Talk:Electromagnetic tensor

System coordinates
How could I express the Electromagnetic tensor in curvilinear coordinates? For example, is it correct:
 * $$F^{\mu \nu} = \begin{vmatrix} 0 & Er/c & E\theta /c & E\phi /c \\ -Er/c & 0 & B\phi & -B\theta \\ -E\theta /c & -B\phi & 0 & Br \\ -E\phi /c & B\theta & -Br & 0 \end{vmatrix}$$?

Maxwell equations must remain the same, i.e., $${F^{\mu\nu}}_{,\nu}=\mu_{0}J^{\mu}$$. But how can I expand it? $${F^{0\nu}}_{,\nu}=\mu_{0}J^{0}$$ must express Gauss's Law in Spherical coordinates and the other equations Faraday's Law --Fgrillopeternella (talk) 01:38, 28 January 2008 (UTC)

Answering your question

 * For answer this question, firstly, we needed to define as the four-vector potential transform itself when it changes of Cartesian to spherical coordinates. Let us consider the following transformation law for four-vector:
 * $${A^\prime}_{\mu}=\frac{\partial{{x}^\nu}}{\partial{{x^\prime}^\mu}}A_\nu.$$
 * You should be considered that $${A^\prime}^{\mu}$$ to be the new coordinates taken in spherical coordinates while that $$A^\nu$$ be the old coordinates taken in Cartesian coordinates. As we know the relationship of transformation between spherical and Cartesian coordinates are given by
 * $$\begin{align}t&=t;\\ x&=r\sin{\theta}\cos{\varphi};\\ y&=r\sin{\theta}\sin{\varphi};\\ z&=r\cos{\theta}. \end{align}$$
 * The transformation matrix will be
 * $$\frac{\partial{{x^{\prime}}^\mu}}{\partial{x^\nu}}=\begin{pmatrix}

1 & 0 & 0 & 0 \\                                                 0 &\sin\theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta \\ 0 & \dfrac{1}{r}\cos\theta\cos\varphi & \dfrac{1}{r}\cos\theta\sin\varphi & -\dfrac{1}{r}\sin\theta \\ 0 & -\dfrac{1}{r}\dfrac{\sin\varphi}{\sin\theta} & \dfrac{1}{r}\dfrac{\cos\varphi}{\sin\theta} & 0 \end{pmatrix}$$
 * that will lead to
 * $$A^\mu=\begin{pmatrix}A^0 & A_r & \dfrac{1}{r}A_\theta & \dfrac{1}{r\sin\theta}A_\varphi\end{pmatrix}.$$
 * In the same way, you can write the transformation law for the partial derivatives as:
 * $${\partial^\prime}^{\mu}=\frac{\partial{x^\mu}}{\partial{{x^\prime}^\nu}}\partial^\nu,$$
 * where, in this case, the matrix transformation is
 * $$\frac{\partial{x^\mu}}{\partial{{x^\prime}^\nu}}=\begin{pmatrix}

1 & 0 & 0 & 0 \\                                                 0 &\sin\theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta \\ 0 & r\cos\theta\cos\varphi & r\cos\theta\sin\varphi & -r\sin\theta \\ 0 & -r\sin\theta\sin\varphi & r\sin\theta\cos\varphi & 0 \end{pmatrix},$$
 * showning that
 * $$\partial_\mu=\begin{pmatrix}\partial_0 & \partial_r & \partial_\theta & \partial_\varphi\end{pmatrix}.$$
 * Now, we are in conditions of to find the $$F^{\mu\nu}$$ tensor correct. To that end, consider the $$F^{\mu\nu}$$ tensor as write in its usual form
 * $$F^{\mu\nu}=\partial^{\mu}A^\nu-\partial^{\nu}A^\mu$$
 * or, after lowering the indices of the partial derivatives,
 * $$F^{\mu\nu}=g^{\mu\rho}\partial_{\rho}A^\nu-g^{\nu\rho}\partial_{\rho}A^\mu,$$
 * we find

SI problems?
As per my comments on http://en.wikipedia.org/wiki/Electromagnetic_four-potential and http://xxx.lanl.gov/abs/physics/0401067 I believe that the defintion of the vector potential in SI units is incorrrect. It should be:
 * $$A^a = \left(\frac{\Phi}{c}, \vec{A} \right) $$
 * $$A_a = \eta_{ab} A^b = \left( -\frac{\Phi}{c}, \vec{A} \right) $$

I will let this stand for comments before editing the page.


