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
:M
B 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[f1→n,f2→k,...,fi→p] | fi→p - equations where fi - form identifier and p is a variable or an integer - the form's degree. |
| Coframe[id1→ expr1,id2→ expr2,...idn→ exprn] | id - identifier for indexed variable - the coframe 1-forms
n - dimension of working manifold (a variable or integer)
idi→ expri - 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:
Total space
Declare total space of the submersion:
| Out[49]= |  |
Declare constants
rg and
a:
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For Kerr metric we use well known aliases
=r2-rg r+ a2,
2=r2+a2-a2(sin(
))2:
| Out[55]= |  |
Now we declare Kerr metric:
| Out[56]= |  |
| Out[57]= |  |
Base space
| Out[58]= |  |
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Mapping[f,m,n,y1→f1(x1,x2...xm),y2→f2(x1,x2...xm),...,
yn→fn(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:
Out[64]//TableForm= |
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Now we can calculate the projectors of the mapping:
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Thus we are on the base manifold. Jumping on the total manifold:
| Out[69]= |  |
Verify that
is vertical vector:
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Let us calculate invariants of the submersion:
The submersion is a riemannian one:
| Out[73]= |  |
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:
| Out[75]= |  |
The mean curvature vectors are horizontal:
| Out[76]= |  |
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The mean curvature vectors are projectable. Realy
where
f=ln(vol(F)):
| Out[78]= |  |
| Out[79]= |  |
| Out[80]= |  |
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:
| Out[81]= |  |
Then for principal connection we obtain:
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Now we can calculate the corresponding curvature of the fibration
=d():
| Out[83]= |  |
For any horizontal vector fields
X and
Y we have
. Thus we can construct the corresponding tensor directly:
| Out[84]= |  |
We can obtain the same tensor from integrability obstruction iO:
| Out[85]= |  |
| Out[86]= |  |
- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
