Something I found very interesting when I learned about it several years ago: it's possible to formulate SR and GR with the Euclidean (++++) metric instead of t

Something I found very interesting when I learned about it several years ago: it's possible to formulate SR and GR with the Euclidean (++++) metric instead of the Minkowski (+---) or (-+++) metric. Such a formulation (there is a variety of them) is sometimes called Euclidean Relativity (ER). See: https://www.euclideanrelativity.com/ https://www.euclideanrelativity.com/ Some ER research is particularly fascinating to me, e.g. Montanus' work on Flat Space Gravitation: https://link.springer.com/article/10.1007/s10701-005-6482-0 https://link.springer.com/article/10.1007/s10701-005-6482-0 A new description of gravitational motion will be proposed. It is part of the proper time formulation of physics as presented on the IARD 2000 conference. According to this formulation the proper time of an object is taken as its fourth coordinate. As a consequence, one obtains a circular space–time diagram where distances are measured with the Euclidean metric. The relativistic factor turns out to be of simple goniometric origin. It further follows that the Lagrangian for gravitational dynamics does not require an interpretation in terms of curvature of space–time. The flat space model for gravitational dynamics leads to the correct predictions for the bending of light, the perihelion shift of Mercury and gravitational red-shift. The new theory is free of singularities.