Abstract
In this chapter we describe the contents of all chapters of the book and their interrelationship.
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Notes
- 1.
By many faces of Maxwell, Dirac and Einstein’s equations, we mean the many different ways in which those equations can be presented using different mathematical theories.
- 2.
For details on the Newtonian spacetime structure see, e.g., [30].
- 3.
Inside a material the equations involve also other fields, the so-called polarization fields. See details in [22].
- 4.
The reader is invited to study Chap. 6 in detail to know the exact meaning of this statement.
- 5.
These mathematical tools are introduced in Chap. 4
- 6.
The elementary approach is related with a choice of a global spin coframe (Chap. 7) in spacetime.
- 7.
The precise mathematical meaning of this statement can be given only within the theory of spin-Clifford bundles, as described in Chap. 6.
- 8.
Soon it became clear that the interaction of the electromagnetic field with the Dirac spinor field could produce pairs (electrons and positrons). Besides that it was known since 1905 that the classical concept of the electromagnetic field was not in accord with experience and that the concept of photons as quanta of the said field needed to be introduced. The theory that deals with the interaction of photons and electrons (and positrons) is a particular case of a second quantized renormalizable quantum field theory and is called quantum electrodynamics. In that theory the electromagnetic and the Dirac spinor fields are interpreted as operator valued distributions [3] acting on the Hilbert space of the state vectors of the system. We shall not discuss further this theory in this book, but will return to some of its issues in a sequel volume [6].
- 9.
- 10.
Indeed, by an equivalence class of diffeomorphic Lorentzian spacetimes.
- 11.
Of course that denomination holds for any manifold M, \(\dim M = n\) equipped with a metric of signature (p, q). 
- 12.
The possible types of different metrics depend on some topological restrictions. This will be discussed at the appropriate place.
- 13.
Bending of a manifold viewed as submanifold of a Euclidean or pseudo-Euclidean space of large dimension is characterized by the shape operator, a concept introduced in Chap. 5
- 14.
Any manifold \(M,\dim M = n\), according to Whitney’s theorem, can be realized as a submanifold of \(\mathbb{R}^{m}\), with m = 2n. However, if M carries additional structure the number m in general must be greater than 2n. Indeed, it has been shown by Eddington [7] that if dim M = 4 and if M carries a Lorentzian metric \(\boldsymbol{g}\), which moreover satisfies Einstein’s equations, then M can be locally embedded in a (pseudo)Euclidean space \(\mathbb{R}^{1,9}\). Also, isometric embeddings of general Lorentzian spacetimes would require a lot of extra dimensions [4]. Indeed, a compact Lorentzian manifold can be embedded isometrically in \(\mathbb{R}^{2,46}\) and a non-compact one can be embedded isometrically in \(\mathbb{R}^{2,87}\)!
- 15.
Spin-Clifford bundles are introduced in Chap. 7.
- 16.
In this book, the metric of the tangent bundle is always denoted by a boldsymbol letter, e.g., \(\boldsymbol{g} \in \mathbf{\sec }T_{2}^{0}M\). The corresponding metric of the cotangent bundle is always represented by a typewriter symbol, in this case, \(\mathtt{g}\ \in \sec T_{0}^{2}M\). Moreover, we represent by \(\mathtt{\underline{g}}: TM \rightarrow TM\) the endomorphism associated with \(\boldsymbol{g}\).We have \(\boldsymbol{g}(\mathbf{u,v}) = \mathtt{\underline{g}}(\mathbf{u})\mathop{\cdot }\limits_{\boldsymbol{g}_{\text{E}}}\mathbf{v}\), for any \(\mathbf{u,v \in }\sec TM\), where \(\boldsymbol{g}_{\mathtt{E}}\) is an appropriate Euclidean metric on TM. The inverse of the endomorphism g is denoted g −1. We represent by \(g: T^{{\ast}}M \rightarrow T^{{\ast}}M\) the endomorphism corresponding to g. Finally, the inverse of g is denoted by g −1. See details in Sect. 2.8
- 17.
The word flat here refers to formulations of the gravitational field, in which this field is a physical field, in the sense of Faraday, living on Minkowski spacetime.
- 18.
We presuppose that the reader of our book knows Relativity Theory at least at the level presented at the classical book [18].
- 19.
A perfect understanding of the Principle of Relativity is also crucial in our forthcoming book [6] which discusses ‘superluminal wave phenomena’.
- 20.
It is important to distinguish between the concept of a frame (which are sections of the frame bundle) introduced in Appendix A.1.1 with the concept of a reference frame to be defined in Chap. 6.
- 21.
See Definition 6.59.
- 22.
See Definition 6.61. γ is a timelike geodesic in the Lorentzian manifold representing spacetime.
- 23.
Here, this principle is a statement about indistinguishable of LLRFγ. It is not to be confused with the imposition of (active) local Lorentz invariance of Lagrangians and field equations discussed in Sect. 10.2
- 24.
This issue is discussed in details in [6].
- 25.
For the genesis of these objects we quote [28].
- 26.
The same as that used in [28].
- 27.
The case of Dirac-Hestenes equation on a Riemann-Cartan manifold is discussed in Sect. 10.1
- 28.
For a description of the gravitational field by a set of 1-forms \(\mathfrak{g}^{\mathbf{a}} \in \sec \bigwedge \nolimits ^{1}T^{{\ast}}M\), a = 0, 1, 2, 3 see Chap. 11.
- 29.
Chapter 10 in the first edition.
- 30.
- 31.
Honestly, we think that gravitation is an emergent macroscopic phenomenon which need not to be quantized and which will eventually find its correct description in a theory about the real structure of the physical vacuum as suggested, e.g., in [33]. However, we are not going to discuss such a possibility in this book.
- 32.
On this issue, see also the book by Kleinert [17], which however describes plastic distortions by means of multivalued functions.
- 33.
\(\mathring{A} =\boldsymbol{\mathring{g}}(\boldsymbol{A}, )\) and \(A =\boldsymbol{ g}(\boldsymbol{A,} )\) with \(\boldsymbol{\mathring{g}}\) and \(\boldsymbol{g}\ \) the metrics of Minkowski spacetime denoted in Chap. 15 by \((M = \mathbb{R}^{4},\boldsymbol{\mathring{g}},\mathring{D},\tau _{\boldsymbol{\mathring{g}}},\uparrow )\) and of the structure \((M = \mathbb{R}^{4},\boldsymbol{g},\boldsymbol{D},\tau _{\boldsymbol{g}},\uparrow )\) describing an effective Lorentzian spacetime.
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Rodrigues, W.A., Capelas de Oliveira, E. (2016). Introduction. 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_1
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