Hyperbolas of constant t^2 - x^2

In summary, "hyperbolas of constant t^2 - x^2" represent a particular type of curve in a spacetime diagram, containing all events that lie at a constant spacetime interval from the origin. This is similar to how circles of constant radius are a particular type of curve in a Euclidean plane, containing all points that are a constant distance from a center point.
  • #1
stglyde
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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!
 
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  • #2
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.
 
  • #3
yenchin said:
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?
 
  • #4
stglyde said:
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]?
 
  • #5
stglyde said:
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.
 
  • #6
stglyde said:
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.
ProperTimeCurves2.jpg
 
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  • #7
stglyde said:
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

[tex]t^{2} - x^{2} = 1[/tex]

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
 

Related to Hyperbolas of constant t^2 - x^2

1. What is a hyperbola of constant t^2 - x^2?

A hyperbola of constant t^2 - x^2 is a type of conic section, or a curve formed by the intersection of a plane and a double cone. It is defined by the equation t^2 - x^2 = k, where k is a constant. This equation describes a set of points in a coordinate system that form a symmetrical curve with two branches that open up and down.

2. What is the significance of the constant k in a hyperbola of constant t^2 - x^2?

The constant k in the equation t^2 - x^2 = k determines the shape and orientation of the hyperbola. It is referred to as the eccentricity of the hyperbola and is equal to the square root of the ratio of the distances from any point on the curve to the two foci. A larger value of k results in a more elongated hyperbola, while a smaller value creates a more circular shape.

3. How are hyperbolas of constant t^2 - x^2 used in real life?

Hyperbolas of constant t^2 - x^2 have many practical applications in fields such as physics, engineering, and astronomy. They can be used to model the orbits of celestial bodies, the paths of projectiles, and the trajectories of particles in electromagnetic fields. They are also useful in designing and optimizing structures such as antennas and reflectors.

4. How do you graph a hyperbola of constant t^2 - x^2?

To graph a hyperbola of constant t^2 - x^2, you can plot a few key points and then connect them with a smooth curve. These points include the two foci, the vertices, and the asymptotes. The foci are located at (0, ±√k) and the vertices are at (±√k, 0). The asymptotes are lines that pass through the foci and intersect at the center of the hyperbola. You can also use the equation t^2 - x^2 = k to plot additional points on the curve.

5. What is the relationship between a hyperbola of constant t^2 - x^2 and its inverse function?

A hyperbola of constant t^2 - x^2 and its inverse function, t = ±√(x^2 + k), are symmetric about the line y = x. This means that if you reflect one branch of the hyperbola over this line, you will get the other branch. The inverse function represents the time it takes for an object to travel a certain distance along the hyperbola, while the original function represents the distance traveled over a certain period of time.

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