### 1.1 Conventions

I will use a 4-metric with signature . Following the MTW [80] conventions, I define
the Riemann tensor as
The Ricci tensor is defined as
and the Einstein tensor as
A spatial 3-metric will be written as and the Riemann and Ricci tensors associated with it will be
defined by (1) and (2). 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 will be written
with over-bars – and . I will also frequently deal with an auxiliary 3-dimensional
space with a metric that is conformally related to the metric of the physical space. The
metric for this space will be denoted . The covariant derivative and Ricci tensor associated
with this metric will be written with tildes – and . Other quantities that have a
conformal relationship to quantities in the physical space will also be written with a tilde over
them.