# Tetrad Formalism

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## Contents |

#### Introduction

The null tetrad formalism allows covariant (coordinate-indepenent) representations of the metric and its first and second derivatives. The metric itself is represented by a null tetrad, i.e., a set of four null vectors. The first derivatives of the metric are expressed by the 12 complex Newman-Penrose spin coefficients, and the second derivatives of the metric are described by 10 real Ricci scalars and 5 complex Weyl scalars.

The values of the Newman-Penrose spin coefficients and the Ricci and Weyl scalars are coordinate independent. Of course, they still depend on the particular choice of tetrad.

#### Null Tetrad

The null tetrad is commonly denoted by the four null vectors , where denotes complex conjugation. Both and are real, is complex. With respect to an event on an (arbitrarily chosen) two-sphere, and are the ingoing and outgoing null normals, respectively, whereas and are tangential null vectors.

(Other notations are common, using e.g. instead of .)

(TODO: Make the notation consistent with the notation in the section explaining the split.)

These null vectors satisfy the following conditions:

(Other normalisations are also used, e.g. , which avoid some of the factors that appear below.)

Remember the following rules for complex conjugation:

The null tetrad vectors can be expressed in terms of timelike and spacelike unit vectors as follows:

These vectors have to be orthogonal to each other and also have to be normalised:

In the following we assume that there is a preferred spacelike hypersurface , with orthogonal to , and that the two-surface spanned by and is contained in . (The timelike normal is sometimes also called .)

With these definitions, the four-metric , the three-metric , and the two-metric are given by:

#### Newman-Penrose Spin Coefficients

The first derivatives of the four-metric can be written as combinations of the Newman-Penrose (NP) spin coefficients. These are 12 complex numbers, written as certain lower-case greek letters. (Don't confuse and with lapse and shift in a setting.)

The NP coefficients are calculated as contractions of (covariant) derivatives of the null tetrad:

Note that there are various commutation identities that follow from the normalisation of the tetrad, for example:

There is also a relation to the extrinsic curvature of the spacelike hypersurface :

since terms like vanish.

The NP coefficients are closely related to the expansion (convergence) of the null normals which define marginally trapped surfaces. Denoting the expansion of the outgoing and ingoing null normals with and , it is:

The expansions can also be written in terms of quantities:

#### Ricci and Weyl Scalars

The second derivatives of the four-metric can be written as combinations of the Ricci scalars and Weyl scalars, which are calculated from the Ricci tensor and Weyl tensor .

The Ricci scalars and are 10 real quantities, defined by certain contractions of the Ricci tensor with tetrad vectors:

(TODO: Why are the contractions involving real?)

The Weyl scalars are 5 complex quantities, defined by certain contractions of the Weyl tensor with tetrad vectors:

If the tetrad is chosen appropriately, then the Weyl scalars satisfy the peeling theorem near infinity:

That means especially that only falls off slowly enough to be non-zero when integrated over a large sphere near infinity.

This leads to the following physical interpretation of the Weyl scalars:

- : transverse radiation along (ingoing gravitational radiation)
- : longitudinal radiation along (ingoing gauge wave)
- : Coulomb field and spin (corresponding to gravitational attraction and frame dragging)
- : longitudinal radiation along (outgoing gauge wave)
- : transverse radiation along (outgoing gravitational radiation)

One has to be careful with this interpretation when the tetrad is not carefully chosen or when one is not near infinity.