 * Looks like the inner product η has signature (-,+,+,+) in Wikipedia. It wasn't clear enough when I changed this page. --TxAlien 18:26, 9 December 2006 (UTC)

All article in SI
I proposed that all article is in the SI, including the lagrangian. The lagragian density for electrodynamics in SI is: $$\mathcal{L}=-\frac{1}{4 \mu_0}F^{\alpha\beta}F_{\alpha\beta},\,$$ i.e., there is the constant $$\mu_0$$ in the lagrangian. —The preceding unsigned comment was added by Lseixas (talk • contribs) 17:55, 24 December 2006 (UTC).
 * Sorry for unsigned my comment. Lseixas 00:03, 27 December 2006 (UTC)

A lot of work will be required, then. On dimensional grounds, then, the following should be noted.

The units for the various quantities are: φ: Volt = Weber/second, 𝐀: Weber/meter, 𝐁: Weber/meter², 𝐄: Webers/(meter·second) = Volt/meter. If using the coordinates (x⁰, x¹, x², x³) = (t, x, y, z), the respective units being t: second, x, y, z: meter; then A_μ dx^μ = 𝐀·d𝐫 - φ dt, where d𝐫 = (dx, dy, dz). Therefore, A₀ = -φ and (A₁, A₂, A₃) = 𝐀. With this coordinate choice, ∂₀ = ∂/∂t, (∂₁, ∂₂, ∂₃) = (∂/∂x, ∂/∂y, ∂/∂z) = ∇. Therefore (F₁₀, F₂₀, F₃₀) = ∇A₀ - ∂𝐀/∂t = -∇φ - ∂𝐀/∂t = 𝐄 and (F₂₃, F₃₁, F₁₂) = ∇×𝐀 = 𝐁.

If using x⁰ = ct, then A₀ = -φ/c, (F₁₀, F₂₀, F₃₀) = 𝐄/c. In all cases, the two-form ½ F_μν dx^μ ∧ dx^ν = 𝐁·d𝐒 - 𝐄·d𝐫∧dt, where d𝐒 = (dy∧dz, dz∧dx, dx∧dy)

The Lagrangian *density* (note the word: DENSITY) is a tensor density, not a tensor! That means it has an extra factor of √|g| and is expressed as 𝔏 = -¼ k √|g| g^{μρ} g^{νσ} F_{μν} F_{ρσ} = -¼ k √|g| F_{μν} F^{μν}, for some constant k. What the constant k is, is independent of what scaling and sign are adopted on the metric g_{μν}. The Euler-Lagrange equations that come out of this Lagrangian density are ∂_ν 𝔊^{μν} = 𝔍^μ, if in the presence of a current source 𝔍^μ. For this Lagrangian density 𝔍^μ = 0 and the "response field" 𝔊^{μν} = k √|g| F^{μν}.

The units for 𝔏 are action per space-time volume. With the convention x⁰ = t, that would be (kilogram·meter²/second)/(second·meter³) = kilogram/(meter·second²). The units for each product 𝔊^{μν} F_{μν} multiply out to this. So, noting that Coulomb·Weber is equal to the unit for action (Coulomb and Weber are conjugate), then (𝔊⁰¹, 𝔊⁰², 𝔊⁰³) has units Coulomb/meter² - the units for the electric displacement 𝐃; while (𝔊²³, 𝔊³¹, 𝔊¹²) has units Coulomb/(meter·second) = Amp/meter, since Amp = Coulomb/meter. Those are the units for 𝐇.

If setting the response field as 𝐃 = (𝔊⁰¹, 𝔊⁰², 𝔊⁰³) and 𝐇 = (𝔊²³, 𝔊³¹, 𝔊¹²), then the Euler-Lagrange equations give rise to 𝔍⁰ = ∂_ν 𝔊^{0ν} = ∇·𝐃 = ρ, the charge density (with units Coulomb/meter³), and 𝔍^i = ∂_ν 𝔊^{iν} = ∂_0 𝔊^{i0} + ∂_j 𝔊^{ij} + ∂_k 𝔊^{ik} = -∂D^i/∂t + ∂_j H_k - ∂_k H_j = (-∂𝐃/∂t + ∇×𝐇)^i = (𝐉)^i, as (i,j,k) range cyclically over (1,2,3), (2,3,1) and (3,1,2). Therefore, 𝐉 = (𝔍¹, 𝔍², 𝔍³), the current density, which has units Coulomb/(meter²·second) = Amp/meter².

