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.
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Initial Data for Numerical Relativity
Gregory B. Cook http://www.livingreviews.org/lrr-2000-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |