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

Question

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 1

A connection on a manifold is used to propagate frames; more precisely we can parallel transport a frame along a path connecting two points, say $p,q$.

This gives a map between all frames at p, to all those at q; and it turns out that this map is linear.

But is linearity enough?

Well recall here Newtons first law - a body is in uniform motion in a straight line when it is not subject to any forces; in GR, we generalise straight (in Euclidean or flat space) to geodesic in (curved space).

So the first law generalises in GR to: parallel transport along a geodesic should mean that physically nothing changes - this generalises inertial motion - and so is the equivalence principle.

But actually it is not 'curved space' but curved spacetime.

Physically this means that there are no local distortions of length or time; thus we require the metric $g$ should be preserved - ie we require an isometry; and it turns out the map above is an isometry when the connection is a Levi-Cevita connection.

Answer 2

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.

Answer 3

I think it helps to study old differential geometry books like Kreyszig to see the development of Levi-Civita, as opposed to the modern formalism, as you show explicitly that plain derivatives of tensors do not transform as tensors, that's why you need the covariant derivative. That is an essential part of defining the covariant derivative (connection). I don't like the mathematician's formulation that is now being popularized. It provides shortcuts but I think they obscure the motivation and purpose of the geometry.

But to answer your question, one needs the crucial fact that inner products of vectors should be preserved as you transport them around space, i.e. the derivative of the inner product vanishes. Infinitesimally, the metric tensor is the inner product of vectors that define the space you are working in, so naturally you want the covariant derivative of the metric tensor to vanish. That defines the property of parallel transport. That property will then get you the Levi-Civita connection. Essentially the physical meaning of the Levi-Civita connection is that it provides the ability to differentiate tensors according to the natural geometry of curved space, which is defined by parallel transport. These circumstances are equivalent to your geodesics, but the vanishing of the metric by the covariant derivative is the local version, whereas geodesic is a global (integral) version, just like we have differential maxwell equations and integral maxwell equations.