|
16 | 16 | "cell_type": "code", |
17 | 17 | "execution_count": 1, |
18 | 18 | "metadata": {}, |
19 | | - "outputs": [ |
20 | | - { |
21 | | - "data": { |
22 | | - "text/markdown": [ |
23 | | - "${}\\text{Attached property Coordinate to~}\\left[r,~ t,~ \\phi,~ \\theta\\right].$" |
24 | | - ] |
25 | | - }, |
26 | | - "execution_count": 1, |
27 | | - "metadata": {}, |
28 | | - "output_type": "execute_result" |
29 | | - }, |
30 | | - { |
31 | | - "data": { |
32 | | - "text/markdown": [ |
33 | | - "${}\\text{Attached property Indices(position=fixed) to~}\\left[\\mu,~ \\nu,~ \\rho,~ \\sigma,~ \\lambda,~ \\kappa,~ \\chi,~ \\gamma\\right].$" |
34 | | - ] |
35 | | - }, |
36 | | - "execution_count": 1, |
37 | | - "metadata": {}, |
38 | | - "output_type": "execute_result" |
39 | | - }, |
40 | | - { |
41 | | - "data": { |
42 | | - "text/markdown": [ |
43 | | - "${}\\text{Attached property PartialDerivative to~}\\partial{\\#}.$" |
44 | | - ] |
45 | | - }, |
46 | | - "execution_count": 1, |
47 | | - "metadata": {}, |
48 | | - "output_type": "execute_result" |
49 | | - } |
50 | | - ], |
| 19 | + "outputs": [], |
51 | 20 | "source": [ |
52 | 21 | "{r,t,\\phi,\\theta}::Coordinate;\n", |
53 | 22 | "{\\mu,\\nu,\\rho,\\sigma,\\lambda,\\kappa,\\chi,\\gamma}::Indices(values={t,r,\\phi,\\theta}, position=fixed);\n", |
|
73 | 42 | { |
74 | 43 | "data": { |
75 | 44 | "text/markdown": [ |
76 | | - "${}\\left[g_{t t} = -1+2M {r}^{-1},~ g_{r r} = {\\left(1-2M {r}^{-1}\\right)}^{-1},~ g_{\\theta \\theta} = {r}^{2},~ g_{\\phi \\phi} = {r}^{2} {\\left(\\sin{\\theta}\\right)}^{2},~ g^{t t} = {\\left(2M {r}^{-1}-1\\right)}^{-1},~ g^{r r} = -2M {r}^{-1}+1,~ g^{\\phi \\phi} = {\\left({r}^{2} {\\left(\\sin{\\theta}\\right)}^{2}\\right)}^{-1},~ g^{\\theta \\theta} = {r}^{-2}\\right]$" |
| 45 | + "${}\\left[g_{t t} = -1+2M {r}^{-1},~\\discretionary{}{}{} g_{r r} = {\\left(1-2M {r}^{-1}\\right)}^{-1},~\\discretionary{}{}{} g_{\\theta \\theta} = {r}^{2},~\\discretionary{}{}{} g_{\\phi \\phi} = {r}^{2} {\\left(\\sin{\\theta}\\right)}^{2},~\\discretionary{}{}{} g^{t t} = {\\left(2M {r}^{-1}-1\\right)}^{-1},~\\discretionary{}{}{} g^{r r} = -2M {r}^{-1}+1,~\\discretionary{}{}{} g^{\\phi \\phi} = {\\left({r}^{2} {\\left(\\sin{\\theta}\\right)}^{2}\\right)}^{-1},~\\discretionary{}{}{} g^{\\theta \\theta} = {r}^{-2}\\right]$" |
77 | 46 | ] |
78 | 47 | }, |
79 | | - "execution_count": 2, |
80 | 48 | "metadata": {}, |
81 | | - "output_type": "execute_result" |
| 49 | + "output_type": "display_data" |
82 | 50 | } |
83 | 51 | ], |
84 | 52 | "source": [ |
|
111 | 79 | "${}\\Gamma^{\\mu}\\,_{\\nu \\rho} = \\frac{1}{2}g^{\\mu \\sigma} \\left(\\partial_{\\rho}{g_{\\nu \\sigma}}+\\partial_{\\nu}{g_{\\rho \\sigma}}-\\partial_{\\sigma}{g_{\\nu \\rho}}\\right)$" |
112 | 80 | ] |
113 | 81 | }, |
114 | | - "execution_count": 3, |
115 | 82 | "metadata": {}, |
116 | | - "output_type": "execute_result" |
