New to the Einstein notation, having trouble with basic calculations

Ineedhelpimbadatphys · Jan 28, 2024

Matrix A with upper indices alpha beta, rows 1 0 -2 -1; -1 0 3 2; 0 0 1 0; -2 1 1 2, and vector v with components (2, 1, -1, 0), followed by a list of index gymnastics tasks a) to h) on A and v.

Handwritten working on squared paper: part a) symmetric and antisymmetric index decomposition with 4x4 matrices of numbers, and part b) a 4x4 matrix product with eta, ending in a final 4x4 matrix.

$Handwritten working on grid paper: part g) v_α A^{αβ} = v_α η^{iα} A_i^β = v^i A_i^β, then a sum of 2[1 0 -2 -1] + [-1 0 3 2] - [0 0 1 0] = [1 0 0 0]; part e) v_{β} A^{αβ} = v_β η^{iβ} A^α_β = v^i A^α_i, then 2 times column [1, -1, 0, -2] plus column [0, 0, 0, 1] minus column [-2, 3, 1, 1] equals column [4, -5, -1, -4].$

Ibix · Jan 28, 2024

I haven't checked your arithmetic, but (c) seems correct, if long-winded. You could just say ##A^{(ab)}=\frac 12(A^{ab}+A^{ba})## by definition.

Similarly (d), where you can just say ##A^{[ab]}=\frac 12(A^{ab}-A^{ba})## by definition. I suspect going the long way round got you into a pickle here, because you've somehow ended up with extra factors of the metric that shouldn't be there. (By the way, don't put commas between indices. Some people use ##V_{a,b}## as shorthand for ##\frac{\partial}{\partial x_b}V_a##, and you're liable to be misinterpreted.)

Your approach to (g) and (h) (did you mean (h) or (e)?) appears correct, but I'm not sure you've lowered indices correctly on the tensor. What are the components of ##\eta_{ab}## in your convention?

New to the Einstein notation, having trouble with basic calculations

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