Einstein, Minkowski - View of 4- and 5-Dimensional Space

In 1906, soon after Albert Einstein announced his special theory of relativity, his former college teacher in mathematics Minkowski developed a new scheme for thinking about space and time that emphasized its geometric qualities24. Because space consists of 3 dimensions (x, y, z) and time (t) is 1 dimensional, a space time continuum is a 4-dimensional object. 


4-dimensional space refers to a Galilean coordinate system in which four coordinates (x, y, z, t) determine an event or, in other words, a point of Minkowski’s four dimensional space time continuum of classical physics.


An object at rest is the observer’s view of an object when the velocity appears to be zero (not moving) in the Galilean frame of reference rigidly attached to the observer. Yet we know from observation that the observer, object and indeed their frame of reference are all still moving in the universe in some way. For example, an observer on the surface of Planet Earth has a motion arising from the sum of the Earth’s rotation, the Earth orbiting the Sun, the Sun moving within the Milky Way, the Milky Way rotating and moving within the Universe. Therefore all observers and all other particles and bodies inevitably have an intrinsic complex dynamic movement within the universe that is ignored in classical physics in 4-Dimensions.


By defining the dynamic movement as a fifth dimension, in the same way as time t is the fourth dimension of a Galilean coordinate system, we convert Classical Physics in 4 Dimensions into Physics in 5 Dimensions.


5-dimensional space refers to a 4-dimensional space with an additional fifth dimension property; the hypothesis giving all particles and bodies their own individual path of common constant velocity as judged from a Galilean coordinate system in 4-dimensional space.


In order to attain the greatest possible clarity25, we consider the example of an observer sitting in the carriage of a train moving with a constant velocity c along a railway track. The observer sees other objects in the carriage to be also at rest with respect to his own local frame of reference within the carriage, yet the observer, carriage and train are all moving with the constant velocity c with respect to a frame of reference rigidly attached to the railway track.

We now extend the example to include an object O in a different railway carriage, as part of a second train, travelling with the same constant velocity c initially on a parallel railway track alongside observer P in his own carriage. For observer P looking across at object O, the object stays in the same place relative to the observer’s carriage and so the observer considers the object to be at rest with respect to his own frame of reference rigidly fixed to his carriage. As long as the two trains carrying the observer P and object O continue to run on parallel tracks both with the constant velocity c, the observer continues to see the object O at rest.


We now consider the case where the object train and carriage carrying object O follow a new track at angle θ relative to the track of the observer’s train. The velocity of the object and its carriage does not change. The object and observer are now following different tracks, subtending an angle θ to each other. 

Object O is now seen by the observer P to have a velocity vector v4 defined in 5-dimensional space26 to be orthogonal to the path of the observer P. In this example, as judged from a Galilean coordinate system rigidly attached to the observer’s track, the observer’s train runs parallel with the y-axis and the object velocity vector v4 is aligned with the x-axis. Object O also has a second velocity vector v5 parallel to the y-axis and we note that from simple geometry we have the scalar relationship c= v4+ v5for the three velocity vectors associated with object O.


In 5-dimensional space the perspective of the observer includes both railway carriages, trains and the tracks carrying the observer and object. Therefore the observer knows about the common velocity c of the observer and object, as well as the object velocity vectors v4 and v5. In any experiments, these velocity vectors can be taken into account when reviewing the experiment and the measured results of momentum and energy.


However in 4-dimensional space, the perspective of the observer is limited and the observer knows nothing about the common velocity c and velocity vector v5. The observer can only register the normal velocity v4. So for experiments in 4-dimensional space, the velocity vectors c and v5 cannot be taken into account when reviewing the experiment and the measured results. Therefore, if the hypothesis of the common constant velocity is true, we must be able to associate some physical effect in 4-dimensional space with the missing knowledge about c and v5. 


From the perspective of Physics in 5 Dimensions, mass m does not vary with velocity v4 as with the perspective of classical physics where a rest mass mo is defined when v4=0. We find that the Physics in 5 Dimensions relationship m v5 = mo c links these two 4- and 5-dimensional perspectives and, using c= v4+ v5to eliminate v5, we get Einstein’s relativistic mass formula where m= m/(1-v42/c2).


The Theory of Physics in 5 Dimensions is the perspective of physics as viewed from 5-dimensional space. In the book on Physics in 5 Dimensions27, we look at a succession of related yet different fields of physics from 5-dimensional space that produces a series of additional hypotheses which are unexpected compared to the classical view of physics however reasonable in that they comply with the fundamentals of physics. 


Compared to classical physics, Physics in 5 Dimensions is a significantly more unified theory. It is a model of the development of the universe, without the need of a Big Bang Theory, as reflected in various examples reviewed.


Minkowski’s space-time as a concept is not required by Physics in 5 Dimensions. The 5-Dimensions can be simply understood in terms of the 3 space dimensions (x, y, z), time t and the common constant velocity c of all particles and objects in the universe. In order to comply with the fundamentals of physics, the common constant velocity c turns out to be the speed of light.



(24) https://einstein.stanford.edu/content/relativity/q411.html

(25) Einstein, Relativity – The Special and General Theory, p12, Translated by Robert W. Lawson, ISBN 0-415-01904-7

(26)The magnitude of v4 is defined to be the scalar velocity v of an object as viewed by an observer in classical physics

(27) Physics in 5 Dimensions ISBN: 978-3-96014-233-1


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The Book by Alan ClarkPhysics in 5 Dimensions - is also available as a PDF file to members of ResearchGate here.