[ { "id": "authors:wh1d1-sas94", "collection": "authors", "collection_id": "wh1d1-sas94", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180823-131125022", "type": "article", "title": "Lagrangian vs Hamiltonian: The best approach to relativistic orbits", "author": [ { "family_name": "Price", "given_name": "Richard H.", "clpid": "Price-R-H" }, { "family_name": "Thorne", "given_name": "Kip S.", "clpid": "Thorne-K-S" } ], "abstract": "In introductory general relativity courses, free particle trajectories, such as astronomical orbits, are generally developed via a Lagrangian and variational calculus, so that physical examples can precede the mathematics of tensor calculus. The use of a Hamiltonian is viewed as more advanced and typically comes later if at all. We suggest here that this might not be the optimal order in a first course in general relativity, especially if orbits are to be solved with numerical methods. We discuss some of the issues that arise in both the Lagrangian and Hamiltonian approaches.", "doi": "10.1119/1.5047439", "issn": "0002-9505", "publisher": "American Association of Physics Teachers", "publication": "American Journal of Physics", "publication_date": "2018-09", "series_number": "9", "volume": "86", "issue": "9", "pages": "678-682" }, { "id": "authors:g04j1-h8g11", "collection": "authors", "collection_id": "g04j1-h8g11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130826-130414353", "type": "article", "title": "Comparison of electromagnetic and gravitational radiation: What we can learn about each from the other", "author": [ { "family_name": "Price", "given_name": "Richard H.", "clpid": "Price-R-H" }, { "family_name": "Belcher", "given_name": "John W.", "clpid": "Belcher-J-W" }, { "family_name": "Nichols", "given_name": "David A.", "clpid": "Nichols-D-A" } ], "abstract": "We compare the nature of electromagnetic fields and gravitational fields in linearized general relativity. We carry out this comparison both mathematically and visually. In particular, the \"lines of force\" visualizations of electromagnetism are contrasted with the recently introduced tendex/vortex eigenline technique for visualizing gravitational fields. Specific solutions, visualizations, and comparisons are given for an oscillating point quadrupole source. Among the similarities illustrated are the quasistatic nature of the near fields, the transverse 1/r nature of the far fields, and the interesting intermediate field structures connecting these two limiting forms. Among the differences illustrated are the meaning of field line motion and of the flow of energy.", "doi": "10.1119/1.4807853", "issn": "0002-9505", "publisher": "American Association of Physics Teachers", "publication": "American Journal of Physics", "publication_date": "2013-08", "series_number": "8", "volume": "81", "issue": "8", "pages": "575-584" }, { "id": "authors:5h23f-bxt39", "collection": "authors", "collection_id": "5h23f-bxt39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:MINprd08b", "type": "article", "title": "Two-timescale adiabatic expansion of a scalar field model", "author": [ { "family_name": "Mino", "given_name": "Yasushi", "clpid": "Mino-Y" }, { "family_name": "Price", "given_name": "Richard H.", "clpid": "Price-R-H" } ], "abstract": "The analysis of gravitational wave data may require greater accuracy than is afforded by the adiabatic approximation to the trajectory of and field produced by a particle moving in curved spacetime. Higher accuracy is available with a two-timescale approach using as an expansion parameter the ratio of orbital time to radiation reaction time. To avoid apparent divergences at large distances, the details of the method are important, especially the choice of the foliation, the spacetime surfaces on which the orbital elements are taken to be constant. Here we apply the two-timescale approach to a simple linear model to demonstrate the details of the method. In particular we use it to show that a null foliation avoids large-distance divergences in the first-order post-adiabatic approximation, and we argue that this will be true more generally for a null foliation.", "doi": "10.1103/PhysRevD.77.064001", "issn": "2470-0010", "publisher": "American Physical Society", "publication": "Physical Review D", "publication_date": "2008-03-15", "series_number": "6", "volume": "77", "issue": "6", "pages": "Art. No. 064001" }, { "id": "authors:33s37-ya665", "collection": "authors", "collection_id": "33s37-ya665", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:ANDprd04", "type": "article", "title": "Periodic standing-wave approximation: Overview and three-dimensional scalar models", "author": [ { "family_name": "Andrade", "given_name": "Zeferino", "clpid": "Andrade-Z" }, { "family_name": "Beetle", "given_name": "Christopher", "clpid": "Beetle-C" }, { "family_name": "Blinov", "given_name": "Alexey", "clpid": "Blinov-A" }, { "family_name": "Bromley", "given_name": "Benjamin", "clpid": "Bromley-B" }, { "family_name": "Burko", "given_name": "Lior M.", "clpid": "Burko-L-M" }, { "family_name": "Cranor", "given_name": "Maria", "clpid": "Cranor-M" }, { "family_name": "Owen", "given_name": "Robert", "clpid": "Owen-R" }, { "family_name": "Price", "given_name": "Richard H.", "clpid": "Price-R-H" } ], "abstract": "The periodic standing-wave method for binary inspiral computes the exact numerical solution for periodic binary motion with standing gravitational waves, and uses it as an approximation to slow binary inspiral with outgoing waves. Important features of this method presented here are: (i) the mathematical nature of the \"mixed\" partial differential equations to be solved, (ii) the meaning of standing waves in the method, (iii) computational difficulties, and (iv) the \"effective linearity\" that ultimately justifies the approximation. The method is applied to three-dimensional nonlinear scalar model problems, and the numerical results are used to demonstrate extraction of the outgoing solution from the standing-wave solution, and the role of effective linearity.", "doi": "10.1103/PhysRevD.70.064001", "issn": "2470-0010", "publisher": "Physical Review D", "publication": "Physical Review D", "publication_date": "2004-09-15", "series_number": "6", "volume": "70", "issue": "6", "pages": "Art. No. 064001" }, { "id": "authors:fs1xq-6z813", "collection": "authors", "collection_id": "fs1xq-6z813", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PRIprd86", "type": "article", "title": "Membrane viewpoint on black holes: Properties and evolution of the stretched horizon", "author": [ { "family_name": "Price", "given_name": "Richard H.", "clpid": "Price-R-H" }, { "family_name": "Thorne", "given_name": "Kip S.", "clpid": "Thorne-K-S" } ], "abstract": "This paper derives the ''membrane formalism'' for black holes. The membrane formalism rewrites the standard mathematical theory of black holes in a language and notation which (we hope) will facilitate research in black-hole astrophysics: The horizon of a black hole is replaced by a surrogate ''stretched horizon,'' which is viewed as a 2-dimensional membrane that resides in 3-dimensional space and evolves in response to driving forces from the external universe. This membrane, following ideas of Damour and Znajek, is regarded as made from a 2-dimensional viscous fluid that is electrically charged and electrically conducting and has finite entropy and temperature, but cannot conduct heat. The interaction of the stretched horizon with the external universe is described in terms of familiar laws for the horizon's fluid, e.g., the Navier-Stokes equation, Ohm's law, a tidal-force equation, and the first and second laws of thermodynamics. Because these laws have familiar forms, they are likely to help astrophysicists understand intuitively and compute quantitatively the behaviors of black holes in complex external environments. Previous papers have developed and elucidated electromagnetic aspects of the membrane formalism for time-independent rotating holes. This paper derives the full formalism for dynamical, evolving holes, with one exception: In its present form the formalism is not equipped to handle horizon caustics, where new generators attach themselves to the horizon.", "doi": "10.1103/PhysRevD.33.915", "issn": "2470-0010", "publisher": "Physical Review D", "publication": "Physical Review D", "publication_date": "1986-02-15", "series_number": "4", "volume": "33", "issue": "4", "pages": "915-941" } ]