Spherical solutions for stars Daniel Wysocki Rochester Institute of - PowerPoint PPT Presentation

Spherical stars 2015-12-14 Spherical solutions for stars Daniel Wysocki Rochester Institute of Technology General Relativity I Presentations December 14th, 2015 Spherical solutions for stars Daniel Wysocki Rochester Institute of Technology General Relativity I Presentations December 14th, 2015 Daniel Wysocki (RIT) Spherical stars December 14th, 2015 1 / 41

Spherical stars Introduction 2015-12-14 Introduction • model stars using spherical symmetry • Schwarzschild metric • T–O–V equation Introduction • real stars • I will model stars using GR assuming spherical symmetry • model stars using spherical symmetry • I will derive the Schwarzschild metric and T–O–V equation • finally I will look into specific types of stars • Schwarzschild metric • T–O–V equation • real stars Daniel Wysocki (RIT) Spherical stars December 14th, 2015 2 / 41

Spherically symmetric coordinates Spherical stars 2015-12-14 Spherically symmetric coordinates Spherically symmetric coordinates • First we need to derive our coordinate system Spherically symmetric coordinates Daniel Wysocki (RIT) Spherical stars December 14th, 2015 3 / 41

Spherically symmetric coordinates Spherical stars Two-sphere in flat spacetime 2015-12-14 Spherically symmetric coordinates Two-sphere in flat spacetime General metric d s 2 = − d t 2 + d r 2 + r 2 (d θ 2 + sin 2 θ d φ 2 ) Metric on 2-sphere d l 2 = r 2 (d θ 2 + sin 2 θ d φ 2 ) ≡ r 2 dΩ 2 Two-sphere in flat spacetime Schutz (2009, p. 256) • we start with the simplest spherically symmetric coordinates General metric – flat spacetime d s 2 = − d t 2 + d r 2 + r 2 (d θ 2 + sin 2 θ d φ 2 ) • 2-sphere in Minkowski spacetime – introduce dΩ 2 for compactness Metric on 2-sphere d l 2 = r 2 (d θ 2 + sin 2 θ d φ 2 ) ≡ r 2 dΩ 2 Schutz (2009, p. 256) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 4 / 41

Spherically symmetric coordinates Spherical stars Two-sphere in flat spacetime 2015-12-14 Spherically symmetric coordinates Two-sphere in flat spacetime General metric d s 2 = − d t 2 + d r 2 + r 2 (d θ 2 + sin 2 θ d φ 2 ) Metric on 2-sphere d l 2 = r 2 (d θ 2 + sin 2 θ d φ 2 ) ≡ r 2 dΩ 2 Two-sphere in flat spacetime Schutz (2009, p. 256) • we start with the simplest spherically symmetric coordinates General metric – flat spacetime d s 2 = − d t 2 + d r 2 + r 2 (d θ 2 + sin 2 θ d φ 2 ) • 2-sphere in Minkowski spacetime – introduce dΩ 2 for compactness Metric on 2-sphere d l 2 = r 2 (d θ 2 + sin 2 θ d φ 2 ) ≡ r 2 dΩ 2 Schutz (2009, p. 256) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 4 / 41

Spherically symmetric coordinates Spherical stars Two-sphere in curved spacetime 2015-12-14 Spherically symmetric coordinates Two-sphere in curved spacetime Metric on 2-sphere d l 2 = f ( r ′ , t )dΩ 2 Relation to r Two-sphere in curved spacetime f ( r ′ , t ) ≡ r 2 Schutz (2009, pp. 256–257) • generalize to 2-sphere in arbitrary curved spherically symmetric spacetime Metric on 2-sphere • inclusion of curvature makes r 2 some function of r ′ and t d l 2 = f ( r ′ , t )dΩ 2 Relation to r f ( r ′ , t ) ≡ r 2 Schutz (2009, pp. 256–257) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 5 / 41

Spherically symmetric coordinates Spherical stars Two-sphere in curved spacetime 2015-12-14 Spherically symmetric coordinates Two-sphere in curved spacetime Metric on 2-sphere d l 2 = f ( r ′ , t )dΩ 2 Relation to r Two-sphere in curved spacetime f ( r ′ , t ) ≡ r 2 Schutz (2009, pp. 256–257) • generalize to 2-sphere in arbitrary curved spherically symmetric spacetime Metric on 2-sphere • inclusion of curvature makes r 2 some function of r ′ and t d l 2 = f ( r ′ , t )dΩ 2 Relation to r f ( r ′ , t ) ≡ r 2 Schutz (2009, pp. 256–257) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 5 / 41

