On one hand, a spacetime $(M,g)$ with the Killing vector $\xi^\mu$ and $x^\mu(\tau)$ a geodesic, we can construct the quantity $$Q = \xi_\mu \frac{dx^\mu}{d\tau}\tag{4.32}$$ that is constant along the geodesic. According to page 164 of D. Tong's lectures notes, Noether's theorem identifies this quantity as being the conserved charge associated the isometry generated by $\xi^\mu$.

On the other hand, Noether's theorem (or at least one statement of Noether's theorem) says that global symmetries of an action are associated to conserved charges, and vice-versa. But in the previous case, the fundamental field of our theory, namely GR, is the metric. Isometries (therefore existence of Killing vectors) heavily depend on the metric we consider. In other words, the symmetries we are considering depend on the explicit form of the field we choose. This is usually not what we do in other field theories, we rarely use explicit expressions for a scalar field for example.

That being said, I don't understand how isometries could qualify as global symmetries and how Noether theorem can be used for isometries.


Actually if you look through Tong's notes, the derivation he is presenting does not consider the metric to be the fundamental variable (it sort of doesn't make sense to talk about isometries of a dynamical metric), but rather the worldline of a particle moving on a fixed background metric.

In particular, after Eq 4.32, Tong writes down the action \begin{equation} S = \int {\rm d} \tau g_{\mu\nu} (x) \frac{{\rm d} x^\mu}{{\rm d} \tau} \frac{{\rm d} x^\nu}{{\rm d} \tau} \end{equation} This action is supposed to be varied with respect to the four functions $x^\mu(\tau)$, with $g_{\mu\nu}(x)$ being a set of functions of $x$. The path that makes the action stationary will obey the geodesic equation.

In this context, an isometry amounts to a change in the coordinates $x^\mu$ that leaves the action invariant. A global symmetry, in this context, is a transformation that does not depend on $\tau$. So for example, a translation in space along the $3$ direction by an amount $a$ would shift $x^3 \rightarrow x^3 + a$ (note that $a$ does not depend on $\tau$), and if $g_{\mu\nu}$ did not depend on $x^3$, the action would clearly remain invariant. This leads to a Noether current, following the logic in Tong's note, which agrees with the equation for the Killing vector associated with translations in the $x^3$ direction.

By contrast, this action is also invariant under reparameterizations $\tau \rightarrow \tilde{\tau}(\tau)$. This time reparameterization is a local / gauge symmetry, since the transformation depends on $\tau$. The interpretation is that this is a 1-dimensional version of diffeomorphism invariance.


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