BadgerBadger92 said:
I am teaching myself math and have a question about cartesian coordinate systems. How is time illustrated in such a graph?
[Moderator's note: Moved from a math forum after post #13.]
The answer depends on within which theory you look.
In non-relativistic physics we use the Galilei-Newton model of spacetime. Mathematically you have time as a oriented one-dimensional real parameter, which you can depict in the usual way as a line. At each point of this line you must think to have put a 3D Euclidean affine space, which describes the physical space at each instance of time. There's no closer relation between time and space in Newtonian physics, which is known since Newton as the idea of absolute time and absolute space, which both are just there unaffected by anything going on in the physical world (mathematically it's a fiber bundle). A non-relativistic space-time diagram is thus just two axis, one labeled with time ##t## and one with a component of Euclidean spatial vectors, describing the value of this component at each instant of time, such as you draw a function graph for a function ##x=x(t)##. There's no geometric meaning of this plane,
In special-relativistic physics things become already more beautiful. There space and time are described by a real space-time continuum as a pseudo-Euclidean affine continuum. It's pretty much like a 4D Euclidean affine continuum with the very important qualification that the "scalar product" is no longer positive definite but a bilinear form of signature (1,3) (or equivalently (3,1) depending on which convention you use; as a relativistic nuclear-particle physicist I prefer the "west-coast convention", which is (1,3)). I.e. the "distance" of spacetime intervals is given by the product of the corresponding four-dimensional vectors with components ##(x^{\mu})=(x^0,\vec{x})## with ##x \cdot y=x^0 y^0 - \vec{x} \cdot \vec{y}##, where ##\vec{x} \cdot \vec{y} = x^1 y^1 + x^2 y^2+x^3 y^3## is the usual Euclidean scalar product for the components with respect to a Cartesian basis system.
Here a Minkowski diagram, describing again the one-dimensional motion of a particle by drawing one axis for time (or more conveniently ##x^0=c t##) and one spatial component, has a geometrical meaning, but not in our usual terms of a Euclidean plane but of a "Minkowski plane", where the metrical properties are given by the pseudo-scalar product rather than the proper scalar product of Euclidean vectors. The most important result to keep in mind is that instead of circles, defining positions of equal distance from a given point, you get hyperbolae, ##(c t)^2-x^2=A=\text{const}##. You have to distinguish time-like "distances" (##A>0##), lightlike ones (##A=0##, for which the hyperbolas degenerate into straight lines, the "light cone, and spacelike (##A<0##).
Changing from one pseudocartesian basis to another and thus using new components ##(c t',x')## in this Minkowski plane describes the transformation of the spacetime vector components of one inertial reference frame to those of the other inertial frame moving with constant velocity in ##x##-direction with respect to the former system. It's a pseudocartesian basis change (i.e., the analogies of rotations in Euclidean space) if and only if the pseudoscalar product looks the same in both sets of coordinates, i.e., the spacetime "distances" don't change. This leads to the Lorentz transformation and the fact that the relative velocity between inertial frames cannot exceed the speed of light. Particularly in the Minkowski diagram the light cone of the origin is unchanged, and it's always given by the bisecting lines between the space-time axes (one cone describing light signals traveling in positive the one those traveling in negative ##x##-direction), which means that for any observer the light signal travels according to ##x=\pm c t##, i.e., for any observer light travels with the same speed ##c##, no matter how the light source might move relative to him or her.
For more details, see my SRT writeup:
https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf
