A note on post-Riemannian structures of spacetime

Friedrich W. Hehl^{∗} Uwe Muench^{†}

Institute for Theoretical Physics, University of Cologne D-50923 Köln, Germany

Abstract

A four-dimensional differentiable manifold is given with an arbitrary linear connection Γ_{α}^{β} = Γ_{iα}^{β} dx^{i}. Megged [1] has claimed that he can define a metric G_{αβ} by means of a certain integral equation such that the connection is compatible with the metric. We point out that Megged’s implicite definition of his metric G_{αβ} is equivalent to the assumption of a vanishing nonmetricity. Thus his result turns out to be trivial.

In the metric-affine theory of gravitation [2, 3], spacetime is assumed to be a four-dimensional differentiable manifold equipped with a linear connection Γ_{α}^{β} and, independently, with a metric g_{αβ}. Ne’eman and one of us proposed methods [4, 5] how one could measure the torsion T^{α} := Dϑ^{α} (here ϑ^{α} is the coframe) and the nonmetricity Q_{αβ} := −Dg_{αβ} of spacetime. In a recent note, Megged [1] has claimed that the use of the nonmetricity Q_{αβ} is misleading in some sense, since one can define a metric, provided the connection is given, such that it is automatically “connection compatible”, if we use his words. We will point out that this rests on the hidden assumption of a vanishing nonmetricity. Since Megged discards the original independent metric g_{αβ} as a meaningful physical field, there is no way that nonmetricity could enter his theory later.

∗E-mail address: hehl@thp.uni-koeln.de

†E-mail address: muench@thp.uni-koeln.de

1 Metric-affine spacetime

If a metric g_{αβ} and a connection Γ_{α}^{β} are given, for the conventions see [2], then we can raise and lower indices by means of g_{αβ}, such as in Γ_{αβ} := Γ_{α}^{γ}g_{γβ}, for example. With the definition of the nonmetricity Q_{αβ} := −Dg_{αβ}, it is straightforward to compute the symmetric part of the connection (see [6] or [2, eq. (3.10.6)]):

2Γ_{(αβ)} = dg_{αβ} + Q_{αβ} .  (1)

These are 40 equations, and no relation between metric and connection has been assumed.

2 Riemann-Cartan spacetime

If we assume the nonmetricity to vanish, then we find a Riemann-Cartan geometry:

Q_{αβ} = 0 ⇒ 2Γ_{(αβ)} = dg_{αβ} .  (2)

We can choose the coframe to be orthonormal (orthonormal gauge),

g_{αβ} =^{∗} o_{αβ} := diag(+1, −1, −1, −1) ,  (3)

then

Γ_{(αβ)} =^{∗} 0 or Γ_{αβ} =^{∗} −Γ_{βα} ,  (4)

i. e., we are left with 24 independent components of a Lorentz connection. Thus the connection one-form is SO(1, 3)-valued, describing the fact, that the scalar product of two vectors is invariant under parallel transport in such a spacetime.

3 Megged’s ansatz

He allows only for a connection to be the primary geometrical quantity. Let us call his connection Γ̂_{α}^{β}. Then he defines implicitly a metric G_{αβ} = G_{βα} by the relation

Γ̂_{α}^{γ}G_{γβ} + Γ̂_{β}^{γ}G_{αγ} = dG_{αβ} ,  (5)

see his equations [1, eq. (7)] and [1, eq. (8)]. The G_{αβ}, defined by (5), can be taken for raising and lowering indices, such as in Γ̂_{αβ} := Γ̂_{α}^{γ}G_{γβ}, for example. Then (5) can be rewritten as

2Γ̂_{(αβ)} = dG_{αβ} .  (6)

If we compare (6) with (1), we recognize that the ansatz (5), which represents 40 independent equations, is equivalent to the assumption

Q̂_{αβ} := −D̂G_{αβ} = 0 .  (7)

Of course, this can also be seen directly from (5), since

dG_{αβ} − Γ̂_{α}^{γ}G_{γβ} − Γ̂_{β}^{γ}G_{αγ} =: D̂G_{αβ} .  (8)

In other words, the equation (5), postulated by Megged, amounts to the assumption of a vanishing nonmetricity Q̂_{αβ} = 0. And then it is not surprising to fall back to the metric-compatible Riemann-Cartan spacetime.

Acknowledgments: We are grateful to Ofer Megged for discussions and for making his paper available to us prior to publication.

References

[1] O. Megged: Post-Riemannian spacetimes admit a causal structure; Los Alamos eprint archive gr-qc/9706068, 22 June 1997.

[2] F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne’eman: Metric-affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance. Physics Reports 258 (1995) 1–171.

[3] F. Gronwald and F. W. Hehl: On the gauge structure of gravity. In Proc. of the 14^{th} Course of the School of Cosmology and Gravitation on Quantum Gravity, held at Erice, Italy, May 1995. P. G. Bergmann, V. de Sabbata, and H.-J. Treder, editors (World Scientific, Singapore 1996).

[4] F. W. Hehl: On the kinematics of the torsion of spacetime. Found. Phys. 15 (1985) 451–471.

[5] Y. Ne’eman and F. W. Hehl: Test matter in a spacetime with nonmetricity. Class. Quantum Grav. 14 (1997) A251–A259.

[6] J. A. Schouten: Ricci-Calculus, 2^{nd} ed. (Springer, Berlin 1954).