Dear Ch. Bayarsaikhan:
You as all physicist have the right to express their opinions as yours that "time is expressed in physics as if a time line is a real number line" before physicists can do that properly, first you all should know what's time, if not I can ask what's a "time line", if you don´t know what's time? what are you going to answer? I should give an example. How physicists should express fuf-fuf in physics?. Certainly they wouldn't be able to do it? Why?, because they don't know what fuf-fuf is. They had many ingenious ways to "pass over" "time definition and empiric meaning", that's why physics become a science, but none of them in the case of "time" are proper enough.
Einstein treat "time" as if he new what it was. He new that "time" hasn't physic existence and that it was a man creation, but he did not know what he was measuring with the clock. ¡ but just defining it, as "time" is what we measure with the clock??????
All this ways and many others that you should know, are desperate efforts of physicists to replace some way, one of the most basic and important quality or properties of nature, movement because this is part of all changes and transformations in the universe, physicists can't give one step without it. Every theory must include directly or indirectly "time" to have sense. Einstein said "time" reconceptualization is the key step forward in developing relativity". So the "pass over" time definition and empiric meaning made possible that each physicist that make a theory, invent their own definition and empiric meaning like on "the loop quantum gravity theory" or "the string theory" The first
"Time flows not like a river but like the ticking of a clock, with "ticks" that are about as long as the Plank time 10-43 second. Or more precisely, time in our universe flows by the ticking of innumerable clocks". Or the second that just "use "time" as an absolute Newton term". Physicist Stephen Hawking said referring to "time definition and empiric meaning" "when the answer is found, it'll sound as obvious as the fact that the earth goes around the sun." like movement I say.
Finally, just around 50 or 60 years ago passing over "time definition and empiric meaning" wasn't enough, we can be find the conscious need to know "time meaning" just to find the absent "time variable", when physicists try to merge the theory of general relativity with quantum mechanics, into a theory of quantum gravity, using the procedure called canonical quantization. which produces an equation without a "time" variable, literally would mean a frozen universe. Professor Christopher Isham from the London Imperial College in one of the possibilities he offer to explain such disappearance he said, "It is possible that time it is not truly lost but merely hidden among the canonical variables that are constrained by the theorema egregium".
said "To obtain a sensible quantum theory of gravity may require identifying such an internal time prior to quantization".
Knowing that "time" is movement, I would say that is not lost or hidden either. I think is there but not recognized as such, because (the time variable) in fact is a "movement variable", this is the lost variable that it would be described by the 3 spatial dimensions that also describe the "event", plus a "time forth dimension" (called imaginary by Einstein) consistent too with a movement, but a "Constant and Uniform" movement, the clock.
Not knowing what "time" is became a big problem for physics. If I were a physicist I would take this as an opportunity.
My very best whishes
Héctor