{ricci} has limited support for tensor fields.
Non-constant tensor fields in {ricci} are modeled by
array()s whose elements are character strings, each string
containing a mathematical expression (which are interpreted as functions
of manifold coordinates).
This approach implies that only subset of all tensor fields can be modeled:
Only mathematical expressions that are understood by R are allowed.
If derivatives are involved only expressions where R’s
deriv() knows the derivative of are allowed.
Not all tensor fields can be expressed in closed-form expressions to begin with.
Nevertheless, the subset of tensor fields that {ricci} does currently support can still be nontrivial and we still consider this feature useful to this extent.
The main usage of tensor fields in {ricci} is to calculate covariant
derivatives, i.e. to make use of the function covd().
Similar to the product operator * which unifies products by
making use of the index structure, covd() unifies various
differential operators, like e.g. the “gradient”, “divergence”, “curl”,
the “Hessian” and the “Laplacian”, in any dimension on any (pseudo-)
Riemannian manifold and in any coordinate system.
covd() requires two main ingredients: a tensor field it
can act upon, and a metric tensor field. This tensor field and the
metric tensor are thought to exist on the same manifold and they are
required to be specified in the same coordinate system.
{ricci} already provides a couple of standard metric tensors, the standard Euclidean metric tensor \(g\) in Cartesian coordinates \((x_1,x_2,x_3)\) is simply the identity matrix, while in spherical coordinates \((r, \phi_1, \phi_2)\) the metric tensor appears more complicated:
library(ricci)
# enable optional simplfying procedures
# (takes a toll on performance)
options(ricci.auto_simplify = TRUE)
g_eucl_cart(3)
#> <Covariant metric tensor field> (x1, x2, x3)
#> [,1] [,2] [,3]
#> [1,] 1 0 0
#> [2,] 0 1 0
#> [3,] 0 0 1
g_eucl_sph(3)
#> <Covariant metric tensor field> (r, ph1, ph2)
#> [,1] [,2] [,3]
#> [1,] "1" "0" "0"
#> [2,] "0" "r^2*1" "0"
#> [3,] "0" "0" "r^2*sin(ph1)^2"Equipped with such metrics, we can for example pick a simple tensor field, namely a scalar function \(f(r, \phi_1, \phi_2) = r^{-1}\), and calculate our first gradient:
The argument .(k) is nothing special but simply defines
a name for the new rank of the array.
The Hessian could be computed in similar fashion:
"1/r" |> covd(.(k, l), g = g_eucl_sph(3))
#> <Labeled Array> [3x3] .(-k, -l)
#> [,1] [,2] [,3]
#> [1,] "2/r^3" "0" "0"
#> [2,] "0" "(-1)/r" "0"
#> [3,] "0" "0" "(-sin(ph1)^2)/r"The operation above can be set into the context of electrodynamics where \(f\) is an electrostatic potential, and its gradient is the electric field. For the same electrostatic potential we can very simply calculate the electromagnetic tensor too after forming the electromagnetic potential \(A_\mu\):
# electromagnetic potential
A <- c("1/r", "0", "0", "0")
A %_% .(m) |>
covd(.(n), g = g_mink_sph(4)) |>
asym(m, n)
#> <Labeled Array> [4x4] .(-m, -n)
#> [,1] [,2] [,3] [,4]
#> [1,] "0" "(-1)/(2*r^2)" "0" "0"
#> [2,] "1/(2*r^2)" "0" "0" "0"
#> [3,] "0" "0" "0" "0"
#> [4,] "0" "0" "0" "0"We can also chain multiple covariant derivatives in any way we’d like. A well known second-order differential operator, the Laplacian \(\Delta = \nabla_k \nabla^k\) can easily be written down:
# on scalar field
"1/r" |> covd(.(k, +k), g = g_mink_sph(4))
#> <Scalar>
#> [1] "0"
# on a vector field
A %_% .(i) |> covd(.(k, +k), g = g_mink_sph(4))
#> <Labeled Array> [4] .(-i)
#> [1] "0" "0" "0" "0"One property of the metric tensor is that the (Levi Civita) covariant derivative of the metric tensor vanishes. We can test this easily:
g <- g_eucl_sph(3)
g %_% .(i, j) |>
covd(.(k), g = g) |>
as_a(i, j, k)
#> , , 1
#>
#> [,1] [,2] [,3]
#> [1,] "0" "0" "0"
#> [2,] "0" "0" "0"
#> [3,] "0" "0" "0"
#>
#> , , 2
#>
#> [,1] [,2] [,3]
#> [1,] "0" "0" "0"
#> [2,] "0" "0" "0"
#> [3,] "0" "0" "0"
#>
#> , , 3
#>
#> [,1] [,2] [,3]
#> [1,] "0" "0" "0"
#> [2,] "0" "0" "0"
#> [3,] "0" "0" "0"