Index notation: Find F_(μν) given F^(μν)?

In summary: You should find A0 = -5 and A3 = 8.Note that the metric is not just any old matrix. It has a very specific form. It has one negative sign and three positive signs. But the sign pattern is the same in each of the four diagonal elements. So, the metric is not just any old symmetric matrix. It has to be a special type of symmetric matrix. It is called a "diagonal matrix".One problem is that the notation is confusing. We use the same symbol "A" to represent all four components of the 4-vector A. We use the same symbol "g" to represent all 16 components of the metric tensor g. So
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Homework Statement


Find Fμν, given Fμν=
(0 Ex Ey Ez)
(-Ex 0 Bz -By)
(-Ey -Bz 0 -Bx)
(-Ez By -Bx 0)

Homework Equations


gμν = Diag( -1, 1, 1, 1)

The Attempt at a Solution


I tried computing Fμν = gμνFμν
but this gave
(0 -Ex -Ey -Ez)
(-Ex 0 Bz -By)
(-Ey -Bz 0 Bx)
(-Ez By -Bx 0)
Which seems wrong to me, since the next question is to find FμνFνμ and this is meant to come out as the Lagrangian and from my notes I don't see this working.
I think I'm messing up my index notation, I think I may need an extra metric in there because there are 2 indices downstairs but I don't have the best understanding.

Any help and advice on dealing with pesky index notation would be greatly appreciated, Thanks!
 
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  • #2
Poirot said:
(0 Ex Ey Ez)
(-Ex 0 Bz -By)
(-Ey -Bz 0 -Bx)
(-Ez By -Bx 0)
There's a typo of a wrong sign for one of the terms.
I tried computing Fμν = gμνFμν
The convention is to sum over repeated indices. So the expression gμνFμνon the right side would imply a sum over μ and a sum over ν. The result would not depend on a specific value of μ and ν; whereas, the left side Fμν represents a specific component for specific values μ and ν. So, something is wrong.

The correct way to lower the two indices is Fμν = gμαgFαβ which involves a sum over α and a sum over β. The μ and ν appearing on the right side have the same values as the μ and ν appearing on the left side.

Because the metric has components gμν = Diag( -1, 1, 1, 1), you should be able to show that whenever you lower a 0 index, you just change the sign. For example A0 = -A0 for any 4-vector. When you lower any of the other indices, there is no change in sign. For example A2 = A2. The same holds for tensors with more than one index. See if you can show, for example, that F03 = -F03 and F23 = F23. Once you get the hang of it, you will be able to lower (or raise) indices very quickly.
 
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  • #3
TSny said:
There's a typo of a wrong sign for one of the terms.

The convention is to sum over repeated indices. So the expression gμνFμνon the right side would imply a sum over μ and a sum over ν. The result would not depend on a specific value of μ and ν; whereas, the left side Fμν represents a specific component for specific values μ and ν. So, something is wrong.

The correct way to lower the two indices is Fμν = gμαgFαβ which involves a sum over α and a sum over β. The μ and ν appearing on the right side have the same values as the μ and ν appearing on the left side.

Because the metric has components gμν = Diag( -1, 1, 1, 1), you should be able to show that whenever you lower a 0 index, you just change the sign. For example A0 = -A0 for any 4-vector. When you lower any of the other indices, there is no change in sign. For example A2 = A2. The same holds for tensors with more than one index. See if you can show, for example, that F03 = -F03 and F23 = F23. Once you get the hang of it, you will be able to lower (or raise) indices very quickly.
Hi and thank you for your response, you've cleared up a lot! Just a quick question, is gμα effectively g, because I know that if for instance you switched the indices you get the -g for example? And how would you know when they are the same or not?
Thanks
 
  • #4
Poirot said:
is gμα effectively g
No, these symbols represent specific components of the the metric tensor and they won't be equal in general.
because I know that if for instance you switched the indices you get the -g for example?
Actually, when you switch the order of the indices on the metric tensor, it does not introduce a negative sign. Instead, g = gβv. The metric is said to be "symmetric". If you put the components of g into a matrix, the matrix will be a symmetric matrix. More specifically, it will be a diagonal matrix where all of the off diagonal terms are zero. So, g = gβv = 0 if v ≠ β.

You should probably practice with some simple examples. Suppose you had a 4-vector A with components A0 = 5, A1 = 6, A2 = 7, and A3 = 8. Then, if I want to find A2 I would apply the general relation Aμ = gμν Aν. Letting μ = 2(on both sides),

A2 = g Aν

The repeated index ν means I have to sum over ν for ν = 0, 1, 2, and 3. So,

A2 = g20A0 + g21A1 + g22A2 + g23A3
= 0⋅A0 + 0⋅A1 +1⋅A2 + 0⋅A3
= A2
= 7

You can try using Aμ = gμν Aν to find A0 and A3.
 

1. What is index notation?

Index notation is a mathematical notation used to represent tensors, which are multidimensional arrays of numbers. It is often used in physics and engineering to represent physical quantities such as vectors and matrices.

2. What does Fμν mean in index notation?

Fμν represents a tensor with two indices, μ and ν. This notation is used to represent a matrix with μ rows and ν columns, where each element in the matrix is represented by the value of the tensor at that particular index.

3. How do you find Fμν given Fμν in index notation?

To find Fμν given Fμν, you can use the inverse operation of the raising or lowering of indices. This involves using the metric tensor to contract the indices and obtain the desired tensor with lowered or raised indices.

4. What is the role of the metric tensor in index notation?

The metric tensor is used to raise or lower indices in index notation. It is a symmetric tensor that defines the inner product of vectors and allows for the conversion between covariant and contravariant indices.

5. Can index notation be used in other fields besides physics and engineering?

Yes, index notation can be used in any field that deals with multidimensional arrays of numbers, such as computer science and mathematics. It provides a concise and consistent way to represent and manipulate these arrays in equations and calculations.

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