General Relativity using Computer AlgebraProblem: for the spacetime metric, 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) Compute the trace of 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where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEmJlBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEoJmVxdWl2O0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJRistSShtZmVuY2VkR0YkNiQtRiw2JVEickYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjItRiw2I1EhRidGMg== is some function of the radial coordinate, LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9JK21zZW1hbnRpY3NHRiRRLlBoeXNpY3NUZW5zb3JGJy1JJW1yb3dHRiQ2Ji1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYlUScmYmV0YTtGJ0Y+RkAvJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GLDYjUSFGJ0ZALUY5NiZGSy1GQzYkRjtGSEZFRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlQvRjZRNVBoeXNpY3NUZW5zb3JJbmRpY2VzRic= is the Ricci tensor, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEnJkRzY3I7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYvNiVRKCZhbHBoYTtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0Y9 is the covariant derivative operator and LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9JK21zZW1hbnRpY3NHRiRRLlBoeXNpY3NUZW5zb3JGJy1JJW1yb3dHRiQ2Ji1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYlUScmYmV0YTtGJ0Y+RkAvJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GLDYjUSFGJ0ZALUY5NiZGSy1GQzYkRjtGSEZFRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlQvRjZRNVBoeXNpY3NUZW5zb3JJbmRpY2VzRic= is the stress-energy tensorLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9JK21zZW1hbnRpY3NHRiRRLlBoeXNpY3NUZW5zb3JGJy1GIzYpLUYvNiVRKCZhbHBoYTtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRictSSNtb0dGJDYtUSgmY29tbWE7RidGQi8lJmZlbmNlR0ZBLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRJyZiZXRhO0YnRkBGQi8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lKXJlYWRvbmx5R0Y0LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGQi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUZFNi1RKSZlcXVhbHM7RidGQkZIL0ZLRkFGTEZORlBGUkZUL0ZXUSwwLjI3Nzc3NzhlbUYnL0ZaRmlvLUkobWZlbmNlZEdGJDYoLUknbXRhYmxlR0YkNjgtSSRtdHJHRiQ2KS1JJG10ZEdGJDYoLUYjNistSSNtbkdGJDYkUSI4RidGQi1GRTYtUTEmSW52aXNpYmxlVGltZXM7RidGQkZIRmdvRkxGTkZQRlJGVEZWL0ZaRlgtSSVtc3VwR0YkNiUtRkU2LVEvJkV4cG9uZW50aWFsRTtGJ0ZCRkhGZ29GTEZORlBGUkZURlYvRlpRLDAuMTExMTExMWVtRictRiM2KS1GLzYlUSkmbGFtYmRhO0YnRkBGQi1GRTYtUTAmQXBwbHlGdW5jdGlvbjtGJ0ZCRkhGZ29GTEZORlBGUkZURlZGYHEtRlxwNiQtRi82JVEickYnRjJGNUZCRmluRlxvRl5vRkIvJTFzdXBlcnNjcmlwdHNoaWZ0R0Zjb0ZdcS1GLzYlUSUmcGk7RidGQEZCRmluRlxvRl5vRkIvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRl1zLyUrZ3JvdXBhbGlnbkdGXXMvJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGZHMtRmVwNigtRmpwNiRGY29GQkZbc0Zec0Zgc0Zic0Zlc0Znc0Znc0Zbc0Zec0Zgcy1GYnA2KUZncy1GZXA2KC1GIzYrRmlwRl1xLUZicTYlRmNyLUZqcDYkUSIyRidGQkZmckZdcUZockZpbkZcb0Zeb0ZCRltzRl5zRmBzRmJzRmVzRmdzRmdzRltzRl5zRmBzLUZicDYpRmdzRmdzLUZlcDYoLUYjNi1GaXBGXXFGYXRGXXEtRmJxNiUtRiM2KS1GLzYlUSRzaW5GJ0ZARkJGXnItRlxwNiQtRi82JVEoJnRoZXRhO0YnRkBGQkZCRmluRlxvRl5vRkJGY3RGZnJGXXFGaHJGaW5GXG9GXm9GQkZbc0Zec0Zgc0Zic0Zlc0Znc0Zbc0Zec0Zgcy1GYnA2KUZnc0Znc0Zncy1GZXA2KC1GIzYtRmlwRl1xLUZicTYlRmRxLUYjNiktRi82JVElJm51O0YnRkBGQkZeckZhckZpbkZcb0Zeb0ZCRmZyRl1xRmhyRl1xLUYvNiVRLSZ2YXJlcHNpbG9uO0YnRkBGQkZpbkZcb0Zeb0ZCRltzRl5zRmBzRmJzRmVzRltzRl5zRmBzLyUmYWxpZ25HUSVheGlzRicvRlxzUSliYXNlbGluZUYnL0Zfc1EnY2VudGVyRicvRmFzUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGNC8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRmV3LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmB4LyUmZnJhbWVHRmB4LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGQS8lLWVxdWFsY29sdW1uc0dGQS8lLWRpc3BsYXlzdHlsZUdGQS8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZdeEZCL0Y5USdNYXRyaXhGJy8lJW9wZW5HUScmbHNxYjtGJy8lJmNsb3NlR1EnJnJzcWI7RidGY3lGaW5GXG9GXm9GQg==b) Compute the components of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRKCZhbHBoYTtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRictSSltcGhhbnRvbUdGJDYkLUYvNiVRJyZiZXRhO0YnRj1GPy8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUYvNiNRIUYnRj8tRiM2JkZKLUZCNiRGOkZHRkRGPy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGUy9JK21zZW1hbnRpY3NHRiRRNVBoeXNpY3NUZW5zb3JJbmRpY2VzRidGPw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYsNi1RKCZlcXVpdjtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGSkYvthe