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Tensor[TraceFreeRicciTensor] - calculate the trace-free Ricci tensor of a metric tensor
Calling Sequences
TraceFreeRicciTensor(g)
TraceFreeRicciTensor(g, C)
TraceFreeRicciTensor(g, R)
Parameters
g - the metric tensor on the tangent bundle of a manifold
C - the curvature tensor of the metric g
R - the Ricci tensor of the metric g
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Description
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This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form TraceFreeRicciTensor(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-TraceFreeRicciTensor.
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Examples
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Example 1.
In this example we calculate the trace-free Ricci tensor for a metric.
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![DGsetup([x, y, z], M)](/support/helpjp/helpview.aspx?si=5630/file05931/math85.png)
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| (2.1) |
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], y], [[2, 2], z], [[3, 3], 1]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math95.png)
| (2.2) |
Calculate the trace-free Ricci tensor for the metric directly.
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], -(1/12)*(2*y^2-z)/(z^2*y)], [[2, 2], (1/12)*(y^2+z)/(z*y^2)], [[2, 3], (1/4)/(y*z)], [[3, 2], (1/4)/(y*z)], [[3, 3], (1/12)*(y^2-2*z)/(z^2*y^2)]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math106.png)
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We check that  is trace-free by computing the inverse metric and using the ContractIndices command.
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![_DG([["tensor", M, [["con_bas", "con_bas"], []]], [[[1, 1], 1/y], [[2, 2], 1/z], [[3, 3], 1]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math128.png)
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![ContractIndices(h, P, [[1, 1], [2, 2]])](/support/helpjp/helpview.aspx?si=5630/file05931/math132.png)
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| (2.5) |
The same calculation can be done with the TensorInnerProduct command.
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Example 2.
The third calling sequence can be applied to any rank 2 symmetric tensor to construct a trace-free, rank 2 symmetric tensor.
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 2], 1], [[2, 1], 1]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math176.png)
| (2.7) |
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], y], [[2, 2], z], [[3, 3], 1]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math183.png)
| (2.8) |
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], 1]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math190.png)
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The tensor ![A[1]](/support/helpjp/helpview.aspx?si=5630/file05931/math201.png) is already trace-free, so its trace-free part is itself.
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 2], 1], [[2, 1], 1]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math209.png)
| (2.10) |
The trace-free part of the metric itself is always 0.
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], 0]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math226.png)
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The trace-free part of ![A[3]](/support/helpjp/helpview.aspx?si=5630/file05931/math237.png) is
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![_DG([["tensor", M, [["cov_bas", "cov_bas"], []]], [[[1, 1], 2/3], [[2, 2], -(1/3)*z/y], [[3, 3], -(1/3)/y]]])](/support/helpjp/helpview.aspx?si=5630/file05931/math245.png)
| (2.12) |
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| (2.13) |
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See Also
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DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], CovariantDerivative, Physics[D_], CurvatureTensor, Physics[Riemann],
SectionalCurvature, RicciScalar, Physics[Ricci], RicciSpinor, NPCurvatureScalars
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