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 .

TỪ KHÓA LIÊN QUAN
crossorigin="anonymous">
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.