Abstract
In this chapter we investigate the nature of the gravitational field. We first give a formulation for the theory of that field as a field in Faraday’s sense (i.e., as of the same nature as the electromagnetic field) on a 4-dimensional parallelizable manifold M. The gravitational field is represented through the 1-form fields \(\{\mathfrak{g}^{\mathbf{a}}\}\) dual to the parallelizable vector fields \(\{\boldsymbol{e}_{\mathbf{a}}\}\). The \(\mathfrak{g}^{\mathbf{a}}\)’s (a = 0, 1, 2, 3) are called gravitational potentials, and it is imposed that at least for one of them, \(d\mathfrak{g}^{\mathbf{a}}\neq 0\). A metric like field \(\boldsymbol{g} =\eta _{\mathbf{ab}}\mathfrak{g}^{\mathbf{a}} \otimes \mathfrak{g}^{\mathbf{b}}\) is introduced in M with the purpose of permitting the construction of the Hodge dual operator and the Clifford bundle of differential forms \(\mathcal{C}\ell(M,\mathtt{g})\), where \(\mathtt{g} =\eta ^{\mathbf{ab}}\boldsymbol{e}_{\mathbf{a}} \otimes e_{\mathbf{b}}\). Next a Lagrangian density for the gravitational potentials is introduced with consists of a Yang-Mills term plus a gauge fixing term and an auto-interacting term. Maxwell like equations for \(F^{\mathbf{a}} = d\mathfrak{g}^{\mathbf{a}}\) are obtained from the variational principle and a legitimate energy-momentum tensor for the gravitational field is identified which is given by a formula that at first look seems very much complicated. Our theory does not uses any connection in M and we clearly demonstrate that representations of the gravitational field as Lorentzian, teleparallel and even general Riemann-Cartan-Weyl geometries depend only on the arbitrary particular connection (which may be or not to be metrical compatible) that we may define on M. When the Levi-Civita connection of \(\boldsymbol{g}\) in M is introduced we prove that the postulated Lagrangian density for the gravitational potentials differs from the Einstein-Hilbert Lagrangian density of General Relativity only by a term that is an exact differential. The theory proceeds choosing the most simple topological structure for M, namely that it is \(\mathbb{R}^{4}\), a choice that is compatible with present experimental data. With the introduction of a Levi-Civita connection for the structure \((M = \mathbb{R}^{4},\boldsymbol{g})\) as a mathematical aid we can exhibit a nice short formula for the genuine energy-momentum of the gravitational field. Next, we introduce the Hamiltonian formalism and discuss possible generalizations of the gravitational field theory (as a field in Faraday’s sense) when the graviton mass is not null. Also we show using the powerful Clifford calculus developed in previous chapters that if the structure \((M = \mathbb{R}^{4},\boldsymbol{g})\) possess at least one Killing vector field, then the gravitational field equations can be written as a single Maxwell like equation, with a well defined current like term (of course, associated to the energy-momentum tensor of matter and the gravitational field). This result is further generalized for arbitrary vector fields generating one-parameter groups of diffeomorphisms of M in Chap. 14 Chapter 11 ends with another possible interpretation of the gravitational field, namely that it is represented by a particular geometry of a brane embedded in a high dimensional pseudo-Euclidean space. Using the theory developed in Chap. 5 we are able to write Einstein equation using the Ricci operator in such a way that its second member (of “wood” nature, according to Einstein) is transformed (also according to Einstein) in the “marble”nature of its first member. Such a form of Einstein equation shows that the energy momentum quantities \(-T^{\mathbf{a}} + \frac{1} {2}T\mathfrak{g}^{\mathbf{a}}\) (where \(T^{\mathbf{a}} = T_{\mathbf{b}}^{\mathbf{a}}\mathfrak{g}^{\mathbf{b}}\) are the energy momentum 1-form fields of matter and T = T a a) which characterize matter is represented by the negative square of the shape operator (\(\mathbf{S}^{2}(\mathfrak{g}^{\mathbf{a}})\)) of the brane. Such a formulation thus give a mathematical expression for the famous Clifford “little hills” as representing matter.