traceless part of LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiWkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYlUScmYmV0YTtGJ0Y7Rj0vJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GLDYjUSFGJ0Y9LUY2NiZGSC1GQDYkRjhGRUZCRj0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlEvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn of item a)c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYlUScmYmV0YTtGJ0Y7Rj0vJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GLDYjUSFGJ0Y9LUY2NiZGSC1GQDYkRjhGRUZCRj0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlEvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn obtained in b)Background: The equations of items a) and b) appear in a paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by Plamen Fiziev, a Maple user. These equations model a problem in the context of a Branse-Dicke theory with vanishing parameter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIi5GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYy The Brans\342\200\223Dicke theory is in many respects similar to Einstein's theory, but the gravitational "constant" is not actually presumed to be constant - it can vary from place to place and with time - and the gravitational interaction is mediated by a scalar field. Both Brans\342\200\223Dicke's and Einstein's theory of general relativity are generally held to be in agreement with observation. The computations below aim at illustrating how this type of computation can be performed using computer algebra, and so they focus only on the algebraic aspects, not the physical interpretation of the results.
<Text-field style="Heading 1" layout="Heading 1">a) The trace of <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>The computations that follow make use of the latest version of the Physics package, available for download on the Maple Physics: Research & Development webpage.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Set the 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Introduce the line element and with it set the 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On the way, keep the display of functions compact and derivatives displayed indexedLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpUERFdG9vbHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6LUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYsNiVRKGRlY2xhcmVGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSRkczJGJ0YvRjJGOQ==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 we got the correct covariant metric componentsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZ19GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYjUSFGJy9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=Define now the stress-energy tensor indicated with the problem. Among the different ways to introduce it (see Physics:-Define), we choose entering it as a matrix for the covariant 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnRGVmaW5lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2I1EhRictSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1FlcC8mSSJURzYiNiRJJmFscGhhRzYiSSViZXRhRzYiLUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHNiIiNXkpNCgpPXlTdVklPUYnLyUrYWN0aW9udHlwZUdRPG1hcGxlc29mdC5jb206bGFiZWwoTDE3ODg5KUYnLyUldmlld0dRJmxhYmVsRidGOi9GM1Enbm9ybWFsRic=After this definition, for example, the component 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 is given 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 the trace of T can be computed viaLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRJmFscGhhRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y6Rj8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVE3U3VtT3ZlclJlcGVhdGVkSW5kaWNlc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiNRIUYnLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRQSZJIlRHNiI2JEkmYWxwaGFHNiJJJ3xncmlyYWxwaGFHNiJGJy8lK2FjdGlvbnR5cGVHUTxtYXBsZXNvZnQuY29tOmxhYmVsKEwxNzg4NylGJy8lJXZpZXdHUSZsYWJlbEYnRjovRjNRJ25vcm1hbEYnHaving defined the metric and the Stress energy