Einstein’s Special Theory of Relativity^{28} gives the relationship between mass and
energy as E = m c^{2 }where E is the total relativistic energy of an object of relativistic mass m (resulting
from its motion with velocity^{29} v4) and c is the speed of light. Einstein’s relativistic mass formula is given by the
expression m=mo /√(1-v4^{2 }/c^{2}) where mo is the “rest” mass of the object when v4 = 0. The momentum of an object
is given by p=mv4 and Einstein’s useful expression linking energy and momentum is given by E^{2 }= c^{2 }p^{2 }+ Eo^{2}.
Using the relativistic kinetic energy term K = p c = m v4 c then gives E^{2 }= K^{2 }+ Eo^{2}. Einstein's theory of
relativity is a generalisation that includes Newton's theory as a special case (in the low velocity limit), so that when v4«c a simpler energy expression results because E ≈ Eo. We find that E ≈ mv4^{2 }/2 + Eo
and using the Newtonian kinetic energy term KE = mv4^{2 }/2 gives E ≈ KE+Eo.

*Physics in 5 Dimensions* is based on the 4-dimensional space of classical physics plus an additional dynamic fifth dimensional 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. From the perspective of *Physics in 5
Dimensions* an object O and observer P are following two different paths with the common constant velocity c, subtending an angle θ to each other.

Object O is seen by observer P to have a velocity vector v4, defined in 5-dimensional space to be orthogonal to the path of the observer P and aligned with the
x-axis in the figure opposite. Object O also has a second velocity vector v5 defined to be parallel to the y-axis. For the three velocity vectors (v4, v5, c) associated with object O, simple
geometry gives the scalar relationship c^{2 }= v4^{2 }+ v5^{2}.

When angle θ=0^{o} degrees, then the object and observer are following parallel paths with their common constant
velocity c so that v4 = 0 and v5 = c. The object will appear to the observer to be at rest. When angle θ=90^{o} degrees, then the object and observer are following paths of common constant velocity c at right angles to each other and v4 = c and v5 = 0.

With Physics in 5 Dimensions the relativistic energy E and the mass of a body are constant^{30} so that E = Eo = mo c^{2 }and m = mo. The kinetic energy terms are given by K
= Ko = mo v4 c and we have the addition of a
potential energy term Vo = mo v5 c. By
multiplying the scalar relationship c^{2 }= v4^{2 }+ v5^{2 }throughout by mo^{2 }c^{2}, we get Eo^{2 }= Ko^{2 } + Vo^{2 }which can be compared with
Einstein’s expression E^{2 }= K^{2 }+ Eo^{2}. A simpler expression
results by defining a new 5-dimensional kinetic energy term where K5 = Eo - Vo = mo v4^{2 }/ (1 + v5/c). K5 is an exact relativistic term however when v4«c then v5≈c, and we get K5 ≈ mov4^{2}/2 = KE; the Newtonian kinetic energy term of
classical
physics.

With the expression m v5 = mo c we can swap between Physics in 5- and 4- Dimensions and used with c^{2 }= v4^{2 }+ v5^{2 }to eliminate v5 we get Einstein’s relativistic mass formula where m = mo /√(1-v4^{2 }/c^{2}). This means that the common constant velocity c of Physics in 5 Dimensions has to take the value of the speed of light in
order to comply with Einstein’s mass formula.

We note that in classical physics, the total relativistic energy of a body increases with increasing relative velocity v4 so that the kinetic energy adds to the rest energy; while in Physics in 5 Dimensions the total energy Eo is constant and can only be shared between the kinetic and potential energy terms. A force must be applied to a particle in order to change the angle θ of its plane of motion with the common constant velocity, but once altered the path stays at the new angle selected and the particle continues on its path with the unchanged common constant velocity c; this represents absolute conservation of energy.

The Schrödinger theory of quantum mechanics^{31} (1933) produced a powerful tool to analyse the behaviour of particles. The
success of Schrödinger’s equation with multi-electron atoms confirmed the validity of treating particles in terms of their wave function and potential energy. We find that Schrödinger’s quantum
theory is important to the Theory of Physics in 5 Dimensions because the form of the wave equation developed fits more comfortably with the expressions of Physics in 5 Dimensions than those of
classical physics. The total particle energy of Schrödinger’s quantum theory is constant and equal to the sum of the kinetic and potential energies; equivalent to the basic relationship of E = K5
+ V of Physics in 5 Dimensions.

**References:**

(28) Quantum Physics, Appendix A “Special Theory of Relativity”, page A-16, Robert Eisberg & Robert Resnick, John Wiley & Sons Inc., ISBN 0-471-87373-X

(29) The magnitude of v4 is defined to be the scalar velocity v of an object as viewed by an observer in the 4-dimensional space of classical physics

(30) The mass of an object doesn’t vary with velocity in Physics in 5 Dimensions and so usually we don’t use the subscript “o” to indicate rest mass. The subscript “o” is only used in text comparing expressions of Physics in 4- and 5- Dimensions

(31) Erwin Schrödinger – Nobel Lecture – The fundamental idea of wave mechanics – December 12. 1933 - https://www.nobelprize.org/uploads/2017/07/schrodinger-lecture.pdf

(32) Page 159 - 5-dimensional space and Schrödinger’s equation, Physics in 5 Dimensions, Alan Clark, 2017, 496 pages, Winterwork, Borsdorf, ISBN: 978-3-96014-233-1

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