Package 'ricci'

Index slot label specification

Description

This function creates a index slot label specification. Any R symbol can serve as a label. .() is typically used in conjunction with %_%.

Usage

.(...)

Arguments

...	Index labels separated by commas optionally prefixed by "+" and "-" to indicate the index position (upper and lower respectively). If no prefix is provided, a lower index ("-") is assumed. This argument uses non-standard evaluation: any R symbol that is not a reserved keyword can be used.

Value

A named list of two character vectors representing the index label names and index position.

Examples

# three lower index slots
.(i, j, k)

# one lower and upper index
.(i, +j)

---

Strip array labels

Description

Converts a tensor() to an array() by stripping the index labels. An index label order needs to be provided so that the array's dim() order is well defined.

Usage

as_a(x, ...)

Arguments

x

A labeled array ("tensor" object) created by %_% or tensor().

...

Index labels separated by commas optionally prefixed by "+" and "-" to indicate the index position (upper and lower respectively). If no prefix is provided, a lower index ("-") is assumed. This argument uses non-standard evaluation: any R symbol that is not a reserved keyword can be used. The specification needs to match all the labels occurring in x. The label order determines the dimension ordering of the resulting array.

Value

A usual array() without attached labels. The dimension order is determined by ....

See Also

The same functionality is implemented in as.array.tensor() but with standard evaluation.

Examples

array(1:8, dim = c(2, 2, 2)) %_% .(i, +i, k) |> as_a(k)

---

Strip array labels

Description

Converts a tensor() to an array() by stripping the index labels. An index label order needs to be provided so that the array's dim() order is well defined.

Usage

## S3 method for class 'tensor'
as.array(
  x,
  index_order = NULL,
  ...,
  arg = "index_order",
  call = rlang::caller_env()
)

Arguments

x	A labeled array ("tensor" object) created by %_% or tensor().
index_order	An index specification created with .(). The specification needs to match all the labels occurring in x. The label order determines the dimension order of the resulting array.
...	Not used.
arg, call	Used for error handling. Can be ignored by the user.

Value

The tensor components as usual array() object without any index labels attached.

Examples

array(1:8, dim = c(2, 2, 2)) %_% .(i, +i, k) |> as.array(.(k))

---

Antisymmetric tensor part

Description

Takes the antisymmetric tensor part of a tensor x with respect to the specified indices .... An error is thrown if the specified indices do not exist.

Usage

asym(x, ...)

Arguments

x	A labeled tensor object, created by %_% or tensor().
...	Any number of index expressions. The indices need to occur in x.

Value

A modified tensor object.

See Also

Wikipedia: Ricci calculus - Symmetric and antisymmetric parts

Other tensor operations: Ops.tensor(), kron(), l(), r(), subst(), sym()

Examples

a <- array(1:4, dim = c(2, 2))
a %_% .(i, j) |> asym(i, j)

---

Evaluate a symbolic array

Description

Evaluates a symbolic array at a particular point in parameter space. Partial evaluation is not allowed, all variables/symbols need to be accounted for. The result is a numeric array.

Usage

at(x, vars)

## S3 method for class 'array'
at(x, vars)

## S3 method for class 'tensor'
at(x, vars)

Arguments

x	A symbolic array() or a tensor().
vars	A named vector with parameter-value assignments. Each named entry represents a substitution of a symbol with the given value.

Value

A numeric array() or tensor().

Examples

g_sph(3) |> at(c(ph1 = 0, ph2 = 0))

---

Christoffel symbols

Description

Provides the Christoffel symbols of the first kind $\Gamma_{ijk}$ with respect to the Levi Civita connection for a given metric tensor.

Usage

christoffel(g)

Arguments

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().

Details

The Christoffel symbols are a rank 3 array of numbers.

Value

Returns the Christoffel symbols of the first kind $\Gamma_{ijk}$ as rank 3 array().

See Also

Wikipedia: Christoffel symbols

Other geometric tensors: ricci(), ricci_sc(), riemann()

Examples

christoffel(g_eucl_sph(3))

---

Covariant Derivative

Description

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.

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 $r = 0$.