tensor, we can solve item a) in one go by defining a single tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiWkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRJSZtdTtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRictSSNtb0dGJDYtUSgmY29tbWE7RidGPy8lJmZlbmNlR0Y+LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRj4vJSpzeW1tZXRyaWNHRj4vJShsYXJnZW9wR0Y+LyUubW92YWJsZWxpbWl0c0dGPi8lJ2FjY2VudEdGPi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYvNiVRJSZudTtGJ0Y9Rj9GPy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj8= as the tensorial equation 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEpUERFdG9vbHNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6LUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZMLUYsNiVRKGRlY2xhcmVGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSRQaGlGJy9GMEY9RjktRlM2JC1GIzYjLUYsNiVRInJGJ0YvRjJGOUY5For illustration purposes we use the inert %D_ command, this also permits to see what we are entering before any computation is 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 covariant derivatives 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 can be activated when desired, for example: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(Note the occurrence of the Christofel symbols in the result above - more about them in the NOTES at the end of the section.)We can now define 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 with the inert LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EnJkRzY3I7RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EuWzEyOCwxMjgsMTI4XUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9JK21zZW1hbnRpY3NHRiRRKlBoeXNpY3NEX0Yn or with the evaluated form . To compare both approaches, we define both 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 and 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The answer to a), that is the trace of 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 is thenLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiWkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRI211RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y6Rj8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUTdTdW1PdmVyUmVwZWF0ZWRJbmRpY2VzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiVGKy1JKG1hY3Rpb25HRiQ2JS1JKm12ZXJiYXRpbUdGJDYjUTsmSSJaRzYiNiRJI211RzYiSSR8Z3Jpcm11RzYiRicvJSthY3Rpb250eXBlR1E8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMTc4NDUpRicvJSV2aWV3R1EmbGFiZWxGJ0YrL0Y4USdub3JtYWxGJ0YrFor a closer look at the representation capabilities using inert functions, compare this result with the equivalent one obtained with the differentiation operations inertLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSMlWkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRI211RicvRjVRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRkQvJSZmZW5jZUdGQy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y/RkQvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGKw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVE3U3VtT3ZlclJlcGVhdGVkSW5kaWNlc0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiNRIUYnLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRPCZJIyVaRzYiNiRJI211RzYiSSR8Z3Jpcm11RzYiRicvJSthY3Rpb250eXBlR1E8bWFwbGVzb2Z0LmNvbTpsYWJlbChMMTc4NDgpRicvJSV2aWV3R1EmbGFiZWxGJ0Y6L0YzUSdub3JtYWxGJw==Activate now the inert covariant 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and contravariant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2J1ErJlBhcnRpYWxEO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRLlsxMjgsMTI4LDEyOF1GJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvSSttc2VtYW50aWNzR0YkUSpQaHlzaWNzZF9GJy1GIzYoLUkpbXBoYW50b21HRiQ2JC1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJy9GOVEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRksvJSlzdHJldGNoeUdGSy8lKnN5bW1ldHJpY0dGSy8lKGxhcmdlb3BHRksvJS5tb3ZhYmxlbGltaXRzR0ZLLyUnYWNjZW50R0ZLLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGWi8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUZBNiQtRi82JVElJm11O0YnL0YzRktGR0Znbi9GNlEqWzAsMCwyNTVdRicvJSlyZWFkb25seUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRkctRiM2KEZARlxvRmBvRmJvRmRvRkcvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRltwL0Y8UTVQaHlzaWNzVGVuc29ySW5kaWNlc0YnRmBvRmJvRmRvRkc= derivatives