26 Chapter 2 The 3+1 decompostion of Einstein’s equations
gain intuition. Specifically, we will cast Maxwell’s equations into 3 + 1 form in Minkowski
spacetime. We will then return to general relativity, introduce a foliation of spacetime, and
define the “intrinsic” and “extrinsic” curvature of spacelike hypersurfaces. Next we will
relate the 3-dimensional curvature intrinsic to these hypersurfaces to the 4-dimensional
curvature of spacetime, and this will give rise to the equations of Gauss, Codazzi and Ricci.
Finally, we will use these equations to rewrite Einstein’s field equations (1.32)intermsof
the 3-dimensional curvatures. The end result will be the complete set of 3 + 1 equations in
standard form, summarized in Box 2.1, and a roadmap for building dynamical spacetimes.
2.1 Notation and conventions
Throughout the remainder of the book we shall, for the most part, adopt abstract index
notation
4
to represent tensors, as is commonly done in numerical relativity. Specifically,
we will use the convention that a variable with Latin indices does not represent a tensor
component, but instead represents the abstract, coordinate-free tensor itself. For example,
the symbol T
ab
no longer stands for the covariant “ab” component in a particular basis of
the tensor heretofore referred to as T. Instead, T
ab
represents the second-rank, coordinate-
free tensor T itself. Likewise, the equation G
ab
= 8π T
ab
is no longer a relation between
tensor components, but is instead a coordinate-independent tensor equation.
5
In fact, many
equations in Chapter 1, including Einstein’s equations (1.32), may be interpreted as tensor
equations in abstract index notation, rather than relations equating tensor components.
In light of our switch in notation, it is useful to revisit some of the other objects and
a few representative equations that we have encountered in Chapter 1. In abstract index
notation, we denote a basis vector e
a
as e
b
(a)
, for example, where the superscript b indicates
that this object is a vector, and the subscript a in parenthesis means that this is the ath basis
vector. In the abstract index notation of Wald (1984) components of tensors in a particular
basis are distinguished from the abstract tensor itself by displaying the components with
Greek indices. For example, the β-component of the ath basis vector, when expanded
in its own basis, is e
β
(a)
= δ
a
β
. Only rarely might we have need to borrow this notation;
hence, our references to components of tensors in a specific basis will appear with Latin
indices, but the meaning should be clear from the context. For example, the dot product
between two vectors can be written as A
a
B
a
or g
ab
A
a
B
b
. In a few, very rare instances in the
following chapters, we may slip back to boldface for clarity or emphasis when representing
a particular tensor.
We shall also adopt the standard convention whereby the letters a − h and o − z are
used for 4-dimensional spacetime indices that run from 0 to 3, while the letters i − n are
reserved for 3-dimensional spatial indices that run from 1 to 3.
4
See, e.g., Wald (1984).
5
Recall, though, that an equality that holds between components of tensors in one frame holds in all frames and in this
sense also constitutes a tensor equation.