Value

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

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))

# laplace operator
covd("sin(x1)", .(-k, +k), g = g_eucl_cart(3))
covd("1/r", .(-k, +k), g = g_eucl_sph(3))

---

Kronecker delta

Description

Provides a labeled generalized Kronecker delta. In the special case of two labels this represents simply the identity matrix. The Kronecker delta always has an even number of indices. Note that the first half of the tensor labels need to be lowered, while the second half needs upper indices. Otherwise an error is thrown.

Usage

d(n)

Arguments

n	The dimension.

Value

A function that expects index labels (see .()) and returns a labeled tensor. The underlying data will differs depending on the number of labels provided.

See Also

Underlying implementation: calculus::delta()

Wikipedia: Generalized Kronecker delta

Other tensor symbols: e()

Examples

d(3)(i, +j)

d(3)(i, j, +k, +l)

---

Levi-Civita epsilon

Description

Provides a labeled Levi-Civita epsilon (pseudo) tensor. The indices are required to be lowered. Otherwise an error is thrown.

Usage

e(...)

Arguments

...	Index labels separated by commas optionally prefixed by "+" and "-" to indicate the index position (upper and lower respectively). If no prefix is provided, a lower index ("-") is assumed. This argument uses non-standard evaluation: any R symbol that is not a reserved keyword can be used.

Value

A labeled tensor object. The underlying data will differs depending on the number of labels provided.

See Also

Underlying implementation: calculus::epsilon()

Wikipedia: Levi-Civita symbol

Other tensor symbols: d()

Examples

e(i, j)

e(i, j, k)

---

Does code return the expected value?

Description

Adds an expectation function that can be used with the testthat package. Compares two tensors and determines whether they are equal or not.

Usage

expect_tensor_equal(object, expected, ...)

Arguments

object, expected

Computation and value to compare it to. Both arguments supports limited unquoting to make it easier to generate readable failures within a function or for loop. See quasi_label for more details.

...

Arguments passed on to waldo::compare tolerance If non-NULL, used as threshold for ignoring small floating point difference when comparing numeric vectors. Using any non-NULL value will cause integer and double vectors to be compared based on their values, not their types, and will ignore the difference between NaN and NA_real_. It uses the same algorithm as all.equal(), i.e., first we generate x_diff and y_diff by subsetting x and y to look only locations with differences. Then we check that mean(abs(x_diff - y_diff)) / mean(abs(y_diff)) (or just mean(abs(x_diff - y_diff)) if y_diff is small) is less than tolerance. max_diffs Control the maximum number of differences shown. The default shows 10 differences when run interactively and all differences when run in CI. Set max_diffs = Inf to see all differences. ignore_srcref Ignore differences in function srcrefs? TRUE by default since the srcref does not change the behaviour of a function, only its printed representation. ignore_encoding Ignore string encoding? TRUE by default, because this is R's default behaviour. Use FALSE when specifically concerned with the encoding, not just the value of the string.

Value

The actual value invisibly.

---

Euclidean metric tensor

Description

Provides the Euclidean metric tensor of $\mathbb{E}^n$. g_eucl_cart() returns a numeric (constant) tensor in Cartesian coordinates,

$ds^2=\sum_{i=1}^n dx_i^2$

while g_eucl_sph() returns a symbolic tensor field in generalized spherical coordinates ${r, \phi_1, \phi_2, ..., \phi_{n-1}}$.

$ds^2=dr^2 + r^2 d\Omega^2$

Usage

g_eucl_cart(n, coords = paste0("x", 1:n))

g_eucl_sph(n, coords = c("r", paste0("ph", 1:(n - 1))))

Arguments

n	The dimension of the metric tensor.
coords	A character vector of coordinate names. The length needs to match the tensor dimensions.

Details

As usual, spherical coordinates are degenerate at $r = 0$ and $\phi_l = 0$, so be careful around those points.

Value

The covariant metric tensor as array imputed with coordinate names.