The corresponding differential forms are ½ 𝔊^{μν} ∂_ν ˩ (∂_μ ˩ dt∧dx∧dy∧dz) = 𝐃·d𝐒 - 𝐇·d𝐫∧dt and 𝔍^μ ∂_μ ˩ dt∧dx∧dy∧dz = ρ dV - 𝐉·d𝐒∧dt, where dV = dx∧dy∧dz.

To find what k is, let β = g₀₀ and -α = g₁₁ = g₂₂ = g₃₃. This is the Minkowski metric up to some scale α, where β = α c² and g_{μν} = 0 if μ ≠ ν. Then √|g| = √|α³β| = α² √|β/α| = α² c. For the inverse metric, we have g⁰⁰ = 1/β, g¹¹ = g²² = g³³ = -1/α and g^{μν} = 0 if μ ≠ ν. Therefore, g⁰⁰ g¹¹ √|g| = α²c/(-αβ) = -αc/β = -1/c and g¹¹ g²² √|g| = α²c/α² = c. Thus (𝔊⁰¹, 𝔊⁰², 𝔊⁰³) = k 𝐄/c and (𝔊²³, 𝔊³¹, 𝔊¹²) = k c 𝐁. For the vacuum, the relevant relations are 𝐃 = ε₀ 𝐄 and 𝐁 = μ₀ 𝐇. Therefore, k = ε₀c = 1/(μ₀c).

Therefore, the Lagrangian density can be written as 𝔏 = -¼ ε₀c √|g| F_{μν} F^{μν}. Substituting for √|g|, the coefficient ε₀c √|g| can also be written as ε₀c √|g| = ε₀c α²c = α²/μ₀, since ε₀c² = 1/μ₀, leading to the following alternate expression: 𝔏 = -¼ α²/μ₀ F_{μν} F^{μν}. Then, either convention g₁₁ = ±1, g₀₀ = ∓c² leads to ε₀c√|g| = 1/μ₀, which is what the article quoted. The values for F^{μν}, unlike those for 𝔊^{μν}, will depend on the scaling of the metric: F^{μν} = 𝔊^{μν}/(ε₀c√|g|) = μ₀𝔊^{μν}/α².

If using x⁰ = ct, instead of x⁰ = t, then c𝐃 = (𝔊⁰¹, 𝔊⁰², 𝔊⁰³) and 𝔍⁰ = ρc and g₀₀ = -g₁₁ = α, √|g| = α² and g⁰⁰ = -g¹¹ = 1/α, instead, and F^{μν} = μ₀c𝔊^{μν}/α². In that case, the coefficient ε₀c √|g| in the Lagrangian density reduces to ε₀cα² = α²/(μ₀c).

The article is getting different conventions mixed up with each other, because it's not properly contextualizing everything; so this is about more than just the choice of SI units. The description above provides the context for everything. — Preceding unsigned comment added by 65.29.226.169 (talk) 07:19, 1 September 2023 (UTC)

About the independence of the components
It is said that the tensor has "Six independent components: which are the three spatial components of the electric field (Ex, Ey, Ez) and magnetic field (Bx, By, Bz)." Ain't those connected by the Maxwell equations ? And in terms of the vector potential, can there be more than 4 independent components ?

Examples of transformations
I have removed the examples section from electromagnetic field (as it certainly doesn't belong there) and placed it here for now. I hope to incorporate it into this article somehow. Here is the section:

Examples
Here are two examples of transformations of the field tensor. Both are transformations due to observers moving with repect to each other on the x-axis. The first transformation shows how the unprimed observer can see an electric field, designated $$E$$, only in the positive z-axis direction, transform such that the primed observer, moving with velocity $$\beta = \frac{v}{c}$$ along the x-axis with respect to the unprimed observer, sees both electric and magnetic fields.


 * $$F^{\mu \nu} = \begin{vmatrix} 0 & 0 & 0 & \frac{E}{c} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ -\frac{E}{c} & 0 & 0 & 0 \end{vmatrix}$$
 * $$\Lambda^\sigma_\mu = \Lambda^\tau_\nu = \begin{vmatrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{vmatrix}$$
 * $$F'^{\sigma \tau} = \Lambda^\sigma_\mu \Lambda^\tau_\nu F^{\mu \nu} $$

So, in the above, it's clear that the field tensor term is zero everywhere, except where $$\mu = 0, \nu = 3$$ or where $$\mu = 3, \nu = 0$$. The results are as below.


