Optical scalars

For geodesic timelike congruences

Denote the tangent vector field of an observer's worldline (in a timelike congruence) as Z a , and then one could construct induced "spatial metrics" that

( 1 ) h a b = g a b + Z a Z b , h a b = g a b + Z a Z b , h b a = g b a + Z a Z b ,

where h b a works as a spatially projecting operator. Use h b a to project the coordinate covariant derivative ∇ b Z a and one obtains the "spatial" auxiliary tensor B a b ,

( 2 ) B a b = h a c h b d ∇ d Z c = ∇ b Z a + A a Z b ,

where A a represents the four-acceleration, and B a b is purely spatial in the sense that B a b Z a = B a b Z b = 0 . Specifically for an observer with a "geodesic" timelike worldline, we have

( 3 ) A a = 0 , ⇒ B a b = ∇ b Z a .

Now decompose B a b into the symmetric part θ a b and ω a b ,

( 4 ) θ a b = B ( a b ) , ω a b = B [ a b ] .

ω a b = B [ a b ] is trace-free ( g a b ω a b = 0 ) while θ a b is of nonzero trace, g a b θ a b = θ . Thus, the symmetric part θ a b can be further rewritten into its trace and trace-free part,

( 5 ) θ a b = 1 3 θ h a b + σ a b .

Hence, all in all we have

( 6 ) B a b = 1 3 θ h a b + σ a b + ω a b , θ = g a b θ a b = g a b B ( a b ) , σ a b = θ a b − 1 3 θ h a b , ω a b = B [ a b ] .

For geodesic null congruences

Now, consider a geodesic null congruence with tangent vector field k a . Similar to the timelike situation, we also define

( 7 ) B ^ a b := ∇ b k a ,

which can be decomposed into

( 8 ) B ^ a b = θ ^ a b + ω ^ a b = 1 2 θ ^ h ^ a b + σ ^ a b + ω ^ a b ,

where

( 9 ) θ ^ a b = B ^ ( a b ) , θ ^ = h ^ a b B ^ a b , σ ^ a b = B ^ ( a b ) − 1 2 θ ^ h ^ a b , ω ^ a b = B ^ [ a b ] .

Here, "hatted" quantities are utilized to stress that these quantities for null congruences are two-dimensional as opposed to the three-dimensional timelike case. However, if we only discuss null congruences in a paper, the hats can be omitted for simplicity.

Definitions: optical scalars for null congruences

The optical scalars { θ ^ , σ ^ , ω ^ } come straightforwardly from "scalarization" of the tensors { θ ^ , σ ^ a b , ω ^ a b } in Eq(9).

The expansion of a geodesic null congruence is defined by (where for clearance we will adopt another standard symbol " ; " to denote the covariant derivative ∇ a )

( 10 ) θ ^ = 1 2 k a ; a .

The shear of a geodesic null congruence is defined by

( 11 ) σ ^ 2 = σ ^ a b σ ¯ ^ a b = 1 2 g c a g d b k ( a ; b ) k c ; d − ( 1 2 k a ; a ) 2 = g c a g d b 1 2 k ( a ; b ) k c ; d − θ ^ 2 .

The twist of a geodesic null congruence is defined by

( 12 ) ω ^ 2 = 1 2 k [ a ; b ] k a ; b = g c a g d b k [ a ; b ] k c ; d .

In practice, a geodesic null congruence is usually defined by either its outgoing ( k a = l a ) or ingoing ( k a = n a ) tangent vector field (which are also its null normals). Thus, we obtain two sets of optical scalars { θ ^ ( ℓ ) , σ ^ ( ℓ ) , ω ^ ( ℓ ) } and { θ ^ ( n ) , σ ^ ( n ) , ω ^ ( n ) } , which are defined with respect to l a and n a , respectively.

For a geodesic timelike congruence

The propagation (or evolution) of B a b for a geodesic timelike congruence along Z c respects the following equation,

( 13 ) Z c ∇ c B a b = − B b c B a c + R c b a d Z c Z d .

