general relativity - What is the physical meaning of the Levi-Civita connection?

I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry:

Let $M$ be a manifold with metric $g$. Then exists an unique torsion-free connection $\nabla$ such that the metric is covariantly constant ($\nabla g=0$).

This is the Levi-Civita connection. In GR we choose to work always with this connection (this works because the difference between two connections is always a tensor field).

I have proven that this implies that geodesics are timelike, nulllike or spacelike meaning that the norm of the tangent vector is conserved along the geodesic.

Are there any more reasons to choose this particular connection? Or is it just used because of its simplicity?

P.S. I have seen What is the physical meaning of the connection and the curvature tensor? but I was wondering why we choose to work with this particular connection, not the physical meaning of a connection.

Answer

Two reasons I can think of really.

One is that it is the connection that requires the least amount of extra data. It is completely determined by the metric, so no additional geometric data is needed to specify it.

However, this doesn't mean we absolutely cannot utilize any additional connections, but consider the fact that since the difference of two connections is a tensor field, we can always choose one connection as a designated connection and represent any other connection that might be needed as tensor fields. So why not choose the connection requiring the least unknown variables as the designated connection?

Two is that when we construct the Riemannian (well, pseudo-Riemannian, but I'd like to ignore the difference now) manifold that models space-time, we want to be as close to euclidean geometry as possible (well, Minkowski-geometry, actually), but still allow for curvature.

Now, if you take a vector space (meaning it is a "flat space"), it naturally admits differentiation of vector fields, so there is a natural connection on it, that also happens to be torsionless. If you put any (algebraic) inner product on the vector space, which we can view as a metric tensor, this natural connection is automatically metric compatible with it. So the natural connection on a flat, euclidean space, is the Levi-Civita connection of its inner product in a natural way.

Furthermore, if we take an embedded hypersurface in $n$-dimensional euclidean space, we can get a relatively natural (although still chosen) connection on the hypersurface, by the following algorithm:

1) Take a vector field tangent to the hypersurface.

2) Extend it arbitrarily into the hypersurface' neighborhood.

3) Differentiate this extension in the direction of a vector field that is fully tangent to the hypersurface (using the connection on the ambient space, obviously).

4) The resulting vector field will be independent of the extension, but won't be tangent to the hypersurface, so substract its normal part to obtain a vector field that is actually tangent to the hypersurface.

The differential operator that does this algorithm is a natural induced connection in the hypersurface. It also happens to be the Levi-Civita connection associated with the induced metric on the hypersurface.

These two examples show that the Levi-Civita connection appears naturally in euclidean geometry, including hypersurfaces of euclidean spaces, therefore, if we wish to construct spacetime as something that is, basically like euclidean space, except curved, there is not much reason to try to construct more alien geometries than those noneuclidean geometries that naturally appear as submanifolds of euclidean spaces.

Edit: For example, building on what 0celo7 said, from euclidean space, we know that the straight line between two points if the one with extremal length.

In Riemannian geometry, we have two different concepts of geodesics. One which is the straightest possible curve (its tangent vector is parallel to itself), and another, which is the curve of extremal length/proper time.

The first concept depends on the connection, the second on the metric. These two will coincide iff the connection if the Levi-Civita connection.

And since in euclidean space, the two coincide, we WANT to construct GR in such a ways that these two concepts will also coincide there.

Edit2: I thought about this stuff a bit, and I'd like to clarify my second point a bit.

What we have here are basically two different, but related concepts. Parallelism, and metricity. The connection $\nabla$ gives us parallelism, and the metric $g$ gives us metricity.

It should not be difficult to convince yourself, that parallelism is not immediately related to metricity. Take for example an arbitrary vector space $V$ over an arbitrary field $\mathbb{F}$. We say that two vectors, $x$ and $y$ are parallel, if there is such an $\alpha\in\mathbb{F}$ scalar that $\alpha x=y$. We haven't put any norm or inner product on this space, so we cannot measure angles or distances. But parallelism makes sense. This is why the vector space in question admits differentiation naturally, if it also has a well-behaved topology, parallelism is needed for differentiation, and a vector space naturally has it.

However, obviously, parallelism can be grasped in terms of angles, so if you have a metric that allows for the measurement of angles, you also have parallelism. The mathematical statement for this is exactly the existence of the Levi-Civita connection - a connection completely determined by the metric.

Obviously, you can, mathematically speaking, disentangle the notion of parallelism from the notion of metricity, by introducing a completely arbitrary connection, but this will result in completely alien geometries that do not match at all what we see in the real world around us.

We might also not immediately see noneuclidean-ity, but just because we allow for curved geometries, it does not mean we should throw away every other previously established geometric property of spacetime (namely, that parallelism is determined by metricity) because we made one modification.