 * $$F'^{0 0} = \Lambda^0_\mu \Lambda^0_\nu F^{\mu \nu} = 0$$
 * $$F'^{0 1} = \Lambda^0_\mu \Lambda^1_\nu F^{\mu \nu} = 0$$
 * $$F'^{0 2} = \Lambda^0_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{0 3} = \Lambda^0_\mu \Lambda^3_\nu F^{\mu \nu} = \gamma \left ( \frac{E}{c} \right )$$


 * $$F'^{1 0} = \Lambda^1_\mu \Lambda^0_\nu F^{\mu \nu} = 0$$
 * $$F'^{1 1} = \Lambda^1_\mu \Lambda^1_\nu F^{\mu \nu} = 0$$
 * $$F'^{1 2} = \Lambda^1_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{1 3} = \Lambda^1_\mu \Lambda^3_\nu F^{\mu \nu} = -\gamma \beta \left ( \frac{E}{c} \right )$$


 * $$F'^{2 0} = \Lambda^2_\mu \Lambda^0_\nu F^{\mu \nu} = 0$$
 * $$F'^{2 1} = \Lambda^2_\mu \Lambda^1_\nu F^{\mu \nu} = 0$$
 * $$F'^{2 2} = \Lambda^2_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{2 3} = \Lambda^2_\mu \Lambda^3_\nu F^{\mu \nu} = 0$$


 * $$F'^{3 0} = \Lambda^3_\mu \Lambda^0_\nu F^{\mu \nu} = -\gamma \left ( \frac{E}{c} \right )$$
 * $$F'^{3 1} = \Lambda^3_\mu \Lambda^1_\nu F^{\mu \nu} = \gamma \beta \left ( \frac{E}{c} \right )$$
 * $$F'^{3 2} = \Lambda^3_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{3 3} = \Lambda^3_\mu \Lambda^3_\nu F^{\mu \nu} = 0$$

The result, in matrix form, looks like this:


 * $$F'^{\sigma \tau} = \begin{vmatrix} 0 & 0 & 0 & \gamma \frac{E}{c} \\ 0 & 0 & 0 & -\gamma \beta \frac{E}{c} \\ 0 & 0 & 0 & 0 \\ -\gamma \frac{E}{c} & \gamma \beta \frac{E}{c} & 0 & 0 \end{vmatrix}$$

As can be seen, if one compares this result with the general form of the field tensor shown above, two things have occurred. Firstly, the primed observer sees the electrical field as being stronger than the unprimed observer. Secondly, the primed observer sees a magnetic field in the positive y-axis direction that the unprimed observer does not see. This hints at the reason that magnetism is sometimes called a relativistic phenomenon.

However, it is not true that all Lorentz transformations on a field tensor with only an electric component will produce a magnetic component. The following example illustrates this, with the same two observers as above, but with the electric field being in the positive x-axis direction instead of the positive z-axis direction. This direction is in the same direction of the relative velocity between the two observers.


 * $$F^{\mu \nu} = \begin{vmatrix} 0 & \frac{E}{c} & 0 & 0 \\ -\frac{E}{c} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{vmatrix}$$
 * $$\Lambda^\sigma_\mu = \Lambda^\tau_\nu = \begin{vmatrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{vmatrix}$$
 * $$F'^{\sigma \tau} = \Lambda^\sigma_\mu \Lambda^\tau_\nu F^{\mu \nu} $$

So, in the above, it's clear that the field tensor term zero everywhere except where $$\mu = 0, \nu = 1$$ or where $$\mu = 1, \nu = 0$$. The results are as below.


 * $$F'^{0 0} = \Lambda^0_\mu \Lambda^0_\nu F^{\mu \nu} = \Lambda^0_0 \Lambda^0_1 F^{0 1} + \Lambda^0_1 \Lambda^0_0 F^{1 0} = -\gamma^2 \beta \frac{E}{c} + \gamma^2 \beta \frac{E}{c} = 0$$
 * $$F'^{0 1} = \Lambda^0_\mu \Lambda^1_\nu F^{\mu \nu} = \Lambda^0_0 \Lambda^1_1 F^{0 1} + \Lambda^0_1 \Lambda^1_0 F^{1 0} = \gamma^2 \frac{E}{c} - \gamma^2 \beta^2 \frac{E}{c} = \frac{E}{c}$$
 * $$F'^{0 2} = \Lambda^0_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{0 3} = \Lambda^0_\mu \Lambda^3_\nu F^{\mu \nu} = 0$$


