Christoffel symbol

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Christoffel symbols are also known as connection coefficients and sometimes the Levi-Civita connection (see below). Historically the symbols come in two forms, known as the second kind (more commonly used) and first kind (less commonly used). Christoffel symbols of the second kind are used to define the affine connection and the covariant derivative.

Given coordinate basis vectors \vec{e}\,, define the Christoffel symbols of the second kind \Gamma^\gamma_{\alpha\beta}\, through the affine connection:

\nabla_{\vec{e}_\beta} ( \vec{e}_\alpha) = \Gamma^\gamma_{\alpha\beta} \vec{e}_\gamma\,

(Note that the differentiation index comes last on \Gamma^\gamma_{\alpha\beta}\,.)

The Christoffel symbols of the first kind are defined by

\Gamma_{\alpha\beta\,\,\gamma} = g_{\gamma\delta} \Gamma^\delta_{\alpha\beta}\,.

It is important to note that these symbols are not tensors as they do not transform as tensors do.

Contents

Compatibility with the metric

Although the Christoffel symbols that define the covariant derivative may be chosen with some degree of freedom, we can choose them in a way that is compatible with the metric g\,. Consider parallel transporting two vectors w^b\, and u^c\, along a curve with tangent vector v^a\,. We first require that v^a \nabla_a w^b = 0\, and v^a \nabla_a u^c = 0\,. Since the dot product of u\, and w\,, g_{ab}u^b w^c\, should remain unaltered by the parallel transport, we then require v^a\nabla_a(g_{bc}u^b w^c) = 0\,. Then:

v^a\nabla_a(g_{bc}u^b w^c) = v^a\nabla_a(g_{bc}) u^b w^c + v^a g_{bc}\nabla_a(u^b) w^c +  v^a g_{bc}u^b\nabla_a( w^c)= 0\,

Since v^a\, was arbitrarily chosen, it must be true that \nabla_a g_{bc} = 0\,. Using this equation, along with the no-torsion condition, \Gamma^\gamma_{\alpha\beta}\, can be determined in terms of g_{\alpha\beta}\,:

Coordinate basis

Let us assume that the Lie bracket \left[\vec{e}_\alpha, \vec{e}_\beta \right] = 0\, vanishes, i.e., that \vec{e}_\alpha\, are a coordinate basis.

First, we expand the components of \nabla_a g_{bc}\,:

\nabla_\alpha g_{\beta\gamma} = \partial_\alpha g_{\beta\gamma} - \Gamma^\kappa_{\beta\alpha}g_{\kappa\gamma} - \Gamma^\kappa_{\gamma\alpha} g_{\beta\kappa} = 0\,

We can then permute the indices \alpha\,, \beta\,, and \gamma\,, and then adding/subtracting gives:

\partial_\alpha g_{\beta\gamma} - \Gamma^\kappa_{\beta\alpha}g_{\kappa\gamma} - \Gamma^\kappa_{\gamma\alpha} g_{\beta\kappa}+\,
\partial_\beta g_{\gamma\alpha} - \Gamma^\kappa_{\gamma\beta}g_{\kappa\alpha} - \Gamma^\kappa_{\alpha\beta} g_{\gamma\kappa} - \,
\partial_\gamma g_{\alpha\beta} + \Gamma^\kappa_{\alpha\gamma}g_{\kappa\beta} - \Gamma^\kappa_{\beta\gamma} g_{\alpha\kappa} = 0\,.

