Euclidean formulation of general relativity
  A variational principle is applied to 4D Euclidean space provided with a tensor refractive index, defining what can be seen as 4-dimensional optics (4DO). The geometry of such space is analysed, making no physical assumptions of any kind. However, by assigning geometric entities to physical quantities the paper allows physical predictions to be made. A mechanism is proposed for translation between 4DO and GR, which involves the null subspace of 5D space with signature $(-++++)$. A tensor equation relating the refractive index to sources is established geometrically and the sources tensor is shown to have close relationship to the stress tensor of GR. This equation is solved for the special case of zero sources but the solution that is found is only applicable to Newton mechanics and is inadequate for such predictions as light bending and perihelium advance. It is then argued that testing gravity in the physical world involves the use of a test charge which is itself a source. Solving the new equation, with consideration of the test particle's inertial mass, produces an exponential refractive index where the Newtonian potential appears in exponent and provides accurate predictions. Resorting to hyperspherical coordinates it becomes possible to show that the Universe's expansion has a purely geometric explanation without appeal to dark matter. 
Jose B. Almeida