Abstract
I consider theories of gravity built not just from the metric and affine connection, but also other (possibly higher rank) symmetric tensor(s). The Lagrangian densities are scalars built from them, and the volume forms are related to Cayley’s hyperdeterminants. The resulting diff-invariant actions give rise to geometric theories that go beyond the metric paradigm (even metric-less theories are possible), and contain Einstein gravity as a special case. Examples contain theories with generalizeations of Riemannian geometry. The 0-tensor case is related to dilaton gravity. These theories can give rise to new types of spontaneous Lorentz breaking and might be relevant for “dark” sector cosmology.
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Notes
I thank M. S. Narasimhan for pointing out that the higher tensor idea dates back to Riemann.
This is an assumption we will make throughout the paper. Torsion is a tensor field on the manifold, and adding it might be interesting for various pruposes, including coupling to spinors [2], but is not crucial for the conceptual line of this paper.
In principle, the connection is a choice, and can even be completely independent of the other fields in the theory, and does not necesarily have to match any number counting constraint (See [3] for a clean discussion of this.). But we would like to find a fairly “natural” generalization of the Levi-Civita prescription, which has some dynamical significance.
Our original motivation to look for generalizations of Levi-Civita was in the context of a spacetime description (as opposed to frame bundle description) of higher spin theories. See for example section 3.1 of [4] for some inspiration in this direction. The theories we will present in this paper only have diffeomorphism invariance, and are built on usual manifolds. Higher spin theories on the other hand have a much bigger gauge invariance [5–7], big enough perhaps to make the notion of singularities and horizons gauge-dependent [8–15]. So it seems unlikely that our theories have a simple relation to higher spin theories. Nonetheless, it will be interesting to see if these theories have any relationship with Chern–Simons theories or Vasiliev theories, at least in specific cases and/or low dimensions.
This viewpoint is hardly original, Albert E himself tried this approach during his many attempts to get to his equations, well before Palatini.
We will mostly be concerned with vacuum situations in this paper (to the extent that the higher rank fields can be thought of us part of geometry), but it is straightforward to include matter by adding matter pieces covariantly coupled to the action. The distinction between matter and geometry is most natural in the cases where the connection is determined algebraically by its EOM, so that one can integrate it out from the action, and then couple the system to matter. Otherwise, the connection EOM will contain matter pieces. As an aside, we observe that in the case of both scalar and vector fields, the structure of the Lagrangian is such that the affine connection doesn’t show up [16]. Also, it is perhaps worth being judicious about distinguishing GR experimental tests that are sensitive to the minimal coupling of the field as opposed to merely its geodesic (aka particle/WKB) limit. As theoretical pastime, one might also wish to study a fully geometric theory of matter, where one treats all fields on an equal footing.
This is easy to see using arguments of symmetry and anti-symmetry. One can also check it trivially for the special case when only the diagonal entries \(\phi _{aa \ldots a}\) (no summation) are no-zero.
This object seems to have got some (but not too much) attention in mathematics, one reference that is quoted in a few places on the web is [19]. But [19] seem to be dealing mostly (exclusively?) with what is called the “geometric” hyperdeterminant, what we are dealing with here is a generalization of the “combinatorial” hyperdeterminant, eg. [20].
We restrict ourselves to covariant symmetric tensors because keeping in touch with Riemann’s original philsophy, we want to interpret them in terms of a notion of distance, eg. \(ds^3 \sim \phi _{abc} dx^a dx^b dx^c\). We will soon see that one can construct contravariant higher rank tensors from covariant ones without resorting to metric, so this restriction is not really a restriction.
We have put \(\phi \)-sub/super-scripts on \(\varepsilon \) to emphasize that they correspond to the volume forms constructed from the higher tensor and not the metric.
The objects defined in (12) can be thought of as inverses in that one can check that \( \phi ^{a_1\dots a_{r_{k-1}}a}\phi _{a_1\dots a_{r_{k-1}}b}=\delta ^{a}_{b}\). Note however that only for the rank 2 case is it a true inverse.
This is legitimate because even though the connection is not a tensor, its variation is, and therefore equations of motion arising from connection variation are tensor equations. This is a familiar fact: even though we vary the Maxwell action with respect to a gauge-dependent quantity (the gauge field \(A_\mu \)), we end up getting Maxwell equations which are gauge-invariant.
It is conceivable that one way to look at our theories is precisely after doing this Palatini to metric-like translation. Note that once this is done, the coupling to non-geometric matter will proceed as usual: via covariantizing with respect to the connection. Of course, this is most natural when the connection is determined algebraically, as in the examples we present here. It will also be interesting to consider theories where all fields are geometrical, as in, they contribute to the volume form. That will be a fully geometrized theory of matter! If one wants to include spinors as well, the natural context to consider such theories would be in the context of (generalized) Riemannian supermanifolds [21]. It is clear that in our set-up, because of the presence of a Lorentzian metric, it is striaghtforward to couple spinors via the intreoduction of a local frame basis [22]. But Cartan’s structure equations will have to be re-considered.
It should be emphasized that affine parellelism and distance minimization give rise to two different notions of geodesics, and Levi-Civita happens to be a case where the two coincide. But it is not clear to me that it is the only possibility. I thank Pallab Basu for an interesting discussion on this.
A potential generalization, for example, could be \(\int _C \ (g_{ab}dx^adx^b)^{1/q}(\phi _{pqrs}dx^pdx^qdx^rdx^s)^{1/p}\).
A bit more severely than in pure GR because in GR we always have the option of turning off the fields other than the metric, and in such solutions local Lorentz invariance is not broken.
This doe not mean however that it is phenomenologically viable. If one plans to use the higher rank fields as some form of dark matter, then either one will have to make sure that the solutions one considers do not allow superluminality, or (to play devil’s advocate) will have to come up with observational evidence that in regions where there is dark matter gradients or boundaries, one has the required (hopefully mild) form of superluminality. To push this last speculation a bit further, one might ask whether the observed bounds on Lorentz invariance and absence of superluminality are comparable in magnitude to the bounds on the dimensionless gradients of dark matter (say, in the solar system).
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Krishnan, C. A Generalization of Gravity. Found Phys 45, 1574–1585 (2015). https://doi.org/10.1007/s10701-015-9941-2
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DOI: https://doi.org/10.1007/s10701-015-9941-2