MODERN COSMOLOGY

1 + 3 Covariant description: equations 119

3.3.3 Bianchi identities

The third set of equations arises from theBianchi identities(3.3). On using
the splitting ofRabcdintoRabandCabcd,the1+3 splitting, (3.21),(3.25) of
those quantities, and the EFE (3.2), these identities give two further propagation
equations and two further constraint equations, which are similar in form to the
Maxwell field equations for the electromagnetic field in an expanding universe
(see [28]).
Thepropagation equationsare:

(E ̇〈ab〉+^12 π ̇〈ab〉)=(curlH)ab−^12 ∇ ̃〈aqb〉−^12 (μ+p)σab−'(Eab+^16 πab)

+ 3 σ〈ac(Eb〉c−^16 πb〉c)− ̇u〈aqb〉

+ηcd〈a[ 2 u ̇cHb〉d+ωc(Eb〉d+^12 πb〉d)], (3.45)

theE ̇-equation,and

H ̇〈ab〉=−(curlE)ab+^12 (curlπ)ab−'Hab+ 3 σ〈acHb〉c

+^32 ω〈aqb〉−ηcd〈a[ 2 u ̇cEb〉d−^12 σb〉cqd−ωcHb〉d], (3.46)

theH ̇-equation, where we have defined the ‘curls’:

(curlH)ab=ηcd〈a∇ ̃cHb〉d,(curlE)ab=ηcd〈a∇ ̃cEb〉d. (3.47)

These equations show how gravitational radiation arises: as in the electromagnetic
case, taking the time derivative of theE ̇-equation gives a term of the form
(curlH) ̇; commuting the derivatives and substituting from the H ̇-equation
eliminatesH, and results in a term inE ̈and a term of the form(curl curlE),
which together give the wave operator acting onE[20, 66]. Similarly the time
derivative of theH ̇-equation gives a wave equation forH,and associated with
these is a wave equation for the shearσ.
Theconstraint equationsare

0 =(C 4 )a=∇ ̃b(Eab+^12 πab)−^13 ∇ ̃aμ+^13 'qa

−^12 σabqb− 3 ωbHab−ηabc[σbdHdc−^32 ωbqc], (3.48)

the(divE)-equationwith its source the spatial gradient of the energy density and

0 =(C 5 )a=∇ ̃bHab+(μ+p)ωa+ 3 ωb(Eab−^16 πab)

+ηabc[^12 ∇ ̃bqc+σbd(Edc+^12 πdc)], (3.49)

the(divH)-equation, with its source the fluid vorticity. The(divE)-equation
can be regarded as a (vector) analogue of the Newtonian Poisson equation [52],
leading to the Newtonian limit and enabling tidal action at a distance. These
equations respectively show that, generically, scalar modes will result in a non-
zero divergence ofEab(and hence a non-zeroE-field) and vector modes in a
non-zero divergence ofHab(and hence a non-zeroH-field).