See Also

Wikipedia: Euclidean metric tensor

Other metric tensors: g_mink_cart(), g_sph(), g_ss(), metric_field()

Examples

g_eucl_cart(3)
g_eucl_cart(3) %_% .(+i, +j)
g_eucl_sph(3)
g_eucl_sph(3) %_% .(+i, +j)

---

Minkowski metric tensor

Description

g_mink_cart() provides the covariant metric tensor in n dimensions in Cartesian coordinates with signature $(-1, 1, 1, ...)$.

$ds^2=-dx_0^2+\sum_{i=1}^{n-1} dx_i^2$

g_mink_sph() provides the same tensor where the spatial part uses spherical coordinates.

$ds^2=-dt^2 + dr^2 + r^2 d\Omega^2$

Usage

g_mink_cart(n, coords = paste0("x", 1:n - 1))

g_mink_sph(n, coords = c("t", "r", paste0("ph", 1:(n - 2))))

Arguments

n	The dimension of the metric tensor.
coords	A character vector of coordinate names. The length needs to match the tensor dimensions.

Value

The covariant metric tensor as array imputed with coordinate names.

See Also

Wikipedia Minkowski metric tensor

Other metric tensors: g_eucl_cart(), g_sph(), g_ss(), metric_field()

Examples

g_mink_cart(4)
g_mink_cart(4) %_% .(+i, +j)
g_mink_sph(4)
g_mink_sph(4) %_% .(+i, +j)

---

Metric tensor of the sphere

Description

Provides the metric tensor of the sphere $S^n$ with radius 1. g_sph() returns a symbolic tensor field in generalized spherical coordinates ${\phi_1, \phi_2, ..., \phi_{n-1}}$.

$d\Omega^2= d\phi_1^2 + \sum_{i=1}^{n-1} \prod_{m=1}^{i-1} sin(\phi_m)^2 d\phi_i^2$

Usage

g_sph(n, coords = paste0("ph", 1:n))

Arguments

n	The dimension of the metric tensor.
coords	A character vector of coordinate names. The length needs to match the tensor dimensions.

Details

As usual, spherical coordinates are degenerate at $\phi_l = 0$, so be careful around those points.

Value

The covariant metric tensor as array imputed with coordinate names.

See Also

Wikipedia: Sphere

Other metric tensors: g_eucl_cart(), g_mink_cart(), g_ss(), metric_field()

Examples

g_sph(3)
g_sph(3) %_% .(+i, +j)

---

Schwarzschild metric tensor

Description

Provides the metric tensor of the Einstein equation's Schwarzschild solution in Schwarzschild coordinates where the Schwarzschild radius $r_s$ is set to 1.

$ds^2 = - \left(1-\frac{r_s}{r}\right) dt^2 + \left(1-\frac{r_s}{r}\right)^{-1} dr^r + r^2 d\Omega^2$

Usage

g_ss(n, coords = c("t", "r", paste0("ph", 1:(n - 2))))

Arguments

n	The dimension of the metric tensor.
coords	A character vector of coordinate names. The length needs to match the tensor dimensions.

Details

Note that Schwarzschild coordinates become singular at the Schwarzschild radius (event horizon) $r=r_s=1$ and at $r=0$.

Value

The covariant metric tensor as array imputed with coordinate names.

See Also

Wikipedia: Schwarzschild metric

Other metric tensors: g_eucl_cart(), g_mink_cart(), g_sph(), metric_field()

Examples

g_ss(4)
g_ss(4) %_% .(+i, +j)

---

Kronecker product

Description

A Kronecker product is simply a tensor product whose underlying vector space basis is relabeled. In the present context this is realized by combining multiple index labels into one. The associated dimension to the new label is then simply the product of the dimensions associated to the old index labels respectively.

Usage

kron(x, ...)

Arguments

x	A labeled tensor object, created by %_% or tensor().
...	Any number of expressions (separated by a comma) of the form ⁠.(<label1>, <label2>, ..., <labeln-1>) -> <labeln+1>⁠.

Value

A modified tensor object.

See Also

Wikipedia: Kronecker Product

Other tensor operations: Ops.tensor(), asym(), l(), r(), subst(), sym()

Examples

a <- array(1:8, dim = c(2, 2, 2))
a %_% .(i, j, k) |> kron(.(i, j) -> l)

---

Lower tensor indices

Description

l() lowers specified tensor indices using a covariant metric tensor provided in g. Note that the order of indices is not preserved due to performance reasons. An error is thrown if the specified indices do not exist or are not in the correct position.