 * $$F'^{1 0} = \Lambda^1_\mu \Lambda^0_\nu F^{\mu \nu} = \Lambda^1_0 \Lambda^0_1 F^{0 1} + \Lambda^1_1 \Lambda^0_0 F^{1 0} = \gamma^2 \beta^2 \frac{E}{c} - \gamma^2 \frac{E}{c} = -\frac{E}{c}$$
 * $$F'^{1 1} = \Lambda^1_\mu \Lambda^1_\nu F^{\mu \nu} = \Lambda^1_0 \Lambda^1_1 F^{0 1} + \Lambda^1_1 \Lambda^1_0 F^{1 0} = -\gamma^2 \beta \frac{E}{c} + \gamma^2 \beta \frac{E}{c} = 0$$
 * $$F'^{1 2} = \Lambda^1_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{1 3} = \Lambda^1_\mu \Lambda^3_\nu F^{\mu \nu} = 0$$


 * $$F'^{2 0} = \Lambda^2_\mu \Lambda^0_\nu F^{\mu \nu} = 0$$
 * $$F'^{2 1} = \Lambda^2_\mu \Lambda^1_\nu F^{\mu \nu} = 0$$
 * $$F'^{2 2} = \Lambda^2_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{2 3} = \Lambda^2_\mu \Lambda^3_\nu F^{\mu \nu} = 0$$


 * $$F'^{3 0} = \Lambda^3_\mu \Lambda^0_\nu F^{\mu \nu} = 0$$
 * $$F'^{3 1} = \Lambda^3_\mu \Lambda^1_\nu F^{\mu \nu} = 0$$
 * $$F'^{3 2} = \Lambda^3_\mu \Lambda^2_\nu F^{\mu \nu} = 0$$
 * $$F'^{3 3} = \Lambda^3_\mu \Lambda^3_\nu F^{\mu \nu} = 0$$

In the above, the following relation was used less explicitly.


 * $$\gamma^2 - \gamma^2 \beta^2 = \frac{1}{\sqrt{1-\beta^2}^2} - \frac{\beta^2}{\sqrt{1-\beta^2}^2} = \frac{1 - \beta^2}{1 - \beta^2} = 1$$

The result, in matrix form, looks like this:


 * $$F'^{\mu \nu} = \begin{vmatrix} 0 & \frac{E}{c} & 0 & 0 \\ -\frac{E}{c} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{vmatrix}$$

Not only does no magnetic component show up, but the whole tensor is unchanged.

Humor?
In the section The field tensor and relativity, somebody had written: "The elegance of these equations stems from the simple replacing of partial with covariant derivatives, a practice sometimes referred to in the parlance of GR as 'replacing partial with covariant derivatives'. "

I assume that this is intended to be some form of ironic humor? Geoffrey.landis 13:49, 17 July 2007 (UTC)

Skipping some steps?
In "Relation to classical electromagnetism", I feel like the transition between eq 5 and 6 (just before "That term in the parenthesis is just the field tensor") lacks justification. One problem is that the same indices mu and nu are double-used, both in the Lagrangian density and in the Euler-Lagrange equation. They should really be separate indices, and then only made equal by noting that only the terms in the sum for the Lagrangian which have the same (or switched) indices contribute. —Preceding unsigned comment added by 71.184.189.66 (talk) 08:08, 11 February 2008 (UTC) i just read ot too. hat a hoot —Preceding unsigned comment added by 209.184.115.181 (talk) 20:07, 27 March 2008 (UTC)

upstairs & downstairs indices
Shouldn't $$F^{\mu \nu} F_{\mu \nu}$$ be $$2 (B^2 + E^2)$$ instead of $$2 (B^2 - E^2)$$? (using $$\hbar = c = 1$$)

I thought the only difference between the components of $$F^{\mu \nu}$$ and $$F_{\mu \nu}$$ are the signs (+-) of the E components, regardless of the metric being -+++ or +---

Immer in Bewegung (talk) 03:08, 7 September 2008 (UTC)


