Uploads from Richard Behiel

Currently Playing: The Four Laws of Black Hole Mechanics

A detailed exploration of Bardeen, Carter, and Hawking’s 1973 paper, “The Four Laws of Black Hole Mechanics”, the paper that established the mathematical analogy between black holes and thermodynamics from general relativity, two years before Hawking radiation made that analogy real. Link to the paper: https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-31/issue-2/The-four-laws-of-black-hole-mechanics/cmp/1103858973.pdf If you enjoyed the video, please consider supporting this channel, for access to Python animation codes, video note PDFs, and your name in the credits: https://www.patreon.com/RichardBehiel A note about sign discrepancies in the paper: https://arxiv.org/abs/2603.25171 Recommended reading: Eric Poisson, A Relativist’s Toolkit (Cambridge University Press). Thank you to everyone who has been supporting my channel on Patreon, and enabling me to make these videos. Special thanks also to Curt Jaimungal, Nirmalya Kajuri, and Will Kinney. The piano is a performance of Black Hole Sun, Westworld piano version, by 88keyz. Downloaded from pixabay.com. Timestamps: 00:00 Video Intro 14:01 Abstract 18:44 Quick Overview of the Four Laws 29:40 Minkowski Spacetime Interval 39:17 Lorentz Boosts 52:02 Lorentzian Spacetime 1:02:29 Derivatives and Curvature 1:17:56 Einstein Field Equations 1:21:14 Stress-Energy Tensor T_ab 1:24:47 Minkowski Metric 1:31:27 Schwarzschild Metric 1:45:25 Kerr Metric 1:50:03 Null Dyad & Null Tetrad 2:13:42 Integral Curves 2:17:13 Killing Vector Fields 2:28:09 P1: Introduction 2:34:48 P2: Killing Field Eqs. 1-5 2:46:52 P2: Stokes' Theorem, Eq. 3 to 5 3:02:28 P2: Komar Mass & Angular Momentum, Eqs. 6-7 3:19:05 P2: Time t and Null Vector l^a, Eq. 8 3:25:18 P2: Surface Gravity Kappa 3:36:38 P2: Proving Kappa Const. on Horizon 4:02:09 P2: Interpreting the Mass Formula, Eq. 13 4:12:49 P3: Going from M to delta M 4:14:34 P3: T_ab for a Perfect Fluid, Eq. 20 4:22:03 P3: Perturbation Setup, Eqs. 21-24 4:28:10 P3: Geometric Constraints on the Horizon, Eq. 25 4:36:32 P3: Calculating delta kappa, Eq. 26-27 5:04:59 P3: Varying M_R, Eq. 28 to 28 to 31 5:23:36 P3: Integrating Eq. 31 over the Infinity Bubble 5:51:17 P3: Integrating Eq. 31 over the Event Horizon 5:59:23 P3: Varying M_T, Eq. 28 to 32 6:12:21 P3: Fluid Variation, Eq. 33 6:28:26 P3: The Final Battle of Part 3 6:35:58 P3: Examining the delta M Formula, Eq. 34 6:39:24 P4: The 2nd Law 6:42:42 P4: The 1st Law 6:55:04 P4: The 0th Law 6:56:11 P4: The 3rd Law