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Notes
- 1.
\((M,\boldsymbol{D},\boldsymbol{g},\tau _{\boldsymbol{g}},\uparrow )\) is said to be equivalent to \((M^{{\prime}},\boldsymbol{D}^{{\prime}},\boldsymbol{g}^{{\prime}},\tau _{\boldsymbol{g}}^{{\prime}},\uparrow ^{{\prime}})\) if there exists a diffeomorphism \(\mathtt{h:}M \rightarrow M^{{\prime}}\) such that \(M = ^{{\prime}}\mathtt{h}M\ \boldsymbol{D}^{{\prime}} = \mathtt{h}^{{\ast}}\boldsymbol{D}\), \(\boldsymbol{g}^{{\prime}} = \mathtt{h}^{{\ast}}\boldsymbol{g}\), \(\tau _{\boldsymbol{g}}^{{\prime}} = \mathtt{h}^{{\ast}}\tau _{\boldsymbol{g}},\uparrow ^{{\prime}} = \mathtt{h}^{{\ast}}\uparrow \) and \(T^{{\prime}} = \mathtt{h}^{{\ast}}T\).
- 2.
- 3.
- 4.
Of course, the true background spacetime may be eventually a more complicated one, since that manifold must represent the global topological structure of the universe, something that is not known at the time of this writing [43]. We do not study this possibility here, but the results we are going to present can be easily generalized for more general spacetime backgrounds.
- 5.
- 6.
In Chap. 15 (see also [33]) we even show that when a Lorentzian spacetime structure \((M,\boldsymbol{D},\ \boldsymbol{g},\tau _{\boldsymbol{g}},\uparrow )\) representing a gravitational field in GRT possess a Killing vector field \(\boldsymbol{K}\), then there are Maxwell like equations with well determined source term satisfied for F = dA with \(A =\boldsymbol{ g}(\boldsymbol{K}, )\) encoding Einstein equation and more, there is a Navier-Stokes equation encoding the Maxwell (like) and Einstein equations.
- 7.
This other possibility does not define in general a legitimate energy-momentum tensor for the gravitational field in GRT, but it defines a legitimate energy-momentum tensor in our theory in which the gravitational field is interpreted as a field in the sense of Faraday living in Minkowski spacetime.
- 8.
We suppose that \(\mathcal{L}_{m}\) does not depend explicitly on the \(d\mathfrak{g}^{\mathbf{a}}\).
- 9.
This will become evident after we present below the nice formula for the t d.
- 10.
Recall that \(\mathfrak{g} = \mathfrak{g}^{\mathbf{a}} \otimes \boldsymbol{ e}_{\mathbf{a}}\) is the identity operator in \(\bigwedge \nolimits ^{1}T^{{\ast}}M\).
- 11.
In general we will also have that \(d\vartheta ^{\mathbf{i}}\neq 0\), i = 1, 2, 3.
- 12.
We use only constrained variations of the \(\mathfrak{g}^{\mathbf{a}}\), which as already recalled in Sect. 11.1 do not change the metric field \(\boldsymbol{g}\).
- 13.
More details on possible choices of the boundary term for different physical situations may be found in [23].
- 14.
See a nice proof in [42].
- 15.
We only observe that Lagrangian density of Logunov’s theory when written in terms of differential forms is not a very elegant expression.
- 16.
In fact, formulation of teleparallel equivalence of GRT is a subject with a old history. See, e.g., [18].
- 17.
See, Eq. (4.197) with Q α β γ = 0.
- 18.
Taking into account, of course, that differently from Clifford’s idea, instead of a space theory of matter, we must talk about a spacetime theory of matter.
- 19.
Recall that \(\mathfrak{R}\) is in general non null even in vacuum.
- 20.
Details about these possibilities are discussed in [11] where a theory of the gravitational field on a brane diffeomorphic to R 4 is discussed.
- 21.
See more details in Chap. 15.
- 22.
