The shortest path among two points inside Earth [closed]

I have this idea and I don't know how to process, explain or question it. I hope you can understand these images and help me formulating a good question. This is like a gravitational train but it doesn't goes to the antipodes, just like a tunnel that cuts a slice of earth. In this image I tried to express a straight path in a round surface. If I walk 1000 meters forward I will go to the shortest way and that way is a straight line. But if instead of walking, I dig a tunnel and, that path or way is still a straight line? Or from my point of view I will see that tunnel curved?

I'm really trying to figure out this. I don't have any scientific education and english is not my first language but I'm very curious and I'm hungry for knowledge :)

• While this might have an application in geophysics, it's not really a physics question. It's actually a spherical geometry question and belongs to the realm of Mathematics SE. Feb 26 '16 at 4:35
• If you like this question you may also enjoy reading this Phys.SE post. Feb 26 '16 at 6:45

The shortest distance between two points in 3D space is a line; if the path is not a line it is called a curve.

If you are confined to a surface, the shortest distance between two points is called a geodesic. For example, the geodesics of a spherical surface are the great circles: circles whose centers pass through the center of the sphere. Lines of longitude on a globe are great circles, as is the equator.

So your proposed tunnels, straight lines, directly connecting two points on the earth's surface, are indeed the shortest possible paths.

• A line is also a curve. Feb 26 '16 at 7:45
• @wythagoras: in older texts they were wordier; I was defining what I meant. See en.wikipedia.org/wiki/Curve Feb 26 '16 at 10:29

The shortest path is simply a straight tunnel !

If you consider the effects of the irregular gravitational field it probably is like a bent wobbly line.

In an Euclidian space (an N-dimensional space that follows a set of laws, which is commonly used to describe our physical world), the shortest path between two points is always a straight line.

If you add another dimension then you can bend that space in this N+1 dimension. In this new space the line can be different from a straight line. That's what happens when you wrap a 2D surface (Earth's surface) around a 3D sphere (Earth globe); the shortest path follows the Earth curvature.

Here, you can consider Gravity as a 4th dimension. The gravitational field bends the geometrical 3D space depending on it's varying strength through the Earth. The shortest tunnel ends up being not straight anymore.