Tetrad Formalism

From GRwiki

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 $l^a, n^a, m^a, \bar{m}^a$ , where $\bar{x}$ denotes complex conjugation. Both $l^a$ and $n^a$ are real, $m^a$ is complex. With respect to an event on an (arbitrarily chosen) two-sphere, $l^a$ and $n^a$ are the ingoing and outgoing null normals, respectively, whereas $m^a$ and $\bar{m}^a$ are tangential null vectors.

(Other notations are common, using e.g. $k^a$ instead of $n^a$ .)

(TODO: Make the notation consistent with the notation in the section explaining the $3+1$ split.)

These null vectors satisfy the following conditions:

$\begin{array}{rcl} l^a l_a = n^a n_a = m^a m_a & = & 0 \\ l^a n_a & = & -1 \\ m^a \bar{m}_a & = & +1 \\ l^a m_a = n^a m_a & = & 0 \end{array}$

(Other normalisations are also used, e.g. $l^a n_a = -2$ , which avoid some of the factors $\sqrt{2}$ that appear below.)

Remember the following rules for complex conjugation:

$\begin{array}{rcl} \overline{a b} & = & \bar{a} \bar{b} \\ a + \bar{a} & = & 2 \,\operatorname{Re}\, a \end{array}$

The null tetrad vectors can be expressed in terms of timelike and spacelike unit vectors $t^a, s^a, e_{\theta}^a, e_{\phi}^a$ as follows:

$\begin{array}{rcl} l^a & = & \frac{1}{\sqrt{2}} (t^a + s^a) \\ n^a & = & \frac{1}{\sqrt{2}} (t^a - s^a) \\ m^a & = & \frac{1}{\sqrt{2}} (e_{\theta}^a + i e_{\phi}^a) \end{array}$

These vectors $t^a, s^a, e_{\theta}^a, e_{\phi}^a$ have to be orthogonal to each other and also have to be normalised:

$\begin{array}{rcl} t^a t_a & = & -1 \\ s^a s_a & = & +1 \\ e_{\theta}^a e_{\theta a} & = & +1 \\ e_{\phi}^a e_{\phi a} & = & +1 \end{array}$

In the following we assume that there is a preferred spacelike hypersurface $\Sigma$ , with $t^a$ orthogonal to $\Sigma$ , and that the two-surface spanned by $e_{\theta}^a$ and $e_{\phi}^a$ is contained in $\Sigma$ . (The timelike normal is sometimes also called $n^a$ .)

With these definitions, the four-metric $g_{ab}$ , the three-metric $\gamma_{ab}$ , and the two-metric $q_{ab}$ are given by:

$\begin{array}{rcl} g^{ab} & = & -l^a n^b - n^a l^b + m^a \bar{m}^b + \bar{m}^a m^b \\ \gamma^{ab} & = & g^{ab} + t^a t^b = s^a s^b + m^a \bar{m}^b + \bar{m}^a m^b \\ q^{ab} & = & \gamma^{ab} - s^a s^b = m^a \bar{m}^b + \bar{m}^a m^b \end{array}$

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 $\alpha$ and $\beta$ with lapse and shift in a $3+1$ setting.)

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

$\begin{array}{rcl} \kappa & = & - m^a l^b D_b l_a \\ \tau & = & - m^a n^b D_b l_a \\ \sigma & = & - m^a m^b D_b l_a \\ \rho & = & - m^a \bar{m}^b D_b l_a \\ \epsilon & = & - \frac{1}{2} ( n^a l^b D_b l_a - \bar{m}^a l^b D_b m_a ) \\ \gamma & = & - \frac{1}{2} ( n^a n^b D_b l_a - \bar{m}^a n^b D_b m_a ) \\ \beta & = & - \frac{1}{2} ( n^a m^b D_b l_a - \bar{m}^a m^b D_b m_a ) \\ \alpha & = & - \frac{1}{2} ( n^a \bar{m}^b D_b l_a - \bar{m}^a \bar{m}^b D_b m_a ) \\ \pi & = & \bar{m}^a l^b D_b n_a \\ \nu & = & \bar{m}^a n^b D_b n_a \\ \mu & = & \bar{m}^a m^b D_b n_a \\ \lambda & = & \bar{m}^a \bar{m}^b D_b n_a \end{array}$