and compare with 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 of the design of the Physics package is that as soon as you define the spacetime metric all the general relativity tensors like the Ricci tensor LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9JK21zZW1hbnRpY3NHRiRRLlBoeXNpY3NUZW5zb3JGJy1JJW1yb3dHRiQ2Ji1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYlUScmYmV0YTtGJ0Y+RkAvJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GLDYjUSFGJ0ZALUY5NiZGSy1GQzYkRjtGSEZFRkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlQvRjZRNVBoeXNpY3NUZW5zb3JJbmRpY2VzRic=, or the Christoffel symbols LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEoJkdhbW1hO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvSSttc2VtYW50aWNzR0YkUTNQaHlzaWNzQ2hyaXN0b2ZmZWxGJy1JJW1yb3dHRiQ2Ky1JKW1waGFudG9tR0YkNiQtSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlIvJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GPDYkLUYsNiVRKCZhbHBoYTtGJ0YvRjJGVS1GLDYlUSUmbXU7RidGL0YyLUY/Ni1RKCZjb21tYTtGJ0YyRkIvRkVRJXRydWVGJ0ZGRkhGSkZMRk5GUC9GVFEsMC4zMzMzMzMzZW1GJy1GLDYlUSUmbnU7RidGL0YyLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUpcmVhZG9ubHlHRl5vLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGMi1GOTYrRjtGWi1GPDYkRmduRlUtRjw2JEZqbkZVLUY8NiRGYW9GVUZkb0Znb0Zpb0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZmcC9GNlE1UGh5c2ljc1RlbnNvckluZGljZXNGJw== and the covariant derivative operator LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEnJkRzY3I7RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRKCZhbHBoYTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== get automatically defined on background. For example, you do not need to additionally compute the Ricci tensor - instead, for instance the value of 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 can be compute directly, 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All the covariant components at once:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmUmljY2lGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiYtRiM2Iy1GLDYjUSFGJy9GM1Enbm9ybWFsRicvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=The same happens with the Riemann, Einsten and Weyl tensors, as well as with the Christoffel symbols of the first and second kind, where the "kind" just depends on whether the indices are covariant or contravariant: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 these tensors, or the ones you define using tensorial equations, admit the keywords nonzero and matrix to display the corresponding components, as well as the shortcut notation where the tensor is called with no indices returning all its covariant components: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Also, when computing with any of these tensors (or the ones you define) without specifying values for the indices, the symmetries under permutation of the indices are automatically taken into account; for example for the Christoffel symbols, permuting the last two indices does not change their value, 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
<Text-field style="Heading 1" layout="Heading 1">b) The components of <Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRKCZhbHBoYTtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRictSSltcGhhbnRvbUdGJDYkLUYvNiVRJyZiZXRhO0YnRj1GPy8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUYvNiNRIUYnRj8tRiM2JkZKLUZCNiRGOkZHRkRGPy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGUy9JK21zZW1hbnRpY3NHRiRRNVBoeXNpY3NUZW5zb3JJbmRpY2VzRidGPw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRKCZhbHBoYTtGJy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRictSSltcGhhbnRvbUdGJDYkLUYvNiVRJyZiZXRhO0YnRj1GPy8lLGNvbnN0cmFpbnRzR1ElYm90aEYnLUYvNiNRIUYnRj8tRiM2JkZKLUZCNiRGOkZHRkRGPy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGUy9JK21zZW1hbnRpY3NHRiRRNVBoeXNpY3NUZW5zb3JJbmRpY2VzRidGPw==</Equation><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYsNi1RKCZlcXVpdjtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGSkYv">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUYsNi1RKCZlcXVpdjtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGSkYv</Equation>the