 * Well $$F^{\mu \nu} F_{\mu \nu} = -F^{\mu \nu} F_{\nu \mu} = -Tr(M_1 M_2)$$ where $$M_1$$ and $$M_2$$ are the matrices given in the first two equations of the article. When I plugged that in just now, I got the 2(B^2-E^2) given in the text. Would you argue that those matrices are not written down correctly? Or do you disagree with this calculation I just did? :-) --Steve (talk) 04:12, 7 September 2008 (UTC)


 * The latter. Should be $$- \text{Tr} (M_1 M_2 ^T)$$
 * Furthermore, the Lagrangian density that describes the Meissner effect / spontaneous symmetry breaking has a term $$\cfrac{1}{4} F_{\mu \nu} F^{\mu \nu}$$ in it which should represent the energy due to the E & M fields. ie. $$\frac{1}{2} (E^2 + B^2)$$. So I think it's a sum, not a difference between the squares of the fields. --Immer in Bewegung (talk) 22:26, 7 September 2008 (UTC)


 * There's plenty of references showing that the signs of this Lagrangian density are opposing (its not the energy density related to the stress tensor which has (E^2 + c^2 B^2) in it). Without invoking trace arguments (which are not obvious to me) or anything else fancy here is a demonstration in terms of the expressions in the article


 * $$F_{\mu\nu} = \begin{bmatrix}

0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & B_z & -B_y \\ E_y/c & -B_z & 0 & B_x \\ E_z/c & B_y & -B_x & 0 \end{bmatrix} $$ Can somebody explain this step here??? -130.149.58.223 (talk) 14:56, 18 November 2010 (UTC)
 * $$F^{\mu\nu} = \begin{bmatrix}

0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & B_z & -B_y \\ -E_y/c & -B_z & 0 & B_x \\ -E_z/c & B_y & -B_x & 0 \end{bmatrix} $$

so
 * $$F_{\mu\nu}F^{\mu\nu} = \begin{bmatrix}

0 & -(E_x)^2/c^2 &-(E_y)^2/c^2 & -(E_y)^2/c^2 \\ -(E_x)^2/c^2 & 0 & (B_z)^2 & (-B_y)^2 \\ -(E_y)^2/c^2 & (-B_z)^2 & 0 & (B_x)^2 \\ -(E_z)^2/c^2 & (B_y)^2 & (-B_x)^2 & 0 \end{bmatrix} $$

Note that $$F^{\mu\nu}F_{\mu\nu}$$ is not a matrix product, but an element by element product with implied sum. Doing that sum one has:

$$ 2( -\mathbf{E}^2/c^2 + \mathbf{B}^2 ) $$

I have fixed the signs in the article.

Peeter.joot (talk) 02:38, 9 September 2008 (UTC)


 * You're right. I stand corrected. $$- \text{Tr} (M_1 M_2 ^T)$$ still shows 2(B^2 - E^2)
 * --Immer in Bewegung (talk) 04:31, 10 September 2008 (UTC)
 * I solved my problem now! (I am the person who asked how to get from $$ F^{\mu\nu}$$ to $$ F_{\mu\nu}$$).
 * The problem is, that one naivly calculates $$F_{\mu\nu}=\eta_{\alpha\nu}F^{\alpha\beta}\eta_{\mu\beta}$$, because one is not used to it, because one has done only exercises with vectors: $$V_{\mu}=\eta_{\alpha\mu}V^{\alpha}$$. Here is only one index ($$\alpha$$), so nothing can go wrong. But! If one has matrices or higher tensors, the order of the indices is very important!!! With high probability one writes down the indices in the wrong order, as I did.
 * The correct order is here: $$F_{\mu\nu}=\eta_{\alpha\nu}F^{\beta\alpha}\eta_{\mu\beta}$$. If one contracts the metrics with F one sees immediately, that this order of indices is the right one and the formula I wrote down first is wrong.
 * Now one can calculate all components of $$F_{\mu\nu}$$ and this leads to the correct sign in the formula of $$ F^{\mu\nu}F_{\mu\nu}$$.
 * Thank you for the attention! --188.103.109.95 (talk) 20:03, 18 November 2010 (UTC)

quantum Lagrangian.
Most of the article uses $$\partial_{\alpha}$$ for the partial derivaties $$\frac{\partial}{\partial x^{\alpha}}$$ but in the quantum Lagrangian is written with $$D_{\alpha}$$. I think that these are intended to be equivalent (and if so think the article probably ought to use consistent notation).

Can anybody verify?

