4  Vector Analysis for Bodies or Particles of Matter
"The whole of science is nothing more than a refinement of everyday thinking"  Albert Einstein ( 1879  1955 )
"Whence arises all that order and beauty we see in the world?"  Issac Newton, Opticks ( 1643  1727 )
"The only thing that interferes with my learning is my education"  Albert Einstein ( 1879  1955 )
The Problem
It's an accepted á priori postulate of all types of physics ( Galileo, Newton, Einstein's Special and General relativity physics ) that the form of all of these mathematical physics laws maintain the same invariant form in all reference frames, whether inertial ( non  accelerating ) as in Special Relativity Physics or the more generalized non  inertial frames ( accelerating ) as in General Relativity Physics.
So it was earlier stated in regards to Special Relativity Physics, the following:
The Principle of Special Relativity Physics  All the laws of physics in their simplest reduced form are Lorentz  transformable and hence form  invariant as between an infinite number of moving reference systems, each one of which is moving uniformly and rectilinearly with respect to any other system and where no one system is privileged or preferred over any other reference ( inertial ) system when measurements of length or time are taken.
However as Einstein proceeded into his General Theory wherein he was dealing with accelerating ( non  inertial ) frames of reference in addition to those of his previous Special Theory, it became necessary for him to propound his General Covariance Principle of Relativity Physics or simply The General Principle of Relativity Physics:
The General Covariance Principle of Relativity Physics  All the laws of physics whether manifested in inertial ( special relativity physics ) or non  inertial ( general relativity physics accelerating ) frames, are form  invariant as regards the Lorentz transformation equations expressed as valid covariant tensors where in the presence of bodies of matter possessing mass the geometry of spacetime becomes geodetic or curved, otherwise identified as "gravity's effect" or simply 'gravity':
"The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion"
 The Foundation of the General Theory of Relativity, 1916, by A. Einstein
So the problem is that in the case of ordinary spacial coordinates for time and velocities, nothing really is Lorentz form  invariant especially as regards time and velocity, all of which will be overcome by enlisting proper time first and foremost in all future special and general relativity equations regarding time, velocity, energy, momentum and acceleration for the motion of bodies of matter possessing mass along world timelines in four  dimensional space and time.
Analysis
§ Define:
§ Lorentz transformations in  space:
However in  space where the invariant properties of space  time are invoked under Lorentz transformation equations, we get
Namely that in Euclidean  space, nothing is Lorentz time  invariant and hence in violation of the Principle of Relativity for form  invariance of the equations of special and general relativity physics!
§ Normalization with The Lorentz ( Correction ) Factor:
That is, with the extra help from the Lorentz ( correction ) Factor, the Euclidean  space can be brought into  space compliance under invariant Lorentz transformations by simply utilizing proper time instead of spacial or coordinate time!
§ The Light Sphere and Invariant Proper Time :
Of course all of the pseudo  Euclidean, Minkowski special relativity equations ultimately derive from the form  invariant light sphere equation under Lorentz transfomation:
where
for which
See: "Minkowski's Space  Time Light Cone"
The Solution
But all of the above  and  space inconsistencies are brought into invariant Lorentz transformation compliance simply by invoking the Lorentz form  invariant proper time where
The 4  Velocity Vector
§ Defining World Timeline Coordinates:
§ Velocity over a world timeline path:
§ Defining the 4  Velocity Vector:
and for which is the 4  velocity vector tangent to the World Timeline path of the body or particle of matter in motion possessing mass, thereby satisfying the Lorentz transformation equations for invariance ( of proper time and invariant distance intervals on the surface of an ever expanding light sphere ) between relatively moving , systems:
§ Corollary:
Case 1: material body of matter possessing mass:
Case 2: zero rest mass, wave  like particle ( photon or graviton ), where :
This is a null 4  velocity vector which is also invariant with respect to proper time .
That is, in Case 2 we can now identify the null 4  velocity vector for light photons comprising a sphere of light such that
where the invariant space  time distance separating light photons on the surface of an ever expanding sphere of light has zero length !
The 4  Momentum Vector
Massless Photons
To the point: only light and gravity are pure energy and travel thru spacetime as null vectors!!
The 4  Acceleration Vector
Please note: for the 4  Acceleration Vector mathematical analysis it's necessary to gain access to the General Relativity Physics parts of this mathematical essay by contacting the author of Relativity Physics and Science Calculator. Thank you for understanding.
The 4  Force Vector
Again please note: for the 4  Force Vector mathematical analysis it's necessary to gain access to the General Relativity Physics parts of this mathematical essay by contacting the author of Relativity Physics and Science Calculator. Thank you for understanding.
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