Index placement on 4-potential

In summary, the conversation discusses the use of the metric diag(1,-1,-1,-1) in relation to the 4-potential (Aμ) = (V/c, A), where V is the scalar potential and A is the vector potential. The potential is being used to calculate the components of the electromagnetic field tensor, where the components of the 3-vector are not necessarily the same as the components of the 4-vector. The conversation also highlights the importance of understanding sign conventions when dealing with covariant and contravariant components.
  • #1

dyn

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Hi.
I am working through some notes which use the following metric diag(1,-1,-1,-1).
They give the 4-potential as ( Aμ ) = ( V/c , A ) where V is the scalar potential and A is the vector potential. This should mean in components A0 = V/c and A1 = A1 and so on but with the metric given shouldn't A1 = - A1 ?
Thanks
 
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  • #2
Be careful not to confuse potential different meanings of ##A_1## and ##A_1##. When you write ##A_1##, do you mean the Cartesian components of the 3-vector ##\vec A## or the covariant components of the 4-vector ##A##?
 
  • #3
When I wrote A1 I meant Ax ie. the x-component of the 3-vector A
 
  • #4
dyn said:
When I wrote A1 I meant Ax ie. the x-component of the 3-vector A
Then no. It is not necessary that ##A^1 = - A_x## as ##A_x## is not the same as ##A_1##. The sign difference is between the covariant and contravariant spatial components of the 4-potential. As I said, do not confuse the components of a 3-vector with the covariant (or contravariant for that matter) components of a 4-vector. Typically, the generalisation of a 3-vector to a 4-vector will be such that the 3-vector components are the same as the covariant components of the 4-vector, but this may sometimes be subject to sign conventions and if the 3-vector is more naturally viewed as having covariant or contravariant components. In some cases, there is no 4-vector generalisation of the 3-vector at all, such as in the case of the electric and magnetic field where their components instead together constitute the components of the electromagnetic field tensor.
 
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  • #5
Thanks. I asked the question because I don't understand the following question. Using E = -∇V - ∂tA the x-component is given as E1/c = -∂x(V/c) -(1/c)∂tA1 = ∂1A0 - ∂0A1
Is this equation correct ? If so , I don't understand the sign change on the 1st term and it seems to me it uses A1 for Ax
 
  • #6
dyn said:
Using E = -∇V - ∂tA the x-component is given as E1/c = -∂x(V/c) -(1/c)∂tA1 = ∂1A0 - ∂0A1

Say ##x^0=ct, V=A^0##,
[tex]E^1=-\frac{\partial V}{\partial x^1}-\frac{\partial A^1}{\partial x^0}[/tex]
[tex]=-\frac{\partial A^0}{\partial x^1}-\frac{\partial A^1}{\partial x^0}[/tex]
[tex]=-\frac{\partial A_0}{\partial x^1}+\frac{\partial A_1}{\partial x^0}[/tex]
[tex]=\frac{\partial A_1}{\partial x^0}-\frac{\partial A_0}{\partial x^1}[/tex]
[tex]=F_{10}=-F^{10}[/tex]
where
[tex]F_{\mu\nu}=\frac{\partial A_\mu}{\partial x^\nu}-\frac{\partial A_\mu}{\partial x^\nu}[/tex]

Similarly ##B^1=F_{23}=F^{23}##
Actually E and B are not vectors but components of antisymmetric electromagnetic tensor F.
 
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1. What is index placement on 4-potential?

Index placement on 4-potential refers to the convention used in theoretical physics to label the components of a 4-vector, which is a mathematical object used to describe physical quantities in four-dimensional spacetime. The index placement indicates the direction of the vector and follows the rules of special relativity.

2. Why is index placement important in 4-potential?

Index placement is important in 4-potential because it allows us to correctly describe the behavior of physical quantities in four-dimensional spacetime. The index placement indicates the direction of the vector and affects the mathematical operations we can perform on the 4-potential.

3. How is index placement determined in 4-potential?

Index placement is determined by the direction of the vector in four-dimensional spacetime. In 4-potential, the index placement follows the rules of special relativity, where one index is raised and one index is lowered. The placement of the indices can also be determined by the symmetries of the vector components.

4. What are the rules for index placement in 4-potential?

The rules for index placement in 4-potential follow the rules of special relativity, where one index is raised and one index is lowered. This ensures that the resulting mathematical operations are independent of the observer's frame of reference. Additionally, the placement of the indices can also be determined by the symmetries of the vector components.

5. How does index placement affect calculations in 4-potential?

Index placement affects calculations in 4-potential because it determines the direction of the vector and affects the mathematical operations that can be performed on the 4-potential. Incorrect index placement can lead to incorrect calculations and interpretations of physical quantities in four-dimensional spacetime.

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