|
60 | 60 | { |
61 | 61 | "cell_origin" : "server", |
62 | 62 | "cell_type" : "latex_view", |
63 | | - "source" : "\\begin{dmath*}{}\\left[g_{t t} = -1+\\frac{2M}{r},~\\discretionary{}{}{} g_{r r} = \\frac{1}{1 - \\frac{2M}{r}},~\\discretionary{}{}{} g_{\\theta \\theta} = r^{2},~\\discretionary{}{}{} g_{\\phi \\phi} = r^{2} \\sin{\\theta}^{2},~\\discretionary{}{}{} g^{t t} = \\frac{r}{2M-r},~\\discretionary{}{}{} g^{r r} = \\frac{-2M+r}{r},~\\discretionary{}{}{} g^{\\phi \\phi} = \\frac{1}{r^{2} \\sin{\\theta}^{2}},~\\discretionary{}{}{} g^{\\theta \\theta} = r^{-2}\\right]\\end{dmath*}" |
| 63 | + "source" : "\\begin{dmath*}{}\\left[g_{t t} = -1+\\frac{2M}{r},~\\discretionary{}{}{} g_{r r} = \\frac{1}{1 - \\frac{2M}{r}},~\\discretionary{}{}{} g_{\\theta \\theta} = {r}^{2},~\\discretionary{}{}{} g_{\\phi \\phi} = {r}^{2} {\\left(\\sin{\\theta}\\right)}^{2},~\\discretionary{}{}{} g^{t t} = \\frac{r}{2M-r},~\\discretionary{}{}{} g^{r r} = \\frac{-2M+r}{r},~\\discretionary{}{}{} g^{\\phi \\phi} = \\frac{1}{{r}^{2} {\\left(\\sin{\\theta}\\right)}^{2}},~\\discretionary{}{}{} g^{\\theta \\theta} = {r}^{-2}\\right]\\end{dmath*}" |
64 | 64 | } |
65 | 65 | ], |
66 | 66 | "source" : "ss:= { g_{t t} = -(1-2 M/r), \n g_{r r} = 1/(1-2 M/r), \n g_{\\theta\\theta} = r**2, \n g_{\\phi\\phi}=r**2 \\sin(\\theta)**2\n }.\n\ncomplete(ss, $g^{\\mu\\nu}$);" |
|
92 | 92 | { |
93 | 93 | "cell_origin" : "server", |
94 | 94 | "cell_type" : "latex_view", |
95 | | - "source" : "\\begin{dmath*}{}\\Gamma^{\\mu}\\,_{\\nu \\rho} = \\left\\{\\begin{aligned}\\square{}_{\\phi}{}_{r}{}^{\\phi}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}= & \\frac{1}{\\tan{\\theta}}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{r}{}^{\\theta}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{r}= & \\frac{M}{r \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{t}{}_{r}{}^{t}= & \\frac{M}{r \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{r}{}_{\\phi}{}^{\\phi}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\phi}{}^{\\phi}= & \\frac{1}{\\tan{\\theta}}\\\\[-.5ex]\n\\square{}_{r}{}_{\\theta}{}^{\\theta}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{t}{}^{t}= & \\frac{M}{r \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{r}= & \\left(2M-r\\right) \\sin{\\theta}^{2}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}= & - \\frac{1}{2}\\sin{2\\theta}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{r}= & 2M-r\\\\[-.5ex]\n\\square{}_{t}{}_{t}{}^{r}= & M \\frac{-2M+r}{r^{3}}\\\\[-.5ex]\n\\end{aligned}\\right.\n\\end{dmath*}" |
| 95 | + "source" : "\\begin{dmath*}{}\\Gamma^{\\mu}\\,_{\\nu \\rho} = \\square{}_{\\nu}{}_{\\rho}{}^{\\mu}\\left\\{\\begin{aligned}\\square{}_{\\phi}{}_{r}{}^{\\phi}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}= & \\frac{1}{\\tan{\\theta}}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{r}{}^{\\theta}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{r}= & \\frac{M}{r \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{t}{}_{r}{}^{t}= & \\frac{M}{r \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{r}{}_{\\phi}{}^{\\phi}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\phi}{}^{\\phi}= & \\frac{1}{\\tan{\\theta}}\\\\[-.5ex]\n\\square{}_{r}{}_{\\theta}{}^{\\theta}= & \\frac{1}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{t}{}^{t}= & \\frac{M}{r \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{r}= & \\left(2M-r\\right) {\\left(\\sin{\\theta}\\right)}^{2}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}= & - \\frac{1}{2}\\sin{2\\theta}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{r}= & 2M-r\\\\[-.5ex]\n\\square{}_{t}{}_{t}{}^{r}= & M \\frac{-2M+r}{{r}^{3}}\\\\[-.5ex]\n\\end{aligned}\\right.\n\\end{dmath*}" |