traceless part of <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field>This problem can be solved in one step, by defining a single tensor 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 with the the traceless part of 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 . We use the standard definition of 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For example, for 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Or the matrix components for the tracless mixed 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Despite that W[mu,nu] is tracless by construction (see ), one can verify that explicitlyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2Ji1GIzYlLUYsNiVRJmFscGhhRicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj8vJSZmZW5jZUdGPi8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJ0Y6Rj8vJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic=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
<Text-field style="Heading 1" layout="Heading 1">c) An exact solution for the nonlinear system of differential equations conformed by the components of <Equation executable="false" style="2D Math" input-equation="" display="LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYlUScmYmV0YTtGJ0Y7Rj0vJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GLDYjUSFGJ0Y9LUY2NiZGSC1GQDYkRjhGRUZCRj0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlEvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn">LUkobXN1YnN1cEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiV0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1GLDYlUSgmYWxwaGE7RicvRjBRJmZhbHNlRicvRjNRJ25vcm1hbEYnLUkpbXBoYW50b21HRiQ2JC1GLDYlUScmYmV0YTtGJ0Y7Rj0vJSxjb25zdHJhaW50c0dRJWJvdGhGJy1GLDYjUSFGJ0Y9LUY2NiZGSC1GQDYkRjhGRUZCRj0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLyUvc3Vic2NyaXB0c2hpZnRHRlEvSSttc2VtYW50aWNzR0YkUTVQaHlzaWNzVGVuc29ySW5kaWNlc0Yn</Equation></Text-field>Create an ODE system with the nonzero components of 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 are three 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 an exact solution for the first two cases is difficult to obain, Case[3] can be tackled and solved to the end in few 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The first equation is just a constraint between the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSkmbGFtYmRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1JKG1mZW5jZWRHRiQ2JC1GLDYlUSJyRicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RitGNw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSYmUGhpO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1JKG1mZW5jZWRHRiQ2JC1GLDYlUSJyRicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RitGNw==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The subsistem of differential equations can now be solved in closed form in terms of three arbitrary constants LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KC1GLDYlUSRfQzFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSgmY29tbWE7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJF9DMkYnRjRGN0Y6LUYsNiVRJF9DM0YnRjRGN0Y+RitGPg==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Introducing this value of Phi(r) into the constraint we specialize one of these constants, we choose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEkX0MxRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0YzUSdub3JtYWxGJw==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The exact solution to the system constructed at the beginning,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1ErYG9kZV9fc3lzYEYnis 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Verifying this 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=TTdSMApJQVJUQUJMRV9TQVZFLzE4NDQ2NzQ0MDc4MTI3MDgzODIyWCwlKWFueXRoaW5nRzYjJSpzeW1tZXRyaWNHNiJbZ2whIiYhISEjKyIlIiUsJC1JJGV4cEc2JCUqcHJvdGVjdGVkRyUoX3N5c2xpYkc2Iy1fSSxUeXBlc2V0dGluZ0c2JEYsRi0lKm1jb21wbGV0ZUc2JC1fRjElI21pRzYjUSkmbGFtYmRhOzYiLV9GMSUlbXJvd0c2JUY1LV9GMSUjbW9HNiNRMCZBcHBseUZ1bmN0aW9uO0Y6LV9GMSUobWZlbmNlZEc2Iy1GNjYjUSJyRjohIiIiIiEsJCokJSJyRyIiI0ZLRkxGTCwkKiZGT0ZQLUkkc2luRzYkRixGLTYjJSZ0aGV0YUdGUEZLRkxGTEZMLUYqNiMtRjA2JC1GNjYjUSUmbnU7RjotRjw2JUZmbkY/RkRGOg==