Euclidean Relativity  (a.k.a. 'proper time physics' or 'proper time geometry')

The Minkowski hyperbolic framework is often hard to grasp for those who get involved in relativity theory. Euclidean relativity proposes an intuitive circular geometry as alternative that uses proper time t as the fourth spatial dimension. Other common elements in Euclidean relativity are the Euclidean (++++) metric as opposed to the traditional Minkowski (+---) or (-+++) metric, and the universal velocity c for all objects in 4D space-time.

The Euclidean metric is derived from the Minkowski metric by rewriting

    

into the equivalent

     .

The roles of time t and proper time   have switched so that proper time   takes the role of the coordinate for the 4th spatial dimension. The universal velocity c appears from the regular time derivative

     .

The approach should not be mixed up with the "Wick rotation" or complex Euclidean relativity. Wick rotation replaces time t by it, which also yields a positive definite metric but it maintains proper time as the Lorentz invariant value whereas in Euclidean relativity   becomes a coordinate.

The Euclidean geometry is consistent with Minkowski based relativity in two reference frames. The hyperbolic Minkowski geometry turns into a rotation in 4D circular geometry where length contraction and time dilation result from the geometric projection of 4D properties to our 3D space. In three reference frames, some interpretations produce a different velocity addition formula, also affecting other formulas that depend on the velocity addition formula. The inconsistency does so far not imply known contradictions with experimental data. This may be due to the uncommon conditions that are required to show measurable deviations.

The Euclidean approach makes relativity accessible in an intuitive way and inspires further development and speculation. Individual authors apply and extrapolate it to various topics, like quantum mechanics, optics, particle physics and so on. Many Euclidean interpretations introduce time t as a parameter for tracking velocity and change. In Jose Almeida's and my own work it is treated as a fifth dimension.

Below are a number of references to articles on Euclidean relativity published by various authors, including my own (see Van Linden). Each author approaches the topic in his own way and individual interpretations were often developed independently without knowledge of the other authors. First fundamental work came from the Dutch mathematician Hans Montanus in the beginning of the 90's. During the last 5-10 years other authors have begun to find each other and have published their articles, initiating an innovative trend in relativity.
 

References:

