Gravity in Complex Hermitian Space-Time

A generalized theory unifying gravity with electromagnetism was proposed by Einstein in 1945. He considered a Hermitian metric on a real space-time. In this work we review Einstein’s idea and generalize it further to consider gravity in a complex Hermitian space-time. email: [email protected] Published in ”Einstein in Alexandria, The Scientific Symposium”, Editor Edward Witten, Publisher Bibilotheca Alexandrina, pages 39-53 (2006) In the year 1945, Albert Einstein [1], [2] attempted to establish a unified field theory by generalizing the relativistic theory of gravitation. At that time it was thought that the only fundamental forces in nature were gravitation and electromagnetism. Einstein proposed to use a Hermitian metric whose real part is symmetric and describes the gravitational field while the imaginary part is antisymmetric and corresponds to the Maxwell field strengths. The Hermitian symmetry of the metric gμν is given by gμν (x) = gνμ (x), where gμν (x) = Gμν (x) + iBμν (x) , so that Gμν (x) = Gνμ (x) and Bμν (x) = −Bνμ (x) . However, the space-time manifold remains real. The connection Γμν on the manifold is not symmetric, and is also not unique. A natural choice, adopted by Einstein, is to impose the hermiticity condition on the connection so that Γνμ = Γ ρ μν , which implies that its antisymmetric part is imaginary. The connection Γ is determined as a function of gμν by defining the covariant derivative of the metric to be zero 0 = gμν,ρ − gμσΓσρν − Γσμρgσν . This gives a set of 64 equations that matches the number of independent components of Γμν which can then be solved uniquely, provided that the metric gμν is not singular. It cannot, however, be expressed in closed form, but only perturbatively in powers of the antisymmetric field Bμν . There are also two possible contractions of the curvature tensor, and therefore, unlike the real case, the action is not unique. Both fields Gμν and Bμν appear explicitly in the action, but the only symmetry present is that of diffeomorphism invariance. Einstein did notice that this unification does not satisfy the criteria that the field gμν should appear as a covariant entity with an underlying symmetry principle. It turned out that although the field Bμν satisfies one equation which is of the Maxwell type, the other equation contains second order derivatives and does not imply that its antisymmetrized field strength ∂μBνρ + ∂νBρμ + ∂ρBμν vanishes. In other words, the theory with Hermitian metric on a real space-time manifold gives the interactions of the gravitational field Gμν and a massless field Bμν . Much later, it was shown that the interactions of the field Bμν are inconsistent at the non-linear level, because one of the degrees of freedom becomes ghost like [3]. There is an