I'm an undergraduate who visited a course on differential manifolds and now I have the task to reformulate the maxwell equations in terms of differential forms. The most obvious question that arises first is: On what manifold are the forms defined?

Unfortunately, it seems to be difficult to get good answers. Some people say Minkowski space is just the $\mathbb R^4$, others completely ignore this point. I know that there are a lot of related questions, but the answers are either not satisfactory (especially for a mathematician studying differential geometry) or contain too advanced maths for an undergraduate.

I assume that it is not easy (or even impossible) to give a definite answer, but I'd like to hear some opinions on this. If you know of a reference where this is discussed in more detail, please let me know.


1 Answer 1


Minkowski space time is a four dimensional affine space whose space of translations is equipped with an indefinite symmetric non-degenerate metric with Lorentzian signature.

Every affine space has a natural smooth (real analytic) manifold structure induced by every arbitrarily fixed Cartesian coordinate system. That is the differentiable structure used in special relativity.

Cartesian coordinate systems associated to pseudo orthonormal bases are interpreted as coordinates at rest with inertial reference frames. Actually,

(1) one considers only basis whose temporal element is future oriented;

(2) two Cartesian systems related by means of a spatial 3-rotation or a space time translation or a combination of both are supposed to define the same inertial reference frame (inertial reference frame are equivalence classes).

  • $\begingroup$ Thank you very much for this answer, I am happy that my guess turned out right (I have a supervisor for the talk about maxwell equations and when I asked him if the Minkowski space is an affine space, he told me that he didn't know what an affine space is). I am not sure that I understand what you mean by a cartesian coordinate system: I learned that if we consider an affine space $M$ equipped with an n-dimensional translation vector space $V$ a tuple $(O,P_1,...,P_n)\in M^{n+1}$ is called coordinate system if $(\vec{OP_1},...,\vec{OP_n})$ is a basis of $V$. $\endgroup$
    – Filippo
    Nov 22, 2020 at 19:44
  • $\begingroup$ Is that the same definition you use? If yes, are cartesian coordinate systems special types of coordinate systems? $\endgroup$
    – Filippo
    Nov 22, 2020 at 19:45
  • $\begingroup$ A Cartesian coordinate system is the map associating each point $P$ of the space to the components of the unique vector joining a fixed origin $O$ with the point $P$. These are very special coordinate systems on the affine space. First of all because every Cartesian system is global: it completely covers the space and identify it with $R^n$. $\endgroup$ Nov 22, 2020 at 20:08
  • $\begingroup$ Are you Italian as your nickname suggests? $\endgroup$ Nov 22, 2020 at 20:11
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    $\begingroup$ Thanks... Indeed, I am considering it. But I am quite busy and these notes should be completed by adding further material before passing to a publication. After all I have published a number of books and I should be content with them :). My next publication will be the English translation of my recent Italian book on Analytical Mechanics... $\endgroup$ Nov 24, 2020 at 15:45

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