Raising and Lowering Vectors/Tensors

[ClaraOxford]

I've been going back over my general relativity problem sheets and have realized I am not raising and lowering vectors/tensors properly. I've written out my calculations in detail below. I've tried to follow exact logical steps, but I must be misunderstanding, because my answers seem wrong (I often end up with just numbers)..

Homework Statement

Imagine we have a tensor X^αβ and a vector V^α with components

X^αβ = \begin{pmatrix} 2 & 0 & 1 & -1 \\-1 & 0 & 3 & 2 \\-1 & 1 & 0 & 0 \\-2 & 1 & 1 & -2 \end{pmatrix}

and V^α = \begin{pmatrix} -1 \\2 \\0 \\-2 \end{pmatrix}

Find X^α_β, X_α^β, X^(αβ), X_[αβ], X^λ_λ, V^αV_α, V_αX^αβ

Homework Equations

I have been trying to use the raising/lowering metric g = \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}

The Attempt at a Solution

X_α^β = g_ααX^αβ = \sum_αg_ααX^αβ = g₀₀X^0β + g₁₁X^1β + g₂₂X^2β + g₃₃X^3β = X^0β - X^1β - X^2β - X^3β = \begin{pmatrix} 2 & 0 & 1 & -1 \end{pmatrix} - \begin{pmatrix} -1 & 0 & 3 & 2 \end{pmatrix} - \begin{pmatrix} -1 & 1 & 0 & 0 \end{pmatrix} - \begin{pmatrix} -2 & 1 & 1 & -2 \end{pmatrix} = \begin{pmatrix} 6 & -2 & -4 & -1 \end{pmatrix}V_α = g_ααV^α = g₀₀V⁰ + g₁₁V¹ + g₂₂V² + g₃₃V³ = V⁰ - V¹ - V² - V³ = -1 -2 -0 -(-2) = -1

→ V^αV_α = -V^αX^λ_λ = g^λαg_λβX^αβ

g_λβX^αβ = g_λ0X^α0 + g_λ1X^α1 + g_λ2X^α2 + g_λ3X^α3 = \begin{pmatrix} 1 \\ 0 \\ 0 \\0 \end{pmatrix}*\begin{pmatrix} 2 \\ -1 \\ -1 \\-2 \end{pmatrix} + \begin{pmatrix} 0 \\ -1 \\ 0 \\0 \end{pmatrix}*\begin{pmatrix} 0 \\ 0 \\ 1 \\1 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ -1 \\0 \end{pmatrix}*\begin{pmatrix} 1 \\ 3 \\ 0 \\1 \end{pmatrix} + \begin{pmatrix} 0 \\ 0 \\ 0 \\-1 \end{pmatrix}*\begin{pmatrix} -1 \\ 2 \\ 0 \\-2 \end{pmatrix} = 2 + 0 + 0 + 2 = 4

I haven't written my attempts to every equation, because I'm pretty sure my method is wrong and it's taking forever to write out the matrices!

If anyone can spot where I'm going wrong that would be really helpful. Seeing as this is the most basic part of my general relativity course, if I can't get this bit right I'll be in trouble in the exam in june!

Thank you.

 

[clamtrox]

Your notation is a little dangerous: you should probably use different dummy indices for summation. So for example

V_{a} = g_{ab} V^b \equiv \sum_{b} g_{ab} V^b

However, this is not your problem.

What you should do is to try to calculate the components explicitly. So for example you know that V_ais a (dual) vector, so it has 4 components. Do them one by one: V_0 = \sum_a g_{0 a} V^a = -1 V^0etc. It seems to me that you keep losing track of what you're calculating, so you're for adding together different vector components, etc.