Kerr black hole

What we do?

Kerr black hole is a 4-dimentional Lorentz manifold M with zero Ricci curvature and group U(1)=S1 as a subgroup of the manifold isometry group.
For Kerr metric calculate the following:
  • connetion 1-forms
  • curvature 2-forms
  • Riemannin tensor field
    • Ricci tensor field
    Verify that are Killing vector vields.

Solution:

Domain[manifold]manifold - string - a manifold name or a name of a manifold domain.
Metric[id→expr]id - variable - metric identifier, expr - expression - metric declaration.
Connection[id]id - variable - connection identifier.

Necessary functions.

Load the Atlas package:
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Just for right simplification:
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Kerr black hole

Declare domain M - black hole space:
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Declare constants rg and a:
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Declare vectors:
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Declare forms:
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Declare coframe:
Declare frame vectors:
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For Kerr metric we use well known aliases =r2-rg r+a2, =r2+a2-a2sin()2:
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Now we declare Kerr metric:
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Connection calculation:
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Let us see a 1-form:
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Curvature calculation:
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Let as see a 2-form:
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Riemannian tensor calculation:
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Ricci tensor calculation:
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E1 is Killing vector field:
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E4 is Killing vector field:
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Killing vector fields:
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