Let’s continue to study the difference between closed, flat and open geometries by computing the ratio between the circumference and radius of a circle.
(a) To compute the radius and circumference of a circle, we look at the spatial part of the metric and concentrate on the two-dimensional part by setting dφ = 0 because a circle encloses a two-dimensional surface. For the flat case, this part is just \[ ds_{2d}^2 = dr^2 + r^2 dθ^2\] The circumference is found by fixing the radial coordinate (r = R and dr = 0) and both sides of the equation (note that θ is integrated from 0 to 2π). The radius is found by fixing the angular coordinate (θ, dθ = 0) and integrating both sides (note that dr is integrated from 0 to R). Compute the circumference and radius to reproduce the famous Euclidean ratio 2π.
With these new definitions of R and dr, our relationship is: \[ ds_{2d}^2 = 0 + R^2 dθ^2\] or \[ ds_{2d} = R dθ\] Taking the integral of each side gives us an expression for circumference: \[ \int_0^C ds = \int_0^{2 \pi} R dθ\] \[ C = 2 \pi R \] This looks pretty familiar, now we can move on to less familiar geometries. \[ \frac{C}{R} = \frac{2 \pi R}{R} = 2 \pi \]
(b) For a closed geometry, we calculated the analogous two-dimensional part of the metric in Problem (1). This can be written as: \[ds_{2d}^2 = dξ^2 + sin(ξ)^2 dθ^2\] Repeat the same calculation above and derive the ratio for the closed geometry. Compare your results to the flat (Euclidean) case; which ratio is larger? (You can try some arbitrary values of ξ to get some examples.)
Radius:
With these new definitions of R and dr, the metric goes from: \[ds_{2d}^2 = dξ^2 + sin(ξ)^2 (dθ^2 + sin(θ)^2 d \phi^2) \] to \[ds_{2d}^2 = dξ^2 \] or \[ ds_{2d} = dξ\] Taking the indefinite integral of each side gives us an expression for radius: \[ \int ds_{2d} = \int dξ\] \[ s = ξ = R \]
Circumference:
Once again, our relationship simplifies nicely. \[ds_{2d}^2 = dξ^2 + sin(ξ)^2 dθ^2\] \[ds_{2d}^2 = sin(ξ)^2 dθ^2 \] \[ds_{2d} = sin(ξ) dθ \] Taking the integral with the outlined bounds to find circumference: \[ \int_0^C ds = \int_0^{2 \pi} sin(ξ) dθ\] \[ C = 2 \pi sin(ξ) \] This expression is less familiar to us, but does resemble \( C = 2 \pi R\) with our definition of R in this case.
Ratio: \[ \frac{C}{R} = \frac{2 \pi sin(ξ)}{ξ} \] Because this circumference is dependent on a sine term, the circumference will never be greater than \( 2 \pi \) or less than \( -2 \pi \) so this ratio will always be equal to or less than the ratio found in Euclidian Space in part a.
(c) Repeat the same analyses for the open geometry, and comparing to the flat case.
Radius:
In an open geometry, the procedure for radius is the same. \[ds_{2d}^2 = dξ^2 + sin(ξ)^2 (dθ^2 + sin(θ)^2 d \phi^2) \] \[ ds_{2d} = dξ\] \[ \int ds_{2d} = \int dξ\] \[ s = ξ = R \] Circumference: In the metric for open geometry is R = sinh(x) so the metric simplifies to \[ds_{2d}^2 = sinh(ξ)^2 dθ^2 \] \[ \int_0^C ds = \int_0^{2 \pi} sinh(ξ) dθ\] \[ C = 2 \pi sinh(ξ) \] Ratio: \[ \frac{C}{R} = \frac{2 \pi sinh(ξ)}{ξ} \] This ratio will always be larger or equal to the ratio in the flat, and closed universe.
(d) You may have noticed that, except for the flat case, this ratio is not a constant value. However, in both the open and closed case, there is a limit where the ratio approaches the flat case. Which limit is that?
The two ratios we found are : \[ \frac{C}{R} = \frac{2 \pi sin(ξ)}{ξ} \] \[ \frac{C}{R} = \frac{2 \pi sinh(ξ)}{ξ} \] As we approach 0, we can use the small angle approximation for both of these functions, (e.g. sin(ξ) is ξ) so as the limit goes to zero, both ratios go to zero, approaching the flat case (this is easy to visualize in the graph above). In class, Ashley compared this geometry to us being small human beings on a large Earth. On the whole, we know the Earth is a three-dimensional sphere, but at small scales, as the limit approaches zero, the Earth looks flat.
Representations of the three types on universes - Source
Good job Danielle! 5/5
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