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Faraday Tensor and Index Notation

  • #1
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Homework Statement



(a) Find faraday tensor in terms of ##\vec E## and ## \vec B ##.
(b) Obtain two of maxwell equations using the field relation. Obtain the other two maxwell equations using 4-potentials.
(c) Find top row of stress-energy tensor. Show how the b=0 component relates to j.

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Homework Equations



The Attempt at a Solution



Part (a)
[/B]
The relations between the potentials and fields are:

[tex] \vec B = \nabla \times \vec A [/tex]
[tex]\vec E = -\nabla \phi - \frac{\partial \vec A}{\partial t} [/tex]

The four-vector potential is given by ## A = \left( \frac{\phi}{c}, \vec A \right)##.

From the relation given: ## F^{ab} = \partial^{a} A ^b - \partial^b A^a ##, it looks something like ##\nabla \times A##. How do I show this? I've read the basics of tensor notation and seems alright, but I can't seem to apply the knowledge.
 

Answers and Replies

  • #2
TSny
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From the relation given: ## F^{ab} = \partial^{a} A ^b - \partial^b A^a ##, it looks something like ##\nabla \times A##. How do I show this? I've read the basics of tensor notation and seems alright, but I can't seem to apply the knowledge.
Just start writing it out explicitly. For example, what do you get for ## F^{01}##?
 
  • #3
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Just start writing it out explicitly. For example, what do you get for ## F^{01}##?
[tex]F^{01} = \frac{\partial A_x}{\partial \phi} - \frac{1}{c} \frac{\partial \phi}{\partial A_x} [/tex]

I don't see the point of writing out everything, as it only gives terms like ##A_x, A_y, A_z## and partial derivatives..
 
  • #4
dextercioby
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You're not calculating the correct derivatives. What does [itex] \partial^{a}[/itex] stand for?
 
  • #5
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You're not calculating the correct derivatives. What does [itex] \partial^{a}[/itex] stand for?
Am I missing out a factor of ##c##? I think ##\partial^a = c \frac{\partial}{\partial \phi}##
 
  • #6
TSny
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Am I missing out a factor of ##c##? I think ##\partial^a = c \frac{\partial}{\partial \phi}##
##\partial^a## is a compact notation for a derivative with respect to a space or time coordinate: ##\frac{\partial}{\partial x^a}##
 
  • #7
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##\partial^a## is a compact notation for a derivative with respect to a space or time coordinate: ##\frac{\partial}{\partial x^a}##
Yes, in this case ##a=0##, so we're taking the first coordinate in the 4-vector ##(\frac{\phi}{c}, \vec A)## which is ##\frac{\phi}{c}##.
 
  • #8
Matterwave
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Yes, in this case ##a=0##, so we're taking the first coordinate in the 4-vector ##(\frac{\phi}{c}, \vec A)## which is ##\frac{\phi}{c}##.
The notation ##\partial^a## means it's a derivative with respect to space-time coordinates. It has nothing to do with the 4-vector potential. It has to do with the space-time coordinates ##(t,\vec{x})##. Specifically ##\partial^a=\eta^{ab}\partial_b\equiv\eta^{ab}\frac{\partial}{\partial x^b}=\eta^{at}\frac{\partial}{\partial t}+\eta^{ax}\frac{\partial}{\partial x}+...##
 
  • #9
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The notation ##\partial^a## means it's a derivative with respect to space-time coordinates. It has nothing to do with the 4-vector potential. It has to do with the space-time coordinates ##(t,\vec{x})##. Specifically ##\partial^a=\eta^{ab}\partial_b\equiv\eta^{ab}\frac{\partial}{\partial x^b}=\eta^{at}\frac{\partial}{\partial t}+\eta^{ax}\frac{\partial}{\partial x}+...##
Alright, so

[tex]F^{01} = \frac{1}{c}\frac{\partial A_x}{\partial t} - c\frac{\partial t}{\partial x} [/tex]
 
  • #10
Matterwave
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Alright, so

[tex]F^{01} = \frac{1}{c}\frac{\partial A_x}{\partial t} - \frac{1}{c}\frac{\partial t}{\partial x} [/tex]
Why are you taking the derivative of ##t## in the second term? Also, your factors of ##c## look off to me. Maybe write it out in index notation first if you are still getting confused. You're almost there though.
 
  • #11
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Why are you taking the derivative of ##t## in the second term? Also, your factors of ##c## look off to me. Maybe write it out in index notation first if you are still getting confused. You're almost there though.
[tex]F^{01} = \partial^0 A^1 - \partial^1 A^0 [/tex]

The space-time four-vector is ## (ct, \vec r)##. The four-vector potential is given by ## (\frac{\phi}{c}, \vec A)##.

[tex]F^{01} = \frac{\partial A_x}{\partial ct} - \frac{\partial \frac{\phi}{c}}{\partial x} [/tex]
[tex] F^{01} = \frac{1}{c}\frac{\partial A_x}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial x} [/tex]
[tex] F^{02} = \frac{1}{c}\frac{\partial A_y}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial y} [/tex]
[tex] F^{03} = \frac{1}{c}\frac{\partial A_z}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial z} [/tex]
 
  • #12
Matterwave
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[tex]F^{01} = \partial^0 A^1 - \partial^1 A^0 [/tex]

The space-time four-vector is ## (ct, \vec r)##. The four-vector potential is given by ## (\frac{\phi}{c}, \vec A)##.

