# What is Field tensor: Definition and 29 Discussions

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely.

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1. ### A Relation of Electromagnetic Field & Field Tensor

hello, 1. according to Robert Wald, General Relativity, equation (4.2.22) the magnetic field as measured by an observer with 4-velocity ## v^b ## is given by ## B_a = - \frac {1}{2} {ϵ_{ab}}^{cd} F_{cd} v^b ## where ## {ϵ_{ab}}^{cd}##, the author says, is the totally antisymmetric tensor (for...
2. ### Tensor help -- Write out this tensor in a simplified sum

I managed to write $$F_{\alpha\beta}F^{\alpha\gamma}=F_{0\beta}F^{0\gamma}+F_{i\beta}F^{i\gamma}$$ where $$i=1,2,3$$ and $$\gamma=0,1,2,3=\beta$$. How do I proceed?
3. ### I Proving Antisymmetry of Electromagnetic Field Tensor with 4-Force

I've already made a post about this topic here, but I realized that I didn't understand the explanation on that post. in Chapter 7 of Rindler's book on relativity, in section about electromagnetic field tensor, he states that _and introducing a factor 1/c for later convenience, we can ‘guess’...
4. ### Antisymmetry of the electromagnetic field tensor

I am trying to answer exercise 5 but I am not sure I understand what the hint is implying, differentiate with respect to ##p_\alpha## and ##p_\beta##, I have done this but nothing is clicking. Also, what is the relevance of the hint "the constraint ##p^\alpha p_\alpha = m^2c^2## can be ignored...
5. ### A How to prove this property of the Dual Strength Field Tensor?

Hi, I've found this property of Strenght Field Tensors: $$F_{\mu}^{\nu}\tilde{F}_{\nu}^{\lambda}=-\frac{1}{4}\delta_{\mu}^{\lambda}F^{\alpha\beta}\tilde{F}_{\alpha\beta}$$ Where $$\tilde{F}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}F^{\alpha\beta}, \qquad \varepsilon_{0123}=1$$ I've...
6. ### I Lorenz gauge, derivative of field tensor

Fμν = ∂μAν- ∂νAμ ∂μFμν = ∂2μAν - ∂ν(∂μAμ) = ∂2μAνWhy ∂ν(∂μAμ) and not ∂μ∂νAμ ? And why does ∂ν(∂μAμ) drop out? thank you
7. ### I Deriving Maxwell's Equations from Field Tensor (Griffith 4ed)

Hello, I am reading Griffith's "Introduction to Electrodynamics" 4ed. I'm in the chapter on relativistic electrodynamics where he develops the electromagnetic field tensor (contravariant matrix form) and then shows how to extract Maxwell's equations by permuting the index μ. I am able to...
8. ### I Relation between Poincare matrix and electromagnetic field t

We know that Poincare matrix which is 0 Kx Ky Kz ( -Kx 0 Jz -Jy ) describes the boost and rotation is very similar to -Ky -Jz 0 Jx...
9. ### Maxwell's Equations from EM field tensor

Hello, I have derived two Maxwell's equations from the electromagnetic field tensor but I have a problem understanding the second formula, which is: \partial_{\lambda} F_{\mu\nu} + \partial_{\mu} F_{\nu\lambda}+\partial_{\nu} F_{\lambda\mu} =0 I have a few questions to help me start: 1) Is...
10. ### EM: Vector potential vs. Field tensor: Which is fundamental?

In my lecture we were discussing the Lagrangian construction of Electromagnetism. We built it from the vector potential ##A^\mu##. We introduced the field tensor ##F^{\mu \nu}##. We could write the Langrangian in a very short fashion as ##-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}## In the end we...
11. ### Gauge Field Tensor from Wilson Loop

It is possible to introduce the gauge field in a QFT purely on geometric arguments. For simplicity, consider QED, only starting with fermions, and seeing how the gauge field naturally emerges. The observation is that the derivative of the Dirac field doesn't have a well-defined transformation...
12. ### Lorentz transformation matrix applied to EM field tensor

In a recent course on special relativity the lecturer derives the Lorentz transformation matrix for the four vector of position and time. Then, apparently without proof, the same matrix is used to transform the EM field tensor to the tensor for the new inertial frame. I am unclear whether it...
13. ### How do Maxwells equations result from the field tensor?

Hi, I've been trying to solve problem 2.1 a in Peskin and schroeder, an introduction to QFT. The problem is to derive Maxwells equations for free space, which I have almost managed to do, using the Euler- lagrange euqation And the definition of the field tensor as F_{μv} = d_μ A_v - d_v A_μ...
14. ### Confusion Field Tensor and derivation of Maxwell's equations

Hi as I'm reading http://www.maths.tcd.ie/~cblair/notes/432.pdf at page 13 I see that he states that the covariant and contravariant field tensors are different. But how can that be? Aren't they related by F_{\mu \nu} = \eta_{\nu \nu'} \eta_{\mu \mu '} F^{\mu ' \nu '} ? and is not the...
15. ### How Can Wedges Help Simplify Tensor Notation for the Electromagnetic Field?