Now, since the metric is symmetric, g_{\alpha\beta} = g_{\beta\alpha}\,, and since we have imposed the torsion free condition that \Gamma^\gamma_{\alpha\beta} = \Gamma^\gamma_{\beta\alpha}\,, we have:

\partial_\alpha g_{\beta\gamma} - \Gamma^\kappa_{\alpha\beta} g_{\kappa\gamma} + \partial_\beta g_{\gamma\alpha} - \Gamma^\kappa_{\beta\alpha} g_{\gamma\kappa} - \partial_\gamma g_{\alpha\beta} = 0\,,

or

2\Gamma^\kappa_{\alpha\beta}g_{\kappa\gamma} = \partial_\alpha g_{\beta\gamma} + \partial_\beta g_{\gamma\alpha} - \partial_\gamma g_{\alpha\beta}\,,

so that the metric compatible connection is

\Gamma^\gamma_{\alpha\beta} = \frac{1}{2}g^{\delta\gamma}\left(\partial_\alpha g_{\delta\beta} + \partial_\beta g_{\delta\alpha} - \partial_\delta g_{\alpha\beta}\right)\,.

Non-coordinate basis

In a non-coordinate basis, such as an orthonormal basis, \left[\vec{e}_\alpha, \vec{e}_\beta \right] \neq 0\,, and the relevant covariant derivative is

\nabla_\alpha g_{\beta\gamma} = \vec{e}_\alpha (g_{\beta\gamma}) - \Gamma^\kappa_{\beta\alpha}g_{\kappa\gamma} - \Gamma^\kappa_{\gamma\alpha} g_{\beta\kappa} = 0\,.

We can again permute the indices \alpha\,, \beta\,, and \gamma\,, and then adding/subtracting gives:

\vec{e}_\alpha ( g_{\beta\gamma}) - \Gamma^\kappa_{\beta\alpha}g_{\kappa\gamma} - \Gamma^\kappa_{\gamma\alpha} g_{\beta\kappa}+\,
\vec{e}_\beta ( g_{\gamma\alpha}) - \Gamma^\kappa_{\gamma\beta}g_{\kappa\alpha} - \Gamma^\kappa_{\alpha\beta} g_{\gamma\kappa} - \,
\vec{e}_\gamma( g_{\alpha\beta}) + \Gamma^\kappa_{\alpha\gamma}g_{\kappa\beta} + \Gamma^\kappa_{\beta\gamma} g_{\alpha\kappa} = 0\,.

The metric is of course symmetric, but now \Gamma^\gamma_{\alpha\beta} \neq \Gamma^\gamma_{\beta\alpha}\,. Instead, the torsion is

\nabla_{\vec{e}_\alpha} \vec{e}_\beta  - \nabla_{\vec{e}_\beta} \vec{e}_\alpha - \left[ \vec{e}_\alpha, \vec{e}_\beta \right] = \left( \Gamma^\gamma_{\beta\alpha}  - \Gamma^\gamma_{\alpha\beta}\right) \vec{e}_\gamma - \left[ \vec{e}_\alpha, \vec{e}_\beta \right] \,,

so that zero torsion implies that

\left( \Gamma^\gamma_{\beta\alpha} - \Gamma^\gamma_{\alpha\beta}\right) = \tilde{e}^\gamma \left( \left[ \vec{e}_\alpha, \vec{e}_\beta \right] \right) = c^\gamma_{\alpha\beta}\,,

where c^\gamma_{\alpha\beta}\, are the commutation coefficients. Then

\vec{e}_\alpha ( g_{\beta\gamma}) - \Gamma^\kappa_{\beta\alpha}g_{\kappa\gamma} - \Gamma^\kappa_{\alpha\gamma} g_{\beta\kappa} - c^\kappa_{\alpha\gamma} g_{\beta\kappa} +\,
\vec{e}_\beta ( g_{\gamma\alpha}) - \Gamma^\kappa_{\gamma\beta}g_{\kappa\alpha} - \Gamma^\kappa_{\beta\alpha} g_{\gamma\kappa} - c^\kappa_{\beta\alpha} g_{\gamma\kappa}  - \,
\vec{e}_\gamma( g_{\alpha\beta}) + \Gamma^\kappa_{\alpha\gamma}g_{\kappa\beta} + \Gamma^\kappa_{\gamma\beta} g_{\alpha\kappa} + c^\kappa_{\gamma\beta} g_{\alpha\kappa}= 0\,,

or \Gamma^\kappa_{\beta\alpha}  = \frac{1}{2}g^{\kappa\gamma}\left( \vec{e}_\alpha ( g_{\beta\gamma})  + \vec{e}_\beta ( g_{\gamma\alpha}) - \vec{e}_\gamma( g_{\alpha\beta})  - c_{\alpha\gamma\,\,\beta} - c_{\beta\alpha\,\,\gamma} + c_{\gamma\beta\,\,\alpha}\right)\,, or more symmetrically,

