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Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes

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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

In this chapter we examine in details the conditions for existence of conservation laws of energy-momentum and angular momentum for the matter fields of on a general Riemann-Cartan spacetime \((M,\boldsymbol{g},\nabla,\tau _{\boldsymbol{g}},\uparrow )\) and also in the particular case of Lorentzian spacetimes \(\mathfrak{M} = (M,\boldsymbol{g},D,\tau _{\boldsymbol{g}},\uparrow )\) which as we already know model gravitational fields in the GRT [3]. Riemann-Cartan spacetimes are supposed to model generalized gravitational fields in so called Riemann-Cartan theories. In what follows, we suppose that in \((M,\boldsymbol{g},\nabla,\tau _{\boldsymbol{g}},\uparrow )\) (or \(\mathfrak{M}\)) a set of dynamic fields live and interact. Of course, we want that the Riemann-Cartan spacetime admits spinor fields, and from what we learned in Chap. 7, this implies that the orthonormal frame bundle must be trivial. This permits a great simplification in our calculations. Moreover, we will suppose, for simplicity that the dynamic fields ϕ A, A = 1, 2, …, n, are in general distinct r-forms, i.e., each \(\phi ^{A} \in \sec \bigwedge ^{r}T^{{\ast}}M\hookrightarrow \mathcal{C}\ell(M,\mathtt{g})\), for some r = 0, 1, …, 4. Before we start our enterprise we think it is useful to recall some results which serve also the purpose to fix the notation for this chapter.

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Notes

  1. 1.

    This is not a serious restriction in the formalism since we already learned how (given a spinorial frame) to represent spinor fields by sums of even multiform fields.

  2. 2.

    For the definition of jet bundles and notations see Sect. A.3.1.

  3. 3.

    We observe that some authors (e.g.,[1, 7, 11]) denote \(\frac{\partial F} {\partial X} - (-1)^{p}d\left ( \frac{\partial F} {\partial dX}\right )\) by \(\frac{\boldsymbol{\delta }F} {\boldsymbol{\delta }X}\), something we also did in the first edition of our book.

  4. 4.

    In this Chapter boldface latin indices, say a, take the values 0, 1, 2, 3. 

  5. 5.

    A Lagrangian density for the \(\{\theta ^{\mathbf{a}}\}\) for the case of GRT will be introduced in Sect. 8.5 and explored in details in the next Chapter.

  6. 6.

    We discuss further the issue of local Lorentz invariance and its hidden consequence in [5, 28].

  7. 7.

    For a derivation using \(\mathcal{L}_{g}\) see Exercise 9.18

  8. 8.

    Recall that from the previous section we learned that energy-momentum conservation law for the matter fields alone exist only when appropriated Killing vector fields exist.

  9. 9.

    More details on this issue may be found in [22].

  10. 10.

    In particular, on this issue the reader should read page 108 of Parrot’s book [23].

  11. 11.

    The reason for the factor 8π in Eq. (9.106) is that we choose units where the numerical value gravitational constant 8π G∕c 4 is 1, where G is Newton gravitational constant.

  12. 12.

    See the details of the calculation, e.g., in [22]

  13. 13.

    This means that the t a are no in this case pseudo 1-forms, as in Einstein’s theory.

  14. 14.

    Note that if we suppose that the universe contains spinor fields, then it must be a spin manifold, i.e., it is parallelizable according to Geroch’s theorem [12, 13], as we already know from Chap. 5

  15. 15.

    A metric is said to be asymptotically flat in given coordinates, if \(g_{\mu \nu } = n_{\mu \nu }(1 +\mathrm{ O}\left (r^{-k}\right ))\), with k = 2 or k = 1 depending on the author. See, e.g., [30, 31, 39].

  16. 16.

    For a Schwarzschild spacetime we have \(g = \left (1 -\frac{2m} {r} \right )dt \otimes dt -\left (1 -\frac{2m} {r} \right )^{-1}dr \otimes dr - r^{2}(d\theta \otimes d\theta +\sin ^{2}\theta d\varphi \otimes d\varphi )\).

  17. 17.

    This observation is true even if we use the so called ADM formalism [2] to be presented in Chap. 11 To be more precise, let us recall that we have a well defined ADM energy only if the fall off rate of the metric is in the interval 1∕2 < k < 1. For details, see [20].

  18. 18.

    The proof also uses as hypothesis the so called energy dominance condition [14].

  19. 19.

    Like, e.g., in [1, 19, 24] and many other textbooks. It is worth to quote here that, at least, Anderson [1] explicitly said: “In an interaction that involves the gravitational field a system can loose energy without this energy being transmitted to the gravitational field.”

  20. 20.

    On this issue, see also [25].

  21. 21.

    Another presentation the theory of the gravitational field in Minkowski spacetime employing Clifford algebra techniques has been given in [15]. However, that work, which contains many interesting ideas, unfortunately contains also some equivocated statements that make (in our opinion) the theory, as originally presented by those authors invalid. This has been discussed with details in [9].

  22. 22.

    Recall that \(\lrcorner\) is an antiderivation.

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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). Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes. 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_9

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