New Insight into the Relation between Torsion and Electromagnetism

In several unified field theories the torsion trace is set equal to the electromagnetic potential. Using fibre bundle techniques we show that this is no leading principle but a formal consequence of another geometric relation between space-time and electromagentism. ∗E-mail: [email protected] Torsion in general relativity is commonly studied within the framework of Einstein–Cartan theory, in which it is related to spin [1]. However, there is also another physical role of torsion suggested in several works on the unification of gravity and electromagnetism [2,3,4,5]. The idea of such a geometric unification is to omit any restrictions on the linear connection Γμβ and to identify its torsion trace Tμ = T α μα = Γ α μα − Γ α αμ with the electromagnetic potential Aμ. It is well-known that Einstein’s so-called non-symmetric unified field theory of gravity and electromagnetism [6] suffered from severe inconsistencies. Subsequently, several authors tried to remedy these drawbacks by changing the employed Lagrangian and by introducing the ansatz Tμ ∼ Aμ in an ad hoc manner [2]. Later on, this ansatz could be motivated by the structure of the field equations, which precisely resembled the Einstein–Maxwell equations [3,4]. Thereby, an arbitrary connection Γμβ was restricted by the field equations to be of the form Γμβ = { α μβ}+ 1 3 δβTμ , (1) where {μβ} is the Christoffel symbol. Despite the formal agreement of the field equations, the proposed identification Tμ ∼ Aμ lacked a clear geometric and physical meaning, because Tμ is only a vector but not an U(1) potential like Aμ and therefore can not be gauged. The so-called λ–transformation, introduced first by Einstein in another context [6], could not substitute the U(1) structure, since its geometric foundation is obscure. The real problem with the ansatz Tμ ∼ Aμ is that no true U(1) fibre bundle structure have been constructed. In [5] such a structure was introduced, but it differed from the common understanding of U(1) gauge theory. For example, charged particles were represented by scalar densities of an “imaginary weight”. A related problem with unified field theories is the lack of a physical interpretation of the resulting connection (1): Since it is not metric, ∇μgαβ = − 3 Tμ · gαβ 6= 0, it must not be applied for the parallel transports of signals on the space-time because this would lead to the dependence of physical invariants upon their histories like in Weyl’s unified theory [7]. Therefore, it is necessary to decompose the whole connection (1) into a metric part and the torsion term. But this can be done in several ways, for example, as Γμβ = [{ α μβ}] + [ 1 3 δβTμ] (2)