| 83 | + "output_type": "display_data" |
117 | 84 | }, |
118 | 85 | { |
119 | 86 | "data": { |
120 | 87 | "text/markdown": [ |
121 | | - "${}\\Gamma^{\\mu}\\,_{\\nu \\rho} = \\square{}_{\\nu}{}_{\\rho}{}^{\\mu}\\left\\{\\begin{aligned}\\square{}_{\\phi}{}_{r}{}^{\\phi}= & {r}^{-1}\\\\[-.5ex]\n", |
122 | | - "\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}= & {\\left(\\tan{\\theta}\\right)}^{-1}\\\\[-.5ex]\n", |
123 | | - "\\square{}_{\\theta}{}_{r}{}^{\\theta}= & {r}^{-1}\\\\[-.5ex]\n", |
124 | | - "\\square{}_{r}{}_{r}{}^{r}= & M {\\left(r \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
125 | | - "\\square{}_{t}{}_{r}{}^{t}= & M {\\left(r \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
126 | | - "\\square{}_{r}{}_{\\phi}{}^{\\phi}= & {r}^{-1}\\\\[-.5ex]\n", |
127 | | - "\\square{}_{\\theta}{}_{\\phi}{}^{\\phi}= & {\\left(\\tan{\\theta}\\right)}^{-1}\\\\[-.5ex]\n", |
128 | | - "\\square{}_{r}{}_{\\theta}{}^{\\theta}= & {r}^{-1}\\\\[-.5ex]\n", |
129 | | - "\\square{}_{r}{}_{t}{}^{t}= & M {\\left(r \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
130 | | - "\\square{}_{\\phi}{}_{\\phi}{}^{r}= & \\left(2M-r\\right) {\\left(\\sin{\\theta}\\right)}^{2}\\\\[-.5ex]\n", |
131 | | - "\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}= & - \\frac{1}{2}\\sin\\left(2\\theta\\right)\\\\[-.5ex]\n", |
132 | | - "\\square{}_{\\theta}{}_{\\theta}{}^{r}= & 2M-r\\\\[-.5ex]\n", |
133 | | - "\\square{}_{t}{}_{t}{}^{r}= & M \\left(-2M+r\\right) {r}^{-3}\\\\[-.5ex]\n", |
| 88 | + "${}\\Gamma^{\\mu}\\,_{\\nu \\rho} = \\square{}_{\\nu}{}_{\\rho}{}^{\\mu}\\left\\{\\begin{aligned}\\square{}_{\\phi}{}_{r}{}^{\\phi}& = {r}^{-1}\\\\[-.5ex]\n", |
| 89 | + "\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}& = {\\left(\\tan{\\theta}\\right)}^{-1}\\\\[-.5ex]\n", |
| 90 | + "\\square{}_{\\theta}{}_{r}{}^{\\theta}& = {r}^{-1}\\\\[-.5ex]\n", |
| 91 | + "\\square{}_{r}{}_{r}{}^{r}& = M {\\left(r \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 92 | + "\\square{}_{t}{}_{r}{}^{t}& = M {\\left(r \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 93 | + "\\square{}_{r}{}_{\\phi}{}^{\\phi}& = {r}^{-1}\\\\[-.5ex]\n", |
| 94 | + "\\square{}_{\\theta}{}_{\\phi}{}^{\\phi}& = {\\left(\\tan{\\theta}\\right)}^{-1}\\\\[-.5ex]\n", |
| 95 | + "\\square{}_{r}{}_{\\theta}{}^{\\theta}& = {r}^{-1}\\\\[-.5ex]\n", |
| 96 | + "\\square{}_{r}{}_{t}{}^{t}& = M {\\left(r \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 97 | + "\\square{}_{\\phi}{}_{\\phi}{}^{r}& = \\left(2M-r\\right) {\\left(\\sin{\\theta}\\right)}^{2}\\\\[-.5ex]\n", |
| 98 | + "\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}& = - \\frac{1}{2}\\sin\\left(2\\theta\\right)\\\\[-.5ex]\n", |
| 99 | + "\\square{}_{\\theta}{}_{\\theta}{}^{r}& = 2M-r\\\\[-.5ex]\n", |
| 100 | + "\\square{}_{t}{}_{t}{}^{r}& = M \\left(-2M+r\\right) {r}^{-3}\\\\[-.5ex]\n", |
134 | 101 | "\\end{aligned}\\right.\n", |
135 | 102 | "$" |
