What's the difference between the diffeomorphism invariance and reparametrization invariance?

Question

Can somebody tell me what's the difference between the diffeomorphism invariance and reparametrization invariance?

Answer 1

Diffeomorphism Invariance

Let $M$ be a smooth manifold. Let $\phi: M \to M$ be a diffeomorphism. A simple property of the Einstein equations is

$$ g \in \otimes^2 TM \text{ is solution to vacuum Einstein equation} \implies \text{ so is } \phi^*g $$

To see that this is true, simply pull back both sides of the Einstein equation by $\phi$, and use the property of the Ricci tensor $\phi^*Ric_{(g)} = Ric_{(\phi^*g)}$.

You have a family $[g] \equiv \{\phi^*g | \phi \in Diff^+(M)\}$ of Riemannian structures on your manifold which are supposed to describe the same physics. This what I would call the notion of diffeomorphism invariance in general relativity.

Reparameterization Invariance

A parametrization of one manifold $M$ by another manifold $N$ is a diffeomorphism $\varphi: M \to N$. $N$ parametrizes $M$ in the sense that a points $p \in M$ smoothly correspond points in $\varphi(p) \in N$.

Reparametrization invariance usually means a scenario where the choice of this diffeomorphism $\varphi \in Diff(M \to N)$ doesn't matter.

Here's a simple example of what people might call reparametrization invariance.

Take a smooth curve $\gamma: [0, 1] \to M$, and suppose that $\gamma$ solves the geodesic equation. Let $\gamma': [0, 1] \to M$ be another curve, we have the following (trivial) fact

$$ \text{ $image(\gamma') = image(\gamma)$ } \implies \gamma' \text{ is also a geodesic} $$

that is, the geodesic equation (length functional, not energy functional) doesn't care how you parameterize your curve.

Sometimes they're the same thing

To summarize, so far, I've presented diffeomorphism invariance as a statement about Riemannian structures on a manifold, and reparametrization invariance as a statement about diffeomorphisms from one manifold to another. There are some scenarios where these seemingly different notions are really the same thing.

Example 1: Curve Reparametrization

Consider our description of diffeomorphism invariance applied to a $1$-manifold. Riemannian structures on $1$-manifolds are equivalent to reprarmetrizations of the curve. Pulling back the Riemannian inner product on the tangent bundle by a smooth map is equivalent to smoothly reparametrizing the curve. So our notion of diffeomorphism invariance and reparametrization invariance are the same thing.

Example 2: Moduli Space of closed orientable surfaces

Let $\Sigma$ be a closed orientable surface. Say we are interested in the space $Mod_{\Sigma, \mathbb R^n}$ of all submanifolds of $\mathbb R^n$ diffeomorphic to $\Sigma$.

One way to make this space explicit is to consider diffeomorphisms

$$e: \Sigma \to \mathbb R^n$$

But some of these diffeomorphisms map into the same submanifold in $\mathbb R^n$. These are exactly the diffeomorphisms related by a precomposition with diffeomorphism on the surface itself. Hence we have a description of our moduli space as equivalence classes of diffeomorphisms

$$ Mod_{\Sigma, \mathbb R^n} \simeq E \equiv \{e: \Sigma \to \mathbb R^n\}/\sim~~~ e \sim f \iff e = f \circ \phi ~~ \text{ for some } \phi \in Diff(\Sigma) $$

Now give $\mathbb R^n$ the usual Euclidean metric $g_{\mathbb R^n}$. A diffeomorphism $e: \Sigma \to \mathbb R^n$ pulls back the metric to $\Sigma$ via $g_e \equiv e^*g_{\mathbb R^n}$. Observe:

$$ g_{e \circ \phi} = (e \circ \phi)^*g = \phi^*g_e $$

In particular, when $\phi \in Diff(\Sigma)$, this says pullbacks by homeomorphisms in the same equivalence class are related by pullbacks by homeomorphisms on the surface. Hence another equivalent description of moduli space is

$$ Mod_{\Sigma, \mathbb R^n} \simeq F \equiv \{g \text{ metric on } \Sigma\}/\sim~~~ g \sim h \iff g = \phi^*h ~~ \text{ for some } \phi \in Diff(\Sigma) $$

In summary, we found

$$ Mod_{\Sigma, \mathbb R^n} \simeq E \simeq F $$

Observe now

• The equivalence relation in $E$ is identification of homeomorphisms that have the same image. We would call this reparametrization invariance.
• The equivalence relation in $F$ is pullback of metric by diffeomorphisms on $\Sigma$. We would call this diffeomorphism invariance.

In other words, in this case diffeomorphism invariance is a manifestation of reparametrization invariance on the space of Riemannian metrics.