Simple S1 - fibration (Kerr black hole)

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.

What we do?

    • In this notebook we construct a riemannian submersion :MB where B=M/U(1) is the corresponding base and S1 is the fiber. The submersion is S1 fibration: where M4 is an open submanifold of M (M4 is just the union of U(1) principal orbits).
    Constants[c1,c2,...,ci,...,cn] c1, c2, ..., ci, ..., cn - constants identifiers.
    Vectors[v1,v2,...,vi,...,vn]vi - vector identivier.
    Forms[f1n,f2k,...,fip]fip - equations where fi - form identifier and p is a variable or an integer - the form's degree.
    Coframe[id1expr1,id2expr2,...idnexprn]id - identifier for indexed variable - the coframe 1-forms n - dimension of working manifold (a variable or integer) idiexpri - equation where idi is indexed variable - coframe 1-form and expri is decomposition of the 1-form on exact 1-forms.
    Connection[id]id - variable - connection identifier.

    Necessary functions.

Kerr black hole

First of all we load atlas package:
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Total space

Declare total space of the submersion:
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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, 2=r2+a2-a2(sin())2:
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Now we declare Kerr metric:
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Connection calculation:
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Base space

Declare base space:
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Declare forms:
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Declare vectors:
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Declare coframe:
Declare frame:
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Mapping[f,m,n,y1f1(x1,x2...xm),y2f2(x1,x2...xm),...,
ynfn(x1,x2...xm)]
f - variable - the mapping identifier i.e. f : m n, m - variable - first domain identifier, n - variable - second domain identifier
Who[l]l -an identifier, list or set of identifies.
Invariants[f]f - mapping identifier

Necessary functions.

The submersion

Declare mapping :
Let us see the attributes of the mapping:
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Now we can calculate the projectors of the mapping:
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Where are we:
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Thus we are on the base manifold. Jumping on the total manifold:
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Verify that is vertical vector:
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Let us calculate invariants of the submersion:
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The submersion is a riemannian one:
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The integrability obstruction is not equal to zero. Thus the corresponding horizontal distribution is not an intagrable one.
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Extraction of the field of mean curvature vectors of the fibers:
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The mean curvature vectors are horizontal:
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The mean curvature vectors are projectable. Realy where f=ln(vol(F)):
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Thus mean curvature K is basic vector field (horizontal and projectable). The principal group U(1)=S1 induces vector field Let us consider principal connection of the fibration. It is easy to see that As soon as:
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Then for principal connection we obtain:
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Now we can calculate the corresponding curvature of the fibration =d():
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For any horizontal vector fields X and Y we have . Thus we can construct the corresponding tensor directly:
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We can obtain the same tensor from integrability obstruction iO:
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Verify the identity:
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