Spherically symmetric coordinates Spherical stars Meaning of r 2015-12-14 Spherically symmetric coordinates Meaning of r • not proper distance from center Mark Hannam • “curvature” or “area” coordinate • radius of curvature and area Meaning of r • r = const, t = const Figure: • A = 4 πr 2 Surface with circular • C = 2 πr symmetry but no coordinate r = 0. Schutz (2009, p. 257) • r is not necessary the “distance from the center” • not proper distance from center • it is merely a coordinate – “curvature” or “area” coordinate Mark Hannam • for instance, we may have a spacetime where the center is missing • “curvature” or “area” coordinate – example: Schwarzschild wormhole spacetime • radius of curvature and area • surface of constant ( r, t ) is a two-sphere of area A and circumference C • r = const, t = const Figure: • A = 4 πr 2 Surface with circular • C = 2 πr symmetry but no coordinate r = 0. Schutz (2009, p. 257) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 6 / 41

Spherically symmetric coordinates Spherical stars Meaning of r 2015-12-14 Spherically symmetric coordinates Meaning of r • not proper distance from center Mark Hannam • “curvature” or “area” coordinate • radius of curvature and area Meaning of r • r = const, t = const Figure: • A = 4 πr 2 Surface with circular • C = 2 πr symmetry but no coordinate r = 0. Schutz (2009, p. 257) • r is not necessary the “distance from the center” • not proper distance from center • it is merely a coordinate – “curvature” or “area” coordinate Mark Hannam • for instance, we may have a spacetime where the center is missing • “curvature” or “area” coordinate – example: Schwarzschild wormhole spacetime • radius of curvature and area • surface of constant ( r, t ) is a two-sphere of area A and circumference C • r = const, t = const Figure: • A = 4 πr 2 Surface with circular • C = 2 πr symmetry but no coordinate r = 0. Schutz (2009, p. 257) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 6 / 41

Spherically symmetric coordinates Spherical stars Meaning of r 2015-12-14 Spherically symmetric coordinates Meaning of r • not proper distance from center Mark Hannam • “curvature” or “area” coordinate • radius of curvature and area Meaning of r • r = const, t = const Figure: • A = 4 πr 2 Surface with circular • C = 2 πr symmetry but no coordinate r = 0. Schutz (2009, p. 257) • r is not necessary the “distance from the center” • not proper distance from center • it is merely a coordinate – “curvature” or “area” coordinate Mark Hannam • for instance, we may have a spacetime where the center is missing • “curvature” or “area” coordinate – example: Schwarzschild wormhole spacetime • radius of curvature and area • surface of constant ( r, t ) is a two-sphere of area A and circumference C • r = const, t = const Figure: • A = 4 πr 2 Surface with circular • C = 2 πr symmetry but no coordinate r = 0. Schutz (2009, p. 257) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 6 / 41

Spherically symmetric coordinates Spherical stars Spherically symmetric spacetime 2015-12-14 Spherically symmetric coordinates Spherically symmetric spacetime General metric d s 2 = g 00 d t 2 + 2 g 0 r d r d t + g rr d r 2 + r 2 dΩ 2 g 00 , g 0 r , and g rr : functions of t and r Spherically symmetric spacetime Schutz (2009, p. 258) • now consider not only surface of 2-sphere, but whole spacetime • now we have some unknown g 00 , g rr , and cross term g 0 r General metric • cross term g 0 r d s 2 = g 00 d t 2 + 2 g 0 r d r d t + g rr d r 2 + r 2 dΩ 2 • cross terms g 0 i for i ∈ { θ, φ } are zero from symmetry g 00 , g 0 r , and g rr : functions of t and r • need more constraints to say anything particular about them Schutz (2009, p. 258) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 7 / 41

Static spacetimes Spherical stars 2015-12-14 Static spacetimes Static spacetimes • now I will impose the static constraint Static spacetimes Daniel Wysocki (RIT) Spherical stars December 14th, 2015 8 / 41

Static spacetimes Spherical stars Motivation 2015-12-14 Static spacetimes Motivation • leads to simple derivation of Schwarzschild metric • unique solution to spherically symmetric, asymptotically flat Einstein vacuum field equations (Birkhoff’s theorem) Motivation Schutz (2009, p. 263) and Misner, Thorne, and Wheeler (1973, p. 843) • we choose the constraint of a static spacetime because – it allows us to easily derive the Schwarzschild metric • leads to simple derivation of Schwarzschild metric – according to Birkhoff’s theorem, this metric is the unique solution to the Einstein vacuum field equations for spherically symmetric, asymptotically flat spacetimes • unique solution to spherically symmetric, asymptotically flat • George David Birkhoff Einstein vacuum field equations (Birkhoff’s theorem) Schutz (2009, p. 263) and Misner, Thorne, and Wheeler (1973, p. 843) Daniel Wysocki (RIT) Spherical stars December 14th, 2015 9 / 41