This isn't actually coursework, I'm doing some studying on my own. These are my very preliminary attempts to wrangle with tensor notation, so please be patient with me. I'm trying to get the components of the electromagnetic field tensor from \partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu} But...
16. ### Electromagnetic Field Tensor in Curvilinear Coordinates

How to express electromagnetic field tensor in curvilinear coordinates, that is given a curvilinear coordinates (t,\alpha,\beta,\gamma) with metric tensor as follows: n_{\mu \nu }= \left[ \begin{array}{cccc}h_0^2& 0 & 0 & 0 \\ 0 & -h_1^2 & 0 & 0 \\ 0 & 0 & -h_2^2 & 0 \\ 0 & 0 & 0 & -h_3^2...
17. ### What kind of tensor is the electromagnetic field tensor?

The covariant form of the Lorentz force can be written as m \ddot x^\mu =q F^{\mu \nu} \dot x_\nu and such a relation should prove by the quotient rule that F is indeed a tensor. But what kind of tensor is it? One can show that it transforms from an unprimed to a primed system like F'^{\mu...
18. ### Construct field tensor an dual tensor

Homework Statement straight wire along z axis carries charge density \lambda traveling in +z direction at speed v. construct field tenor and dual at point (x,0,0_ Homework Equations E=(2\lambda /4\pi\epsilono r)r^ B=(\muo I/2\pir)\phi^ The Attempt at a Solution I just don't get...
19. ### Transforming the EM field tensor

Homework Statement So, I'm asked to find how the fields (E, and B) transform by transforming the electromagnetic field tensor. The transformations are a) rotation around y axis, and b) boost along z. Homework Equations F'_{\mu\nu}=\Lambda^\mu_\rho \Lambda^\nu_\sigma F_{\rho\sigma} The...
20. ### Gauss' Law as a derivative of the electromagnetic field tensor

Homework Statement Prove \nabla \bullet E =4 \pi \rho from \partial_{\beta}F^{\alpha \beta}=4 \pi J^{\alpha} where J^{\alpha}=(\rho, J^{1}, J^{2}, J^{3}). Homework Equations We are given that F_{\alpha \beta} is 0~~~~E_x~~~E_y~~~E_z -E_x~~~0~~~~-B_z~~B_y -E_y~~B_z~~~~0~~~-B_x...
21. ### Rank 3 tensor created by taking the derivative of electromagnetic field tensor

Homework Statement Show that the rank 3 tensor S_{\alpha \beta \gamma}=F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta} is completely antisymmetric. I just don't feel comfortable doing this stuff at all. Each of the three terms seems like they should be...
22. ### Motivation for electromagnetic field tensor

Is there any deep reason for introducing the electromagnetic field tensor other than the fact that Maxwell's equations can be written in a very succinct form in terms of it? Would it be possible to write down a lagrangian involving a normal kinetic term for A^{\mu} that reproduces the physics...
23. ### Dual of electromagnetic field tensor.

What is the importance of dual electromagnetic field tensor? Generally this is not included in the action. What will be the advantage/disadvantage if I include terms like F_{\mu \nu}\tilde{F}^{\mu \nu}, \tilde{F}_{\mu \nu}\tilde{F}^{\mu \nu} in the action? (The tilde denotes the dual tensor.)...
24. ### Weak Field Tensor: Explained by Ben

Hi, After actually struggling to find anything relevant in books/google/this forum I'd really appreciate if someone could enlighten me: W^{\mu\nu} What is meant by this exactly? Can I write this down in matrix form like the EM tensor? Thanks Ben
25. ### Geometric Algebra: Signs of electromagnetic field tensor components?

[SOLVED] Geometric Algebra: Signs of electromagnetic field tensor components? Here's a question that may look like an E&M question, but is really just a geometric algebra question. In particular, I've got a sign off by 1 somewhere I think and I wonder if somebody can spot it. PF isn't...
26. ### Understanding and Proving the Antisymmetry of the Electromagnetic Field Tensor

Homework Statement Zwiebach 44 My book defines T_{\lambda \mu \nu} = \partial_{\lambda} F_{\mu \nu} + \partial_{\mu} F_{ \nu \lambda} + \partial_{\nu} F_{\lambda \mu } where F is the electromagnetic field tensor and says that it is identically zero due to Maxwell's. It then asks me to...
27. ### How do we express Maxwell's equations in terms of the field tensor?

Hi, could someone show me how to express \frac{\partial G^{\mu\nu}}{\partial x^\nu} = 0 which are Maxwell's equations, G is the dual tensor, in terms of the field tensor F: \frac{\partial F_{\mu\nu}}{\partial x^\lambda} + \frac{\partial F_{\nu\lambda}}{\partial x^\mu} + \frac{\partial...
28. ### Derivative of Electromagnetic Field tensor

Hello, i had a question (as many do on these forums, it appears ;). I know E_{k}=F_{0k} I also know B_{k}=(1/2)*(\epsilon_{klm}*F_{lm}) EM field tensor F^{uv} defined as: (I put tildes (~) into make it more like a matrix form) 0~~~~E_x~~~E_y~~~E_z -E_x~~~0~~~~-B_z~~B_y...
29. ### Why is the Electromagnetic Field Tensor in the QED Lagrangian?

The QED Lagrangian is given by \mathcal{L}_{\hbox{QED}} = \bar{\psi}(i\partial - m)\psi - \frac{1}{4}(F_{\mu\nu})^2 - e\bar{\psi}\gamma^\mu\psi A_\mu What is the purpose of the middle term. I know that it represents the energy of the E and B fields. However is that due to the external...