Hyperbolas of constant t^2 - x^2

[stglyde]

I know hyberbola in spacetime diagram is that curve-like line below the light cone much like the bottom of a bath tub. But what does "hyperbolas of constant t^2 - x^2" represent? Can anyone point to a spacetime diagram or draw it? Thanks. Happy New Year!

 

[yenchin]

It means t^2 - x^2 = C^2 for some constant C^2. For different value of C you should a different curve, just like the circle x^2 + y^2 = R^2, for different R you get circle of different radius.

 

[stglyde]

It means t^2 - x^2 = C^2 for some constant C^2. For different value of C you should a different curve, just like the circle x^2 + y^2 = R^2, for different R you get circle of different radius.

t is time, x is horizontal coordinate.. what is C? is it the speed of light? or vertical? or other points?

 

[yenchin]

t is time, x is horizontal coordinate.. what is C? is it the speed of light? or vertical? or other points?

I did say C is some constant, just a number. You can ask the same question in the case of a circle, what is R? This question really has nothing to do with special relativity, it's just mathematics of a hyperbola. Just consider the hyperbola in a more familiar Euclidean setting: x^2 - y^2 = 1, x^2 - y^2 = 2, x^2 - y^2 = 3, etc do you see that they are different hyperbola [different curve]?

 

[Dale]

I know hyberbola in spacetime diagram is that curve-like line below the light cone much like the bottom of a bath tub. But what does "hyperbolas of constant t^2 - x^2" represent? Can anyone point to a spacetime diagram or draw it? Thanks. Happy New Year!

They are all of the points which are separated from the origin by the same spacetime interval. I.e. an inertial clock starting at the origin would show the same time at any of those events.

 

[bobc2]

I know hyberbola in spacetime diagram is that curve-like line below the light cone much like the bottom of a bath tub. But what does "hyperbolas of constant t^2 - x^2" represent? Can anyone point to a spacetime diagram or draw it? Thanks. Happy New Year!

Hi stglyde, and Happy New Year to you. Here are a couple of space-time diagram sketches along with a development of the algebra associated with the hyperbolic curves in the diagrams.

 

[PeterDonis]

I know hyberbola in spacetime diagram is that curve-like line below the light cone much like the bottom of a bath tub.

The term "hyperbola" refers to a general type of curve. Hyperbolas of constant t^2 - x^2 are a particular subset of all possible hyperbolas. As yenchin said, just pick any constant number for the "constant"; I'll choose 1. Then the curve

$$t^{2} - x^{2} = 1$$

is a hyperbola, a curve containing all events that lie at a spacetime interval of 1 from the origin (t=0, x=0).

If you're not practiced at being able to draw (or get the computer to draw) and visualize curves from their equations, it would be well worth taking some time to learn. This Wikipedia page is a good place to start looking for software that can help:

http://en.wikipedia.org/wiki/List_of_information_graphics_software