Wave Forms

Assume a spacetime $g_{\alpha\beta}$ which can be written as a Schwarzschild background $g_{\alpha\beta}^{Schwarz}$ with perturbations $h_{\alpha\beta}$:

$$g_{\alpha\beta} = g^{Schwarz}_{\alpha\beta} + h_{\alpha\beta}$$ (part311)

with

$$\{g^{Schwarz}_{\alpha\beta}\}(t,r,\theta,\phi) = \left( \be... ...r^2 \sin^2\theta \end{array}\right) \qquad S(r)=1-\frac{2M}{r}$$ (part312)

The 3-metric perturbations $\gamma_{ij}$ can be decomposed using tensor harmonics into $\gamma_{ij}^{lm}(t,r)$ where

$$\gamma_{ij}(t,r,\theta,\phi)=\sum_{l=0}^\infty \sum_{m=-l}^l \gamma_{ij}^{lm}(t,r)$$

and

$$\gamma_{ij}(t,r,\theta ,\phi ) = \sum_{k=0}^6 p_k(t,r) {\bf V}_k(\theta ,\phi )$$

where $\{{\bf V}_k\}$ is some basis for tensors on a 2-sphere in 3-D Euclidean space. Working with the Regge-Wheeler basis (see Section D11.7) the 3-metric is then expanded in terms of the (six) standard Regge-Wheeler functions $\{c_1^{\times lm}, c_2^{\times lm}, h_1^{+lm}, H_2^{+lm}, K^{+lm}, G^{+lm}\}$ [26], [23]. Where each of the functions is either odd ($\times$) or even ($+$) parity. The decomposition is then written

$$\gamma_{ij}^{lm} = c_1^{\times lm}(\hat{e}_1)_{ij}^{lm} + c_2^{\times lm}(\hat{e}_2)_{ij}^{lm}$$

$$+ h_1^{+lm}(\hat{f}_1)_{ij}^{lm} + A^2 H_2^{+lm}(\hat{f}_2)_{ij}^{lm} + R^2 K^{+lm}(\hat{f}_3)_{ij}^{lm} + R^2 G^{+lm}(\hat{f}_4)_{ij}^{lm}$$ (part313)

which we can write in an expanded form as

$$\gamma_{rr}^{lm} = A^2 H_2^{+lm} Y_{lm}$$ (part314)

$$\gamma_{r\theta }^{lm} = - c_1^{\times lm} \frac{1}{\sin\theta } Y_{lm,\phi}+h_1^{+lm}Y_{lm,\theta}$$ (part315)

$$\gamma_{r\phi }^{lm} = c_1^{\times lm} \sin\theta Y_{lm,\theta}+ h_1^{+lm}Y_{lm,\phi}$$ (part316)

$$\gamma_{\theta \theta }^{lm} = c_2^{\times lm}\frac{1}{\sin\theta }(Y_{lm,\theta\phi}-\cot\theta Y_{lm,\phi}) + R^2 K^{+lm}Y_{lm}+ R^2 G^{+lm} Y_{lm,\theta\theta}$$ (part317)

$$\gamma_{\theta \phi }^{lm} = -c_2^{\times lm}\sin\theta \frac{1}{2} \left( Y_{lm,\theta\theta}... ...n^2\theta}Y_{lm}\right) + R^2 G^{+lm}(Y_{lm,\theta\phi}-\cot\theta Y_{lm,\phi})$$ (part318)

$$\gamma_{\phi \phi }^{lm} = -\sin\theta c_2^{\times lm} (Y_{lm,\theta\phi}- \cot\theta Y_{lm,... ...theta Y_{lm} +R^2 G^{+lm} (Y_{lm,\phi\phi}+\sin\theta \cos\theta Y_{lm,\theta})$$ (part319)

A similar decomposition allows the four gauge components of the 4-metric to be written in terms of three even-parity variables $\{H_0,H_1,h_0\}$ and the one odd-parity variable $\{c_0\}$

$$g_{tt}^{lm} = N^2 H_0^{+lm} Y_{lm}$$ (part3110)

$$g_{tr}^{lm} = H_1^{+lm} Y_{lm}$$ (part3111)

$$g_{t\theta }^{lm} = h_0^{+lm} Y_{lm,\theta}- c_0^{\times lm}\frac{1}{\sin\theta }Y_{lm,\phi}$$ (part3112)

$$g_{t\phi }^{lm} = h_0^{+lm} Y_{lm,\phi}+ c_0^{\times lm} \sin\theta Y_{lm,\theta}$$ (part3113)

