Vector notation / manipulation question

[tourjete]

Homework Statement

X^uv is a 4x4 tensor and V^u is a vector.

V^u = (-1, 2, 0, -2) (i.e. it is a 1x4 vector).

Find the quantities V^u V_u and V_u X^uv

Homework Equations

Given above

The Attempt at a Solution

I'm having trouble finding V_u. Initially I thought that it should be the transpose of V^u (so a 4x1 vector) but then you can't multiply a 4x1 matrix with a 4x4 matrix, which the tensor is.

Also, my book says that you can turn a vector into a dual vector by doing
V_u = η_uv V^v. However, I don't kow V^v; I only know V^u

Can somebody point me in the right direction? I've taken linear algebra before so I should understand this but my teacher used a different notation than the general relativity class I'm taking right now uses.

 

[gabbagabbahey]

Also, my book says that you can turn a vector into a dual vector by doing
V_u = η_uv V^v. However, I don't kow V^v; I only know V^u

$\nu$ is just an index, it will range from 0 to 3 (or 1 to 4) just like $\mu$. If you know $V^{\mu}$, then you know $V^{\nu}$. The components of the vector do not change just because one uses a different index to refer to them.

What is the metric tensor in this case (i.e. what type of space-time are you using)? What is your tensor $X^{\mu\nu}$?

 

[Muphrid]

Superscripts and subscripts range from 0-3 (or 1-4, whatever your preference). Calling a superscript a different letter has no significance. It's only when an index is repeated that you should interpret this to mean that the index should be the same for both.

 

[tourjete]

I'm in Minkowski flat space-time so the metric tensor is \begin{array}{ccc}
-1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array}

and the tensor is
\begin{array}{ccc}
2 & 0 & 1 & -1 \\
-1 & 0 & 3 & 2 \\
-1 & 1 & 0 & 0 \\
-2 & 1 & 1 & -2 \end{array}

I'm afraid I'm still confused. So
V_u = η_uvV^u? This still makes no sense to me because you can't multiply a 4x4 by a 1x4 matrix.

 

[Muphrid]

This still makes no sense to me because you can't multiply a 4x4 by a 1x4 matrix.

Sure you can. You just multiply the 1x4 on the left of the matrix. (Whether this is what your instructor wants you to do is a different story. It's a matter of convention whether up-index vectors represent row matrices multiplied on the left of the transformation matrix or column matrices multplied on the right.)


Really, though, you're trying to jump into representing things with matrices when you don't have a grasp of the underlying math yet. Index notation frees you from dealing with actual matrix multiplication at all.

Expand the implied summation to see what you need to do.

$$v_\mu = \eta_{\mu \nu} v^\nu = \eta_{\mu 1} v^1 + \eta_{\mu 2} v^2 + \eta_{\mu 3} v^3 + \eta_{\mu 4} v^4$$