Prof. Robert d'E Atkinson  Probably the first exploration of Euclidean relativity in history. General Relativity in Euclidean Terms (Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Volume 272, Issue 1348, pp. 60-78, 02-1963(!)). Does not yet use proper time    as fourth spatial dimension because it only deals with general relativity with stationary mass particles. Dr. R. G. Newburgh, Dr. T. E. Phipps  US Air Force research paper. A Space-Proper Time Formulation of Relativistic Geometry (Air Force Cambridge Research Laboratories, Office of Aerospace Research, U.S. Air Force, 1969) Seems like the first proposal to use proper time   as fourth spatial dimension. See also this document for a complete listing and biography of Dr. Phipps. Drs. Hans Montanus  Introduces the difference between Relative and Absolute Euclidean space-times (REST versus AEST). He favors the latter (my own articles build on REST). Special relativity in an absolute Euclidean Space-Time (Physics Essays, vol 4, nr 3, 1991) The Fizeau experiment in an absolute Euclidean Space-time (Physics Essays, vol 5, nr 4, 1992) A new concept of time (Physics Essays, vol 6  nr 4, 1993) General relativity in an absolute Euclidean space-time (Physics Essays, vol 8, nr 4, 1995) Electrodynamics in an absolute Euclidean space-time (Physics Essays, vol 10, nr 1, 1997) Arguments against the general theory of relativity and for a flat alternative (Physics Essays, vol 10 nr 4, 1997) Compton scattering, pair annihilation, and Pion decay in an absolute Euclidean space-time (Physics Essays, vol 11, nr 2, 1998) A Geometrical Explanation for the Deflection of Light (Physics Essays, vol 11, nr 3, 1998) Hyperbolic Orbits in an Absolute Euclidean Space-time (Physics Essays, vol 11 nr 4, 1998) Proper Time Physics (PDF 5MB) (Hadronic J.22:625-673,1999) Galactic Rotation and Dark Matter in an Absolute Euclidean Space-time (Physics Essays, vol 12 nr 2, 1999) Proper-Time Formulation of Relativistic Dynamics (Found. Phys. 31, Issue 9, Sep 2001, Pages 1357 - 1400) Flat Space Gravitation (Found. Phys. 35, Issue 9, Sep 2005, Pages 1543 - 1562) Talk at the IARD conference 2004. Prof. Jose Almeida  An Euclidean extrapolation to general relativity, explaining geodesic motion of objects as a result of a 4D refractive index, hence the alternative name '4D Optics'. Almeida considers 4D space-time as a Euclidean null-subspace of a 5D space-time with metric (-++++). The approach allows a treatment of  mass particles in 4D equivalent to photons in 3D, which is supplemented by considering particle worldlines as normals to wavefronts. An alternative to Minkowski space-time (arXiv:gr-qc/0104029, 2001) 4-Dimensional optics, an alternative to relativity (arXiv:gr-qc/0107083, 2001); A theory of mass and gravity in 4-dimensional optics (arXiv:physics/0109027, 2001) K-calculus in 4-dimensional optics (arXiv:physics/0201002, 2002) Prospects for unification under 4-dimensional optics (arXiv:hep-th/0201264, 2002) Unification of classic and quantum mechanics (arXiv:physics/0211056 ,2002) Maxwell's equations in 4-dimensional Euclidean space (arXiv:physics/0403058, 2004) Euclidean formulation of general relativity (arXiv:physics/0406026, 2004) The null subspace of G(4,1) as source of the main physical theories (arXiv:physics/0410035, 2004) Talk at the Moscow conference Number Time Relativity 2004 Talk at the PIRT IX conference Londen 2004. Choice of the best geometry to explain physics (arXiv:physics/0510179, 2005) Monogenic functions in 5-dimensional spacetime used as first principle: gravitational dynamics, electromagnetism and quantum mechanics (arXiv:physics/0601078, 2006) How much in the Universe can be explained by geometry? (arXiv:0801.4089, 2008) Prof. Alexander Gersten  Uses the term 'Mixed Space' to refer to the space where time t and proper time have changed place. Probably the first one to recognize the value of Montanus' work. Talk at the IARD conference 2002. Euclidean Special Relativity (PDF file) (Found. Phys. 33, 2003, Pages 1237-1251) Carl Brannen MSc. Emphasis on the geometry and mathematics (geometric or Clifford algebra) that could be used as a basis for Euclidean relativity. The Proper Time Geometry (pdf, ver 1.0 10/19/2004) Phase Velocity of de Broglie Waves (pdf, ver 1.0 11/20/2004) The Geometry of Fermions (pdf, ver 1.01, 10/21/2004) The Geometric Speed of Light (pdf, ver 1.0 11/07/2004) Nonlinear Waves on the Geometric Algebra (pdf, ver 1.1 12/02/2004) Homepage of Carl with various other papers on particle physics. Dr. Giorgio Fontana  Summarizes the results of Montanus, Gersten and Almeida in his first article and extends this with some more speculative thoughts The Four Space-times Model of Reality (arXiv.org, physics/0410054A) Hyperspace for Space Travel, Video of presentation at the STAIF 2007 by Dr. Eric Davis (American Institute of Physics, C.P. 880, pp. 1117-1124) Gravitational Waves in Euclidean Space (Excerpt from AIP Conference Proceedings 969, 1055 (2008)) On the foundations of Gravitation Inertia and Gravitational Waves (Scribd) Extending Maxwell's equations to Euclidean relativity in 5D Towards an Unified Engineering Model for Long (and short?) Range Forces and Wave Propagation (Powerpoint presentation) Homepage of Giorgio Fontana. Dr. Anthony Crabbe  As an alternative to the traditional Minkowski hyperbolic geometry the author uses 'Circular Function Geometry' (CFG), which is natural for many Euclidean interpretations of special relativity. Alternative conventions and geometry for Special Relativity (Annales de la Fondation Louis de Broglie Vol 29 no 4, 2004) The Limitations of the Minkowski Model of Space-time, talk at the 13th Triennial Conference of the International Society for the Study of Time, (Monterery, CA July 28-Aug 3 2007) Dr. Phillips V. Bradford  Characteristic elements of Euclidean relativity, using proper time and universal velocity c for all objects in space-time. Alternative ways of looking at physics, with amongst others A space-time, geometric interpretation of the beta factor in Special Relativity. Dr. Witold Nawrot, Is the space-time reality Euclidean? Another Euclidean interpretation, comparing Four-dimensional Euclidean Reality (FER) with Lorentzian space-time. Again a similar approach with   as fourth dimension. Richard D. Stafford Ph.D., Resolution of the Relativity/Quantum Mechanics Conflict  (on Web Archive) Uses Euclidean space-time with  as fourth dimension to solve a common problem with the perception of reality. Subramaniam Kanagaraj, Euclidean Special Relativity Personal website, presenting articles based on a Euclidean interpretation of special relativity. A velocity vector 4-Euclidean Space-Time (EST) geometrical model governed by the functions of a circle is formulated with the (++++) Euclidean signature. Rob F.J. Van Linden BSc. (i.e., my own work, also accessible through the buttons in the navigation menu) Dimensions in special relativity theory (Galilean Electrodynamics Vol 18 nr 1, Jan/Feb-2007); A Euclidean interpretation of special relativity providing arguments for a geometrical unification of gravity and electromagnetism in five dimensions. Mass particles as bosons in five dimensional Euclidean gravity (web, May 2007); Extending the ideas of the previous article to gravity, proposing to regard mass particles themselves, instead of gravitons, as bosons following null geodesics in 5D gravity. Minkowski versus Euclidean 4-vectors (web, feb 2006); Associating 4-vectors with geometric properties in Euclidean space-time. Propulsion without propellant using four-momentum of photons in Euclidean special relativity (web, apr 2008); describing an alternative method to accelerate particles or objects, using principles of 4D momentum that follow from Euclidean special relativity. Relativity Simplified; Simplified and popularized description of of the essentials of Euclidean relativity. The Universe as a Multi-Dimensional Fractal; Speculations on a fractal-like universe, based on the geometry of Euclidean relativity. It suggests a hierarchical ordering of the four forces of Nature together with their fermions and bosons through their number of dimensions. Note on articles by Dr. Frans-Guenter Winkler (website, arXiv): although the same terms Euclidean special and general relativity are used, the geometry of the model is different. It maintains t as fourth dimension and   as invariant, yet uses a (++++) metric. The approach falls outside the scope of this page.

3D graph of classical equation for relativistic addition of velocities           3D graph of new equation for relativistic addition of velocities as derived by Van Linden (Dimensions in special relativity theory, Galilean Electrodynamics Vol 18 nr 1, Jan/Feb-2007)       Hans Montanus' visualization of the relation between Minkowski and Euclidean diagrams (From: Proper Time Physics, Hadronic Journal 22, 625-673, 1999)