Sometimes The Shortest Distance Between Two Points is NOT a Straight Line: GEODESICS by Parth G

What happens when the shortest distance between two points is NOT a straight line, and exactly what is a geodesic?

Hey everyone, in this video we'll be looking at how the surface we happen to be studying impacts the definition of the "shortest" distance between two points on that surface. A geodesic is exactly that, the shortest distance on a specific surface.

On a flat surface like a piece of paper or a computer screen, the shortest distance is always a straight line. This can be proved rigorously using calculus of variations, but is also quite intuitive. For example, we can try to imagine any path that isn't a straight line between two points, and then laying a piece of string along it. The string can then be straightened and compared to the straight line path - and the not-straight path will always be longer. This also applies in 3D space, assuming the space is "flat", or "Euclidean" as it's known.

However, if we now consider two points on the surface of a sphere, then the shortest distance between them (if we are restricted to move along the surface) is a curve. And yet we can imagine digging into the surface along a straight line to take an even shorter path. But this only makes sense if we consider a 2D sphere "embedded" in 3D space.

In the study of relativity, embedding is just a useful visualization tool - but it doesn't necessarily refer to anything physical. Using our sphere example, it's a 2D surface that NEED NOT be embedded in 3D space, and so it makes no physical sense to dig into the sphere as we only have access to the two dimensions of the surface.

Similarly, our universe can be described by 4D spacetime in relativity. And it need not be embedded in anything 5D. So when spacetime itself is curved, the concept of a straight line does not have any physical meaning. Two points in this curved spacetime may have a curved geodesic between them.

It's important to note that light travels along these geodesics, which is why it is important that we study them. And it's also worth nothing that spacetime curvature is caused by mass or energy found within the spacetime, as explained by Einstein's Field Equations.

So the point is that the shortest distance between two points need not be a straight line, especially if the surface that the points are on is curved in some way. And the shortest distance for a given surface is known as a geodesic.

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