Peeter.joot (talk) 17:22, 7 September 2008 (UTC)


 * I thought $$D_\alpha$$ is meant to be a covariant derivative... ie. $$D_\alpha = \partial_\alpha + ie A_\alpha$$ --Immer in Bewegung (talk) 22:57, 7 September 2008 (UTC)

Metric tensor definition?
In the article http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism

the matrix of the metric tensor is written:


 * $$F^{\alpha\beta} = \left(

\begin{matrix} 0 & -E_x &  -E_y &  -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{matrix} \right) .$$

whereas in this article:


 * $$F^{\mu\nu} = \begin{bmatrix}

0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & B_z & -B_y \\ -E_y/c & -B_z & 0 & B_x \\ -E_z/c & B_y & -B_x & 0 \end{bmatrix} $$

Other than the different stylistic issues, the signs of the electric field coordinates are inverted despite both claiming to be associated with a -+++ metric. One of them looks wrong. What I get working out the tensor in terms of the bivector equation:



F = E + icB = E^i \gamma_{i0} + \gamma_{0123} c B^{i} \gamma_{i0} $$ where

F^{\mu\nu} = F \cdot \gamma^{\nu\mu} $$

agrees with the covariant article, but it's fairly easy to get indexes messed up in these tensor expressions, so perhaps somebody familiar with the traditional non-clifford-algebra formulation of the em tensor equations can comment.

Peeter.joot (talk) 12:39, 9 September 2008 (UTC)

Sign Errors!
Given the choice of metric: $$g_{\mu\nu}\sim \left( \begin{array}{cccc} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{array}\right)$$

and given the raised index definition:

$$A^\mu\sim (A^0,A^1,A^2,A^3) = (c^{-1}\phi,\mathbf{A})$$ then the article should read: $$A_\mu\sim (A_0,A_1,A_2,A_3) = (c^{-1}\phi,-\mathbf{A})$$

This in turn changes the form of the Electromagnetic tensor.

$$F_{01} = \partial_0 A_1 - \partial_1 A_0 = \partial_{ct}(-\mathbf{A}_x) - \partial_x(c^{-1}\phi) = c^{-1} E_x$$

$$F_{23} = \partial_2 A_3 - \partial_3 A_2 = \partial_y(-\mathbf{A}_z) - \partial_z(-\mathbf{A}_y)=[\nabla\times\mathbf{A}]_x=B_x$$

Using $$F_{row\, column}$$ convention this gives:

$$F_{\mu\nu} \sim \left( \begin{array}{cccc} 0&c^{-1}E_x&c^{-1}E_y&c^{-1}E_z\\ -c^{-1}E_x&0&B_z&-B_y\\ -c^{-1}E_y&-B_z&0&B_x\\ -c^{-1}E_z&B_y&-B_x&0 \end{array}\right)$$

and hence:

$$F^{\mu\nu} \sim \left( \begin{array}{cccc} 0&-c^{-1}E_x&-c^{-1}E_y&-c^{-1}E_z\\ c^{-1}E_x&0&B_z&-B_y\\ c^{-1}E_y&-B_z&0&B_x\\ c^{-1}E_z&B_y&-B_x&0 \end{array}\right)$$

These are the negatives of what appears in the article. I'm going to do some double checking and then edit it.

This of course does not change the quadratic $$ F^{\mu\nu}F_{\mu\nu}$$.

Regards, James Baugh (talk) 18:09, 30 December 2008 (UTC)

P.S. It looks to me like the metric got changed without changing the definitions of $$F_{\mu\nu}$$ and $$A_\mu$$. I'll consider editing to correct and put in notes on what changes with each metric convention. Before I do I want to work through the full derivation and that of the Lorentz-Coulomb forces being very careful of signs.

Regards, James Baugh (talk) 18:28, 30 December 2008 (UTC)


