Gravitational collapse of the dust sphere

What we do?

Energy-momentum tensor for spherically symmetric matter is , here is energy density function and pressure (just dust). Einstein equations are
Energy-momentum tensor for spherically symmetric matter is , here is energy density function and pressure p = 0 (just dust). Einstein equations are r - = 8kT, where r - Ricci tensor, s - Ricci scalar, k - gravitational constant (here light velocity c = 1).
Calculate Einstein tensor and corresponding equations for dust sphere metric. Verify that corresponding space-time has axial symmetry and it is not stationary.

Solution:

Connection[id] id - variable - connection identifier.
Coframe[id1expr1,id2expr2,...idnexprn]id - identifier for indexed variable - the coframe 1-forms, n - dimension of working manifold (a variable or integer), idi
expri - rules where idi is indexed variable - coframe 1-form and expri is decomposition of the 1-form on exact 1-forms.

Necessary functions.

Load the Atlas package:
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Dust sphere metric:

Constants:
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Functions:
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Vector fields:
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Differential p-forms:
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Coframe 1-forms:
Frame vector fields:
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Metric tensor field:
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Connection 1-forms:
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Curvature 2-forms:
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Curvature tensor field:
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Ricci tensor field calculation:
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Energy-momentum tensor for dust sphere:
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"eqs" tensor components:
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Verify that are Killing vector field but is not:
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Using covariant derivative:
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