I have read so many complicated description of whats theory of relativity that I tend to think such complicacy serves two purposes;
Misleading: it takes us away from the real concepts.
Complicated: It does not make the concepts easy at all to grasp.
Ineffective: It ill-prepares even very good students — as their precious time is wasted and the mediocre rejoice.
Here is an appropriate description of relativity theory.
Relative as suggests hinges on in reference to what? So you may call it a reference-dependent theory. But if you understand what theory of relativity is you may as well call it an equivalence theory.
This theory establishes for the first time in greater depth the equivalence of many ideas, concepts, physical laws and methods. This is the work of Albert Einstein. But if you dig deeper you will see that most of the concepts were known before, and classical physics is capable of producing these ideas 1st hand.
It is for no strange reason relativity theory is included as classical physics, in greater conceptual frameworks. Its because its merely a more refined calculation of certain problems, some of which were known and some which were not, but never to the extent and power of the methods of Einstein.
Einstein eg solved the problem of perihelion of mercury very exactly which was not as exactly understood in the canonical formulations of the old classical theory, so theory of relativity is the new classical theory.
But there are two episodes of this story.
1. Special theory of relativity.
This is a regime of the theory of relativity where it pertains exclusively to kinetic energy: .
We should note one thing about T, the kinetic energy, which is also sometimes written K or K.E. Its a very simple variable, depending on v only, — sans the theory of relativity idea that mass m itself, which appears in T here, depends on v. But the idea is v is the only kinematic variable here. What that does is makes life simple to study energy in one of the simplest ways possible in mathematics.
As you remember Physics is all calculus. Physics in its essence is differential equations, which therefore involves differentiation, integration and calculus of variation. And since our problems in special theory of relativity now, depend only on v, the transformation of physical laws are described by what are called as inertial frames of references.
This is nothing but a way to assign v to a physical apparatus. Potential energy is much more complicated. Which is also why space and time do not seem to be equivalents in the old classical theory, it has to do with the simpler form of energy known as T.
In the more complicated forms of energy viz. V or U, one may find space and time are equivalents or correlated — i.e. not separable in a differential equation, that follows from principle of least action, with the most general form of action defined as function of all kinematic variables, and so on.
2. General theory of relativity.
While its called a general theory most under-prepared experts relate this theory exclusively to gravity. Its a severe misunderstanding. 1st of all gravity is a very simpler form of potential energy.
Potential energy involves, in the old classical theory, only a distance or space variable and in the new classical theory — see point 1 above, mass terms as well which are not stationary in time, but based on what frame of reference we are using on depends on the speed of that frame.
In general any potential energy is a function of x, the location, 1st time differential of x, the speed, 2nd time differential of x, the acceleration and time itself. This makes the differential equations more complicated and now there is even acceleration in gravitational energy.
This fact is called the weak equivalence principle as when suitably expressed with consistence, the energy from acceleration is not distinguishable from the energy of gravitational origin.
Most people think this is called as general theory of relativity and this equivalence is the central point. Thats simply not true. It would be like genitalia is the central organ of any living organism.
But some good old styled people like me realize its our mind which is an abstract and ever evolving concept, which has any central significance, if at all. Gravity is only partially general.
But here as noted in point 1, the forms of S, the action — or simply U, the potential energy, which makes the Lagrangian L or Hamiltonian H, x, the space variable and t = time may be correlated or in other words equivalenced.
Whats general then? Well Potential energy is general. Gravity is only a special case of that. So we can call gravity as a weak general theory of relativity.
But the concepts remain that general theory of relativity establishes the equivalence of
variables: the equivalence of various variables — such as x and t, T and U, E(U) and E(a), E and B,
frames of reference: the equivalence of all frames of reference with the same speed v,
energy forms: the equivalence of different forms of energy — E(U) and E(a), E(heat), E(E,B) etc.
Its this equivalence for which the theory is called a relativity theory.
So now we know that general theory predicts * more exactly that space-time is warped. If we can show from any general form of potential energy or action principle, space and time are correlated in an equation, we would know that equivalence was already there in old classical theory.
* But it was known that potential energy is a complicated function of space, therefore not flat but warped.
What happened in new classical theory of Einstein is that the ideas got exact formulation and some more specific problems were solved. Also the so-called Galilean transformation or Galilean relativity that we learn from our text-books where the concept of kinetic energy and uniform kinetic energy — or uniform speed, are used, what if we use general forms of potential energy into the transformation or action Principle: may be the equivalence and time dilation etc were known in old classical theory.
Just from a simpler form of gravity I have proved in an article written couple days ago thats to be completed with these derivations but the ideas have been described, that gravitational time dilation can be derived from old classical theory, without using any Einsteinian concept — such as changing mass with speed of frame of reference or universal speed limit.