Relativity Calculator
"Reality is merely an illusion, albeit a very persistent one" - Albert Einstein ( 1879 - 1955 )
Invariance of Newton's 2nd Law of Motion under Galilean Transformation: No Acceleration
§ Case 1 — 
is constant:
Hence, the laws of classical physics are the same whether stationary or in relative inertial motion!!
§ Case 2 — 
is a function of constant uniform velocity:
That is, is distance traveled of system 
relative to stationary system 
as a function of 
, a constant uniform velocity between system relative to stationary system .
This again demonstrates that the laws of Galileo and Newton of motion physics are invariant. In other words, the laws of classical physics are the same whether stationary or in relative inertial motion!
Invariance of Newton's 2nd Law of Motion under Galilean Transformation: Acceleration is taken into account
§ Case 3 — 
:
That is, is the distance traveled between system relative to stationary system as a function of 
, a constant acceleration between system relative to stationary system , especially when initial velocity 
is set to zero.
Unsurprisingly the laws of classical physics are the same whether stationary or in relative non - inertial motion.
§ Case 3a — 
:∗
∗since we can always choose an appropriate coordinate system to set initial velocity to zero.
Pictorially the above equations are shown as follows:∗∗
∗∗note : this example is used in the future upcoming Relativity Calculator Mac application for Galilean Relativity.
In other words, Galilean Relativity simply reflects ordinary and intuitive human common sense, namely the simple addition of distances, velocities, accelerations and forces, although upon deeper mathematical reflection these non - relativistic quantities are valid only for velocities significantly less than the speed of light.
Nevertheless, in conclusion, the laws of classical physics are the same whether stationary or in relative non - inertial motion !!
Question: What is 
?
Tentative answer: is the "pseudo force" applied to system in order to create relative motion between system and relatively stationary system . This pseudo force accounting is necessary in order to make the laws of Galileo and Newton of motion physics invariant; hence, the laws of classical physics are the same whether occurring in a stationary frame of reference or in a relative inertial [ uniform straight - line motion ] frame of reference.
Summary
The Relativity Principle of Galileo and Newton simply states that the laws of classical physics are invariant under conditions of uniform inertial motion, whether stationary or in relative motion.
"It's often a mistake to accept something as true, merely because it's obvious." - Basil Rathbone as Sherlock Holmes in the movie "Dressed to Kill" ( 1946 )
Again Question: What is this "pseudo force" ??
Tentative answer: Gravity is the "pseudo force" !
How so? Hidden deep inside of but in plain view of
is this mysterious new "pseudo force" of unknown origin which Einstein ingeniously explained as follows:
This latter #2). quote by Einstein is simply a recapitulation of Galileo's Law of Falling Bodies.
Now by Einstein's own genius, he reformulated the above #2). proposition as
Then by simple algebra, Einstein wrote
since we ourselves can easily derive Einstein's simplified equation combining Newton's 2nd law of motion and Galileo Galilei's Law of Falling Bodies
Einstein then perceived intuitively that "fields of acceleration" can be created or destroyed by appropriate frames of relative motion; and that forces proportional to given amounts of mass can also be created or destroyed by designing appropriate frames of relative motion. It therefore all depends upon finding the correct coordinate system for there to be "fields of acceleration" and hence extant forces!!
Furthermore according to Einstein's thinking, "By a suitable choice of units we can thus make [ the ratio of the gravitational mass to the inertial mass ] equal to unity", whereby Einstein stated the following law:
"The gravitational mass of a body is equal to its inertial mass."
Einstein further believes that the mass of a body is an inherent quality which is made manifest
"according to circumstances as 'inertia' or as 'weight' ( lit. 'heaviness' )."
And, therefore,
Einstein's diagram for acceleration's gravity field: Einstein first creates an imaginary pure Galilean "world" in which there are no other masses of bodies or stars to effect anyone or anything, except that there does exist an observer floating freely in this pure in vacuo Galilean space. Einstein then places an imaginary "chest" enclosing this free floating observer which in turn serves as the observer's inertial frame of reference. However by some external "being", using Einstein's terminology, the enclosing chest for our free floating observer is converted to a non - inertial ( accelerating ) frame of reference by the accelerating pull or force made possible by this same external "being". Naturally our free floating observer will experience an accelerating field of gravity and the inherent quality of mass for the enclosed observer is made obvious in the following pictorial:
By giving a physical interpretation to Einstein's "Law of Equivalence of Acceleration and Gravity", Einstein also demonstrates the "Law of Equivalence of Inertial and Gravitational Mass" and, hence, does so within a powerfully new generalized postulate of relativity!!
∗In all of Einstein's other writings, it is spacetime geometry itself which is pulling and pushing gravitational masses [ not mysterious "beings"! ] which in turn are deforming spacetime's malleable fabric; in this way, according to Einstein, gravitational masses are best understood in terms of non - Euclidian metric geometries for deformable spacetime fabrics and therefore General Relativity mathematics employs 4 - dimensional tensors, a specialized matrix algebra and calculus for curvilinear Riemannian surface geometries of which Euclid's cartesian geometry is a unique case.
Einstein's General Relativity answer: From before, "Galilean - Newton" Relativity reveals
for a non - inertial accelerating system relatively moving in respect to stationary system and for which an appropriately chosen "metric geometry" and related coordinate system will determine the amount of "force" involved in any so - called "pseudo force":
by Einstein's general relativity postulate of the equivalence of "fields of acceleration" and "fields of gravity".
In essence, therefore, an appropriately chosen geometry of spacetime metric coordinates will reveal hidden forces of gravity. In fact, gravity forces and spacetime coordinates ( metric ) geometries are intimately connected !!
§ References:
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