136 | 103 | ] |
137 | 104 | }, |
138 | | - "execution_count": 3, |
139 | 105 | "metadata": {}, |
140 | | - "output_type": "execute_result" |
| 106 | + "output_type": "display_data" |
141 | 107 | } |
142 | 108 | ], |
143 | 109 | "source": [ |
|
171 | 137 | "${}R^{\\rho}\\,_{\\sigma \\mu \\nu} = \\partial_{\\mu}{\\Gamma^{\\rho}\\,_{\\nu \\sigma}}-\\partial_{\\nu}{\\Gamma^{\\rho}\\,_{\\mu \\sigma}}+\\Gamma^{\\rho}\\,_{\\mu \\lambda} \\Gamma^{\\lambda}\\,_{\\nu \\sigma}-\\Gamma^{\\rho}\\,_{\\nu \\lambda} \\Gamma^{\\lambda}\\,_{\\mu \\sigma}$" |
172 | 138 | ] |
173 | 139 | }, |
174 | | - "execution_count": 4, |
175 | 140 | "metadata": {}, |
176 | | - "output_type": "execute_result" |
| 141 | + "output_type": "display_data" |
177 | 142 | } |
178 | 143 | ], |
179 | 144 | "source": [ |
|
191 | 156 | { |
192 | 157 | "data": { |
193 | 158 | "text/markdown": [ |
194 | | - "${}R^{\\rho}\\,_{\\sigma \\mu \\nu} = \\square{}_{\\nu}{}_{\\sigma}{}^{\\rho}{}_{\\mu}\\left\\{\\begin{aligned}\\square{}_{t}{}_{t}{}^{r}{}_{r}= & 2M \\left(2M-r\\right) {r}^{-4}\\\\[-.5ex]\n", |
195 | | - "\\square{}_{\\theta}{}_{\\theta}{}^{r}{}_{r}= & -M {r}^{-1}\\\\[-.5ex]\n", |
196 | | - "\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}{}_{\\theta}= & 2M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
197 | | - "\\square{}_{\\phi}{}_{\\phi}{}^{r}{}_{r}= & -M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
198 | | - "\\square{}_{t}{}_{r}{}^{t}{}_{r}= & 2M {\\left({r}^{2} \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
199 | | - "\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}{}_{\\theta}= & -2M {r}^{-1}\\\\[-.5ex]\n", |
200 | | - "\\square{}_{r}{}_{t}{}^{r}{}_{t}= & 2M \\left(-2M+r\\right) {r}^{-4}\\\\[-.5ex]\n", |
201 | | - "\\square{}_{r}{}_{\\theta}{}^{r}{}_{\\theta}= & M {r}^{-1}\\\\[-.5ex]\n", |
202 | | - "\\square{}_{\\theta}{}_{\\phi}{}^{\\theta}{}_{\\phi}= & -2M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
203 | | - "\\square{}_{r}{}_{\\phi}{}^{r}{}_{\\phi}= & M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
204 | | - "\\square{}_{r}{}_{r}{}^{t}{}_{t}= & 2M {\\left({r}^{2} \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
205 | | - "\\square{}_{r}{}_{r}{}^{\\theta}{}_{\\theta}= & M {\\left({r}^{2} \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
206 | | - "\\square{}_{\\theta}{}_{\\theta}{}^{\\phi}{}_{\\phi}= & 2M {r}^{-1}\\\\[-.5ex]\n", |
207 | | - "\\square{}_{r}{}_{r}{}^{\\phi}{}_{\\phi}= & M {\\left({r}^{2} \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
208 | | - "\\square{}_{t}{}_{t}{}^{\\phi}{}_{\\phi}= & M \\left(-2M+r\\right) {r}^{-4}\\\\[-.5ex]\n", |
209 | | - "\\square{}_{t}{}_{t}{}^{\\theta}{}_{\\theta}= & M \\left(-2M+r\\right) {r}^{-4}\\\\[-.5ex]\n", |
210 | | - "\\square{}_{\\phi}{}_{\\phi}{}^{t}{}_{t}= & -M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
211 | | - "\\square{}_{\\theta}{}_{\\theta}{}^{t}{}_{t}= & -M {r}^{-1}\\\\[-.5ex]\n", |