96 | 96 | } |
97 | 97 | ], |
98 | 98 | "source" : "ch:= \\Gamma^{\\mu}_{\\nu\\rho} = 1/2 g^{\\mu\\sigma} ( \n \\partial_{\\rho}{g_{\\nu\\sigma}} \n +\\partial_{\\nu}{g_{\\rho\\sigma}}\n -\\partial_{\\sigma}{g_{\\nu\\rho}} ):\n \nevaluate(ch, ss, rhsonly=True);" |
|
132 | 132 | { |
133 | 133 | "cell_origin" : "server", |
134 | 134 | "cell_type" : "latex_view", |
135 | | - "source" : "\\begin{dmath*}{}R^{\\rho}\\,_{\\sigma \\mu \\nu} = \\left\\{\\begin{aligned}\\square{}_{t}{}_{t}{}^{r}{}_{r}= & 2M \\frac{2M-r}{r^{4}}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{r}{}_{r}= & - \\frac{M}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}{}_{\\theta}= & 2M \\frac{\\sin{\\theta}^{2}}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{r}{}_{r}= & -M \\frac{\\sin{\\theta}^{2}}{r}\\\\[-.5ex]\n\\square{}_{t}{}_{r}{}^{t}{}_{r}= & \\frac{2M}{r^{2} \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}{}_{\\theta}= & - \\frac{2M}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{t}{}^{r}{}_{t}= & 2M \\frac{-2M+r}{r^{4}}\\\\[-.5ex]\n\\square{}_{r}{}_{\\theta}{}^{r}{}_{\\theta}= & \\frac{M}{r}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\phi}{}^{\\theta}{}_{\\phi}= & -2M \\frac{\\sin{\\theta}^{2}}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{\\phi}{}^{r}{}_{\\phi}= & M \\frac{\\sin{\\theta}^{2}}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{t}{}_{t}= & \\frac{2M}{r^{2} \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{\\theta}{}_{\\theta}= & \\frac{M}{r^{2} \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{\\phi}{}_{\\phi}= & \\frac{2M}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{\\phi}{}_{\\phi}= & \\frac{M}{r^{2} \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{t}{}_{t}{}^{\\phi}{}_{\\phi}= & M \\frac{-2M+r}{r^{4}}\\\\[-.5ex]\n\\square{}_{t}{}_{t}{}^{\\theta}{}_{\\theta}= & M \\frac{-2M+r}{r^{4}}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{t}{}_{t}= & -M \\frac{\\sin{\\theta}^{2}}{r}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{t}{}_{t}= & - \\frac{M}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{r}{}^{\\phi}{}_{r}= & \\frac{M}{r^{2} \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{t}{}^{\\phi}{}_{t}= & M \\frac{2M-r}{r^{4}}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{r}{}^{\\theta}{}_{r}= & \\frac{M}{r^{2} \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{t}{}^{\\theta}{}_{t}= & M \\frac{2M-r}{r^{4}}\\\\[-.5ex]\n\\square{}_{t}{}_{\\phi}{}^{t}{}_{\\phi}= & M \\frac{\\sin{\\theta}^{2}}{r}\\\\[-.5ex]\n\\square{}_{t}{}_{\\theta}{}^{t}{}_{\\theta}= & \\frac{M}{r}\\\\[-.5ex]\n\\end{aligned}\\right.\n\\end{dmath*}" |
