I read so many complicated description of whats theory of Relativity that I think such complicacy serves two purposes:
1. Takes us away from the real concepts
2. Does not make the concepts easy at all to grasp
3. ill-prepares even very good students [as their precisous 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 as well 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 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 not to the extent and power of the methods of Einstein.
Einstein eg solved the perihelion of Mercury very exactly which was not as exactly understood in the canonical formulations [the old classical theory, so Relativity is the new classical theory]
But here are two episodes of this story.
1. Special Theory: here it pertains only and only Kinetic Energy. T=mvv/2=pp/2m [as obvious vv=square of v and pp=square of p: momentum]
You should note one thing about T which is also sometimes written K or K.E. Its a very simple variable depending on v only. 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. [differential equations which therefore involves integration and calculus of variation] And since it depends only on v the transformation of physical laws are described by what are called inertial frames of references: which 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 old classical theory, it has to do with the simpler form of energy known as T. In the more complicated forms of energy = V/U, one may find space and time are equivalents or correlated [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, x, x-dot, x-dot-dot, t and so on]
2. General Theory of Relativity: While its called a general theory most under-prepared experts relate this theory to Gravity. Its a severe misunderstanding. 1st of all Gravity is a very simpler form of Potential energy. It involves in the old classical theory only a distance or space variable and in the new classical theory [point 1 above] mass terms as well which are not stationary in time but depending on what frame of reference you are depends on the speed of that frame. In general any potential energy is a function of x=location, 1st time differential of x= speed, 2nd time differential of x=acceleration and time itself. This makes the differential equations more complicated: you see and now there is even acceleration in gravitational energy, this is called weak equivalence principle as suitably expressed with consistence the energy from acceleration is not differentiable from the energy of gravity. Most people think this is called General Theory of Relativity and this equivalence is the central point. Thats simply not true. It would be like your genitalia is your central organ. But some good old styled people like me realize its your mind which is an abstract and ever evolving concept. Gravity is only partially general. But here as noted in 1 the forms of S=action [or simply U=potential energy, which makes the Lagrangian L or Hamiltonian H] space=x and time=t 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 you can call Gravity as a weak General Theory of Relativity.
But the concepts remain that its the equivalence of various variables [such as x & t, T & U, E(U) & E(a), E&B], equivalence of frames of reference with same speed v, equivalence of different forms of energy [E(U) & E(a), E(heat), E(E,B) etc]. Its this equivalence for which the theory is called a Relativity Theory, and the Hindi/Odia etc for this theory can be called “samyata tatwa” rather than the more complicated and less obvious “sapekhyata, apekhyika” etc. [less obvious because the words themselves do not get much used, they are obscure for the tastes of a native like me]
Special Relativity Theory: nirdista samyata
General Relativity Theory: Anirdista samyata [or bahubidhi samyata]
And for space-time dik-kal is fine but lok-kal is more appropriate I think. [desha-kal is fine too, desha maay have come from chinese/sanscrit: dikusa, disha all of them point to region, space, direction, but dik should be used for dimension, das-dik tatwa=10-dim theory]. Why invent fictitious words rather than words that are used far more in day to usage and easy to follow for students.
So now you know that General Theory predicts [but it was known that potential energy is a complicated function of space therefore not flat but warped] more exactly that space-time is warped. If you can show from any general form of potential energy or Action Principle, space and time are correlated in an equation you would know that equivalence was already there in old classical theory.
What happened in new classical theory [Einstein] that they got exact formulation and some more specific problems were solved. Also the so-called Galilean transformation or Relativity that you are told is as per your text-book where it uses the concept of kinetic energy and uniform kinetic energy [or uniform speed], what if you 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]