On this issue recall Sect. 6.9 keeping in mind that there are articles criticizing the notion that black holes are predictions of GRT due mainly to some mathematical misunderstandings as, e.g.[1, 4, 36] and/or physical grounds. When thinking on this issue take also into account the ‘pasticcio’ concerning the black hole information ‘paradox’ (see, [15, 17]) and its possible resolutions with the suggested existence of a “complementarity principle” [37] or existence of firewalls [2] as an indication that the foundations of GRT and its relation to other theories of Physics are not well understood as some people would like us to think. Recently adding stuff to the “pasticcio” Hawking [16] is claiming that “The absence of event horizons mean that there are no black holes—in the sense of regimes from which light can’t escape to infinity”. But this statement seems to be already an old idea. More information at http://asymptotia.com/2014/01/30/hawking-an-old-idea/ and http://www.physics.ohio-state.edu/~mathur/.
- 23.
The possibility for time machines arises when closed timelike curves exist in a Lorentzian manifold. Such exotic configurations, it is said, already appears in Gödel’s universe model. However, a recent thoughtful analysis by Cooperstock and Tieu (which we endorse) shows that the old claim is wrong. Authors like, e.g, Davies [9] (which are proposing to build time machines even at home), Gott [14] and Novikov [28] are invited to read [7] and find a error in the argument of those authors.
- 24.
For those people in that class we offer Chap. 12
- 25.
References
Abrams, L.S.: Alternative space-time for the point mass. Phys. Rev. D 20, 2474 (1979) [arXiv:gr-qc/0201044]
Almheiri, A., Marolf, D., Polchinski, J., Sully, J.: Black holes: complementarity or firewalls? (2012) [arXiv:1207.3223v2 [hep-th]]
Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten L. (ed.) Gravitation: An Introduction to Current Research. Wiley, New York (1962) [gr-qc/0405109]
Chapline, G.: Dark energy stars. In: Proceeding of the Texas Conference on Relativistic Astrophysics, Stanford (2004) [astro-ph/0503200]
Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics, revisited edition. North Holland, Amsterdam (1982)
Clifford, W.K.: On the space-theory of matter. In: Proceedings of the Cambridge Philosophical Society, vol. 2, pp. 157–158 (1864–1876 - Printed 1876)
Cooperstock, F.I., Tieu, S.: Closed timelike curves and time travel: dispelling the myth Found. Phys. 35, 1497–1509 (2005)
da Rocha, R., Rodrigues, W.A. Jr.: The Einstein-Hilbert lagrangian density in a 2-dimensional spacetime is an exact differential. Mod. Phys. Lett. A 21, 1519–1527 (2006) [hep-th/0512168]
Davies, P.: How to Build Time Machines, Allen Lane. The Penguin, London (2001)
de Andrade, V.C., Guillen, L.C.T., Pereira, J.G.: Gravitational energy-momentum density in teleparallel gravity. Phys. Rev. Lett. 84, 4533–4536 (2000)
Fernández, V.V, Rodrigues, W.A. Jr.:Gravitation as plastic distortion of the Lorentz vacuum. Fundamental Theories of Physics, vol. 168. Springer, Heidelberg (2010) [errata for the book at: http://www.ime.unicamp.br/~walrod/errataplastic]
Geroch, R.: Spinor structure of space-times in general relativity I. J. Math. Phys. 9, 1739–1744 (1968)
Geroch, R.: Spinor structure of space-times in general relativity II. J. Math. Phys. 11, 343–348 (1970)
Gott, J.R.: Time Travel in Einstein’s Universe. Weidenfeld & Nicolson, London (2001)
Hawking, S.W.: The information paradox for black holes. Lecture at the 17th International Conference on General Relativity and Gravitation. Dublin (2004). http://www.gr17.com
Hawking, S.W.: Information preservation and weather forecasting for black holes (2014) [arXiv:1401.5761]
