Abstract
This chapter gives a Clifford bundle approach to a theory of the gravitational field where this field is interpreted as the curvature of a Sl\((2, \mathbb{C})\) gauge theory. The proposal of this presentation is to emphasize the many faces of Einstein equation and in the development of the theory a collection of some non trivial and useful formulas is derived and some misconceptions in presentations of the theory of the gravitational field as a Sl\((2, \mathbb{C})\) gauge theory is discussed and fixed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Of course, this will not exhaust the possible faces of Einstein’s equations, for finding new faces depends mainly on authors mathematical knowledge and imagination. Also, the reader is here advised that this chapter do not intend to be a presentation of the many developments that goes under the name of gauge theories of gravitation. The interested reader on this issue may consult, e.g., [9, 12].
- 2.
The name is due to the fact that \(\mathcal{D}\) does not always satisfy a Leibniz type rule when applied to the ⊗∧product of arbitrary cliforms.
- 3.
- 4.
See also Chap. 15
- 5.
We are not going to discuss this issue here. Such a deficiency may be supplied by a careful and lucid analysis of the problem of formulating GRT as a possible gauge theory with gauge groups \(\mathrm{Sl}(2, \mathbb{C})\) or T(4) as done by Wallner [20].
- 6.
In words, \(\mathop{\boldsymbol{\omega }}\limits^{\blacktriangle }\) is a 1-form in the cotangent space of the bundle of orthonormal frames with values in the Lie algebra \(\mathrm{so}_{1,3}^{e} \simeq \mathrm{ sl}(2, \mathbb{C})\) of the group SO1, 3 e.
- 7.
Note that this tensor is the identity tensor acting on the space of vector fields on \(U \subset M\). We denoted the identity tensor by \(\mathfrak{g}\) in Chap. 11.
- 8.
Recall that the product ⊗∧is given in Definition A.68.
- 9.
Note that \(\mathop{A}\limits^{m}\) and \(\mathop{B}\limits^{m}\) are general non-homogeneous multivector fields.
- 10.
The result printed in [15] is wrong.
- 11.
For a Clifford algebra formula for the calculation of \(\nabla _{e_{\mathbf{r}}}\mathcal{A}\), \(\mathcal{A}\in \sec \bigwedge \nolimits ^{p}T^{{\ast}}M\) recall Eq. (7.44).
- 12.
These objects have been introduced in Definition 4.89.
- 13.
- 14.
The value obtained for \(\mathcal{D}^{2}A\) in [15] is wrong.
- 15.
Recall that \(\mathbf{J}_{\nu } \in \sec \bigwedge \nolimits ^{2}TM\hookrightarrow \sec \mathcal{C}\ell(M,\boldsymbol{g})\).
- 16.
Note that we could also produce another Maxwell-like equation, by using the usual Levi-Civita covariant derivative operator D in the definition of the current, i.e., we can put \(\mathcal{J}_{\mathbf{b}} = D_{\mathbf{e}_{\mathbf{a}}}(\mathsf{T}^{\mathbf{a}}\mathbf{e}_{\mathbf{b}} -\mathbf{e}_{\mathbf{b}}\mathsf{T}^{\mathbf{a}})\), and in that case we obtain \(D_{\mathbf{e}_{\mathbf{a}}}\mathcal{F}_{\mathbf{b}}^{\mathbf{a}} = \mathcal{J}_{\mathbf{b}}\). An equation of this form appears in [17, 18].
References
Baez, J., Muniain, J.P.: Gauge Fields, Knots and Gravity. World Scientific, Singapore (1994)
Benn, I.M., Tucker, R.W.: An Introduction to Spinors and Geometry with Applications in Physics. Adam Hilger, Bristol (1987)
Bleecker, D.: Gauge Theory and Variational Principles. Addison-Wesley, Reading (1981)
Carmeli, M.: Group Theory and Relativity. McGraw-Hill, New York (1977)
Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics, revisited edition. North Holland Publ. Co., Amsterdam (1982)
de Oliveira, E.C., Rodrigues, W.A. Jr.: Dotted and undotted spinor fields in general relativity. Int. J. Mod. Phys. D13, 1637–1659 (2004)
Frankel, T.: The Geometry of Physics. Cambridge University Press, Cambridge (1997)
Göckler, M., Schücker, T.: Differential Geometry, Gauge Theories and Gravity. Cambridge University Press, Cambridge (1987)
Gronwald, F., Hehl, F.W.: On the Gauge aspects of gravity. In: Bergmann, P.G., et al. (eds.) International School of Cosmology and Gravitation: 14thCourse: Quantum Gravity, Erice (1995). The Science and Culture Series, vol. 10, pp. 148–198. World Sci. Publ., River Edge (1996) [gr-qc/9602013]
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1. Interscience, New York (1963)
Lounesto, P.: Scalar product of spinors and an extension of the Brauer-Wall groups. Found. Phys. 11, 721–740 (1981)
Mielke, E.W.: Geometrodynamics of Gauge Fields. Physical Research, vol. 3. Akademie, Berlin (1987)
Nash, C., Sen, S.: Topology and Geometry for Physicists. Academic, London (1983)
Palais, R.S.: The Geometrization of Physics. Lecture Notes from a Course at the National Tsing Hua University. Hsinchu, Taiwan (1981)
Rodrigues, W.A. Jr., de Oliveira, E.C.: Clifford valued differential forms, and some issues in gravitation, electromagnetism and “unified” theories. Int. J. Mod. Phys. D 13, 1879–1915 (2004) [math-ph/0407025]
Ryder, L.H.: Quantum Field Theory, 2nd edn. Cambridge University Press, Cambridge (1996)
Sachs, M.: General Relativity and Matter. D. Reidel, Dordrecht (1982)
Sachs, M.: Quantum Mechanics and Gravity. The Frontiers Science IV. Springer, Berlin (2004)
Sternberg, S.: Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs (1964)
Wallner, R.P.: Notes on the Gauge theory of gravitation. Acta Phys. Austriaca 54, 165–189 (1982)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Rodrigues, W.A., Capelas de Oliveira, E. (2016). On the Many Faces of Einstein Equations. 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_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-27637-3_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-27636-6
Online ISBN: 978-3-319-27637-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)