I am having a hard time understanding the physical meaning of asymptotic symmetries in General relativity. I think I understand the mathematics, but the meaning eludes me. I'll try to write the things I do understand, with an emphasis on an analogy with electromagnetism, and then I'll ask what I don't get.
Gauge symmetry in electromagnetism
The field equations of EM $$\partial_\mu F^{\mu\nu}=J^\nu$$ and $$\partial_{[\mu}F_{\alpha\beta]}=0$$ are invariant under the gauge transformation $$A_\mu \rightarrow A_\mu + \partial_\mu \lambda (x)$$.
What I mean by invariant is specifically that the form of the equations remains the same and that the numerical value of the fields doesn't change. If I were to measure the electric field at some point in spacetime, its value would be the same in all gauges.
It is possible to use Noether's theorem to find the conserved quantity arising from this symmetry. One finds essentially two types of conservation laws:
- Case I: The gauge parameter $\lambda(x)$ vanishes on the boundaries of the spacetime domain of the theory. Then, the Noether conserved charge vanishes identically on shell. This means that these transformations are not "true symmetries" but simply redundancies of the theory, since they don't give nontrivial conservation laws.
- Case II: The gauge parameter doesn't vanish at the boundary. In this case, the Noether charge is a surface integral at the boundary and it gives nontrivial conservation laws.
A particular example of the second case is when $\lambda(x)=$constant. Then we get conservation of electric charge. Furthermore, when $\lambda(x)$ goes to a non-constant value at the boundary, we get the conservation laws associated with "asymptotic symmetries". These are described, for example, in Strominger's lecture notes https://arxiv.org/abs/1703.05448.
What I think I get
- We have a symmetry transformation (in the sense that the equations remain unchanged)
- that is trivial when the transformation vanishes at the boundaries of the theory (In the sense that the Noether charge vanishes on shell)
- and that gives nontrivial conservation laws when the transformation doesn't vanish at the boundaries (These conservations include the usual charge conservation when $\lambda=$constant and more general "large gauge transformations" when it's something else).
Diffeomorphism invariance General relativity
In GR we work with a geometric theory of gravity. Since the theory is geometrical and all equations and objects are as well, the theory is invariant under diffeomorphisms. Already at this level, something is different to the EM case, since the meaning of "invariant" is only that the equations will be unchanged, but the values of the fields at different points. This is not really a problem for me, it seems that the fact that the electric field is trully numerically invariant under gauge transformations is more of an extra perk of EM. Indeed, in Yang-mills theories, even the field strength $F_{\mu\nu}$ is gauge-dependent and it's only the form of the equations of motion that will remain unchanged.
Isometries in General Relativity
There is a sense in which we can ask about symmetries that truly leave the numerical components of objects unchanged. Any diffeomorphism that takes points $x^\mu$ to $x^{\mu'}$ will transform the metric as
$$ g_{\mu\nu}(x)\rightarrow g'_{\mu\nu}(x)=\frac{\partial x^{\mu'}}{\partial x^{\mu}}\frac{\partial x^{\nu'}}{\partial x^{\nu}}g_{\mu'\nu'}(x')$$
This is simply the statement that $g_{\mu\nu}$ transforms as a (0 2) tensor under diffeomorphisms. Alternatively, I'm just stating active transformations $g_{\mu\nu}(x)\rightarrow g'_{\mu\nu}(x)$ and passive transformations $g_{\mu\nu}(x)\rightarrow g_{\mu'\nu'}(x')$ should be equivalent. Now, expanding the coordinate transformation as $ x^{\mu'}=x^{\mu}+\xi^\mu$ for small $\xi$ we get
$$ g'_{\mu\nu}(x)=g_{\mu\nu}(x)+\mathcal{L}_\xi g_{\mu\nu}(x)$$
This means that when $\mathcal{L}_\xi g_{\mu\nu}=0$, $\xi$ is an isometry of the spacetime and not only the equations remain unchanged (which is always true for diffeomorphisms) but even the numerical components of the metric in a coordinate system don't change under diffeomorphisms generated by $\xi$.
asymptotic symmetries in General relativity
One can also try to find diffeomorphisms that solve $\mathcal{L}_\xi g_{\mu\nu}=0$ asymptotically. That is, to first order in some suitable limit as you take $r\rightarrow \infty$. This gives the BMS group, which can also be understood as the symmetries that preserve the structure of the boundary of spacetime. The BMS group can also be understood in analogy with the EM case as large gauge transformations in the sense that they are diffeomorphisms that don't vanish on the boundary, hence being physical transformations and not redundancies.
Things I don't get
The fact that asymptotic symmetries are large gauge transformations makes me want to think of them as true symmetries. However, the fact that they are obtained as approximate isometries as one takes an asymptotic limit to infinity makes me think of them as approximate symmetries. Which way of thinking about them is more accurate?
Following this line of thought, what would the analogue of isometries be in electromagnetism (or, more generally, in Yang-Mills theories)?
