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I've been interested in expressing the metric tensor $g$ in terms of it's harmonic expansions. In particular I'm interested in writing the tetrad/frame-fields in terms of such expansions.

For simplicity I'm considering a compact, closed, simply connected spacelike 3-manifold (the inclusion of time is rather straightforward, and can be dealt with later). Due to the Poincaire Conjecture, such a space is homeomorphic to the three-sphere $S^{3}$.

Such “ultra-spherical” harmonic expansions have been dealt with in many papers. One of the most readable such papers (not having the strongest background in group theory) I've found is by Lindblom, Taylor, and Zhang. Therein, they first derive the scalar harmonics $Y^{klm}$ on the three sphere where $0\leq k\leq\infty,0\leq l\leq k,-l\leq m\leq l$.

They use these harmonics to then define three classes of vector harmonics $Y_{(0)}^{klm},Y_{(1)}^{klm},Y_{(2)}^{klm}$ (section 3, equations 12,13,14). Now I'm no expert in group theory; however I understand that $S^{3}$ is diffeomorphic to $SU(2)$ and that vector field basis on $S^{3}$ can be viewed as elements of $SU(2)$ (or perhaps more properly it's Lie algebra $su(2)$). I'm also aware of the relationship between special functions (ie. those in harmonic expansions) and associated lie groups. I'm therefore expecting that these three classes of vector harmonics are each related to an element of $SU(2)$. In particular I'm thinking each k corresponds to an irreducible representation of $SU(2)$ with matrix entries given by the $l$ and $m$s? This point isn't required for my end point which is:

I can, in principle represent a basis $e^{\mu}$ for this general metric as:

$$e^{\mu}=e_{a}^{\mu}e^{a}=\sum_{A=0}^{2}\sum_{k=0}^{\infty}\sum_{l=0}^{k}\sum_{m=-l}^{l}A_{(A)}^{klm}Y_{(A)}^{klm}$$

As I understand it, I can now form ladder operators using linear combinations of the fundamental representation of the $SU(2)$ basis (which, as an aside, correspond to null tetrads on the three-sphere ). Such ladder operators $J_{\pm}$ in this context are discussed at length in (https://aip.scitation.org/doi/10.1063/1.523778).

In this manner our harmonics may be expressed as something like (I'm being sloppy but I'm just trying to get the point across):

$$Y_{(A)}^{kl(m\pm1)}=J_{\pm}Y_{(A)}^{klm}$$

Now, if we choose our latin indices to be basis on $S^{3}$ or rather $SU(2)$ (instead of the usual Minkowskian basis), we may write a general frame field, (greek indices) as a composition (call it $a(J_{\pm})$) of such ladder operators:

$$e^{\mu}=e_{a}^{\mu}e^{a}=J_{\pm}....J_{\pm}e^{a}=a(J_{\pm})e^{a}$$

Such operators are Hermitian conjugates of one another $J_{+}=J_{-}^{\dagger}$, therefore we also have our inverse verbeins:

$$e^{a}=e_{\mu}^{a}e^{\mu}=a(J_{\pm})^{\dagger}e^{\mu}$$ I haven't bothered with normalization of the operators, should be such that $$aa^{\dagger}=e_{a}^{\mu}e_{\mu}^{a}=1$$, but that's rather straightforward. I've been a bit sloppy with my terminology, but I was really just trying to get the point across, the formalism is so similar to QFT, that I found it fascinating. Anyway, do people do general relativity this way? is it valid? It seems like the Einstein equations would take on particularly simple forms.

The build up to my question was already soo long I couldn't get into spin weighted and spinor spherical harmonics, or things like an expanding universe (once time is included in) leading to breaking of certain Lie group symmetries but please feel free to talk about it in your answer.

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  • $\begingroup$ In your first equation, what happened to the $\mu$ index? Shouldn't it be somewhere on the right-hand side? $\endgroup$ – G. Smith Jan 29 at 2:58
  • $\begingroup$ In your third and fourth equations, why does the $\mu$ index turn into an $a$ index and vice versa? $\endgroup$ – G. Smith Jan 29 at 3:00
  • $\begingroup$ @G._Smith apologies, you are right that the vector expansions should have indices, I'll fix that and the others $\endgroup$ – R. Rankin Jan 29 at 4:51
  • $\begingroup$ So what would be the ultimate goal? Are you expecting to find a metric of, say, Kerr solution by action of some sequence of operators on some initial seed? Or is it more of notational excercise, by simplifying large indicial expressions behind operator algebra? $\endgroup$ – A.V.S. Jan 29 at 5:06
  • $\begingroup$ @A.V.S. I was interested in how the eigenvalues (k,l,m) of these eigenfunction/harmonics would appear in the trace reversed Ricci (or Einstein) tensor, and consequently the energy momentum tensor of the sources, certainly these integers would have to appear there, the similarity to quantization I thought was interesting, or at least worthy of further examination. $\endgroup$ – R. Rankin Jan 29 at 5:23
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No: the expansion in harmonics you sketch is unavailable outside of the very specific case of $S^3$ you are considering and this is why people do not do GR this way to my knowledge.

Having said that it's not entirely clear to me how your sketch works for $S^3$; perhaps you should reconsider it.

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  • $\begingroup$ Just like a closed 1d shape can be represented by a fourier series of U(1) irreducible representations on $S^1$, and likewise in 2d with spherical harmonics on $S^2$, the harmonics on $S^3$ can represent a closed three space of essentially arbitrary shape, the basis of which can be expressed in terms of the vector harmonics. This all comes from the Peter-Weyl theorem. $\endgroup$ – R. Rankin Jan 29 at 4:46
  • $\begingroup$ Yes, a sort of multipole expansion should in principle capture the information you want, but I don't think I can make your specific sketch with the ladder operators and such work. If you're broadly speaking interested in gravitational multipole expansions, there is indeed literature originally due to Geroch I think. A recent paper is link.springer.com/article/10.1007/JHEP05(2018)054 $\endgroup$ – alexarvanitakis Jan 29 at 15:29

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