1.1 Conventions

I will use a 4-metric gμν with signature (− ,+, +, +). Following the MTW [80Jump To The Next Citation Point] conventions, I define the Riemann tensor as
R μναβ ≡ ∂αΓ μνβ − ∂βΓ μνα + Γ μσαΓ σνβ − Γ μ σβΓ σνα. (1 )
The Ricci tensor is defined as
σ R μν ≡ R μσν, (2 )
and the Einstein tensor as
G μν ≡ R μν − 1gμνR. (3 ) 2
A spatial 3-metric will be written as γij and the Riemann and Ricci tensors associated with it will be defined by (1View Equation) and (2View Equation). Four-dimensional indices will be denoted by Greek letters, while 3-dimensional indices will be denoted by Latin letters.

To avoid confusion, the covariant derivative and Ricci tensor associated with γ ij will be written with over-bars – ¯ ∇j and ¯ Rij. I will also frequently deal with an auxiliary 3-dimensional space with a metric that is conformally related to the metric γij of the physical space. The metric for this space will be denoted &tidle;γij. The covariant derivative and Ricci tensor associated with this metric will be written with tildes – ∇&tidle;j and R&tidle;ij. Other quantities that have a conformal relationship to quantities in the physical space will also be written with a tilde over them.


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