Take the trace of Eq(13) by contracting it with g a b , and Eq(13) becomes

( 14 ) Z c ∇ c θ = θ , τ = − 1 3 θ 2 − σ a b σ a b + ω a b ω a b − R a b Z a Z b

in terms of the quantities in Eq(6). Moreover, the trace-free, symmetric part of Eq(13) is

( 15 ) Z c ∇ c σ a b = − 2 3 θ σ a b − σ a c σ b c − ω a c ω b c + 1 3 h a b ( σ c d σ c d − ω c d ω c d ) + C c b a d Z c Z d + 1 2 R ~ a b .

Finally, the antisymmetric component of Eq(13) yields

( 16 ) Z c ∇ c ω a b = − 2 3 θ ω a b − 2 σ [ b c ω a ] c .

For a geodesic null congruence

A (generic) geodesic null congruence obeys the following propagation equation,

( 16 ) k c ∇ c B ^ a b = − B ^ b c B ^ a c + R c b a d k c k d ^ .

With the definitions summarized in Eq(9), Eq(14) could be rewritten into the following componential equations,

( 17 ) k c ∇ c θ ^ = θ ^ , λ = − 1 2 θ ^ 2 − σ ^ a b σ ^ a b + ω ^ a b ω ^ a b − R c d k c k d ^ ,

( 18 ) k c ∇ c σ ^ a b = − θ ^ σ ^ a b + C c b a d k c k d ^ ,

( 19 ) k c ∇ c ω ^ a b = − θ ^ ω ^ a b .

For a restricted geodesic null congruence

For a geodesic null congruence restricted on a null hypersurface, we have

( 20 ) k c ∇ c θ = θ ^ , λ = − 1 2 θ ^ 2 − σ ^ a b σ ^ a b − R c d k c k d ^ + κ ( ℓ ) θ ^ ,

( 21 ) k c ∇ c σ ^ a b = − θ ^ σ ^ a b + C c b a d k c k d ^ + κ ( ℓ ) σ ^ a b ,

( 22 ) k c ∇ c ω ^ a b = 0 .

Spin coefficients, Raychaudhuri's equation and optical scalars

For a better understanding of the previous section, we will briefly review the meanings of relevant NP spin coefficients in depicting null congruences. The tensor form of Raychaudhuri's equation governing null flows reads

( 23 ) L ℓ θ ( ℓ ) = − 1 2 θ ( ℓ ) 2 + κ ~ ( ℓ ) θ ( ℓ ) − σ a b σ a b + ω ~ a b ω ~ a b − R a b l a l b ,

where κ ~ ( ℓ ) is defined such that κ ~ ( ℓ ) l b := l a ∇ a l b . The quantities in Raychaudhuri's equation are related with the spin coefficients via

( 24 ) θ ( ℓ ) = − ( ρ + ρ ¯ ) = − 2 Re ( ρ ) , θ ( n ) = μ + μ ¯ = 2 Re ( μ ) ,

( 25 ) σ a b = − σ m ¯ a m ¯ b − σ ¯ m a m b ,

( 26 ) ω ~ a b = 1 2 ( ρ − ρ ¯ ) ( m a m ¯ b − m ¯ a m b ) = Im ( ρ ) ⋅ ( m a m ¯ b − m ¯ a m b ) ,

where Eq(24) follows directly from h ^ a b = h ^ b a = m b m ¯ a + m ¯ b m a and

( 27 ) θ ( ℓ ) = h ^ b a ∇ a l b = m b m ¯ a ∇ a l b + m ¯ b m a ∇ a l b = m b δ ¯ l b + m ¯ b δ l b = − ( ρ + ρ ¯ ) ,

( 28 ) θ ( n ) = h ^ b a ∇ a n b = m ¯ b m a ∇ a n b + m b m ¯ a ∇ a n b = m ¯ b δ n b + m b δ ¯ n b = μ + μ ¯ .