Usage

l(x, ..., g = NULL)

Arguments

x	A labeled tensor object, created by %_% or tensor().
...	Any number of index expressions. The indices need to occur in x.
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(). If no metric tensor is provided, indices are raised/lowered with the identity matrix.

Value

A modified tensor object.

See Also

Other tensor operations: Ops.tensor(), asym(), kron(), r(), subst(), sym()

---

Create a metric tensor field

Description

Metric tensors are an essential ingredient of (Pseudo-) Riemannian manifolds and define distance relations between points. They are used to define geometric tensors such as e.g. the Ricci curvature ricci(), and a metric connection, i.e. a covariant derivative. They are also essential for raising and lowering indices of tensor fields correctly when using non-flat coordinates.

Usage

metric_field(metric, metric_inv, coords)

Arguments

metric	A nxn matrix / array representing the covariant metric tensor components. The components are usually expressions as character strings formed from coordinates, since numeric values can only represent constant tensor fields.
metric_inv	A nxn matrix / array representing the contraviant metric tensor components, i.e. the inverse matrix of the covariant metric tensor component matrix.
coords	A character vector of n coordinate names that are used in the component expressions. This information is essential for forming symbolic derivatives.

Value

An object of class c("metric_field", "array") that represents the components of a metric tensor on a (Pseudo-) Riemannian manifold in a certain coordinate system specified by coords.

See Also

Wikipedia: Metric tensor

Other metric tensors: g_eucl_cart(), g_mink_cart(), g_sph(), g_ss()

---

Arithmetic tensor operations

Description

Once a labeled array (tensor) has been defined, tensor arithmetic operations can be carried out with the usual +, -, *, /, and == symbols.

Usage

## S3 method for class 'tensor'
Ops(e1, e2)

Arguments

e1, e2

Labeled arrays created with %_%.

Value

A resulting labeled array in case of +, -, *, /. TRUE or FALSE in case of ==.

Addition and Subtraction

Addition and subtraction requires the two tensors to have an equal index structure, i.e. the index names their position and the dimensions associated to the index names have to agree. The index order does not matter, the operation will match dimensions by index name.

Multiplication

Tensor multiplication takes into account implicit Ricci calculus rules depending on index placement.

• Equal-named and opposite-positioned dimensions are contracted.

• Equal-named and equal-positioned dimensions are subsetted.

• The result is an outer product for distinct index names.

Division

Division performs element-wise division. If the second argument is a scalar, each element is simply divided by the scalar. Similar to addition and subtraction, division requires the two tensors to have an equal index structure, i.e. the index names their position and the dimensions associated to the index names have to agree.

Equality check

A tensor $a_{i_1 i_2 ...}$ is equal to a tensor $b_{j_1 j_2 ...}$ if and only if the index structure agrees and all components are equal.

See Also

Other tensor operations: asym(), kron(), l(), r(), subst(), sym()

Examples

a <- array(1:4, c(2, 2))
b <- array(3 + 1:4, c(2, 2))

# addition
a %_% .(i, j) + b %_% .(j, i)

# multiplication
a %_% .(i, j) * b %_% .(+i, k)

# division
a %_% .(i, j) / 10

# equality check
a %_% .(i, j) == a %_% .(i, j)
a %_% .(i, j) == a %_% .(j, i)
a %_% .(i, j) == b %_% .(i, j)

# this will err because index structure does not agree
try(a %_% .(i, j) == a %_% .(k, j))

---

Raise tensor indices

Description

r() raises specified tensor indices using a covariant metric tensor provided in g. Note that the order of indices is not preserved due to performance reasons. An error is thrown if the specified indices do not exist or are not in the correct position.

Usage

r(x, ..., g = NULL)

Arguments

x	A labeled tensor object, created by %_% or tensor().
...	Any number of index expressions. The indices need to occur in x.
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(). If no metric tensor is provided, indices are raised/lowered with the identity matrix.

Value

A modified tensor object.

