First a little motivation: Over the last two years I have been reading about jet bundles and higher order differential geometry and learned that the frame bundle of a spacetime $M$ is formally seen as the bundle of invertible 1-jets of local charts (i.e. local diffeomorphisms) $$\Phi:M\rightarrow\mathbb{R}^{1,3}$$ That is to say a section of $FM$ or linear frame is literally the linearization (or first order Taylor expansion) of such coordinate charts.
In a sense then we can view the structure group $GL(4,\mathbb{R})$ of the frame bundle as the linearization of the group of local diffeomorphisms fixing a point on $M$ (This is explained here).
Since the Einstein-Hilbert lagrangian is the scalar curvature of a connection on this bundle (or an associated bundle or some reduction thereof) we are in a sense gauging only the linearization of all possible coordinate charts. While understandable at the time it doesn't seem entirely within the spirit of general covariance. So I am wondering what happens if we consider the scalar curvature of a connection on bundles whose sections are higher order approximations to such coordinate charts (looking at second order seems a reasonable place to start). Could we possibly find physically measurable phenomena from “higher order covariance”?
From another point of view, the diffeomorphism group of a smooth manifold encodes it's topology and smooth structure. To use only a linear coordinate charts essentially truncates this information. I'd like to find anyone utilizing higher order information on spacetime.
To this end I found the book by Hawking and Ellis "The large scale structure of spacetime" And it turns out that to define/describe genuine spacetime singularities they utilize (section 8.3) the "b-metric" which is essentially a positive definite metric on the oriented orthonormal frame bundle of spacetime (this method was pioneered by Schmidt "A New Definition of Singular Points in General Relativity").
A frame on the total space of the frame bundle can be seen as a linear combination of a (lifted) frame on the tangent space of $M$ and frame on $T(SO(1,3))$. Because the connection in GR is a compatible one, it is given by the partial derivative of the frame. Then a frame on the tangent space of $FM$ can be viewed as a second order Taylor expansion of the local charts on $M$. This bundle or “b” metric then provides the kind of higher order approximation to local charts I am interested in.
Assuming metric structure and orientability of our spacetime $M$ the orthonormal frame bundle $FM$ is a principal $SO(1,3)$ bundle and a ten dimensional manifold. Then a positive definite bundle metric here (as used by Hawking, Ellis and schmidt) reduces the structure group of $FFM$, the frame bundle of the frame bundle to an $SO(10)$ bundle over $FM$. Just as orientability and metricity reduce $FM$ to an $SO(1,3)$ manifold, let us consider the reduction of $FFM$ completely using the induced geometric structures. In general such higher order bundles have much richer structures even at second order (see for example Saunders “Geometry of Jet bundles” or Kolar, Michor and Slovaks' “Natural operations in Differential Geometry”).
Beginning with our $SO(10)$ bundle over $FM$ we have that the linear connection on $FM$ is equivalent to a splitting or Whitney sum of $TFM$ into horizontal and vertical sub-bundles respectivly: $$TFM=TM\oplus T(SO(1,3))$$ reducing our structure group of $FFM$ over $FM$: $$SO(10)\rightarrow SO(4)\times SO(6)$$ Moreover the frame bundle of any manifold is itself a parallelizable manifold (Kobayashi) and therefore always admits an almost complex structure and therefore we have no obstruction to choosing a hermitian b-metric, this yields the further reduction: $$SO(4)\times SO(6)\rightarrow U(2)\times U(3)$$ We now note that the determinant line bundle of the frame bundle metric of an orientable manifold is always trivial, giving one final reduction $$U(2)\times U(3)\rightarrow S(U(2)\times U(3))$$ We can write this in a more recognizable form (see HERE ): $$S(U(2)\times U(3))=(SU(2)\times U(1)\times SU(3))/\mathbb{Z}_{6}$$ Just as $FM$ reduces to a principal $SO(1,3)$ bundle for an orientable spacetime we can say that $FFM$ also reduces to a principal $(SU(2)\times U(1)\times SU(3))/\mathbb{Z}_{6}$ bundle over $FM$.
Moreover, One might note that we can ALWAYS define Spin structures (i.e. spinors) on the tangent space of $FM$ due to it's parallelizability which carry a $16$-dimensional projective (Spinor) representation of $SO(10)$. Let us examine again the series of group reductions going in a slightly different order than before: $$SO(10)\rightarrow U(5)\rightarrow SU(5)\rightarrow S(U(2)\times U(3))=(SU(2)\times U(1)\times SU(3))/\mathbb{Z}_{6}$$ it is well-known that under such a group reduction the $16$ dimensional spinor representation breaks up accordingly:
$$16^{+}\rightarrow\cdots\rightarrow1\oplus10\oplus5^{*}\rightarrow\left(3,2,\frac{1}{6}\right)\oplus\left(3^{*},1,-\frac{2}{3}\right)\oplus\left(3^{*},1,-\frac{1}{3}\right)\oplus\left(1,2,-\frac{1}{2}\right)\oplus\left(1,1,1\right)$$
Where following Zee (“Group theory in a nutshell for physicists chapter” IX.) we are using the notation $\left(a,b,c\right),a\in SU(3),b\in SU(2),c\in U(1)$. The spinor fields on TFM are precisely those
of one generation of the standard model fermions (plus what is usually interpreted as a sterile neutrino). That is to say, that spinors on the tangent space of $FM$ appear to carry precisely the representations of standard model fermions.
Can that possibly be a coincidence? Can we interpret frames on the b-metric or frame bundle metric of spacetime physically? Also note that such frames transform externally under the Poincare group (as that is literally the tangent space of orthonormal frame bundle we are dealing with). Someone might recognize the two different group reduction routes above as being related to the Pati-Salam and Geordi-Glashow group reductions respectively (also interesting)
Can someone versed in this weigh in? Finally note that if we examine the curvature on such bundle metrics on the base spacetime we get Kaluza-Klein type Yang mills terms. I just wanted to look at taking general covariance beyond gauging linearization of local charts and The b-metric seemed the way to begin... I know this might be outside of mainstream; however, it directly applies to Hawking and Elliss book regarding the bundle metric of Schmidt and... is very fascinating.
