Curvature 2-forms and tetrad method
In differential geometry, and especially in general relativity, the Riemann curvature tensor is an important object for the geometric property of a manifold, solving the Enstein's equation one always needs to calculate the curvature tensor. But the component expression of Riemann tensor is too complicated, making it painful to calculate it by hand, and less powerful when dealing with various problems. Instead, using curvature 2-form would somehow be better.
Among textbooks and lectures, there're two different yet equivalent definitions of Riemann curvature tensor:
From which we can write their component expressions:
Since the second is more common, we will use it in the rest of this article.
Instead of work in the coordinate tetrad, as one usually did in general relativity, we choose an arbitary tetrad , in which case the connection is , satisfies
Where is the dual basis. Define connection 1-form by
which can be regarded as an matrix with 1-form entries. If the connection is torsion-free, it then satisfies
This is the Cartan's first structural equation.
The last step of above isn't true when the connection has a non-vanishing torsion, and will contribute an additional torsion term. We can derive it in the following way:
where we have used the definition of torsion tensor
So can be written as
is the torsion 2-from.
Written in matrix form, the above result is simply
where and are column vectors whose entries are and .
It might be suprising at this point, since the exterier derivative has no dependency on the connection. In fact, the above result doesn't fully determine the connection 1-form, it's just a constraint on it, more equations are required if we wish to calculate the connection.
Now consider the Levi-Civita connection. If is rigid tetrad, i.e., the metric's components are all constant on this tetrad, then satisfies
This property, together with the previous one, completely determines the connection 1-form. So if we choose the tetrad so that is diagonal, then we have . In the Riemannian case (not pseudo-Riemannian), for instance, this means is antisymmetric.
As an exmple, let us calculate the connection 1-form of :
We first choose an othonormal basis
So we have
since is antisymmetric in this case, let
substitute into above, we have
So that .
Curvature 2-form is defined by
which can also be regarded as an matrix. This relation is the Cartan's second structural equation. Direct calculation shows its relation with the Riemann curvature tensor:
where is the components of the Riemann curvature tensor in the tetrad . It is easy to verify that is antisymmetric in rigid tetrad:
Matrix form of the curvature 2-form is simply
This expression is very similiar to the field strength of Yang-Mills gauge theory
In fact, the gauge field is also a kind of connection, it's the connection on the principal bundle, and can also be regarded as curvature.
The Ricci tensor can also be calculated from the curvature 2-form. Since
(we add a bar to distinguish it with the scalar curvature). In matrix form
here should be a row vector, means also contracts with when doing matrix multiplication.
Again, as an example, we calculate the curvature 2-form and Ricci tensor of .
Notice that .
When calculating curvature tensors in spherical coordinate, sometimes we actually don't need to compute all the components of the Riemann curvature tensor. For example, when solving Einstein's equation in 4 dimensional spacetime, we see that the Ricci tensor components and gives the same equation. This indicates the spherical part of the coordinate can be treated as a whole. The tetrad method above gives a (maybe) useful trick to these problems.
We'll explain this method by an example. Consider -dimensional spherical symmetric static spacetime, metric in spherical coordinate has the form
is the spherical part. In higher dimensions only the form of will be different. We may choose a rigid tetrad on the sphere, so that the spherical part of the metric can be written as
where is a column vector with 1-form entries , they can be regarded as the dual basis of the sphere, and is the metric of unit sphere in this tetrad. In the case above we have
and is a row vector whose components are given by
Let be the dual of , i.e. , we can write the othonormal basis as
The Cantan's first structural equation reads
which solves to
that is, it's the connection of . A similiar relation holds for
The two terms of are
where , .
Summing these terms gives the curvature 3-form. But we still have an undetermined matrix . To find an expression for , consider the metric of Euclidean space in spherical coordinate
A similiar calculation gives the curvature 2-form
which should vanish since it's euclidean space, so we get
Now the curvature 2-form reads
To calculate the Ricci tensor, note that
From which we can calculate the Ricci tensor
The vacuum Einstein's equation then reads
which solves to
where and are two constants. When , we get the well-known Schwartchild solution.