[tex]F^{01} = \frac{\partial A_x}{\partial ct} - \frac{\partial \frac{\phi}{c}}{\partial x} [/tex]
[tex] F^{01} = \frac{1}{c}\frac{\partial A_x}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial x} [/tex]
Ok. I think all that's left is you are missing a negative sign that should have came in when you raised the indices in ##\partial^a=\eta^{ab}\partial_b##. See post #8 above. Depending on what metric signature you are using, one or the other term should have an additional negative in it.

After that, ask yourself "what is ##E^x##?"
 
  • #13
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Ok. I think all that's left is you are missing a negative sign that should have came in when you raised the indices in ##\partial^a=\eta^{ab}\partial_b##. See post #8 above. Depending on what metric signature you are using, one or the other term should have an additional negative in it.

After that, ask yourself "what is ##E^x##?"
[tex] \vec E = -\nabla \phi - \frac{\partial \vec A}{\partial t} [/tex]

[tex]E_x = -\frac{\partial \phi}{\partial x} - \frac{\partial A_x}{\partial t} [/tex]
[tex]F^{01} = \frac{1}{c}\frac{\partial A_x}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial x} [/tex]
 
  • #14
TSny
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As matterwave has pointed out, I was wrong when I said in post #8 that ##\partial^a = \frac{\partial}{\partial x^a}##. I should have said ##\partial_a = \frac{\partial}{\partial x^a}## with the index in the lower position on the left side of the equation. Matterwave has shown how to relate ##\partial_a## and ##\partial^a##. Sorry for the confusion.
 
  • #15
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As matterwave has pointed out, I was wrong when I said in post #8 that ##\partial^a = \frac{\partial}{\partial x^a}##. I should have said ##\partial_a = \frac{\partial}{\partial x^a}## with the index in the lower position on the left side of the equation. Matterwave has shown how to relate ##\partial_a## and ##\partial^a##. Sorry for the confusion.
[tex]E_x = -\frac{\partial \phi}{\partial x} - \frac{\partial A_x}{\partial t}[/tex]
[tex]F^{01} = - \frac{1}{c}\frac{\partial \phi}{\partial x} -\frac{1}{c}\frac{\partial A_x}{\partial t} [/tex]

This implies that ##F^{01} = \frac{E_x}{c} ##.
 
  • #16
Matterwave
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[tex]E_x = -\frac{\partial \phi}{\partial x} - \frac{\partial A_x}{\partial t}[/tex]
[tex]F^{01} = - \frac{1}{c}\frac{\partial \phi}{\partial x} -\frac{1}{c}\frac{\partial A_x}{\partial t} [/tex]

This implies that ##F^{01} = \frac{E_x}{c} ##.
Ok, now do this for ##F^{02},~F^{03},~F^{12},...## and you will have your answer. Notice that ##F^{ab}## should have 6 independent components because it is anti-symmetric, so only the top right (or bottom left) triangle is independent. Be careful with the negative signs though.
 
  • #17
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Ok, now do this for ##F^{02},~F^{03},~F^{12},...## and you will have your answer. Notice that ##F^{ab}## should have 6 independent components because it is anti-symmetric, so only the top right (or bottom left) triangle is independent.
[tex]F^{01} = -\frac{1}{c}\frac{\partial A_x}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial x}[/tex]

[tex]F^{02} = -\frac{1}{c}\frac{\partial A_y}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial y}[/tex]

[tex]F^{03} = -\frac{1}{c}\frac{\partial A_z}{\partial t} - \frac{1}{c}\frac{\partial \phi}{\partial z}[/tex]

[tex]F^{12} = \frac{\partial A_y}{\partial x} - \frac{A_x}{\partial y} [/tex]

[tex]\vec B = \nabla \times \vec A[/tex]
[tex]\vec E = -\nabla \phi - \frac{\partial \vec A}{\partial t}[/tex]
 
  • #18
Matterwave
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[tex]F^{12} = -\frac{\partial A_y}{\partial x} - \frac{A_x}{\partial y} [/tex]
Like I said, be careful with the negative signs. Why do you have a negative sign in the first term here?

There's 2 other components of ##F^{ab}## that matters. Now match these expressions to ##\vec{E},\vec{B}##.
 
  • #19
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Like I said, be careful with the negative signs. Why do you have a negative sign in the first term here?

There's 2 other components of ##F^{ab}## that matters. Now match these expressions to ##\vec{E},\vec{B}##.
By inspection, the time-like part is electric in nature, while the space-like part is magnetic in nature?

[tex]F = ( \frac{1}{c}\vec E, \vec B) [/tex]
 
  • #20
Matterwave
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By inspection, the time-like part is electric in nature, while the space-like part is magnetic in nature?

[tex]F = ( \frac{1}{c}\vec E, \vec B) [/tex]
The statement is correct, but this mathematical notation doesn't really mean anything...
 
  • #21
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The statement is correct, but this mathematical notation doesn't really mean anything...
How do I express it mathematically? I'm having some trouble translating this concept into some notation.
 
  • #22
Matterwave
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How do I express it mathematically? I'm having some trouble translating this concept into some notation.
You can either simply enumerate ##F^{01}=E^x, F^{02}=E^y,...## or, usually for simplicity, you can write out a 4x4 matrix and then put in the corresponding ##\vec{E},\vec{B}## components.
 
  • #23
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You can either simply enumerate ##F^{01}=E^x, F^{02}=E^y,...## or, usually for simplicity, you can write out a 4x4 matrix and then put in the corresponding ##\vec{E},\vec{B}## components.
I think writing out the 4x4 matrix is simpler and better, since this is the presentation adopted in the text as well.

I'll have a go at parts (b), (c) and (d) meanwhile!
 

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