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Some Differential Geometry

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The Many Faces of Maxwell, Dirac and Einstein Equations

Part of the book series: Lecture Notes in Physics ((LNP,volume 922))

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Abstract

The main objective of this chapter is to present a Clifford bundle formalism for the formulation of the differential geometry of a manifold M, equipped with metric fields \(\boldsymbol{g} \in \sec T_{2}^{0}M\) and \(\mathtt{g} \in \sec T_{0}^{2}M\) for the tangent and cotangent bundles. We start by first recalling the standard formulation and main concepts of the differential geometry of a differential manifold M. We introduce in M the Cartan bundle of differential forms, define the exterior derivative, Lie derivatives, and also briefly review concepts as chains, homology and cohomology groups, de Rham periods, the integration of form fields and Stokes theorem. Next, after introducing the metric fields \(\boldsymbol{g}\) and g in M we introduce the Hodge bundle presenting the Hodge star and the Hodge coderivative operators acting on sections of this bundle. We moreover recall concepts as the pullback and the differential of maps, connections and covariant derivatives, Cartan’s structure equations, the exterior covariant differential of ( p + q)-indexed r-forms, Bianchi identities and the classification of geometries on M when it is equipped with a metric field and a particular connection. The spacetime concept is rigorously defined. We introduce and scrutinized the structure of the Clifford bundle of differential forms (\(\mathcal{C}\ell(M,\mathtt{g})\)) of M and introduce the fundamental concept of the Dirac operator (associated to a given particular connection defined in M) acting on Clifford fields (sections of \(\mathcal{C}\ell(M,\mathtt{g)}\)). We show that the square of the Dirac operator (associated to a Levi-Civita connection in M) has two fundamental decompositions, one in terms of the derivative and Hodge codifferential operators and other in terms of the so-called Ricci and D’Alembertian operators. A so-called Einstein operator is also introduced in this context. These decompositions of the square of the Dirac operator are crucial for the formulation of important ideas concerning the construction of gravitational theories as discussed in particular in Chaps. 9, 11, 15. The Dirac operator associated to an arbitrary (metrical compatible) connection defined in M and its relation with the Dirac operator associated to the Levi-Civita connection of the pair \((M,\boldsymbol{g})\) is discussed in details and some important formulas are obtained. The chapter also discuss some applications of the formalism, e.g., the formulation of Maxwell equations in the Hodge and Clifford bundles and formulation of Einstein equation in the Clifford bundle using the concept of the Ricci and Einstein operators. A preliminary account of the crucial difference between the concepts of curvature of a connection in M and the concept of bending of M as a hypersurface embedded in a (pseudo)-Euclidean space of high dimension (a property characterized by the concept of the shape tensor, discussed in details in Chap. 5) is given by analyzing a specific example, namely the one involving the Levi-Civita and the Nunes connections defined in a punctured 2-dimensional sphere. The chapter ends analyzing a statement referred in most physical textbooks as “tetrad postulate” and shows how not properly defining concepts can produce a lot of misunderstanding and invalid statements.

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Notes

  1. 1.

    In general we are not going to present proofs of the propositions, except for a few cases, which may considered as exercises. If you need further details, consult e.g., [3, 11, 25].

  2. 2.

    We remark that some authors (see, e.g., [25]) call sometimes the coordinate function x i simply by coordinate. Also, some authors (see, e.g., [11]) call sometimes {x i} a coordinate system (for \(U \subset M\)). We eventually also use these terminologies.

  3. 3.

    In Appendix we list the main concepts concerning fiber bundle theory that we need for the purposes of this book.

  4. 4.

    See details in Notation A.6 in the Appendix.

  5. 5.

    See Remark 4.41 for the reason of the notation dx i.

  6. 6.

    Note that in general \(\boldsymbol{y}^{\mu }(\mathtt{h}(\mathfrak{e}))\neq y^{\mu }\).

  7. 7.

    Eventually a more rigorously notation for a basis of T ∗ U should be {d x i}. 

  8. 8.

    See Definition 4.65.

  9. 9.

    In Chap. 6 we will learn that a spacetime manifold admitting spinor fields must necessarily be orientable.

  10. 10.

    Of course, we should write \(\tau =\varphi _{ \alpha }^{{\ast}}(dx^{1} \wedge \cdots \wedge dx^{n})\) since dx i are 1-forms in \(T_{\varphi _{\alpha (U)}}\mathbb{R}^{n}\). So, ours is a sloppy (universally used) notation.

  11. 11.

    When Lorentzian manifolds serve as models of spacetimes it is also imposed that M is noncompact. See Sect. 4.7.1.

  12. 12.

    When there is no chance of confusion we eventually used the symbol ⋅ instead of the symbol \(\mathop{\cdot }\limits_{\boldsymbol{g}}\) in order to simplify the notation.

  13. 13.

    For the exact meaning of the concept of diffeomorphism invariance of a spacetime physical theory (as used in this text) see Sect. 6.6.3

  14. 14.

    EndTM means the set of endomorphisms \(TM \rightarrow TM\).

  15. 15.

    Sometimes also called exterior covariant derivative.

  16. 16.

    As usual the inverted hat over a symbol (in Eq. (4.117)) means that the corresponding symbol is missing in the expression.

  17. 17.

    Multi indices are here represented by J and K.

  18. 18.

