Find Area of Curved Space | 2D, Positive Geometry

In summary, the conversation discusses finding the area of a curved space described by a 2D metric, specifically a sphere with a radius of R. The solution involves finding the circumference of the sphere and then summing up all the circumferences within a specific interval. The final solution is to multiply by 2 to account for the other half of the sphere.
  • #1
Niles
1,866
0
[SOLVED] Area of curved space

Homework Statement


I have the following metric, which describes a 2D, positive curved space with flat geometry (ie. a sphere): [tex]

ds^2\,=\,dr^2\,+\,R^2 \sin ^2 (r/R)d\theta ^2

[/tex]

Here ds is the distance between two points (r, theta) and (r + dr, theta + dtheta), R is the radius of the sphere.

I want to find the area of the sphere using this metric.

The Attempt at a Solution


Using the metric, I have found the circumference of the sphere to be 2*Pi*R (big surprise). Now I want sum up "all the circumferences" on the sphere. Is that possible?
 
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  • #2
I solved it!

When finding the circumference, I must find all the other circumferences, and r is in the interval [0; Pi*R/2]. Just multiply with 2 in the end (since we only found the first half), and you're set!
 

FAQ: Find Area of Curved Space | 2D, Positive Geometry

What is the concept of "curved space" in 2D geometry?

The concept of "curved space" in 2D geometry refers to the idea that space can be bent or curved in a two-dimensional plane. This is different from the traditional Euclidean geometry, which assumes that space is flat and follows straight lines.

How is the area of curved space measured?

The area of curved space can be measured using different mathematical methods, depending on the type of curve involved. For example, the area of a circle can be calculated using the formula A = πr^2, where r is the radius of the circle. Other curves may require more complex mathematical equations to determine their area.

What is positive geometry and how does it relate to curved space?

Positive geometry is a branch of mathematics that deals with the study of shapes and spaces that have positive curvature. In the context of curved space, this means that the space is bending in a way that creates a convex shape. Positive geometry is important in understanding and calculating the area of curved space.

Can the area of curved space be negative?

No, the area of curved space cannot be negative. The concept of area always refers to a positive value, representing the amount of space within a given shape. Even in a negatively curved space, the area would still be considered positive.

How is the concept of curved space applied in real-world situations?

The concept of curved space has many practical applications in the fields of physics, engineering, and architecture. For example, understanding the area of a curved surface is crucial in designing and building structures such as bridges and tunnels. It is also important in understanding the behavior of objects in space, as the curvature of space can affect the trajectory of objects moving through it.

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