Vector notation / manipulation question

In summary, Vu = (-1, 2, 0, -2) and Vu Xuv = (-1, 2, 0, 2). The quantities Vu Vu and Vu Xuv are found by trying to expand the implied summation. The Attempt at a Solution gives some more instructions on how to do this.
  • #1
tourjete
25
0

Homework Statement


Xuv is a 4x4 tensor and Vu is a vector.

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

Find the quantities Vu Vu and Vu Xuv



Homework Equations


Given above

The Attempt at a Solution


I'm having trouble finding Vu. Initially I thought that it should be the transpose of Vu (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
Vu = ηuv Vv. However, I don't kow Vv; I only know Vu

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.
 
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  • #2
tourjete said:
Also, my book says that you can turn a vector into a dual vector by doing
Vu = ηuv Vv. However, I don't kow Vv; I only know Vu

[itex]\nu[/itex] is just an index, it will range from 0 to 3 (or 1 to 4) just like [itex]\mu[/itex]. If you know [itex]V^{\mu}[/itex], then you know [itex]V^{\nu}[/itex]. 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 [itex]X^{\mu\nu}[/itex]?
 
  • #3
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.
 
  • #4
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
Vu = ηuvVu? This still makes no sense to me because you can't multiply a 4x4 by a 1x4 matrix.
 
  • #5
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.

[tex]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[/tex]
 

What is vector notation?

Vector notation is a mathematical representation used to describe a vector, which is a quantity that has both magnitude and direction. It typically involves using a letter with an arrow on top to indicate the vector, such as v. The length of the arrow represents the magnitude and the direction of the arrow represents the direction of the vector.

How is vector notation used in physics?

Vector notation is used in physics to represent physical quantities that have both magnitude and direction, such as velocity, acceleration, and force. It allows for easy manipulation and calculation of these quantities, as well as graphical representation of their relationships.

What is a unit vector?

A unit vector is a vector with a magnitude of 1. It is often represented as a letter with a caret on top, such as u. Unit vectors are commonly used in vector notation to indicate the direction of a vector without changing its magnitude.

How do you add or subtract vectors using vector notation?

To add or subtract vectors using vector notation, you simply add or subtract the corresponding components of the vectors. For example, if v = (2,3) and w = (1,4), then v + w = (2+1, 3+4) = (3,7) and v - w = (2-1, 3-4) = (1,-1).

How do you find the magnitude and direction of a vector using vector notation?

To find the magnitude of a vector using vector notation, you can use the Pythagorean theorem, which states that the magnitude of a vector v = (a,b) is given by √(a² + b²). To find the direction, you can use trigonometric functions such as sine and cosine. The direction is typically given in degrees or radians.

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