| … | … | @@ -6966,33 +6966,46 @@ or |
| 6966 | 6966 | |
| 6967 | 6967 | \end_inset |
| 6968 | 6968 | |
| 6969 | | which smells OK. |
| 6970 | | \end_layout |
| 6969 | which is probably right but smells bad. |
| 6970 | It's probably right because I think Claude copied it out of a book. |
| 6971 | It smells bad because it is putting tangent–space indexes (the latin letters) on coordinate–space partial derivatives. |
| 6972 | So let me see if I can patch this up by hand. |
| 6973 | Let me try to patch this by writing |
| 6974 | \begin_inset Formula $\partial^{a}=e^{a\mu}\partial_{\mu}$ |
| 6975 | \end_inset |
| 6971 | 6976 | |
| 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 |
| 6977 | . |
| 6978 | This is an abuse of notation, |
| 6979 | but I think we can get away with it if we are careful. |
| 6980 | This gives |
| 6981 | \begin_inset Formula |
| 6982 | \[ |
| 6983 | \omega_{\;\;\mu}^{ab}=\frac{1}{\lambda}\left(\delta_{\mu}^{a}e^{b\nu}-\delta_{\mu}^{b}e^{a\nu}\right)\partial_{\nu}\lambda |
| 6984 | \] |
| 6978 | 6985 | |
| 6979 | | \begin_layout Standard |
| 6980 | | First, |
| 6981 | | |
| 6982 | | \begin_inset Formula $\partial^{a}=e^{a\mu}\partial_{\mu}$ |
| 6983 | 6986 | \end_inset |
| 6984 | 6987 | |
| 6985 | | with |
| 6988 | Then for |
| 6986 | 6989 | \begin_inset Formula $e^{a\mu}=\delta^{a\mu}/\lambda$ |
| 6987 | 6990 | \end_inset |
| 6988 | 6991 | |
| 6989 | | . |
| 6990 | | The calculation in greek indexes, |
| 6992 | (another abuse of notation) this becomes |
| 6993 | \begin_inset Formula |
| 6994 | \[ |
| 6995 | \omega_{\;\;\mu}^{ab}=\frac{1}{\lambda^{2}}\left(\delta_{\mu}^{a}\delta^{b\nu}-\delta_{\mu}^{b}\delta^{a\nu}\right)\partial_{\nu}\lambda |
| 6996 | \] |
| 6997 | |
| 6998 | \end_inset |
| 6999 | |
| 7000 | The partial derivative gets done in greek indexes, |
| 7001 | i.e. |
| 7002 | in coordinate space, |
| 6991 | 7003 | for |
| 6992 | 7004 | \begin_inset Formula $\lambda=2A/\left(1-r^{2}\right)$ |
| 6993 | 7005 | \end_inset |
| 6994 | 7006 | |
| 6995 | | gives |
| 7007 | . |
| 7008 | This gives |
| 6996 | 7009 | \begin_inset Formula |
| 6997 | 7010 | \[ |
| 6998 | 7011 | \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} |
| … | … | @@ -7000,35 +7013,41 @@ First, |
| 7000 | 7013 | |
| 7001 | 7014 | \end_inset |
| 7002 | 7015 | |
| 7003 | | So converting to latin: |
| 7016 | So |
| 7004 | 7017 | \begin_inset Formula |
| 7005 | 7018 | \[ |
| 7006 | | \partial^{a}\lambda=\frac{\lambda}{A}x^{a} |
| 7019 | \omega_{\;\;\mu}^{ab}=\frac{1}{A}\left(\delta_{\mu}^{a}\delta^{b\nu}-\delta_{\mu}^{b}\delta^{a\nu}\right)x_{\nu} |
| 7007 | 7020 | \] |
| 7008 | 7021 | |
| 7009 | 7022 | \end_inset |
| 7010 | 7023 | |
| 7011 | | and so now comes more confusion. |
| 7012 | | Earlier, |
| 7013 | | Claude claimed, |
| 7014 | | in Cartesian: |
| 7024 | Drop mu next, |
| 7025 | to get |
| 7015 | 7026 | \begin_inset Formula |
| 7016 | | \[ |
| 7017 | | \omega^{ab}=\frac{\lambda}{A}\left(x^{b}dx^{a}-x^{a}dx^{b}\right) |
| 7018 | | \] |
| 7019 | | |
| 7020 | | \end_inset |
| 7027 | \begin{align*} |
| 7028 | \omega^{ab} & =\frac{1}{A}x_{\nu}\left(\delta^{b\nu}dx^{a}-\delta^{a\nu}dx^{b}\right)\\ |
| 7029 | & =\frac{\lambda}{A}x_{\nu}\left(e^{b\nu}dx^{a}-e^{a\nu}dx^{b}\right)\\ |
| 7030 | & =\frac{\lambda}{A}\left(x^{b}dx^{a}-x^{a}dx^{b}\right) |
| 7031 | \end{align*} |
| 7021 | 7032 | |
| 7022 | | but this is off by a factor of |
| 7023 | | \begin_inset Formula $\lambda$ |
| 7024 | 7033 | \end_inset |
| 7025 | 7034 | |
| 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. |
| 7035 | with this third line being the abuse of notion. |
| 7036 | This need a big ***caution*** sticker on it: |
| 7037 | its dangerous but OK, |
| 7038 | as long as one consistently abuses the notation. |
| 7039 | So we plow ahead, |
| 7040 | damn the torpedoes. |
| 7041 | Not sure who ever launched a torpedo towards a plow, |
| 7042 | but we're not exactly sailing here, |
| 7043 | either. |
| 7044 | Claude's wordcel trickery has slowed me down. |
| 7045 | That it is not a shape rotator becomes exposed very clearly in exercises like this. |
| 7046 | Alas. |
| 7047 | For me, |
| 7048 | alas. |
| 7049 | I can't trust it. |
| 7050 | Lets plow some more ravines. |
| 7032 | 7051 | \end_layout |
| 7033 | 7052 | |
| 7034 | 7053 | \begin_layout Paragraph* |
| … | … | @@ -7130,43 +7149,6 @@ e_{3} & =\frac{1}{\lambda r\sin\theta}\partial_{\phi} |
| 7130 | 7149 | |
| 7131 | 7150 | The spin connection. |
| 7132 | 7151 | |
| 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: |
| 7157 | | \begin_inset Formula |
| 7158 | | \begin{align*} |
| 7159 | | \omega^{0i} & =0\\ |
| 7160 | | \omega^{12} & =-\frac{1}{\lambda r}d\theta\\ |
| 7161 | | \omega^{13} & =-\frac{1}{\lambda r}\sin\theta\,d\phi\\ |
| 7162 | | \omega^{23} & =-\cos\theta\,d\phi |
| 7163 | | \end{align*} |
| 7164 | | |
| 7165 | | \end_inset |
| 7166 | | |
| 7167 | | I copied these from Claude, |
| 7168 | | but they look maybe not right. |
| 7169 | | Let me check. |
| 7170 | 7152 | \end_layout |
| 7171 | 7153 | |
| 7172 | 7154 | \begin_layout Section* |