"But we must always be cognizant of the fact that the idealization that lies in the fiction of a rigid ( measuring ) body as a natural object might someday be shown to be unjustified or else justified only in relation to certain natural phenomena. General relativity has already shown the illegitimacy of this concept for spaces that are not small in the astronomical sense." - from "Non - Euclidean Geometry and Physics", 1925, by Albert Einstein ( 1879 - 1955 )
At velocities approaching a fractional but noticeable amount of the speed of light, , it's incumbent to maintain both the law of conservation of mass as well as the law of conservation of momentum under conditions utilizing the law of additions of relativistic velocities.
§ Assume frames of reference for systems , , and with the following stipulations:
is stationary relative to
is moving away from with relative velocity
, and that each is flying towards the other with a combined total relative velocity in which is wholly contained in
§ Assume further that immediately after the collision both coalesce into one larger mass-particle, - i.e., an inelastic collision.
 Einstein's Solution
Einstein devised a relativity mathematics such that mass is a variable quantity and is speed - dependent on velocity ( and hence ultimately on its energy content ) as follows:
§ Law of Conservation of Mass:
§ Law of Conservation of Momentum:
§ 1st Derivation:
Here we can observe that the conserved mass, , is certainly not equal to the sum of rest masses . That is,
§ Final Mass Derivation for Dilation ( expansion ):
Summary Meaning of Mass As A Variable
Remember this: if you stay with a body ( or particle ) of mass, called proper or rest mass , the amount of mass is constant! Common sense.
It is only when you are stationary and stand back, so to speak, and then measure the amount of mass a moving body ( or particle ) has, that you will both observe and hence measure different amounts of mass directly related to the velocity of the passing body of mass!!
Different observers moving at relatively different velocities to the passing body ( or particle ) of mass, will therefore observe and measure different amounts of mass .
Different velocities == differently observed and measured masses
The constancy of rest ( or proper ) mass is otherwise proven due to the "Invariance of the energy - momentum vector" which is another way of saying that the total energy of a system is always constant.
And, therefore, rest or proper mass is always a constant!
But, mass ( of body or particle ) in motion is always a speed - dependent variable!!
Directly dependent upon relative motion to an observer or observers, plural.
Question: what therefore exactly is 'mass'?
In fact, the entire physics world is awaiting the LHC - Large Hadron Collider - to eventually answer this question: what is 'mass' ??
Also, what makes 'gravity' ?
Ditto ... dunno.
Well, not exactly ... See: Einstein's "Principle of Equivalence" as between acceleration and gravity at 'Reinterpreting Galileo and Newton's Principle of Relativity'.
Again: the LHC will hopefully answer this profound physics and philosophical question during the next decade.
The name 'Einstein' in the following passages of this web page essay is to be understood more generally as encompassing the mathematical works of Max Planck, especially as regards "relativistic mass" which Einstein himself avoided in favor of inertial or, equivalently, rest ( proper ) mass, as revealed down below in the Epilogue: Some Final Questions regarding what did Einstein attach to e=mc2? Please see: Epilogue: Some Final Questions at e=mc2 page
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