I am studying approaches to gravity where the Poincaré group is “gauged”. The original motivation of this is to understand what is meant on the statement that “Teleparallel gravity is a gauge theory of the translation group”. The standard references are highly confusing and imprecise.
The situation with Poincaré-based theories is also messy with lots of papers using very “physicist-y” math where the geometric meaning or even the validity of construction is questionable. I have also found some papers where more rigorous mathematics is employed, however in this case, I have difficulty translating between the two languages.
I suspect I can clarify a great deal of my (mis)understanding if I understand properly what a “Cartan radius vector” is.
- I am doing a bit of a “translation work” here, so it is also possible I completely misunderstand my references, but it seems to me a “naive” approach to gauging the Poincaré group is to work (at least initially) in flat Minkowski spacetime, (with general curvilinear coordinates $ x^\mu$ is necessary) and we are given four functions $ y^a$ on the space, which are interpreted as flat/inertial/Cartesian coordinates. In this case a holonomic, orthonormal vielbein is given by $ $ \theta^a=\mathrm dy^a. $ $ Under a Poincaré transformation with constant coefficients $ y^{\prime a}=\Lambda^a_{\ b}y^b+\tau^a$ , the vielbein transforms as $ $ \theta^{\prime a}=\Lambda^a_{\ b}\theta^b, $ $ however under a Poincaré transformation with point-dependent coefficients, this is not the case. $ $ \$ $ We can save the day however by considering the inertial coordinates $ y^a$ as some kind of section of an affine bundle, and introduce the affine connection $ $ \mathscr Dy^a=\mathrm dy^a+\Gamma^a_{\ b}y^b+B^a, $ $ and define $ \theta^a=\mathscr Dy^a$ . $ $ \ $ $ From this point on, however it gets fuzzy, because teleparallel gravitists (see for ex. Aldrovandi, Pereira) tend to use this expression to define the vielbein. $ $ \ $ $ But in for example Metric-affine gauge theory of gravity by Hehl et al. it is stated that $ y^a$ is the “Cartan radius vector” if $ B^a=0$ , and also that in order to have the $ (\theta^a,\Gamma^a_{\ b})$ double as a Cartan-connection, we must have (here apparantly only the linear part of the conenction is used) $ $ Dy^a=\mathrm dy^a+\Gamma^a_{\ b}y^b=0. $ $
- A bit later in the same Hehl paper, it is stated that the Cartan radius vector is defined by the equation (they used the notation $ \xi$ for what I called $ y$ before as well) $ $ D\xi^a=\theta^a. $ $ Here apprantly it is a linear object, not an affine one, and the covariant derivative $ D$ is linear, and is claimed that the above equation is not totally integrable in general, but if integrated along an infinitesimal loop, it gives essentially affine holonomy of the form $ $ \Delta\xi^a=\frac{1}{2}\left(R^a_{\ b\mu\nu}\xi^b+T^a_{\ \mu\nu}\right)\mathrm dx^\mu\wedge\mathrm dx^\nu. $ $
- Based on what I have read about Cartan connections, one can describe a Cartan connection modelled on $ G/H$ by having a $ G$ -fiber bundle $ (E,\pi,M,G/H,G)$ with typical fiber $ G/H$ , an Ehresmann $ G$ -connection on $ E$ specified by a vertical projector $ \mathrm v:TE\rightarrow VE$ , and a section $ s:M\rightarrow E$ such that the pullback $ $ s^\ast\mathrm v|_x:T_xM\rightarrow V_{s(x)}E\simeq\mathfrak g/\mathfrak h $ $ is an isomorphism. $ $ \$ $ Here the section $ s$ has interpretation of specifying the point of contact between the model geometry $ E_x$ and the manifold $ M$ , and the last condition states that at the point of contact the tangent space of the model geometry must be isomorphic to the tangent space of the base geometry. $ $ \ $ $ In fibred coordinates $ (x^\mu,y^a)$ for $ E$ , we can write the connection as $ $ \mathrm v=\partial_a\otimes\left( \mathrm dy^a+\Gamma^a(x,y) \right). $ $ In case we have $ G=\text{ISO}(3,1)$ and $ H=\text{SO}(3,1)$ , the model space is $ G/H\simeq\mathbb R^4$ the affine Minkowski space, and the connection is $ $ \mathrm v=\partial_a\otimes(\mathrm dy^a+\Gamma^a_{\ b}(x)y^b+B^a(x)), $ $ since $ G$ is an affine group, and the pullback condition is that $ $ s^\ast\mathrm v=\partial_a\otimes(\mathrm ds^a(x)+\Gamma^a_{\ b}(x)s^b(x)+B^a(x)) $ $ is nondegenerate. But this is basically the affine covariant derivative of $ s$ .
So my question is, how are the objects $ y^a$ , $ \xi^a$ , $ s$ defined in my bullet points related? What is it we actually mean under a Cartan radius vector? What is its interpretation?
It is clear to me that my $ y^a$ in the first bullet point is basically $ s$ (in the last bullet point), however confusingly, Hehl says that our affine connection is a Cartan connection if $ dy^a+\Gamma^a_{\ b}y^b=0$ , which seems to me that i) is impossible to be integrated in general, ii) is in conflict with the more abstract definition in the third bullet point, where for the connection to be Cartan it is enough that $ dy^a+\Gamma^a_{\ b}y^b+B^a$ is nondegenerate (which is consistent with the interpretation of $ \mathscr D y^a$ as a vielbein).
But I also know that a Cartan connection is, from another point of view, basically a coframe and a linear connection together, and $ B^a$ is not in general a coframe in terms of transformation properties, as it has been elucidated by Hehl.
I basically would like to clarify this mess into something coherent. References for papers treating Poincaré gauge gravity with mathematic rigour, consistency and geometric clarity is something I also would like.