212 | | - "\\square{}_{\\phi}{}_{r}{}^{\\phi}{}_{r}= & M {\\left({r}^{2} \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
213 | | - "\\square{}_{\\phi}{}_{t}{}^{\\phi}{}_{t}= & M \\left(2M-r\\right) {r}^{-4}\\\\[-.5ex]\n", |
214 | | - "\\square{}_{\\theta}{}_{r}{}^{\\theta}{}_{r}= & M {\\left({r}^{2} \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
215 | | - "\\square{}_{\\theta}{}_{t}{}^{\\theta}{}_{t}= & M \\left(2M-r\\right) {r}^{-4}\\\\[-.5ex]\n", |
216 | | - "\\square{}_{t}{}_{\\phi}{}^{t}{}_{\\phi}= & M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
217 | | - "\\square{}_{t}{}_{\\theta}{}^{t}{}_{\\theta}= & M {r}^{-1}\\\\[-.5ex]\n", |
| 159 | + "${}R^{\\rho}\\,_{\\sigma \\mu \\nu} = \\square{}_{\\nu}{}_{\\sigma}{}^{\\rho}{}_{\\mu}\\left\\{\\begin{aligned}\\square{}_{t}{}_{t}{}^{r}{}_{r}& = 2M \\left(2M-r\\right) {r}^{-4}\\\\[-.5ex]\n", |
| 160 | + "\\square{}_{\\theta}{}_{\\theta}{}^{r}{}_{r}& = -M {r}^{-1}\\\\[-.5ex]\n", |
| 161 | + "\\square{}_{\\phi}{}_{\\phi}{}^{\\theta}{}_{\\theta}& = 2M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
| 162 | + "\\square{}_{\\phi}{}_{\\phi}{}^{r}{}_{r}& = -M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
| 163 | + "\\square{}_{t}{}_{r}{}^{t}{}_{r}& = 2M {\\left({r}^{2} \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 164 | + "\\square{}_{\\phi}{}_{\\theta}{}^{\\phi}{}_{\\theta}& = -2M {r}^{-1}\\\\[-.5ex]\n", |
| 165 | + "\\square{}_{r}{}_{t}{}^{r}{}_{t}& = 2M \\left(-2M+r\\right) {r}^{-4}\\\\[-.5ex]\n", |
| 166 | + "\\square{}_{r}{}_{\\theta}{}^{r}{}_{\\theta}& = M {r}^{-1}\\\\[-.5ex]\n", |
| 167 | + "\\square{}_{\\theta}{}_{\\phi}{}^{\\theta}{}_{\\phi}& = -2M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
| 168 | + "\\square{}_{r}{}_{\\phi}{}^{r}{}_{\\phi}& = M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
| 169 | + "\\square{}_{r}{}_{r}{}^{t}{}_{t}& = 2M {\\left({r}^{2} \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 170 | + "\\square{}_{r}{}_{r}{}^{\\theta}{}_{\\theta}& = M {\\left({r}^{2} \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 171 | + "\\square{}_{\\theta}{}_{\\theta}{}^{\\phi}{}_{\\phi}& = 2M {r}^{-1}\\\\[-.5ex]\n", |
| 172 | + "\\square{}_{r}{}_{r}{}^{\\phi}{}_{\\phi}& = M {\\left({r}^{2} \\left(2M-r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 173 | + "\\square{}_{t}{}_{t}{}^{\\phi}{}_{\\phi}& = M \\left(-2M+r\\right) {r}^{-4}\\\\[-.5ex]\n", |
| 174 | + "\\square{}_{t}{}_{t}{}^{\\theta}{}_{\\theta}& = M \\left(-2M+r\\right) {r}^{-4}\\\\[-.5ex]\n", |
| 175 | + "\\square{}_{\\phi}{}_{\\phi}{}^{t}{}_{t}& = -M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
| 176 | + "\\square{}_{\\theta}{}_{\\theta}{}^{t}{}_{t}& = -M {r}^{-1}\\\\[-.5ex]\n", |
| 177 | + "\\square{}_{\\phi}{}_{r}{}^{\\phi}{}_{r}& = M {\\left({r}^{2} \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 178 | + "\\square{}_{\\phi}{}_{t}{}^{\\phi}{}_{t}& = M \\left(2M-r\\right) {r}^{-4}\\\\[-.5ex]\n", |
