| … | … | @@ -6934,7 +6934,8 @@ and |
| 6934 | 6934 | |
| 6935 | 6935 | \end_inset |
| 6936 | 6936 | |
| 6937 | | |
| 6937 | Again, |
| 6938 | this looks correct to me. |
| 6938 | 6939 | \end_layout |
| 6939 | 6940 | |
| 6940 | 6941 | \begin_layout Paragraph* |
| … | … | @@ -6946,7 +6947,7 @@ For |
| 6946 | 6947 | \begin_inset Formula $e_{\;\mu}^{a}=\lambda\,\delta_{\mu}^{a}$ |
| 6947 | 6948 | \end_inset |
| 6948 | 6949 | |
| 6949 | | this works out to |
| 6950 | Claude tells me that this works out to |
| 6950 | 6951 | \begin_inset Formula |
| 6951 | 6952 | \[ |
| 6952 | 6953 | \omega_{\;\;\mu}^{ab}=\frac{1}{\lambda}\left(\delta_{\mu}^{a}\partial^{b}\lambda-\delta_{\mu}^{b}\partial^{a}\lambda\right) |
| … | … | @@ -6965,27 +6966,51 @@ or |
| 6965 | 6966 | |
| 6966 | 6967 | \end_inset |
| 6967 | 6968 | |
| 6968 | | so for |
| 6969 | which smells OK. |
| 6970 | \end_layout |
| 6971 | |
| 6972 | \begin_layout Standard |
| 6973 | But now there is stumbling on notational issues. |
| 6974 | The greek letters are for coordinate spaces; |
| 6975 | the latin letters are for the tangent spaces. |
| 6976 | Lots of abuse of notation ensues. |
| 6977 | \end_layout |
| 6978 | |
| 6979 | \begin_layout Standard |
| 6980 | First, |
| 6981 | |
| 6982 | \begin_inset Formula $\partial^{a}=e^{a\mu}\partial_{\mu}$ |
| 6983 | \end_inset |
| 6984 | |
| 6985 | with |
| 6986 | \begin_inset Formula $e^{a\mu}=\delta^{a\mu}/\lambda$ |
| 6987 | \end_inset |
| 6988 | |
| 6989 | . |
| 6990 | The calculation in greek indexes, |
| 6991 | for |
| 6969 | 6992 | \begin_inset Formula $\lambda=2A/\left(1-r^{2}\right)$ |
| 6970 | 6993 | \end_inset |
| 6971 | 6994 | |
| 6972 | | we get |
| 6995 | gives |
| 6973 | 6996 | \begin_inset Formula |
| 6974 | 6997 | \[ |
| 6975 | | \partial^{a}\lambda=\frac{4A}{\left(1-r^{2}\right)^{2}}r\partial^{a}r=\frac{\lambda^{2}}{A}r\partial^{a}r=\frac{\lambda^{2}}{A}x^{a} |
| 6998 | \partial_{\mu}\lambda=\frac{4A}{\left(1-r^{2}\right)^{2}}r\partial_{\mu}r=\frac{\lambda^{2}}{A}r\partial_{\mu}r=\frac{\lambda^{2}}{A}x_{\mu} |
| 6976 | 6999 | \] |
| 6977 | 7000 | |
| 6978 | 7001 | \end_inset |
| 6979 | 7002 | |
| 6980 | | Equivalently: |
| 7003 | So converting to latin: |
| 6981 | 7004 | \begin_inset Formula |
| 6982 | 7005 | \[ |
| 6983 | | \partial_{\mu}\lambda=\frac{\lambda^{2}}{A}x_{\mu} |
| 7006 | \partial^{a}\lambda=\frac{\lambda}{A}x^{a} |
| 6984 | 7007 | \] |
| 6985 | 7008 | |
| 6986 | 7009 | \end_inset |
| 6987 | 7010 | |
| 6988 | | and so as before, |
| 7011 | and so now comes more confusion. |
| 7012 | Earlier, |
| 7013 | Claude claimed, |
| 6989 | 7014 | in Cartesian: |
| 6990 | 7015 | \begin_inset Formula |
| 6991 | 7016 | \[ |
| … | … | @@ -6994,7 +7019,16 @@ and so as before, |
| 6994 | 7019 | |
| 6995 | 7020 | \end_inset |
| 6996 | 7021 | |
| 7022 | but this is off by a factor of |
| 7023 | \begin_inset Formula $\lambda$ |
| 7024 | \end_inset |
| 6997 | 7025 | |
| 7026 | overall. |
| 7027 | Fuck me. |
| 7028 | It is better to avoid using expressions like this. |
| 7029 | I leave this here, |
| 7030 | for now, |
| 7031 | with a ***caution*** sticker on it. |
| 6998 | 7032 | \end_layout |
| 6999 | 7033 | |
| 7000 | 7034 | \begin_layout Paragraph* |
| … | … | @@ -7095,7 +7129,31 @@ e_{3} & =\frac{1}{\lambda r\sin\theta}\partial_{\phi} |
| 7095 | 7129 | \end_inset |
| 7096 | 7130 | |
| 7097 | 7131 | The spin connection. |
| 7098 | | No mixed time–space: |
| 7132 | |
| 7133 | \begin_inset Formula |
| 7134 | \[ |
| 7135 | \omega^{ab}=\frac{1}{\lambda}\left(dx^{a}\,\partial^{b}\lambda-dx^{b}\,\partial^{a}\lambda\right) |
| 7136 | \] |
| 7137 | |
| 7138 | \end_inset |
| 7139 | |
| 7140 | so for for |
| 7141 | \begin_inset Formula $\lambda=2A/\left(1-r^{2}\right)$ |
| 7142 | \end_inset |
| 7143 | |
| 7144 | we get |
| 7145 | \begin_inset Formula $\partial^{2}\lambda=\partial^{3}\lambda=0$ |
| 7146 | \end_inset |
| 7147 | |
| 7148 | and |
| 7149 | \begin_inset Formula $\partial^{1}\lambda=\lambda^{-1}\partial_{1}\lambda=\lambda r/A$ |
| 7150 | \end_inset |
| 7151 | |
| 7152 | and so ... |
| 7153 | \end_layout |
| 7154 | |
| 7155 | \begin_layout Standard |
| 7156 | No mixed time–space: |
| 7099 | 7157 | \begin_inset Formula |
| 7100 | 7158 | \begin{align*} |
| 7101 | 7159 | \omega^{0i} & =0\\ |