Lanczos-Lovelock and f (R) Gravity from Clifford Space Geometry

A rigorous construction of Clifford-space Gravity is presented which is compatible with the Clifford algebraic structure and permits the derivation of the generalized connections in Clifford spaces ( C-space) in terms of derivatives of the C-space metric. We continue by arguing how Lanczos-Lovelock higher curvature gravity can be embedded into gravity in Clifford spaces and suggest how this might also occur for extended gravitational theories based on f(R), f(Rµν),.... actions, for polynomial-valued functions. Black-strings and black-brane metric solutions in higher dimensions D> 4 play an important role in finding specific examples. 1 The construction of Clifford-space Gravity At the beginning of this section we follow closely the work in [1] and then we depart from it by constructing Clifford space (C-space) gravity without making any a priori assumptions on the C-space connections. Let the vector fields γµ, µ = 1, 2,..., n be a coordinate basis in Vn satisfying the Clifford algebra relation γµ · γν ≡ 1 2 (γµγν + γνγµ) = gµν (1) where gµν is the metric of Vn. In curved space γµ and gµν cannot be constant but neces-sarily depend on position xµ. An arbitrary vector is a linear superposition [2] a = aµγµ (2) where the components aµ are scalars from the geometric point of view, whilst γµ are vectors. 1 Besides the basis {γµ} we can introduce the reciprocal basis1 {γµ} satisfying γµ · γν ≡ 1 2 (γµγν + γνγµ) = gµν (3) where gµν is the covariant metric tensor such that gµαgαν = δ µ ν, γ µγν + γνγ µ = 2δµν and γµ = gµνγν. Following ref.[2] (see also [3]) we consider the vector derivative or gradient defined according to ∂ ≡ γµ∂µ (4) where ∂µ is an operator whose action depends on the quantity it acts on. Applying the vector derivative ∂ on a scalar field φ we have ∂φ = γµ∂µφ (5) where ∂µφ ≡ (∂/∂xµ)φ coincides with the partial derivative of φ. But if we apply it on a vector field a we have ∂a = γµ∂µ(a νγν) = γ µ(∂µa νγν + a ν∂µγν) (6) In general γν is not constant; it satisfies the relation [2], [3