Also from $g_{tt}=-\alpha^2+\beta_i\beta^i$ we have

$$\alpha^{lm} = -\frac{1}{2}NH_0^{+lm}Y_{lm}$$ (part3114)

It is useful to also write this with the perturbation split into even and odd parity parts:

$$g_{\alpha\beta} = {g}^{background}_{\alpha\beta} + \sum_{l,m} g^{lm,odd}_{\alpha\beta} +\sum_{l,m} g^{lm,even}_{\alpha\beta}$$

where (dropping some superscripts)

$$\begin{eqnarray*} \{g_{\alpha\beta}^{odd}\} &=& \left( \begin{array}{cccc} 0 & ... ...phi\phi}+\sin\theta \cos\theta Y_{lm,\theta}) \end{array}\right) \end{eqnarray*}$$

Now, for such a Schwarzschild background we can define two (and only two) unconstrained gauge invariant quantities $Q^{\times}_{lm}=Q^{\times}_{lm}(c_1^{\times lm},c_2^{\times lm})$ and $Q^{+}_{lm}=Q^{+}_{lm}(K^{+ lm},G^{+ lm},H_2^{+lm},h_1^{+lm})$, which from [10] are

$$Q^{\times}_{lm} = \sqrt{\frac{2(l+2)!}{(l-2)!}}\left[c_1^{\times lm} + \frac{1}{2}\... ...rtial_r c_2^{\times lm} - \frac{2}{r} c_2^{\times lm}\right)\right] \frac{S}{r}$$ (part3115)

$$Q^{+}_{lm} = \frac{1}{\Lambda}\sqrt{\frac{2(l-1)(l+2)}{l(l+1)}} (4rS^2 k_2+l(l+1)r k_1)$$ (part3116)

$$\equiv \frac{1}{\Lambda}\sqrt{\frac{2(l-1)(l+2)}{l(l+1)}} \left(l(l+1)S(... ...^{+lm}-2h_1^{+lm})+ 2rS(H_2^{+lm}-r\partial_r K^{+lm})+\Lambda r K^{+lm}\right)$$ (part3117)

where

$$k_1 = K^{+lm} + \frac{S}{r}(r^2\partial_r G^{+lm} - 2h^{+lm}_1)$$ (part3118)

$$k_2 = \frac{1}{2S} \left[H^{+lm}_2-r\partial_r k_1-\left(1-\frac{M}{rS}... ...1 + S^{1/2}\partial_r (r^2 S^{1/2} \partial_r G^{+lm}-2S^{1/2}h_1^{+lm})\right]$$ (part3119)

$$\equiv \frac{1}{2S}\left[H_2-rK_{,r}-\frac{r-3M}{r-2M}K\right]$$ (part3120)

NOTE: These quantities compare with those in Moncrief [23] by

$$\begin{eqnarray*} \mbox{Moncriefs odd parity Q: }\qquad Q^\times_{lm} &=& \sqrt... ...ity Q: } \qquad Q^+_{lm} &=& \sqrt{\frac{2(l-1)(l+2)}{l(l+1)}}Q \end{eqnarray*}$$

Note that these quantities only depend on the purely spatial Regge-Wheeler functions, and not the gauge parts. (In the Regge-Wheeler and Zerilli gauges, these are just respectively (up to a rescaling) the Regge-Wheeler and Zerilli functions). These quantities satisfy the wave equations

$$\begin{eqnarray*} &&(\partial^2_t-\partial^2_{r^*})Q^\times_{lm}+S\left[\frac{l... ...t) \right)+\frac{l(l-1)(l+1)(l+2)}{r^2\Lambda}\right]Q^+_{lm}=0 \end{eqnarray*}$$

where

$$\begin{eqnarray*} \Lambda &=& (l-1)(l+2)+6M/r \\ r^* &=& r+2M\ln(r/2M-1) \end{eqnarray*}$$