In this paper we present a theory of the gravitational field where this field, represented by a (1, 1)-extensor field h describing a plastic distortion of the Lorentz vacuum (a real substance that lives in a Minkowski spacetime) due to the presence of matter. The field h distorts the Minkowski metric extensor η generating what may be interpreted as an effective Lorentzian metric extensor g = h † ηh and also it permits the introduction of different kinds of parallelism rules on the world manifold, which may be interpreted as distortions of the parallelism structure of Minkowski spacetime and which may have non null curvature and/or torsion and/or non metricity tensors. We thus have different possible effective geometries which may be associated to the gravitational field and thus its description by a Lorentzian geometry is only a possibility, not an imposition from Nature. Moreover, we developed with enough details the theory of multiform functions and multiform functionals that permitted us to successfully write a Lagrangian for h and to obtain its equations of motion, that results equivalent to Einstein field equations of General Relativity (for all those solutions where the manifold M is diffeomorphic to R 4 ). However, in our theory, differently from the case of General Relativity, trustful energy-momentum and angular momentum conservation laws exist. We express also the results of our theory in terms of the gravitational potentials g µ = h † (ϑ µ ) where {ϑ µ } is an orthonormal basis of Minkowski spacetime in order to have results which may be easily expressed with the * Some (odd) misprints and typos have been corrected, some sentences have been changed for better intelligibility and Appendix F has new important remarks which result from discussions that W.A .R. had with A. Lasenby at ICCA10 (Tartu) in August 2014.

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cℓ(V, G E ) and the theory of its deformations leading to metric geometric algebras Cℓ(V, G) and some special types of extensors. Those tools permit obtaining, the remarkable golden formula relating calculations in Cℓ(V, G) with easier ones in Cℓ(V, G E ) ( e.g., a noticeable relation between the Hodge star operators associated to G and G E ). Several useful examples are worked in details fo the purpose of transmitting the "tricks of the trade".

In this paper we develop with considerable details a theory of multivector functions of a p-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. Several important types of derivatives for these multivector functions are introduced, as e.g., the A-directional derivative (where A is a p-vector) and the generalized concepts of curl, divergence and gradient. The derivation rules for different types of products of multivector functions and for compositon of multivector functions are proved.

Let V be an n-dimensional real vector space. In this paper we introduce the concept of euclidean Clifford algebra C (V, G E ) for a given euclidean structure on V , i.e., a pair (V, G E ) where GE is an euclidean metric for V (also called an euclidean scalar product). Our construction of C (V, GE) has been designed to produce a powerful computational tool. We start introducing the concept of multivectors over V . These objects are elements of a linear space over the real field, denoted by V. We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of two contraction operators on V, and the concept of euclidean interior algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develope a skill on the subject, preparing himself for the reading of the following papers in this series.