galgebra.metric¶
Metric Tensor and Derivatives of Basis Vectors.
Members¶

class
galgebra.metric.
Metric
(basis, **kwargs)[source]¶ Metric specification

g
¶ metric tensor
Type: sympy matrix[,]

g_inv
¶ inverse of metric tensor
Type: sympy matrix[,]

norm
¶ normalized diagonal metric tensor
Type: list of sympy numbers

coords
¶ coordinate variables
Type: list[] of sympy symbols

is_ortho
¶ True if basis is orthogonal
Type: bool

connect_flg
¶ True if connection is nonzero
Type: bool

basis
¶ basis vector symbols
Type: list[] of noncommutative sympy variables

r_symbols
¶ reciprocal basis vector symbols
Type: list[] of noncommutative sympy variables

n
¶ dimension of vector space/manifold
Type: integer

n_range
¶ list of basis indices

de
¶ derivatives of basis functions. Two dimensional list. First entry is differentiating coordiate. Second entry is basis vector. Quantities are linear combinations of basis vector symbols.
Type: list[][]

sig
¶ Signature of metric
(p,q)
wheren = p+q
. If metric tensor is numerical and orthogonal it is calculated. Otherwise the following inputs are used:Input Signature Type "e"
(n,0)
Euclidean "m+"
(n1,1)
Minkowski (One negative square) "m"
(1,n1)
Minkowski (One positive square) p
(p,np)
General (integer not string input) Type: Tuple[int, int]

gsym
¶ String for symbolic metric determinant. If self.gsym = ‘g’ then det(g) is sympy scalar function of coordinates with name ‘det(g)’. Useful for complex nonorthogonal coordinate systems or for calculations with general metric.
Type: str

Christoffel_symbols
(mode=1)[source]¶ mode = 1 Christoffel symbols of the first kind mode = 2 Christoffel symbols of the second kind

Isq
= None¶ Sign of I**2, only needed if I**2 not a number

detg
= None¶ Determinant of g

static
dot_orthogonal
(V1, V2, g=None)[source]¶ Returns the dot product of two vectors in an orthogonal coordinate system. V1 and V2 are lists of sympy expressions. g is a list of constants that gives the signature of the vector space to allow for noneuclidian vector spaces.
This function is only used to form the dot product of vectors in the embedding space of a vector manifold or in the case where the basis vectors are explicitly defined by vector fields in the embedding space.
A g of None is for a Euclidian embedding space.

g_adj
= None¶ Adjugate of g

g_inv
= None Inverse of g


galgebra.metric.
collect
(A, nc_list)[source]¶ Parameters:  A – a linear combination of noncommutative symbols with scalar expressions as coefficients
 nc_list – noncommutative symbols in A to combine
Returns: A sum of the terms containing the noncommutative symbols in nc_list such that no elements of nc_list appear more than once in the sum. All coefficients of a given element of nc_list are combined into a single coefficient.
Return type: sympy.Basic

galgebra.metric.
square_root_of_expr
(expr)[source]¶ If expression is product of even powers then every power is divided by two and the product is returned. If some terms in product are not even powers the sqrt of the absolute value of the expression is returned. If the expression is a number the sqrt of the absolute value of the number is returned.

galgebra.metric.
test_init_slots
(init_slots, **kwargs)[source]¶ Tests kwargs for allowed keyword arguments as defined by dictionary init_slots. If keyword argument defined by init_slots is not present set default value asdefined by init_slots. Allow for backward compatible keyword arguments by equivalencing keywords by setting default value of backward compatible keyword to new keyword and then referencing new keywork (see init_slots for Metric class and equivalence between keywords ‘g’ and ‘metric’)