    We use the notation \(\nabla _{\!\sigma }t_{\nu \ldots }^{\mu \ldots } \equiv (\nabla _{e_{\sigma }}t)_{\nu \ldots }^{\mu \ldots } \equiv (\nabla t)_{\sigma \nu \ldots }^{\mu \ldots }\) for the components of the covariant derivative of a tensor field t. This is not to be confused with \(\nabla _{e_{\sigma }}t_{\nu \ldots }^{\mu \ldots } \equiv e_{\sigma }(t_{\nu \ldots }^{\mu \ldots })\), the derivative of the components of t in the direction of \(e_{\sigma }\).

  19. 19.

    To our knowledge, Eqs. (4.125) and (4.126c) are not found anywhere in the literature, although they appear to be the most natural extension of the structure equations for metric manifolds.

  20. 20.

    See, e.g., [3, p. 135].

  21. 21.

    Or Riemann space.

  22. 22.

    Recall again that the symbol A ↪ B means that A is embedded in B and \(A \subseteq B\).

  23. 23.

    It is crucial to distinguish the Dirac operators introduced in this chapter and which act on sections of Clifford bundles with the spin Dirac operator introduced in Chap. 7 and which act on sections of spin-Clifford bundles.

  24. 24.

    And more generally, to any metric compatible connection.

  25. 25.

    We note that the possibility of decomposing the connection coefficients into rotation (torsion), shear and dilation has already been suggested in a Physics paper by Baekler et al. [1] but in their work they do not arrive at the identification of a tensor-like quantity associated to these last two objects. The idea of the decompositions already appeared in [40].

  26. 26.

    Equations (4.196) and (4.197) have appeared in the literature in two different contexts: with \(\nabla \boldsymbol{\mathring{g}} = 0\), they have been used in the formulations of the theory of the spinor fields in Riemann-Cartan spaces [15, 46] and with \(\Theta [\nabla ] = 0\) they have been used in the formulations of the gravitational theory in a space endowed with a background metric [8, 13, 23, 35, 36].

  27. 27.

    Besides the ones presented in this chapter, others will be exhibited in Chap. 13

  28. 28.

    Thirring [44] said that the two equations in Eq. (4.251) is the twentieth Century presentation of Maxwell equations.

  29. 29.

    See however the news in [31] where it is claimed that magnetic monopoles have been observed in a synthetic magnetic field.

  30. 30.

    See also [7].

  31. 31.

    Also called by some authors pseudo forms.

  32. 32.

    We shall see in Chap. 6 that any Lorentzian spacetime admitting spinor fields must have a global tetrad.

  33. 33.

    Sometimes in the written of some formulas in the next chapters it is convenient to use the notation \(\mathcal{T}^{\mathbf{a}} = -T^{\mathbf{a}}\).

  34. 34.

    Pedro Salacience Nunes (1502–1578) was one of the leading mathematicians and cosmographers of Portugal during the Age of Discoveries. He is well known for his studies in Cosmography, Spherical Geometry, Astronomic Navigation, and Algebra, and particularly known for his discovery of loxodromic curves and the nonius. Loxodromic curves, also called rhumb lines, are spirals that converge to the poles. They are lines that maintain a fixed angle with the meridians. In other words, loxodromic curves directly related to the construction of the Nunes connection. A ship following a fixed compass direction travels along a loxodromic, this being the reason why Nunes connection is also known as navigator connection. Nunes discovered the loxodromic lines and advocated the drawing of maps in which loxodromic spirals would appear as straight lines. This led to the celebrated Mercator projection, constructed along these recommendations. Nunes invented also the Nonius scales which allow a more precise reading of the height of stars on a quadrant. The device was used and perfected at the time by several people, including Tycho Brahe, Jacob Kurtz, Christopher Clavius and further by Pierre Vernier who in 1630 constructed a practical device for navigation. For some centuries, this device was called nonius. During the nineteenth century, many countries, most notably France, started to call it vernier. More details in http://www.mlahanas.de/Stamps/Data/Mathematician/N.htm.

  35. 35.

    Some authors call the Columbus connection the Nunes connection. Such name is clearly unappropriated.

  36. 36.

    This wording, of course, means that this vectors are identified as elements of the appropriate tangent spaces.

  37. 37.

    See, e.g., [17, 27, 34, 41] for details.

  38. 38.

    \(\mathbf{P}_{\mathrm{SO}_{1,3}^{e}}(M)\) is the orthonormal frame bundle (see Appendix A.1.2).

  39. 39.

    Recall that other authors prefer the notations \((\boldsymbol{\nabla }_{\partial _{\mu }}\boldsymbol{V })^{\alpha }:= V _{:\mu }^{\alpha }\) and \((\boldsymbol{\nabla }_{\partial _{\mu }}C)_{\alpha }:= C_{\alpha:\mu }\). What is important is always to have in mind the meaning of the symbols.

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  1. IMECC, Universidade Estadual de Campinas, Campinas, São Paulo, Brazil

    Waldyr A. Rodrigues Jr & Edmundo Capelas de Oliveira

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Rodrigues, W.A., Capelas de Oliveira, E. (2016). Some Differential Geometry. In: The Many Faces of Maxwell, Dirac and Einstein Equations. Lecture Notes in Physics, vol 922. Springer, Cham. https://doi.org/10.1007/978-3-319-27637-3_4

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