Hayard, S.A.: The disinformation problem for black holes. 14th Workshop on General Relativity and Gravitation. Kyoto University (2004) [gr-qc/0504037]
Hayashi, K., Shirafuji. T.: New general relativity. Phys. Rev. D 19, 3542–3553 (1979)
Laughlin, R.B.: A Different Universe: Reinventing Physics from the Bottom Down. Basic Books, New York (2005)
Logunov, A.A.: Relativistic Theory of Gravity. Nova Science Publication, New York (1999)
Logunov, A.A., Mestvirishvili, M.A.: The Relativistic Theory of Gravitation. Mir Publications, Moscow (1989)
Maluf, J.W.: Hamiltonian formulation of the teleparallel description of general relativity. J. Math. Phys. 35, 335–343 (1994)
Meng, F.-F.: Quasilocal center-of-mass moment in general relativity. M.Sc. Thesis, National Central University, Chungli (2001). http://thesis.lib.ncu.edu.tw/ETD-db/ETD-search/view_etd?URN=89222030
Misner, C.M., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman and Co., San Francisco (1973)
Nester, J.M., Yo, H.-J.: Symmetric teleparallel general relativity. Chin. J. Phys. 37, 113–117 (1999) [arXiv:gr-qc/909049v2]
Notte-Cuello, E., Rodrigues, W.A. Jr.: Freud’s identity of differential geometry, the Einstein-Hilbert equations and the vexatious problem of the energy-momentum conservation in GR. Adv. Appl. Clifford Algebras 19, 113–145 (2009) [arXiv:0801.2559v4 [math-ph]]
Notte-Cuello, E.A., da Rocha, R., Rodrigues, W.A. Jr.: Some thoughts on geometries and on the nature of the gravitational field. J. Phys. Math. 2, 20–40 (2010)
Novikov, I.D.: The River of Time. Cambridge University Press, Cambridge (1998)
Ohanian, H.C., Ruffini, R.: Gravitation and Spacetime, 2nd edn. W.W. Norton & Co., New York (1994)
Rodrigues, W.A. Jr.: Killing vector fields, Maxwell equations and Lorentzian spacetimes. Adv. Appl. Clifford Algebras 20, 871–884 (2010)
Rodrigues, W.A. Jr.: The nature of the gravitational field and its Legitimate energy-momentum tensor. Rep. Math. Phys. 69, 265–279 (2012)
Rodrigues, W.A. Jr., Sharif, M.: Equivalence principle and the principle of local Lorentz invariance. Found. Phys. 31, 1785–1806 (2001). Corrigenda: Found. Phys. 32, 811–812 (2002)
Rodrigues, F.G., da Rocha, R., Rodrigues, W.A. Jr.: The Maxwell and Navier-Stokes equations equivalent to Einstein equation in a spacetime containing a killing vector field. In: AIP Conference Proceedings, vol. 1483, pp. 277–295 (2012) [arXiv:1109.5274v1 [math-ph]]
Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Springer, New York (1977)
Schücking, E.L.: Einstein’s apple and relativity’s gravitational field (2009) [arXiv:0903.3768]
Stravroulakis, N.: Vérité Scientifique et Trous Noirs. Le Abus du Formalism (premiere part). Ann. Fond. L. de Broglie 24, 67–108 (1999)
Susskind, L.,Thorlacius, L., Uglum, J.: The stretched horizon and black hole complementarity. Phys. Rev. D 48, 3743–3761 (1993) [hep-th/9306069]
Synge, J.L.: Relativity: The General Theory. North Holland, Amsterdam (1960)
Szabados, L.B.: Quasi-local energy-momentum and angular momentum in GR: a review article. Living Rev. Relativ. (2004) http://www.livingreviews.org/lrr-2004-4
Thirring, W.: Classical Field Theory, vol. 2, Springer, New York (1980)
Volovik, G.E.: The Universe in a Liquid Droplet. Clarendon, Oxford (2003)
Wallner, R.P. Asthekar’s variables reexamined. Phys. Rev. D 46, 4263–4285 (1992)
Weeks, J.R.: The Shape of Space, 2nd edn. Marcel Decker, New York (2012)
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Rodrigues, W.A., Capelas de Oliveira, E. (2016). On the Nature of the Gravitational Field. 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_11
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