Vaidya metric

In general relativity, the Vaidya metric describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes the simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the 'radiating(shining) Schwarzschild metric'.Suppose F := 1 − 2 M ( u ) r {displaystyle F:=1-{frac {2M(u)}{r}}} , then the Lagrangian for null radial geodesics ( L = 0 , θ ˙ = 0 , ϕ ˙ = 0 ) {displaystyle (L=0,{dot { heta }}=0,{dot {phi }}=0)} of the 'retarded(/outgoing)' Vaidya spacetime Eq(6) isFor the 'retarded(/outgoing)' Schwarzschild metric Eq(3), let G := 1 − 2 M r {displaystyle G:=1-{frac {2M}{r}}} , and then the Lagrangian for null radial geodesics will have an outgoing solution u ˙ = 0 {displaystyle {dot {u}}=0} and an ingoing solution r ˙ = − G 2 u ˙ {displaystyle {dot {r}}=-{frac {G}{2}}{dot {u}}} . Similar to Box A, now set up the adapted outgoing tetrad bySuppose F ~ := 1 − 2 M ( v ) r {displaystyle { ilde {F}}:=1-{frac {2M(v)}{r}}} , then the Lagrangian for null radial geodesics of the 'advanced(/ingoing)' Vaidya spacetime Eq(7) isFor the 'advanced(/ingoing)' Schwarzschild metric Eq(5), still let G := 1 − 2 M r {displaystyle G:=1-{frac {2M}{r}}} , and then the Lagrangian for the null radial geodesics will have an ingoing solution v ˙ = 0 {displaystyle {dot {v}}=0} and an outgoing solution r ˙ = G 2 v ˙ {displaystyle {dot {r}}={frac {G}{2}}{dot {v}}} . Similar to Box C, now set up the adapted ingoing tetrad by In general relativity, the Vaidya metric describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes the simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the 'radiating(shining) Schwarzschild metric'.