 * As you can see from previous talk-page discussions above (which you should read if you haven't), the signs have been debated on a number of occasions in the past. I haven't checked your work, but if the signs are wrong I wouldn't be surprised. I'm very happy for you to be checking. I think you'd be well-advised to not only check the details yourself, but also find reliable sources to back yourself up and cite them in the article. (Try google book search, for example, that way everyone else can check too). That will somewhat (but not entirely) dissuade people from changing it back erroneously in the future. I'd also encourage you, while you're at it, to lend your expertise to covariant formulation of classical electromagnetism and other articles whose sign conventions are tied to this one. In fact, I think there's some sort of sign-error debate at Talk:covariant formulation of classical electromagnetism going on right now. Best wishes! :-) --Steve (talk) 01:31, 31 December 2008 (UTC)
 * Yes. I started in www.physicsforums.com and in trying to answer that post looked at covariant formulation of classical electromagnetism and thence here.  I've been playing in my own sandbox User:Jambaugh/Sandbox/EM trying to get a consistent parallel writeup covering both topics.  I'd like to include sufficient footnotes to cover all the possible convention choices (2 or 3 independent ones)  I think it might be best, long-term to merge the two articles since they are so interdependent. I'll see what I can get done before the semester starts.  The main reference I started with is
 * W. Rindler, Introduction to Special Relativity, 2nd edition, Oxford Science Publications, 1991, ISBN 0-19-853952-5.
 * which also includes an extra factor of c in the E-M tensor.
 * Regards, James Baugh (talk) 15:48, 3 January 2009 (UTC)

Factor of 2 Error
I may be mistaken, but I'm fairly certain that there is a factor of 2 error in the identity:
 * $$ \epsilon_{\alpha\beta\gamma\delta}F^{\alpha\beta} F^{\gamma\delta} = \frac{4}{c} \left( \vec B \cdot \vec E \right) = \mathrm{invariant} \,$$

I've worked it out several times, and I think it should be:
 * $$ \epsilon_{\alpha\beta\gamma\delta}F^{\alpha\beta} F^{\gamma\delta} = \frac{8}{c} \left( \vec B \cdot \vec E \right) = \mathrm{invariant} \,$$

Hopefully someone is willing to check me on this. Thanks! Also, please forgive me if I'm not following protocol correctly. This is my first time ever editing something on Wikipedia.WikiCamBob (talk) 16:09, 24 February 2009 (UTC)

clean up
Main changes:


 * clean up
 * add the many missing links that should be there, a reader will have no clue where to read further (will they know that "det" means determinant of a matrix?? will they know what a differential 2-form is??)
 * re-word some of the text, it was prosy and hype defeating WP:NPOV, such as
 * "Hidden beneath the surface of this complex mathematical equation is an ingenious unification of Maxwell's equations for electromagnetism."......

F = q(E+v×B) ⇄ ∑ici 18:14, 9 April 2012 (UTC)

Different sign conventions in different textbooks
Wheeler, Misner, Thorne in their "Gravitation" and Griffiths in his "Introduction to Electrodynamics" use different sign in definition of $$F^{\mu \nu}$$ than in this article.

Look at page 99 in Wheeler, Misner, Thorne "Gravitation" (1973) and page 536 in Griffiths "Introduction to Electrodynamics" (2nd edition).

The one that is used here is from Landau and Lifshitz, "Classical Theory of Fields".

However, the tensor with mixed components $$F^{\mu}_{\; \ \nu}$$ is the same in both texbooks:

$$ F^{\mu}_{\; \ \nu}(Wheeler, Misner, Thorne) = F^{\mu}_{\; \ \nu}(Landau, Lifshitz) $$

One can view this difference as a consequence of difference in signature ({-,+,+,+} or {+,-,-,-}). If you want the formula for Lorentz force $$ \frac{d p^\alpha}{d \tau} = q F^{\alpha}_{\; \ \beta} u^\beta$$ to look the same in both signatures, then you should start with $$ F^{\mu}_{\; \ \nu} $$ and only then 	lower and raise indices with metric tensor.

Extra sign errors
An additional sign error appears in the tensor expression for the inhomogeneous Maxwell equation $$\partial_{\beta} F^{\alpha\beta} = \mu_0 J^{\alpha}$$.

Given that we are using the $$F_{row\, column}$$ convention and the metric (+---), the correct expression should be

$$\partial_{\beta} F^{\alpha\beta} = -\mu_0 J^{\alpha}$$. OR $$\partial_{\alpha} F^{\alpha\beta} = \mu_0 J^{\beta}$$.

Either works and I prefer the latter because of the lack minus sign. I've changed it because it is consistent with the metric used in the rest of this article, and because I've personally made this sign mistake before. Fastman99 (talk) 02:44, 17 February 2013 (UTC)

Units
In a very tiny font, this article indicates that it uses SI units. I suppose that is fine, but the EM tensor is slightly nicer in Gaussian units, were there are no /c in the E components. Would it be too much to explain this, somewhere in the article? Gah4 (talk) 18:56, 10 May 2019 (UTC)