See Also

Other tensor operations: Ops.tensor(), asym(), kron(), l(), subst(), sym()

---

Ricci curvature tensor

Description

Provides the covariant Ricci curvature tensor $R_{ij}=R^{s}_{i s j}$.

Usage

ricci(g)

Arguments

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().

Value

Returns the covariant Ricci curvature tensor $R_{ij}$ as rank 2 array().

See Also

Wikipedia: Ricci curvature tensor

Other geometric tensors: christoffel(), ricci_sc(), riemann()

Examples

ricci(g_eucl_sph(3))

---

Ricci scalar

Description

Provides the Ricci scalar $R$.

Usage

ricci_sc(g)

Arguments

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().

Value

Returns the Ricci scalar $R$ as single number/expression.

See Also

Wikipedia: Ricci scalar

Other geometric tensors: christoffel(), ricci(), riemann()

Examples

ricci_sc(g_eucl_sph(3))

---

Riemann curvature tensor

Description

Provides the covariant Riemann curvature tensor $R_{ijkl}$.

Usage

riemann(g)

Arguments

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().

Value

Returns the covariant Riemann curvature tensor $R_{ijkl}$ as rank 4 array().

See Also

Wikipedia: Riemann curvature tensor

Other geometric tensors: christoffel(), ricci(), ricci_sc()

Examples

riemann(g_eucl_sph(3))

---

Simplify symbolic expressions

Description

Attempts to simplify expressions in an array or tensor. Non-array objects are coerced to arrays with as.array().

Usage

simplify(x)

Arguments

x	A character array() or tensor() consisting of mathematical expressions.

Details

Instead of using an explicit call to simplify() you also have the option to enable automatic simplification via option(ricci.auto_simplify = TRUE). Note however that this comes at a significant performance cost.

This operation requires the Ryacas package.

Value

A character array() or tensor() of the same form, potentially with simplified expressions.

Examples

simplify("x + y - x")

---

Substitute tensor labels

Description

Substitutes tensor labels with other labels. This is simply a renaming procedure. The operation might trigger implicit diagonal subsetting. An error is thrown if the specified indices do not exist.

Usage

subst(x, ...)

Arguments

x	A labeled tensor object, created by %_% or tensor().
...	Any number of expressions (separated by a comma) of the form ⁠<label1> -> <label2>⁠.

Value

A modified tensor object.

See Also

Other tensor operations: Ops.tensor(), asym(), kron(), l(), r(), sym()

---

Symmetric tensor part

Description

Takes the symmetric tensor part of a tensor x with respect to the specified indices .... An error is thrown if the specified indices do not exist.

Usage

sym(x, ...)

Arguments

x	A labeled tensor object, created by %_% or tensor().
...	Any number of index expressions. The indices need to occur in x.

Value

A modified tensor object.

See Also

Wikipedia: Ricci calculus - Symmetric and antisymmetric parts

Other tensor operations: Ops.tensor(), asym(), kron(), l(), r(), subst()

Examples

a <- array(1:4, dim = c(2, 2))
a %_% .(i, j) |> sym(i, j)

---

Create a labeled array (tensor)

Description

Creates a labeled array (tensor) from an array. ⁠%_%⁠ and tensor() serve the same purpose, but typically usage of ⁠%_%⁠ is preferred due to brevity. tensor() is exported to provide a standard-evaluation interface as well which might be useful under some circumstances.

Usage

tensor(a, index_names, index_positions, call = NULL)

a %_% i

Arguments

a	An array or any object that can be coerced to an array via as.array().
index_names	A character vector of index names / labels.
index_positions	A character vector of index positions with two allowed values "+" and "-", for "upper" and "lower" position respectively. The length of index_positions needs to agree with the length of index_names.
call	For internal use only.
i	An index slot label specification created with .().

Value

A labeled tensor object of class "tensor", an array() with attached dimension labels. Note that the index structure of the resulting tensor does not necessarily have to match i. In case implicit calculations are already triggered (e.g. contractions) the index structure reflects the resulting tensor.

Examples

a <- array(1:4, dim = c(2, 2))
a %_% .(i, j)

---