"Sir, an equation has no meaning for me unless it expresses a thought of GOD."  Srinivasa Aiyangar Ramanujan ( 1887  1920, India )
( beyond brilliant mathematical number theorist, selftaught, exceedingly poor to periods of literal starvation )
1). Laws of Conservation of Relativistic Momentum and Energy:
1a). Example of Relativistic ( or Effective ) Momentum:^{∗}
Let rest mass in , achieve a relative velocity of , say 0.88, to a stationary observer in .
Therefore, the relativistic or effective mass calculated by our stationary observer in is given by
And, relativistic momentum in terms of "classical momentum" of is simply derived as follows:
^{∗} note: this example is used in the future upcoming Relativity Science Calculator Mac application
The Invariant Energy  Momentum Vector
Because of different relative velocities for different observers in different frames of reference, the values of and ( momentum and energy respectively ) will accordingly be different for different observers residing in different systems.
However,
will always have the same value for all observers in all moving frames of reference. This essentially states that the magnitude of the energy  momentum vector is equal to the mass rest energy, where in particle physics rest mass, also known as proper mass,
,
is invariant for all observers in all frames of reference.
Here's a simple heuristic for this vital relationship:
Another important energy  momentum relationship is given by
and appears in nature for a quanta of light energy where as follows:
2a). Derivation of :
2b). Derivation of :
Also please notice that the rest mass of a quanta of light energy is always zero since there is no reference frame in which the light photon is at rest and hence the following is further confirmation of nature's reality:
This last equation expresses the relationship between the momentum of a quanta light photon and the energy of a flash of light. This relationship was also used by Einstein in his proof of the fundamental law of the inertia of energy.
2c). Much Simpler Derivation of :
2d). Example for :^{∗∗∗∗}
The classical electron radius is
According to T. Jacobson, Department of Physics, University of Oslo, whose paper in pdf, "An empirical mass formula for and leptons and some remarks on trident production", February 2, 2008, suggests the electron may itself possess an inner constituent structure, therefore for the sake of argument let's say that some unknown constituent of the electron has rest mass of
Using Heisenberg's Uncertainty Principle, the unknown constituent must reside somewhere within the classical electron radius and hence
^{∗∗∗∗}note: this example is used in the future upcoming Relativity Science Calculator Mac application
2e). Now we'll prove the :
i). Let have relative velocity as it moves away from system . Also assume that a mass  particle has velocity in . Then by the addition of relativistic velocities we also know that
will be the velocity of a mass  particle for an observer in .
Finally, assume the following:
Here we go:
Let = 1, unity, where velocities , are simply some fraction of in both and systems. [ note: we make as a unity in order to simplify our equations and will bring it back at the end of our mathematical endeavors as will shortly be seen. ] Hence the following is also true:
.
ii). Continuing ...
iii). We can thus see that
Hence the Lorentz transformations for these energies and momenta for a mass  particle moving with velocity in as they are viewed in are indeed invariant!
iv). A corollary: assuming that is moving not strictly parallel to the x  axis of , then we have these vector components respectively
End of proof.
But more importantly, the invariance of the "energy  momentum" vector determines the fact that rest mass is always a constant notwithstanding that between each respective moving reference frame, differing amounts of relativistic mass will be observed and hence measured!
Common sense, of course, dictates that whenever you as the observer are stationary relative to a body of mass that that mass will not only be at rest but also that the "amount" of rest mass remains constant ! It is only when a body of mass begins to move at a significant fraction of the speed of light relative to you as the observer, that the body of mass begins to "grow" in quantity!! Still, relativity mathematics confirms any person's common sense understanding that rest ( or proper ) mass is always a constant.
Finally, the invariance of the "energy  momentum" vector simply expresses the conservation of total energy precisely because it is invariant under Lorentz transformation!
3). Computing or :
If we know either or of a mass  particle or a quanta of energy, the obverse is easily derived.
For example, a flash of light is relativistically pure kinetic energy with rest mass = 0 ( since it is meaningless to speak of a frame of reference where light photons or "light quanta" are ever at rest ) and velocity .
Hence,
which is what Einstein used in his proof of the fundamental law of the inertia of light energy ! That is, massless light photons and gravitons possess only relativistically pure kinetic energy and momentum.
