tailieunhanh - Intro to Differential Geometry and General Relativity - S. Warner Episode 12

Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | gradient. Ifp is large to begin with big stars this vicious cycle diverges ending in the gravitational collapse of a star leading to neutron stars or in extreme cases black holes. b Einstein Field Equations Gab 8nTab Looking at the 4 4 component first and substituting from the expressions for G and T we find 1 e-2od- r r i-e-2A 8_we 20 r2 e dr lr 1 e 8npe . If we define 1 r 1-e-2A m r then the equation becomes 1 e-2odm r 4npe 20 or 4 . I This looks like an equation for classical mass since classically M R Jj R 4nr2p r dr where the integrand is the mass of a shell whose thickness is dr. Thus dM R 4n2p r . Here p is energy density and by our choice of units energy is equal to rest mass so we interpret m r as the total mass of the star enclosed by a sphere of radius r. Now look at the 1 1 component o e 4A - 1-e 2A 8npe 2A rr 2 e -2A _ o2A - o - 1-e 8npe 111 2A Z1 -2Ax o_ 2 2A 2r - e 1-e 8nr pe h 2A 1-e-2A 8nr2p e--------2r In the expression for m solve for e2A to get 2A _ 1 e 1-2m r giving d 8nr2p 2m r dr 2r 1-2m r or d 4nr3p m 5 L_0 a . II dr r r-2m It can be checked using the Bianchi identities that we in fact get no additional information from the 2 2 and 3 3 components so we ignore them. Consequences of the Field Equations Outside the Star Outside the star we take p p 0 and m r M the total stellar mass getting I 0 nothing new since m M constant II d dr r r-2M which is a separable first order differential equation with solution e2 1 - 2M r if we impose the boundary condition 0 as r . See the exercise Set . Recalling from the definition of m that 2A1 e 1-2M r we can now express the metric outside a star as follows 112 Schwarzschild Metric g-- g 0 0 0 0 r2 0 0 0 0 r2sin20 0 _ 0 0 0 - 1-2M r J In the exercise set you will see how this leads to Newton s Law of Gravity. Exercise Set 14 1. Use v20 4nGp and the divergence theorem to deduce Newton s law V0 GM r2 for a spherical mass of uniform density p. _2A _2O . Calculate the Einstein tensor tor the metric g .