2 The Einstein equations require that the matter distri- bution be spherically symmetric, i.e. , that the derivatives with respect to θ and ϕ of ρ, J$_{(}$$_{r}$$_{)}$ , and P$_{(}$$_{r}$$_{)}$ vanish. A spherically symmetric T ( m ) ab is said to describe a static matter distribution iff [61] where T ( m ) ab is the matter energy-momentum tensor, yield Eq. (3) with the assumption J$_{(}$$_{r}$$_{)}$ = 0 (equivalent to T ( m ) 01 = 0 in a region where A and B are finite and positive) guarantees that ˙ B = 0, i.e. , B = B ( r ). Then eq. (5) with the assumption ∂P$_{(}$$_{r}$$_{)}$/∂t = 0 guarantees that A $^{′}$/A is time-independent, which only leaves the possi- bility that A depends on time through a multiplicative factor, A ( t, r ) = f ( t ) a ( r ). But then the line element as- sumes the form Au $^{c}$∇ $_{c}$ρ = ∂ρ ∂t = 0 , Au $^{c}$∇ $_{c}$P$_{(}$$_{r}$$_{)}$ = ∂P$_{(}$$_{r}$$_{)}$ ∂t = 0 , J$_{(}$$_{r}$$_{)}$ = 0 . (11) ds 2 = − a $^{2}$( r ) f $^{2}$( t ) dt 2 + B $^{2}$( r ) dr 2 + r $^{2}$d Ω 2 (12) and the redefinition of the time coordinate t → ¯ t with d ¯ t ≡ f ( t ) dt then absorbs the factor f ( t ) into ¯ t and casts the metric into locally static form. Then the Einstein equations (6) and (7) imply that also T ( m ) 22 and T ( m ) 33 are time-independent, and this is true for the tangential pressures where a prime and an overdot denote differentiation with respect to r and t , respectively. Let us consider a timelike observer at rest in the co- ordinate system ( t, r, θ, ϕ ) adapted to the spherical sym- metry of the metric (1), i.e. , one with 4-velocity u µ = ( A − $^{1}$, 0 , 0 , 0 ) . The matter energy density relative to this observer “at rest” is as well, where e a ( θ ) and e a ( ϕ ) are spacelike unit vectors in the angular directions. We have therefore the Jebsen-Birkhoff theorem (version 1): If a so- lution of the Einstein equations is spherically symmetric and the matter distribution is static ( ∂ρ ∂t = ∂P$_{(}$ r ) ∂t = 0 and J$_{(}$$_{r}$$_{)}$ = 0 ), then the metric is static in a region in which t remains timelike and ( r, θ, ϕ ) stay spacelike. and coincides with the right hand side of the Hamiltonian constraint (4) apart from the factor 8 πG . The radial energy current relative to this observer is where e b ( r ) is the spacelike unit vector in the radial direc- tion with components e µ ( r ) = ( 0 , B − $^{1}$, 0 , 0 ) . Finally, the radial pressure relative to this observer is The restriction to the region where the coordinates preserve their timelike or spacelike character is neces- sary for the validity of the theorem, as originally noted by Ehlers and Krasinski [24]. For example, this restric- tion is not satisfied in the region inside the Schwarzschild black hole horizon or outside the de Sitter cosmological horizon, where these metrics become time-dependent. A better statement of the Jebsen-Birkhoff theorem is that, under the conditions stated “a spherically symmetric so- lution admits, besides the SO(3) generators, an addi- tional hypersurface-orthogonal Killing vector field” ([24], see also [12, 25, 26]). However, the metric is static only where this additional Killing field remains timelike, which excludes the black hole horizon (if this exists and coin- cides with the Killing horizon) where the Killing field The non-radial stresses T ( m ) ij with i = j ( i, j = 1 , 2 , 3) vanish identically because of the Einstein equations and the fact that the components G$_{ij}$ of the Einstein tensor with i = j vanish in spherical symmetry. Furthermore, eqs. (6) and (7) imply that T ( m ) 33 sin $^{2}$θ = T ( m ) 22 . P$_{(}$$_{r}$$_{)}$ ≡ T ( m ) ab e a ( r $_{)}$e b ( r ) = T ( m ) 11 B 2 . (10) J$_{(}$$_{r}$$_{)}$ ≡ − T ( m ) ab u $^{a}$eb ( r ) = T ( m ) 01 AB , (9) ρ ≡ T ( m ) ab u $^{a}$ub = T ( m ) 00 A 2 (8) P$_{(}$$_{θ}$$_{)}$ ≡ T ( m ) ab e a ( θ $_{)}$e b ( θ ) = T ( m ) 22 r 2 = P$_{(}$$_{ϕ}$$_{)}$ ≡ T ( m ) ab e a ( ϕ $_{)}$e b ( ϕ ) = T ( m ) 33 r 2 sin $^{2}$θ (13) r sin $^{2}$θ B 3 ( A $^{′}$B A − B ′ − rB 2 ¨ B A 2 + r ˙ AB 2 ˙ B A 3 − rA $^{′}$B ′ A + rA $^{′′}$B A ) = 8 πG T ( m ) 33 , (7) The non-vacuum case ˙ B B = 4 πGr T ( m ) 01 , (3) 1 r 2 + 2 B ′ B $^{3}$r − 1 B $^{2}$r2 = 8 πG T ( m ) 00 A 2 , (4) 2 A ′ Ar − B 2 r 2 + 1 r 2 = 8 πG T ( m ) 11 , (5) r B 3 ( A $^{′}$B A − B ′ − rB 2 ¨ B A 2 + r ˙ AB 2 ˙ B A 3 − rA $^{′}$B ′ A + rA $^{′′}$B A ) = 8 πG T ( m ) 22 , (6)