Talk:Special relativity: electromagnetism

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    Reviewer A

    This is an excellent continuation of Professor Rindler's sequence of articles on special relativity, which follows the kinematic and mechanics sections. As with the previous sections the material is presented clearly, concisely, and attractively. Before the article is posted I would recommend a thorough round of proofreading; I noticed some typos and a few problems with the cross-referencing of equations.

    Reviewer B

    This is an almost ideal example of a Scholarpedia article.

    I have greatly appreciated the introduction to tensors which are very often kind of a mystery for students who are not strong enough, or else have not time enough, to read Differential Geometry text books.

    I have read with great pleasure the study of the electromagnetic field tensor whose nature and properties are directly deduced from Lorentz’s force law, contrary to what is often done when one gets information on the electromagnetic field tensor, which is a physical quantity, starting from potentials in Lorenz’s gauge choice which, of course is just a non-physical mathematical trick.

    The analysis of the physical consequences of the transformation laws of the electromagnetic field is extremely interesting and shows a clear explicit example of the role of tensors in relativity. The same holds true for the analysis of the energy-momentum balance in the interaction of charges and fields.

    I would like however to add a simple suggestion for an improvement of the mathematical introduction to tensors. My point is that the word “field” being a dominant word through the whole article it would be natural to introduce in the simplest possible way the idea of “tensor field”. For example one could insert in the first chapter (whose title could be changed into “Four-Tensors and Tensor Fields”) in the last lines of the chapter after the sentence: “In general one-index tensors are called vectors. ” The few following lines:

    ++ Until now the elements of tensors correspond to constants, However one can easily generalize the idea of tensor identifying its elements with space-time functions, or, more generally functions of some parameter. In the case of space-time function tensors one defines Tensor Field a tensor \(A^{\mu…\rho}_{\sigma…\tau}(x,y,z,ct)\) which from S to S’ transforms according to the rule: \[ \tag{1} A_{\sigma^{'}\dots \tau^{'}}^{\mu^{'}\dots \rho^{'}}(x^{'}, y^{'}, z^{'}, ct^{'}) = A_{\sigma \dots \tau}^{\mu \dots \rho}(x, y, z, ct) \;p_{\mu}^{\mu^{'}} \dots p_{\rho}^{\rho^{'}}p_{\sigma^{'}}^{\sigma} \dots p_{\tau^{'}}^{\tau}.\]

    In particular a tensor field without indices is called Scalar Field. The gradient of a tensor field is a new tensor field with a further covariant index. Let us see this in the simplest case of a scalar field, if we write

    \[ \tag{2} \frac{\partial \phi}{\partial x^{\mu}}=\phi_{,\mu},\]

    we have, again by the chain rule,

    \[ \tag{3} \phi_{, \mu^{'}} = \phi_{,\mu}\,p^{\mu}_{\mu^{'}}. \]

    A different situation is when the tensor components depend on some parameter, the most interesting case being when the components of a given tensor associated with a particle depend on the proper time of the particle which is an invariant (scalar) quantity. In this case the derivative of a tensor with respect to a an invariant parameter is a tensor of the same type. ++

    This insertion corresponds to adding 3-4 text lines and one formula.

    The last lines above should replace the same comment inserted at the end of the chapter “Tensor Algebra (and Differentiation)”. The idea is that once one starts considering derivatives it is convenient to discuss the different situations altogether.

    A further possible advice could be the insertion on a short discussion), together with the metric tensor, of the anti-symmetric rank 4 tensor which is invariant in much the same way as the metric. This could allow putting into evidence the tensorial character of the invariants in Eq. (181). This belongs to the aims of the article. It is just an exercise and takes 2-3 more lines and a formula.

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