CoolTensors

The easiest way to create Tensor objects is using IndexPos objects. They can easily created with the custom T"..." string literal. ' denotes contravariant indices and , covariant indices:

julia> x = T"'"[√2, 1]
2-element Tensor{Float64,1,T"'",Array{Float64,1}}:
 1.4142135623730951
 1.0

julia> Λ(ψ) = T"',"[cosh(ψ) -sinh(ψ); -sinh(ψ) cosh(ψ)]
Λ (generic function with 1 method)

julia> using LinearAlgebra

julia> g = T",,"(Diagonal([1, -1]))
2×2 Tensor{Int64,2,T",,",Diagonal{Int64,Array{Int64,1}}}:
 1   0
 0  -1

lower and raise lower and raise indices: (The metric is assumed to be euclidian.)

julia> lower(x, 1)
2-element Tensor{Float64,1,T",",Array{Float64,1}}:
 1.4142135623730951
 1.0

julia> 𝔤 = raise(g, 1, 2)
2×2 Tensor{Int64,2,T"''",Diagonal{Int64,Array{Int64,1}}}:
 1   0
 0  -1

Indexing can be done just like a regular array, using TIndex, which stores an additional IndexPos in the type parameter, or by using hvcat syntax. When using the latter, indices are separated by either whitespace or ;, ; switches the contra-/covariance of the following indices, whitespace means an index has the same contra-/covariance as the last index. To specify that the first index is covariant, write \; before the first index.

julia> x[1]
1.4142135623730951

julia> lower(x, 1)[\; 1]
1.4142135623730951

julia> Λ(.5)[1; 2]
-0.5210953054937474

julia> g[\; 2 2]
-1

julia> g[1; 2]
ERROR: ArgumentError: Index positions T"'," don't match indices of Tensor T",,"
Stacktrace:
[...]

Tensors can be called with vectors and covectors. The rightmost index is always contracted first. If called with multiple (co-)vectors, the first (co-)vector is contracted with the last index of the tensor, the second with the second-last index, and so on. : can be used to skip the contraction of an index.

julia> Λ(.5)(x)
2-element Tensor{Float64,1,T"'",Array{Float64,1}}:
 1.073608627785168
 0.3906859168881719

julia> Λ(.5)(lower(x, 1))
ERROR: ArgumentError: Tensor indices don't match
Stacktrace:
[...]

julia> Λ(.5)(:, lower(x, 1))
2-element Tensor{Float64,1,T",",Array{Float64,1}}:
 1.073608627785168
 0.3906859168881719

julia> lower(x, 1)(x)
3.0000000000000004

Let's prove that g(x, x) is Lorentz invariant!

julia> g(x, x)
1.0000000000000004

julia> g(Λ(.5)(x), Λ(.5)(x))
0.9999999999999998

QED.

⊗ (\otimes) from TensorCore.jl is overloaded and exported to calculate the tensor product of two Tensors:

julia> x ⊗ lower(x, 1)
2×2 Tensor{Float64,2,T"',",Array{Float64,2}}:
 2.0      1.41421
 1.41421  1.0

Disclaimer: This is currently only a prototype. It is still missing a lot of features. Performance should be pretty bad. There will be bugs. Don't use this for any productive work.