Mathematics of General Relativity - Tensorial Derivatives - The Covariant Derivative

The Covariant Derivative

Let be a point, a vector located at, and a vector field. The idea of differentiating at along the direction of in a physically meaningful way can be made sense of by choosing an affine connection and a parameterized smooth curve such that and . The formula

for a covariant derivative of along associated with connection turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative.

It can be expressed using connection coefficients:

The expression in brackets, called a covariant derivative of (with respect to the connection) and denoted by, is more often used in calculations:

A covariant derivative of X can thus be viewed as a differential operator acting on a vector field sending it to a type (1, 1) tensor ('increasing the covariant index by 1') and can be generalised to act on type (r, s) tensor fields sending them to type (r, s + 1) tensor fields. Notions of parallel transport can then be defined similarly as for the case of vector fields. By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field.

In the literature, there are three common methods of denoting covariant differentiation:

Many standard properties of regular partial derivatives also apply to covariant derivatives:

\begin{align} \nabla_a (X^b + Y^b) &= \nabla_a X^b + \nabla_a Y^b \\ \nabla_a (X^b Y^c) &= Y^c (\nabla_a X^b) + X^b (\nabla_a Y^c) \\ \nabla_a (f(x) X^b) &= f \nabla_a X^b + X^b \nabla_a f = f \nabla_a X^b + X^b {\partial f \over \partial x^a} \\ \nabla_a (c X^b) &= c \nabla_a X^b, \quad c \text{ is constant} \end{align}

In General Relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. By definition, Levi-Civita connection preserves the metric under parallel transport, therefore, the covariant derivative gives zero when acting on a metric tensor (as well as its inverse). It means that we can take the (inverse) metric tensor in and out of the derivative and use it to raise and lower indices:

Read more about this topic:  Mathematics Of General Relativity, Tensorial Derivatives

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