\Gamma^\kappa_{\alpha\beta}  = \frac{1}{2}g^{\kappa\gamma}\left( \vec{e}_\alpha ( g_{\gamma\beta})  + \vec{e}_\beta ( g_{\gamma\alpha}) - \vec{e}_\gamma( g_{\alpha\beta})  +  c_{\gamma\beta\,\,\alpha}+c_{\gamma\alpha\,\,\beta} - c_{\alpha\beta\,\,\gamma}\right)\,,

where c_{\alpha\beta\,\,\gamma} \equiv g_{\gamma\lambda} c^\lambda_{\alpha\beta}\,. In particular, in an orthogonal basis, only the commutation coefficients contribute.

Transformation under diffeomorphism

Under a change of coordinates from (x^1,...,x^n)\, to (y^1,...,y^n)\,, we find that

{\Gamma^\prime}^k {}_{ij}  =
\frac{\partial x^p}{\partial y^i}\,
\frac{\partial x^q}{\partial y^j}\,
\Gamma^r {}_{pq}\,
\frac{\partial y^k}{\partial x^r}
+ 
\frac{\partial y^k}{\partial x^m}\, 
\frac{\partial^2 x^m}{\partial y^i \partial y^j}  
\,

showing that {\Gamma^\gamma}_{\alpha\beta}\, does not transform as a tensor.

As an aside, note that the difference between two connections does transform as a tensor. In other words, suppose we consider a one-parameter family of connections {\Gamma^\gamma}_{\alpha\beta}(s)\, parametrized by s\,. Then

\frac{d{\Gamma^\gamma}_{\alpha\beta} } { d s }\,

transforms as a tensor. This can also be seen from Lifshitz' formula.

Lifshitz' formula

The variation of {\Gamma^\gamma}_{\alpha\beta}\, due to a variation in the metric h_{\alpha\beta} = \delta g_{\alpha\beta}\, can be found as follows[1]:

\delta \Gamma^\gamma_{\alpha\beta}\, = \delta( g^{\gamma\delta } g_{\delta \sigma} \Gamma^\sigma_{\alpha\beta} )\,,
= g^{\gamma\delta } \delta(  g_{\delta \sigma} \Gamma^\sigma_{\alpha\beta} ) - \Gamma^\sigma_{\alpha\beta} {h^\gamma}_\sigma \,,
= \frac{1}{2} g^{\gamma\delta }  \left(\partial_\alpha h_{\delta \beta} + \partial_\beta h_{\delta \alpha} - \partial_\delta h_{\alpha\beta}\right) - \Gamma^\sigma_{\alpha\beta} {h^\gamma}_\sigma \,,

which gives Lifshitz' formula:

\delta \Gamma^\gamma_{\alpha\beta} = \frac{1}{2}g^{\gamma\delta}\left(\nabla_\beta h_{\alpha\delta} + \nabla_\alpha h_{\beta\delta} - \nabla_\delta h_{\alpha\beta}\right)\,.

This also implies that

\delta \Gamma^\gamma_{\alpha\gamma}  = \frac{1}{2} \nabla_\alpha ( {h^\gamma}_\gamma ) = \frac{1}{2} \partial_\alpha ( {h^\gamma}_\gamma ) \,.
back to covariant derivative
on to divergence

References

  1. E. Lifshitz (1946). "On the Gravitational Stability of the Expanding Universe". J. Phys. (USSR) 10: 116. 
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