Covariant Derivative

Calculates the (symbolic) covariant derivative \(\nabla_\rho a_{\mu_{1} \mu_{2} ...}^{\nu_{1}\nu_{2}...}\) with respect to the Levi Civita connection of any (symbolic) tensor field. The result is a new tensor of one rank higher than the original tensor rank.

Usage

covd(x, i, g, act_on = NULL)

Arguments

x

A labeled tensor object, created by %_% or tensor(). covd() only handles symbolic derivatives, i.e. the tensor components are required to be character()-valued and consist of mathematical expressions in terms of coordinates identical to the coordinates used by g.

i

An index slot label specification created with .(). The number of indices specify the number of covariant derivatives taken in the same order. Each covariant derivative adds one index each with the specified names. After the covariant derivatives are calculated, implicit contraction rules are applied (in case of reoccurring index label names).

g

A covariant metric tensor, a "metric_field" object. See metric_field() to create a new metric tensor, or use predefined metrics, e.g. g_eucl_cart().

act_on

An optional index slot label specification created with .() that specifies on which indices the covariant derivative should act on. This might be useful if not all tensor factors are elements of the tangent space of the underlying manifold. If not provided the covariant derivative acts on all indices. If no indices are selected explicitly (with .()), the covariant derivative acts like it would on a scalar.

Value

The covariant derivative: a new labeled array with one or more additional indices (depending on i).

Details

Note that symbolic derivatives are not always completely trustworthy. They usually ignore subtle issues like undefined expressions at certain points. The example \(\nabla_a \nabla^a r^{-1}\) from below is telling: The symbolic derivative evaluates to zero identically, although strictly speaking the derivative is not defined at \(r = 0\).

See also

Wikipedia: Covariant Derivative

Examples

options(ricci.auto_simplify = TRUE)

# gradient of "sin(sqrt(x1^2+x2^2+x3^2))" in 3-dimensional euclidean space
covd("sin(x1)", .(k), g = g_eucl_cart(3))
#> <Labeled Array> [3] .(-k)
#> [1] "cos(x1)" "0"       "0"      

# laplace operator
covd("sin(x1)", .(-k, +k), g = g_eucl_cart(3))
#> <Scalar>
#> [1] "-sin(x1)"
covd("1/r", .(-k, +k), g = g_eucl_sph(3))
#> <Scalar>
#> [1] "0"