Draw Minkowski Diagram: x-axis 0-5m, ct-axis 0-5m

[Pixter]

draw a minkowski space-time diagram for s and s' showing x- and ctaxis as well ass the x'- and ct'-axis. the x-axis and ct axis will span from 0-5m. everwhere x'-axis ct'=0 and likewise for ct'-axis x'=0. [(v=0.6c)(t=t'=0, x=x'=0)]

so any way I start out with drawing ct and x. like a normal x&y diagram.

Then i have the formulas x'=gamma(x-vt)
t'=gamma(t-(vx/c^2))

gamma= yeah you know. =)

oki so i can get ct' by using the formula tan(theta) = c/v

then i can get x' by taking 45(angle for c)-[(tan^-1(c/v))-45(angle for c)]
.

okey i done that. now i have to calibrate the x'- and ct'-axis by using the relations x^2 - (ct)^2 = 1..

now how do I do that? and did I do it correctly in the first part?

 

[SN1987a]

I believe you did part 1 correctly. Not too sure about how you calculated the angle for x', but it should be the same(from the x axis) as the angle for ct' (from the ct axis), and this one you got alright.

As for the calibration, you can do it using the lorentz transforms, which also work with intervals
$$\Delta x'=\gamma(\Delta x-v \Delta t)$$

But to calibrate the x' axis, find $\Delta x'$ for $\Delta t=0$.

By a similar reasoning, you can find $\Delta ct'$ for a ct interval of 1 in the stationary frame.

You can of course also use the invariant. $\Delta x^2- \Delta ct^2=1$ describes a hyperbola on the minkowski diagram, and this value is the same in all frames of reference, so you can just "read" off any axis.

 

[robphy]

draw a minkowski space-time diagram for s and s' showing x- and ctaxis as well ass the x'- and ct'-axis. the x-axis and ct axis will span from 0-5m. everwhere x'-axis ct'=0 and likewise for ct'-axis x'=0. [(v=0.6c)(t=t'=0, x=x'=0)]
so any way I start out with drawing ct and x. like a normal x&y diagram.
Then i have the formulas x'=gamma(x-vt)
t'=gamma(t-(vx/c^2))
gamma= yeah you know. =)
oki so i can get ct' by using the formula tan(theta) = c/v
then i can get x' by taking 45(angle for c)-[(tan^-1(c/v))-45(angle for c)]
.
okey i done that. now i have to calibrate the x'- and ct'-axis by using the relations x^2 - (ct)^2 = 1..
now how do I do that? and did I do it correctly in the first part?

Great question... about calibrating the primed-axes. [What text are you using? If solutions are later given, I'd be curious to know how it was solved.]


As SN1987a said, you could use the Lorentz transformations explicitly or recognize that x^2 - (ct)^2 = 1 is a hyperbola.
There is a conceptually simpler construction using triangles of equal area with sides parallel to the light cone. (This has connections to the Bondi k-calculus and the Lorentz transformations in light-cone coordinates.)


See these references:
http://arxiv.org/abs/gr-qc/0407022 Brill and Jacobson
http://www.aip.de/~lie/GEOMETRIE/GeomZeit.html Liebscher
http://www.arxiv.org/abs/gr-qc/0411069 Mermin
http://www.lassp.cornell.edu/~cew2/P209/P209_home.html Mermin notes
http://arxiv.org/abs/physics/0505134 mine