Question: My question is whether or not science has a physical or mathematical model that proves the roundness of the earth beyond what our senses tell us.
Answer: Yes, one can determine whether the surface he/she is being on (including the surface of the earth) is flat or curved just by measuring some parameters of the surface, and he/she can do this on the surface itself without having to leave/go above the surface to see/check its curvature or seek other physical evidence. This procedure is the determination of the curvature of a manifold, which in this case is a 3-dimensional surface (the surface of the earth).
The determination of the curvature of a manifold (a 3-dimensional surface in this case) is a mathematically complex subject that involves tensor calculus. It's difficult to explain it completely in a short page. But to just get a general idea of how this can be done, examples can be demonstrated as follows:
Consider a rectangular coordinate with axes X and Y on a flat plane. The square of the distance between 2 points of coordinates (x,y) and (x+∆x, y+∆y) is ∆x∆x + ∆y∆y everywhere on the plane. It’s also important to note that the unit vector (the vector that points in the direction of the axis and that is of 1 unit of that axis) of X and the unit vector Y are constant in length and direction everywhere on the plane.
Now consider a common coordinate system on the earth surface that’s made up of latitude and longitude (X and Y) axes, that are perpendicular to each other at every point of their intersections and that are parameterized at equal intervals into 360 units (this is arbitrary and may be other numbers) on each axis (see Figure 1).
If x and y are the coordinates of axis X and axis Y, respectively, then the square of the distance between 2 points of coordinates (x,y) and (x+∆x, y+∆y) is not ∆x∆x + ∆y∆y everywhere on the surface because, at higher y, one unit of x has shorter length (as you can see from the figure). That is, the unit vector of X gets shorter at higher y, while the unit vector of Y is constant in length everywhere. Notably, this is different from the unit vectors of the flat surface (see above).
By tedious measurement at a large number of points on the surface, one will be able to determine that the square of the distance between 2 points of coordinates (x,y) and (x+∆x, y+∆y) is [∆xcos(y/r)][∆xcos(y/r)] + ∆y∆y, where r is some constant that can be determined from the actual measurements. (It will turn out that r is, in fact, the length of the earth’s radius!) Thus, just by making measurements on the surface itself, one can determine the differences between a flat surface and a seemingly flat surface (which is, in fact, a spherical surface) of the earth.
Mathematically, there is a tensor that contains all the information that specifies the curvature of a surface. It’s called Riemann curvature tensor. This tensor expresses the curvature of a surface in tensor components. If we put the values obtained from the measurements above (the way the unit vector changes and the way the distance between 2 points on the surface is calculated, which is related to the metric tensor of that surface) into the formula that calculates the Riemann curvature tensor, we will get the Riemann curvature tensor of that surface. And we will find that the Riemann curvature tensor’s components are all 0 for the flat surface while the Riemann curvature tensor’ s components are not all 0 for the earth surface, which means the earth’s surface is not flat, but curved. More specifically, we will find that the tensor's components of the earth's surface are those of a spherical surface as you'd like to have the proof.
The above is the simplest way I can explain this matter. For more comprehensive and additional explanations, I suggest you put this question at the Mathematics Forum. There are a lot of people who are very mathematically savvy there.
There are no references that can explain curvature of a surface completely but are simple and math-free (esp. tensor-free). Some references from Wikipedia here are probably not-so-difficult starters:
- Curvature of Riemannian manifolds
- Unit vector
- Metric tensor