Considerations on Space-time Geometry

Einstein-Cartan Theory: Don't dismiss but put in proper perspective instead

(Updated and published on talk page of Einstein-Cartan theory.)

General suggestions

Primarily present Riemann-Cartan theory as a bona fide mathematical theory of geometry, more general than (pseudo-)Riemannian geometry (which it is).

Present it subsequently as a candidate for generalization of General Relativity (see next section for some hints); the references indicated below already go in this direction.

In order to mitigate the justified suspicion of self-indulgence on te part of the principal author, I suggest at least to add some further references, which might include:

- Ne'eman, Y.; Hehl, F.W.: Test Matter in a Space-time with Nonmetricity. Classical and Quantum Gravity, 14 (1997), A251-A259.
- Hehl, F.W.; Mielke, E.W.; Tresguerres, R.: Skaleninvarianz und Raumzeitstruktur. In Geyer, B.; Herwig, H., Rechenberg, H. (Eds.): Werner Heisenberg. Physiker und Philosoph, p 299-306. Spektrum Akademie Verlag, Heidelberg, 1993.
- Hehl, F.W.; McCrea, J.D., Mielke, E.W.; Ne'eman, Y.: Progress in Metric-affine Gauge Theories of Gravity: Field Equations, Noether Identities, World Spinors and Breaking of Dilation Invariance. Physics Reports, 258 (1995), 1-171.

+ others. In fact, there is substantial body of work on this by Friedrich Hehl, with various co-authers, in the period 1970 up to this day.

And also:

- Debeyer, R.: Elie Cartan - Albert Einstein. Lettres sur le Parallelisme Absolu 1929 - 1932. Académie Royale de Belgique et Princeton University Press, Brussels, 1979.
- Vargas, J.G.; Torr, D.G.: The Cornerstone Role of the Torsion in Finslerian Physical Worlds. General Relativity and Gravitation, 27 (1995), 629-644.
- Trautman, A.: Foundations and Current Problems of General Relativity. In: Trautman, A.; Pirani, F.A.E.; Bondi, H. (Eds.): Brandeis Summer Institute: Lectures on General Relativity, p 1-248. Prentice Hall, 1964.
- Trautman, A.: On the Einstein-Cartan Equations, I - III. Bulletin de l' Académie Polonaise des Sciences. Série des Sciences math., astr. et phys., 20 (1972), 185-190, 503-506, 895-896.

etc.

This said, the article should also make clear that while (as argued below) a worthwhile area of complementary research etc. etc., Riemann-Cartan geometry has not superseded Riemannian General Relativity.

Value of the theory in a broader perspective

Ideally, this topic should be presented as part of a more general overview of "space-time geometries" (including Weyl and Finsler geometries), indicating motivations, merits and failures of such attempts. Failure typically being lack of experimental evidence or indeed evidence to the contrary.

A scheme of such general geometries is given by Proberii (Metric-affine Scale-covariant Gravity. General Relativity and Gravitation, 26 (1994), 1011-1054) as: general Affine / Weyl-Cartan ("geodesic lightcone") / Weyl (zero torsion) / Riemann-Cartan (zero Weyl vectorfield) / Riemann (no torsion, no Weyl) / Minkowski (zero curvature).

Space-time geometries with torsion (as is the case with Riemann-Cartan geometry) have been investigated not only with the aim of including a spin contribution to energy-momentum, but also to check for space-time models suitable at the nano-scale, which is dominated by quantum effects, and even as an approach to the quantization of space-time itself.

Further theoretical interest in this model resides in the fact that a (Poincaré) gauge field formulation is possible, which necessarily leads to torsion (the gauge potentials may be interpreted as curvature resp. torsion of a Riemann-Cartan space. These models are sometimes referred to as Einstein-Cartan-Sciama-Kibble theories.

One series of variants of space-time geometries with torsion, has been initiated by Trautman. A spin density is introduced, with the explicit aim of avoiding singularities. If not a "flaw", these are at least a nuisance in standard Riemannian spacetimes, where "physics breaks down".

Based on correspondence between Einstein and Cartan, Torr/Vargas have also pointed out the fact that Finsler geometries generally exhibit non-zero torsion.

Apart from all this, there is an important foundational aspect to considering more general space-time geometries: the choice of a 4-dimensional pseudo-Riemannian manifold as "the" model for space-time leaves many physical questions unanswered.

There have been many attempts to give a more explicit, physically motivated axiomatics of space-time, in which the Riemannian choice is then derived (instead of postulated at the outset). Famous contributions have been made by Reichenbach, Synge, Ehlers-Pirani-Schild and many others.

As "physical" posulates are added, the generality of the geometry is restricted further and further. It appeared especially difficult to motivate why a Weyl structure should ultimately "collapse" to the Riemannian one, without recourse to quantum mechanical considerations. This was finally achieved in the period 1995-2000 by Schröter and Schelb. (Schröter, J.; Schelb, U.: Remarks concerning the Notion of Free Fall in Axiomatic Space-Time Theory. General Relativity and Gravitation, 27, 1995, 605 + further publications by these authors.)

Conclusion

From the above, it should at least be clear that the topic is meaningful also in mathematical and foundational physics, and a part of the ongoing research on the nature and modeling of space-time. So definitely not nonsense.

And let's not forget that several (often more radical) generalizations / modifications of the classical space-time model are currently being investigated actively and in earnest by many renowned scientists. They do not shun such things as complexifications ("imaginary time"), change of metric signature, change of topology, twistors etc.

--Marc Goossens 13:41, 2 December 2007 (UTC)