The scenario where a sphere with a larger radius lies inside a sphere with a smaller radius might initially seem counterintuitive in the Euclidean metric. However, in non-Euclidean geometries, such as those involving warped spaces or other exotic metrics, this can occur due to how distances are measured.

Here’s an example of such a metric:

The Construction of a Warped Metric

Consider a 3D space with a warped metric defined as:

$ds^2 = dr^2 + f(r)^2 \, d\Omega^2,$

where:

• $r$ is the radial coordinate,
• $f(r)$ is a smooth function of $r$,
• $d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2$ is the standard metric on a 2-sphere.

Now, define $f(r)$ such that $f(r)$ shrinks as $r$ increases beyond some threshold. For example:

$f(r) = \frac{1}{r},$

for $r > 1$. In this case, as $r \to \infty$, the radius of the sphere defined by the function $f(r)$ becomes smaller. This means that a 2-sphere located at a large $r$ in this space can have a smaller physical radius than a 2-sphere located at a smaller $r$.

How the Sphere "Lies Inside"

Using this warped metric, a sphere at $r = 2$ will have a physical radius of $f(2) = \frac{1}{2}$. Meanwhile, a sphere at $r = 1$ has a physical radius of $f(1) = 1$.

In this geometry:

• The sphere at $r = 2$ (large $r$) has a smaller physical radius ($\frac{1}{2}$).
• The sphere at $r = 1$ (small $r$) has a larger physical radius ($1$).

Thus, the "large-$r$" sphere lies entirely within the "small-$r$" sphere.

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This metric appears naturally in some models of general relativity and warped spacetimes. It’s a non-intuitive property resulting from the fact that the coordinate distance $r$ doesn’t correspond to the physical radius in this warped space.

Questions for Further Exploration

1. How does this warped metric differ from the Euclidean metric?
2. What physical systems or theories might give rise to such metrics?
3. Can this concept be related to higher-dimensional spacetimes, such as in string theory?
4. How do geodesics behave in such a warped space?
5. Can you provide an example of a real-world system where warped metrics are applied?

Tip

When working with non-Euclidean metrics, always visualize the relationship between coordinate distances and physical distances to understand how geometrical properties like radii and volumes behave.