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xem_007
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What exactly the "time" is?
Is it just like other spatial dimension?
It is very confusing for me to understand.
Is it just like other spatial dimension?
It is very confusing for me to understand.
We could define spacetime to be the set [itex]\mathbb R^4[/itex] and let functions of the form [itex]C:\mathbb R\rightarrow\mathbb R^4[/itex] represent an object's motion, even if we knew nothing about special relativity. It's this definition that turns time into a "dimension". Time is a dimension in that model, for the reasons DaleSpam explained.
In this model of spacetime, "space" is a subset of spacetime such that all the members of it are simultaneous (i.e. have the same time coordinate). The difference between SR and pre-relativistic physics is what atyy said: Each inertial observer would call a different 3-dimensional slice of spacetime "space" (assuming that they assign coordinates to events using a pretty obvious definition of simultaneity involving light, a mirror and a clock).
Regarding the definition of time...
We can define a coordinate system in Newtonian mechanics, SR and GR as a function [itex]x:M\rightarrow\mathbb R^4[/itex], where M is spacetime, and then define "coordinate time" as a component of that function. In SR and GR it's also necessary to define "proper time", which is the integral of [tex]\sqrt{-g_{\mu\nu}dx^\mu dx^\nu}[/tex] along a curve.
That takes care of the definitions in the mathematical models used in these three theories, but the theories must still include postulates that tell us how these things are related to what clock's measure. In Newtonian mechanics, clocks measure coordinate time. In SR and GR, a clock measures the proper time of the curve that represents its motion.
Maybe this essay will deconfuse you:Is it just like other spatial dimension?
It is very confusing for me to understand.
Is it just like other spatial dimension?
It is very confusing for me to understand.