Schwarzschild metric in General Relativity
In this worksheet the Schwarzschild metric is used to generate the components of different tensors used in general relativity. The following expressions are calculated automatically by Maple, whereas for convenience only the non zero components are shown:The covariant metric tensorIts determinantBoth Christoffel symbols of first and second kindRiemann tensorRicci tensor and Ricci scalarEinstein tensorWeyl tensorLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTA==QyQtSSV3aXRoRzYiNiNJJ3RlbnNvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJSEiIg==QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJNURpZmZlcmVudGlhbEdlb21ldHJ5R0YnISIiJSFHPrepare the metricHere we define the coordinate system.QyQ+SSdjb29yZHNHNiI3JkkidEdGJUkickdGJUkmdGhldGFHRiVJJHBoaUdGJSEiIg==Tell how the metric looks like.QyQ+SSlnX2NvbXB0c0c2Ii1JJmFycmF5RyUqcHJvdGVjdGVkRzYmSSpzeW1tZXRyaWNHRiVJJ3NwYXJzZUdGJTsiIiIiIiVGLCEiIg==Fill the diagonal components of the Schwarzschild metric.QyQ+JkkpZ19jb21wdHNHNiI2JCIiIkYoLUkkZXhwR0YmNiMtSSNudUdGJjYjSSJyR0YmISIiQyQ+JkkpZ19jb21wdHNHNiI2JCIiI0YoLCQtSSRleHBHRiY2Iy1JJ2xhbWJkYUdGJjYjSSJyR0YmISIiRjE=QyQ+JkkpZ19jb21wdHNHNiI2JCIiJEYoLCQqJEkickdGJiIiIyEiIkYtQyQ+JkkpZ19jb21wdHNHNiI2JCIiJUYoLCQqJkkickdGJiIiIy1JJHNpbkdGJjYjSSZ0aGV0YUdGJkYsISIiRjE=JSFHJSFHFinally create the metric tensor.PkkiZ0c2Ii1JJ2NyZWF0ZUdGJDYkNyQhIiJGKS1JJWV2YWxHJSpwcm90ZWN0ZWRHNiNJKWdfY29tcHRzR0Ykprint(); # input placeholderCalculate all tensorsNow let Maple calculate everything. Of course it is also possible to do every step separately and change accordingly.LUkqdGVuc29yc0dSRzYiNi1JJ2Nvb3Jkc0dGJEkiZ0dGJEkuY29udHJhX21ldHJpY0dGJEkoZGV0X21ldEdGJEkjQzFHRiRJI0MyR0YkSSNSbUdGJEkjUmNHRiRJIlJHRiRJIkdHRiRJIkNHRiQ=LUkuZGlzcGxheV9hbGxHUkc2IjYtSSdjb29yZHNHRiRJImdHRiRJLmNvbnRyYV9tZXRyaWNHRiRJKGRldF9tZXRHRiRJI0MxR0YkSSNDMkdGJEkjUm1HRiRJI1JjR0YkSSJSR0YkSSJHR0YkSSJDR0Ykprint(); # input placeholderJSFHJSFH