| 135 | + "source" : "\\begin{dmath*}{}R^{\\rho}\\,_{\\sigma \\mu \\nu} = \\square{}_{\\nu}{}_{\\sigma}{}^{\\rho}{}_{\\mu}\\left\\{\\begin{aligned}\\square{}_{t}{}_{t}{}^{r}{}_{r}= & 2M \\frac{2M-r}{{r}^{4}}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{r}{}_{r}= & - \\frac{M}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}{}_{\\theta}= & 2M \\frac{{\\left(\\sin{\\theta}\\right)}^{2}}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{r}{}_{r}= & -M \\frac{{\\left(\\sin{\\theta}\\right)}^{2}}{r}\\\\[-.5ex]\n\\square{}_{t}{}_{r}{}^{t}{}_{r}= & \\frac{2M}{{r}^{2} \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}{}_{\\theta}= & - \\frac{2M}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{t}{}^{r}{}_{t}= & 2M \\frac{-2M+r}{{r}^{4}}\\\\[-.5ex]\n\\square{}_{r}{}_{\\theta}{}^{r}{}_{\\theta}= & \\frac{M}{r}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\phi}{}^{\\theta}{}_{\\phi}= & -2M \\frac{{\\left(\\sin{\\theta}\\right)}^{2}}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{\\phi}{}^{r}{}_{\\phi}= & M \\frac{{\\left(\\sin{\\theta}\\right)}^{2}}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{t}{}_{t}= & \\frac{2M}{{r}^{2} \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{\\theta}{}_{\\theta}= & \\frac{M}{{r}^{2} \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{\\phi}{}_{\\phi}= & \\frac{2M}{r}\\\\[-.5ex]\n\\square{}_{r}{}_{r}{}^{\\phi}{}_{\\phi}= & \\frac{M}{{r}^{2} \\left(2M-r\\right)}\\\\[-.5ex]\n\\square{}_{t}{}_{t}{}^{\\phi}{}_{\\phi}= & M \\frac{-2M+r}{{r}^{4}}\\\\[-.5ex]\n\\square{}_{t}{}_{t}{}^{\\theta}{}_{\\theta}= & M \\frac{-2M+r}{{r}^{4}}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{\\phi}{}^{t}{}_{t}= & -M \\frac{{\\left(\\sin{\\theta}\\right)}^{2}}{r}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{\\theta}{}^{t}{}_{t}= & - \\frac{M}{r}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{r}{}^{\\phi}{}_{r}= & \\frac{M}{{r}^{2} \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{\\phi}{}_{t}{}^{\\phi}{}_{t}= & M \\frac{2M-r}{{r}^{4}}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{r}{}^{\\theta}{}_{r}= & \\frac{M}{{r}^{2} \\left(-2M+r\\right)}\\\\[-.5ex]\n\\square{}_{\\theta}{}_{t}{}^{\\theta}{}_{t}= & M \\frac{2M-r}{{r}^{4}}\\\\[-.5ex]\n\\square{}_{t}{}_{\\phi}{}^{t}{}_{\\phi}= & M \\frac{{\\left(\\sin{\\theta}\\right)}^{2}}{r}\\\\[-.5ex]\n\\square{}_{t}{}_{\\theta}{}^{t}{}_{\\theta}= & \\frac{M}{r}\\\\[-.5ex]\n\\end{aligned}\\right.\n\\end{dmath*}" |
136 | 136 | } |
137 | 137 | ], |
138 | 138 | "source" : "substitute(rm, ch)\nevaluate(rm, ss, rhsonly=True);" |
|
204 | 204 | { |
205 | 205 | "cell_origin" : "server", |
206 | 206 | "cell_type" : "latex_view", |
207 | | - "source" : "\\begin{dmath*}{}K = \\frac{48M^{2}}{r^{6}}\\end{dmath*}" |
| 207 | + "source" : "\\begin{dmath*}{}K = \\frac{48{M}^{2}}{{r}^{6}}\\end{dmath*}" |
208 | 208 | } |
209 | 209 | ], |
210 | 210 | "source" : "substitute(K, rm)\nevaluate(K, ss, rhsonly=True);" |
|
302 | 302 | { |
303 | 303 | "cell_origin" : "server", |
304 | 304 | "cell_type" : "latex_view", |
305 | | - "source" : "\\begin{dmath*}{}K = \\frac{48M^{2}}{r^{6}}\\end{dmath*}" |
| 305 | + "source" : "\\begin{dmath*}{}K = \\frac{48{M}^{2}}{{r}^{6}}\\end{dmath*}" |
306 | 306 | } |
307 | 307 | ], |
308 | 308 | "source" : "substitute(K, rm)\nevaluate(K, ss);" |
|
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