2 The Initial-Value Equations1 Introduction1 Introduction

1.1 Conventions 

I will use a 4-metric tex2html_wrap_inline2975 with signature (-,+,+,+). Following the MTW [79Jump To The Next Citation Point In The Article] conventions, I define the Riemann tensor as

  equation31

The Ricci tensor is defined as

  equation48

and the Einstein tensor as

  equation54

A spatial 3-metric will be written as tex2html_wrap_inline2979 and the Riemann and Ricci tensors associated with it will be defined by (1Popup Equation) and (2Popup 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 tex2html_wrap_inline2979 will be written with over-bars - tex2html_wrap_inline2983 and tex2html_wrap_inline2985 . I will also frequently deal with an auxiliary 3-dimensional space with a metric that is conformally related to the metric tex2html_wrap_inline2979 of the physical space. The metric for this space will be denoted tex2html_wrap_inline2989 . The covariant derivative and Ricci tensor associated with this metric will be written with tildes - tex2html_wrap_inline2991 and tex2html_wrap_inline2993 . Other quantities that have a conformal relationship to quantities in the physical space will also be written with a tilde over them.



2 The Initial-Value Equations1 Introduction1 Introduction

image Initial Data for Numerical Relativity
Gregory B. Cook
http://www.livingreviews.org/lrr-2000-5
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