of Kerr geometry
One of the more conceptually interesting developments in gravitational wave astronomy is the inspiral-merger-ringdown (IMR) consistency test. At a heuristic level, the idea is rather simple: different sectors of a binary black hole coalescence should reconstruct the same final spacetime … Continue reading
Posted in Expository, Notes
Tagged Astrophysics, Black holes, consistency tests, general relativity
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A Consistency Test for Kerr Black Holes via Orbital Motion, Ringdown, and Imaging
Kerr Trisector Closure (KTC) is a consistency test for the Kerr hypothesis that tries to stay honest about what is actually being inferred from data. The guiding principle is simple: if the exterior spacetime of an astrophysical, stationary, uncharged black … Continue reading
Posted in Expository, Notes
Tagged consistency tests, general relativity, gravitational waves, KTC, theoretical physics
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