| 179 | + "\\square{}_{\\theta}{}_{r}{}^{\\theta}{}_{r}& = M {\\left({r}^{2} \\left(-2M+r\\right)\\right)}^{-1}\\\\[-.5ex]\n", |
| 180 | + "\\square{}_{\\theta}{}_{t}{}^{\\theta}{}_{t}& = M \\left(2M-r\\right) {r}^{-4}\\\\[-.5ex]\n", |
| 181 | + "\\square{}_{t}{}_{\\phi}{}^{t}{}_{\\phi}& = M {\\left(\\sin{\\theta}\\right)}^{2} {r}^{-1}\\\\[-.5ex]\n", |
| 182 | + "\\square{}_{t}{}_{\\theta}{}^{t}{}_{\\theta}& = M {r}^{-1}\\\\[-.5ex]\n", |
218 | 183 | "\\end{aligned}\\right.\n", |
219 | 184 | "$" |
220 | 185 | ] |
221 | 186 | }, |
222 | | - "execution_count": 5, |
223 | 187 | "metadata": {}, |
224 | | - "output_type": "execute_result" |
| 188 | + "output_type": "display_data" |
225 | 189 | } |
226 | 190 | ], |
227 | 191 | "source": [ |
|
248 | 212 | "${}R_{\\sigma \\nu} = R^{\\rho}\\,_{\\sigma \\rho \\nu}$" |
249 | 213 | ] |
250 | 214 | }, |
251 | | - "execution_count": 6, |
252 | 215 | "metadata": {}, |
253 | | - "output_type": "execute_result" |
| 216 | + "output_type": "display_data" |
254 | 217 | }, |
255 | 218 | { |
256 | 219 | "data": { |
257 | 220 | "text/markdown": [ |
258 | 221 | "${}R_{\\sigma \\nu} = 0$" |
259 | 222 | ] |
260 | 223 | }, |
261 | | - "execution_count": 6, |
262 | 224 | "metadata": {}, |
263 | | - "output_type": "execute_result" |
| 225 | + "output_type": "display_data" |
264 | 226 | } |
265 | 227 | ], |
266 | 228 | "source": [ |
|
287 | 249 | "${}K = R^{\\mu}\\,_{\\nu \\rho \\sigma} R^{\\lambda}\\,_{\\kappa \\gamma \\chi} g_{\\mu \\lambda} g^{\\nu \\kappa} g^{\\rho \\gamma} g^{\\sigma \\chi}$" |
288 | 250 | ] |
289 | 251 | }, |
290 | | - "execution_count": 7, |
291 | 252 | "metadata": {}, |
292 | | - "output_type": "execute_result" |
| 253 | + "output_type": "display_data" |
293 | 254 | } |
294 | 255 | ], |
295 | 256 | "source": [ |
|
308 | 269 | "${}K = 48{M}^{2} {r}^{-6}$" |
309 | 270 | ] |
310 | 271 | }, |
311 | | - "execution_count": 8, |
312 | 272 | "metadata": {}, |
313 | | - "output_type": "execute_result" |
| 273 | + "output_type": "display_data" |
314 | 274 | } |
315 | 275 | ], |
316 | 276 | "source": [ |
|
335 | 295 | ], |
336 | 296 | "metadata": { |
337 | 297 | "kernelspec": { |
338 | | - "display_name": "Cadabra", |
| 298 | + "display_name": "Cadabra2", |
339 | 299 | "language": "python", |
340 | | - "name": "cadabra" |
| 300 | + "name": "cadabra2" |
341 | 301 | }, |
342 | 302 | "language_info": { |
343 | | - "file_extension": ".cdb", |
344 | | - "mimetype": "text/cadabra", |
345 | | - "name": "cadabra", |
346 | | - "version": "2.0.0" |
| 303 | + "codemirror_mode": "python", |
| 304 | + "file_extension": ".ipynb", |
| 305 | + "mimetype": "text/python", |
| 306 | + "name": "cadabra2", |
| 307 | + "pygments_lexer": "python" |
347 | 308 | } |
348 | 309 | }, |
349 | 310 | "nbformat": 4, |
|
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