Suppose that we are two dimensional beings living on this surface.
Suppose that we want to move forward in straight lines.
The equivalent of a straight line on a curved surface is a Geodesic.
Here, each color represents a different Geodesic.
In this first example, we are dealing with only a small amount of curvature.
Here, each path is deflected by only a small amount.
For example, the green and red paths start out close together,
and they are deflected by slightly different amounts.
The results will be completely different if we have a large amount of curvature.
Here, notice how the red path and the green path start out very close together,
but they end up leaving in completely opposite directions.
This is shown for a curved two dimensional space and this can also be extended to higher dimensions.
However, there are important considerations that need to be taken into account
when applying this to the four dimensional space-time of General Relativity,
and this will be discussed later in this video.
Here, each color represents a different Geodesic.
Suppose we want to get from one side of this hill to the other.
The path going directly over the hill is a geodesic,
but going around the hill involves a much shorter distance.
One common misconception is that every Geodesic is always the shortest path between two points,
but this example clearly shows that this is not true.
It is only when we zoom in and look at two very closely neighboring points on a Geodesic that the Geodesic is always the shortest path between them.
Here is another way to think about Geodesics.
A Geodesic is the path that would be followed by a small car stuck to the surface,
with its wheels always pointing directly forward,
and with the wheels on the left and right side of the car always rotating at the exact same speed.
A Geodesic is the path that would be followed by a small car stuck to the surface,
with its wheels always pointing directly forward,
and with the wheels on the left and right side of the car always rotating at the exact same speed.
In this next example, the different colors are all just different parts of the same Geodesic.
The red arrows are always the first to wrap around and create a new path that all the other arrows will follow.
Neighboring lines that start out parallel on this surface end up converging.
Neighboring lines that start out parallel on this other surface end up diverging.
Here, every color once again represents a different Geodesic.
Let’s now consider yet another example.
The red and green arrows are not perfectly aligned with this donut.
Although the red and green Geodesics have very different starting conditions,
they both eventually end up preferring to travel around the “outside” of the donut.
And now, the point of no return.
In this example, the geodesics that venture close to the center spiral inward never to be heard from again,
but the geodesics that stay further away manage to escape.
The geodesics that are swallowed continually spiral downward,
whereas the geodesics that escape spiral back upwards.
Here is another way to think about Geodesics.
Suppose we zoom in very closely so that the curved space can be approximated as being flat.
Suppose we use this approximation to move our arrow forward by a tiny amount.
We now correct for the fact that the space is not flat
by rotating our arrow along the line perpendicular to the surface.
If we repeat this over and over again, we get the path of the Geodesic.
One thing that all these examples have in common is that
the curved space is represented as a two dimensional surface
embedded in a three dimensional space that obeys Euclidean Geometry.
This can be misleading,
as the curved four dimensional space-time of our Universe
does not necessarily need to be embedded in a space of more than four dimensions
in order for General Relativity to work.
It could just be that the Universe we live in is simply not governed by Euclidean Geometry.
In cases where we are travelling at a speed far less than the speed of light
and Newtonian physics gives us accurate approximations,
contrary to popular belief,
the apparent force of gravity has little to do with the curvature of the three spatial dimensions relative to each other.
In these cases, the apparent force of gravity is primarily due to the curvature of the “time” dimension, relative to the three spatial dimensions.
This is explained and visualized in the video titled
“Gravitational Time Dilation causes Gravitational Attraction.”
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