On the other hand, for a body or particle of matter possessing mass but no momentum ( i.e., ) where the object is simply residing in its rest frame, the "Energy  Momentum Equation" reduces to the famous Einstein equation for total relativistic energy as given by:
4). light as a "packet" or quanta of energy :
Because light is composed of relativistically pure kinetic energy with velocity always in all frames of reference, and
this formulation breaks down completely and is in fact irrelevant for light energy as follows:
Moreover, we do have the following quantum or wave relationship between light energy and frequency:
But nevertheless from before
4a). Proof of photon rest ( or proper ) mass = 0 :
Analogous to Newton's Law of Universal Gravitational Force Attraction, we also have Coulomb's Law of Force Attraction between charged electrical particles:
For any photon, therefore, to be at rest, it will have to exist at an infinite distance in space  time from any other [ arbitrary ] photon in space  time in order that
By applying Heisenberg's Uncertainty Principle
4b). If the photon rest ( or proper ) mass = 0, how come light is bent in a gravity field? :
Newton's Theory of Gravity requires that there be two masses at a given distance ( Einstein: spooky 'action at a distance' ) from each other for a force of attraction to exist between them:
The answer lies in Einstein's General Theory of Relativity which differs from Newton's concept of space and time as follows:
In other words, it is not that one body of mass directly attracts another body of mass, but rather that bodies of mass deform the fabric of spacetime and therefore bodies of mass follow a path of least energy expenditure according to both the mathematics of Hamilton's Principle and Einstein's General Relativity mathematics which Einstein interestingly and recursively derived from William Rowan Hamilton ( 1805  1865 )'s Principle! See: "Hamilton's Principle And The General Theory of Relativity", by A. Einstein, translated from "Hamiltonsches Princip und allgemeine Relativitätstheorie", found in "The Principle of Relativity", Dover Publications, Inc.
Therefore, The bending of light as it passes close by a gravity field induced by a body of mass is not that there's a direct Newtonian mutual force of attraction between massless photons and a body of mass such as the sun, but rather that a relativistic beam of light photons in motion ( i.e., photons not at rest! ), therefore possessing relativistic or effective mass, follows a path of least energy expenditure as laid down by the fabric of spacetime prescribed by the nearby body of mass!
It was Sir Arthur Eddington ( 1882  1944 ) who famously travelled to Portuguese  speaking Isle Principe, an island in the Gulf of Guinea off the west coast of Africa, on May 29, 1919 during a total solar eclipse, to tentatively confirm Einstein's 1915 General Theory of Relativity:
source: Google maps
The final and strongest confirmation of General Relativity mathematics for the bending of starlight across the sun's corona came during a solar eclipse in Australia when American astronomer William Wallace Campbell ( 1862  1938, Director Lick Observatory from 1901 to 1930 ) in 1922 provided the final and most definitive empirical confirmation for Einstein's general relativity mathematics.
See: "On the influence of Gravitation on the Propagation of Light" ( "Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes" ), by Albert Einstein, Annalen der Physik 35(10): 898  908, 1911, in the original German.
Also: "On the influence of Gravitation on the Propagation of Light", by Albert Einstein, Annalen der Physik 35(10): 898  908, 1911, English version, translator Michael D. Godfrey, Information Systems Lab, Stanford University. Click here for a 2nd English translation version.
5). Example of light as a "packet" or quanta of energy from Israeli  Weapons YouTube site :
Click here for US and Israel Weapons systems development: MTHEL ( Mobile Tactical High Energy Laser ) movie
Also, Boeing's 2013 HEL MD example:
Boeing's HEL MD ( High Energy Laser Mobile Demonstrator )© 2013: "... putting photons on target!"
source: http://bcove.me/ts6kzcmg
We are therefore left with the following conclusions regarding photon light:
▶ The explanation of light as a mass  particle or photon, where rest mass = 0, is not an entirely satisfactory concept nor even as a fully comprehensive explanation; rather light is somewhat more akin to a probabilistic wave function where exact location and momentum for the "light quanta" are inherently and simultaneously "uncertain".
The better equations for a "packet" or quanta of light energy thus becomes
▶ Corollaries to all of this:
(i). If so  called light mass  particles or photons exhibit probabilistic wave function characteristics, then why not other mass  particles? Sub  atomic particles for instance?? In other words, all matter and not just light possesses a wave  particle duality which is precisely the de Broglie Hypothesis of Louis de Broglie for which he was awarded the 1929 Nobel Prize for Physics.
(ii). Heisenberg Uncertainty Principle  Werner Heisenberg was a celebrated German physicist who received the Nobel Prize in Physics in 1932 for Quantum Mechanics ( he also was a loyal Nazi in Hitler's Germany who was working on Hitler's "secret projects" [ or was he slowling them down according to some historians? ] as well as [ perhaps? this part is not so certain ] on the "Final Solution" to the "Jewish Question" and the "American Question" ), nevertheless his major contribution to atomic physics can be somewhat easily described as follows:
Summary
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