(a.k.a. 'proper time physics' or 'proper time geometry')
Relativity theory traditionally uses the Minkowski hyperbolic framework. Euclidean relativity
proposes a circular geometry as alternative that uses proper time
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
becomes the coordinate for the 4th spatial dimension. The universal
appears from the regular time derivative
The switch impacts all relativistic formulas for
displacement, velocity, acceleration and so on in a similar way; Lorentz
invariants are based on t while vector components representing the
4th dimension are based on
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.
reference frames, some interpretations produce a different velocity addition formula, also
affecting other formulas that depend on the velocity addition formula.
inconsistency does so far not imply known contradictions with experimental data.
Extreme conditions are required to show measurable deviations
The justification for the Euclidean approach is twofold: it makes relativity
accessible in an intuitive way and it opens new opportunities to further develop
Individual authors apply and extrapolate it in various topics, like
quantum mechanics, optics, particle physics and so on. Many Euclidean
interpretations introduce time
t as a
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.
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
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.
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)
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)
in an absolute Euclidean space-time
(Physics Essays, vol 10, nr 1, 1997)
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
Proper Time Physics (PDF
Galactic Rotation and Dark Matter in an Absolute Euclidean Space-time (Physics Essays, vol 12 nr
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.
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
4-Dimensional optics, an alternative to relativity
A theory of mass and gravity in 4-dimensional optics
K-calculus in 4-dimensional optics
Prospects for unification under 4-dimensional optics
Unification of classic and quantum mechanics
Maxwell's equations in 4-dimensional Euclidean space
Euclidean formulation of general relativity
The null subspace of G(4,1) as source of the main physical theories
Talk at the Moscow conference Number Time Relativity 2004
the PIRT IX conference Londen 2004.
Choice of the best geometry to explain physics
Monogenic functions in 5-dimensional spacetime used as first principle: gravitational dynamics, electromagnetism and quantum mechanics
How much in the Universe can be explained by geometry?
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
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
Space-times Model of Reality (arXiv.org, physics/0410054A)
for Space Travel,
Video of presentation at the STAIF
2007 by Dr. Eric Davis (American Institute of Physics, C.P. 880,
Gravitational Waves in Euclidean Space (Excerpt
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
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
c for all objects in space-time.
Alternative ways of
looking at physics,
with amongst others
geometric interpretation of the beta factor in Special Relativity.
Dr. Witold Nawrot,
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
Web Archive) Uses Euclidean
as fourth dimension to solve a
common problem with the perception of reality.
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
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
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.
(web, feb 2006);
Associating 4-vectors with geometric properties in Euclidean space-time.
Propulsion without propellant using four-momentum of photons in Euclidean
(web, apr 2008); describing an alternative method to accelerate
particles or objects, using principles of 4D momentum that follow
from Euclidean special relativity.
Simplified and popularized description of
of the essentials of Euclidean relativity.
The Universe as a Multi-Dimensional Fractal;
Speculations on a
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
Note on articles by Dr. Frans-Guenter Winkler (website,
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)
Space-Time-Matter consortium, coordinated by Prof. Paul Wesson.
Not so much identifiable as Euclidean relativity but proposals very
similar to mine regarding the application of a fifth dimension, based on the
Campbell-Magaard embedding theorem.
A quote from an article of Paul Wesson,
In Defense of Campbell's Theorem as a Frame for New Physics
(arXiv.org gr-qc/0507107, July 25th 2005)
reflects one of the key elements of my own article, "Mass Particles as
Bosons in Five Dimensional Euclidean Gravity" :
implication of this for particles is clear: they should travel on null 5D
geodesics. This idea has recently been taken up in the literature, and has a
considerable future. It means that what we perceive as massive particles in 4D
are akin to photons in 5D."
Suggestions for additional sources are welcome.
Updated januari 25th, 2015
Other interesting science/physics/math links:
How to become a good theoretical physicist?
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Accredited Online Colleges A list of online colleges with academic and
professional resources in the area of theoretical physics.
Eric Weisstein's World of Physics
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Einstein archives online
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Walter Babin's page on special relativity and theoretical physics
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Applets A collection of visualizations of various Physics and Math
topics by Paul Falstad.
Andrew Hamilton's homepage Curved spacetime visualized. Watch yourself falling into a black hole.
Interpretations of Relativity Theory (PIRT, Moscow 2011)
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Wikipedia: Special relativity (alternative formulations)
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Relativity links in the dmoz Open Directory Project The Open Directory is an attempt to build a global catalogue of the web.