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

$\begin{array}{rcl} l^b D_a n_b + n^b D_a l_b & = & D_a l^b n_b = D_a (-1) = 0 \\ t^b D_a t_b & = & \frac{1}{2} D_a t^b t_b = \frac{1}{2} D_a (-1) = 0 \end{array}$

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

$\begin{array}{rcl} K_{ab} & = & - \gamma^c_a \gamma^d_b D_c t_d \\ & = & - (g^c_a + t^c t_a) (g^d_b + t^d t_b) D_c t_d \\ & = & - D_a t_b - t^d t_b D_a t_d - t^c t_a D_c t_b - t^c t_a t^d t_b D_c t_d \\ & = & - D_a t_b - t^c t_a D_c t_b \end{array}$

since terms like $t^b D_a t_b$ 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 $\Theta_{(l)}$ and $\Theta_{(n)}$ , it is:

$\begin{array}{rcl} \Theta_{(l)} & = & q^{ab} D_a l_b \\ & = & (m^a \bar{m}^b + \bar{m}^a m^b) D_a l_b \\ & = & - \rho - \bar{\rho} \\ \Theta_{(n)} & = & q^{ab} D_a n_b \\ & = & - \gamma^{ab} D_a s_b - (\gamma^{ab} - s^a s^b) K_{ab} \\ & = & - \mu - \bar{\mu} \end{array}$

The expansions can also be written in terms of $3+1$ quantities:

$\begin{array}{rcl} \Theta_{(l)} & = & q^{ab} D_a l_b \\ & = & (\gamma^{ab} - s^a s^b) (D_a s_b + D_a t_b) \\ & = & + (\gamma^{ab} - s^a s^b) D_a s_b + (\gamma^{ab} - s^a s^b) D_a t_b \\ & = & + \gamma^{ab} D_a s_b - (\gamma^{ab} - s^a s^b) K_{ab} \\ \Theta_{(n)} & = & - \gamma^{ab} D_a s_b - (\gamma^{ab} - s^a s^b) K_{ab} \end{array}$

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 $R_{ab}$ and Weyl tensor $C_{abcd}$ .

The Ricci scalars $\Lambda$ and $\Phi_{nm}$ are 10 real quantities, defined by certain contractions of the Ricci tensor with tetrad vectors:

$\begin{array}{rcl} \Lambda & = & \frac{1}{24} R \\ \Phi_{00} & = & - \frac{1}{2} R_{ab} l^a l^b \\ \Phi_{11} & = & - \frac{1}{4} R_{ab} (l^a n^b + m^a \bar{m}^b) \\ \Phi_{01} & = & - \frac{1}{2} R_{ab} l^a m^b \\ \Phi_{12} & = & - \frac{1}{2} R_{ab} n^a m^b \\ \Phi_{10} & = & - \frac{1}{2} R_{ab} l^a \bar{m}^b \\ \Phi_{21} & = & - \frac{1}{2} R_{ab} n^a \bar{m}^b \\ \Phi_{02} & = & - \frac{1}{2} R_{ab} m^a m^b \\ \Phi_{22} & = & - \frac{1}{2} R_{ab} n^a n^b \\ \Phi_{20} & = & - \frac{1}{2} R_{ab} \bar{m}^a \bar{m}^b \end{array}$

(TODO: Why are the contractions involving $m^a$ real?)

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

$\begin{array}{rcl} \Psi_0 & = & C_{abcd} l^a m^b l^c m^d \\ \Psi_1 & = & C_{abcd} l^a m^b l^c n^d \\ \Psi_2 & = & C_{abcd} l^a m^b \bar{m}^c n^d \\ \Psi_3 & = & C_{abcd} l^a n^b \bar{m}^c n^d \\ \Psi_4 & = & C_{abcd} \bar{m}^a n^b \bar{m}^c n^d \end{array}$

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

$\begin{array}{rcl} \lim_{r\to\infty} \Psi_n & \propto & \frac{1}{r^{5-n}} \end{array}$

That means especially that only $\Psi_4$ 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:

$\Psi_0$ : transverse radiation along $n$ (ingoing gravitational radiation)
$\Psi_1$ : longitudinal radiation along $n$ (ingoing gauge wave)
$\Psi_2$ : Coulomb field and spin (corresponding to gravitational attraction and frame dragging)
$\Psi_3$ : longitudinal radiation along $l$ (outgoing gauge wave)
$\Psi_4$ : transverse radiation along $l$ (outgoing gravitational radiation)