# The spectrum of symmetric teleparallel gravity

## Abstract

General Relativity and its higher derivative extensions have *symmetric teleparallel* reformulations in terms of the non-metricity tensor within a torsion-free and flat geometry. These notes present a derivation of the exact propagator for the most general infinite-derivative, even-parity and generally covariant theory in the symmetric teleparallel spacetime. The action made up of the non-metricity tensor and its contractions is decomposed into terms involving the metric and a gauge vector field and is found to complement the previously known non-local ghost- and singularity-free theories.

Recent detections of gravitational waves [1, 2], or fluctuations in the gravitational field, fully agree with the predictions of General Relativity (GR). As a theory of the metric gravitational field, however, GR remains incomplete in the ultra-violet. Simple but infinite-derivative-order actions that alleviate the singular structure of GR without introducing new degrees of freedom [3, 4, 5, 6], have lead to promising results in recent investigations into e.g. quantum loops [7, 8], scattering amplitudes [9, 10], inflation [11, 12], bouncing cosmology [13, 14, 15] and black holes [16, 17].

The purpose of this note is to generalise the classification of metric theories from Riemannian (see Ref. [6]) to a more general geometry.

Recently it has been suggested that a reconciliation between gravitation as a gauge theory of translations [18, 19] and as a gauge theory of the general linear transformation *GL*(4) [20, 21] could be achieved by stipulating that the former group of transformations should be the unbroken remainder of the latter, in the frame where inertial effects are absent [22] (where by ‘unbroken’ we mean that the gauge is fixed in such a way that the affine connection always remains a translation). This logic leads us to the symmetric teleparallel spacetime [23], see also [22, 24, 25, 26].

*teleparallel*spacetime, where \(R^a{}_{bcd}=0\), the inertial connection is given by a general linear transformation \(J^{a}{}_{b}\) of the trivial vanishing connection solution or “coincident gauge” [22], as \(\Gamma _{\phantom {a}bc}^{a} =(J^{-1})^{a}{}_{d}\partial _{b}J^{d}{}_{c}\), where \((J^{-1})^{a}{}_{d}\) are the components of the inverse matrix that parameterises the

*GL*(4) transformation. In a

*symmetric teleparallel*spacetime, the torsion also vanishes \(T^a_{\phantom {a}bc}=0\). It follows that \((J^{-1})^{a}{}_{d}\partial _{[b}J^{d}{}_{c]}=0\), which indeed leaves us with the coordinate-changing diffeomorphism \(J^{a}{}_{b} =\partial _b \xi ^a\) which was identified with translations (by a vector \(\xi ^a\) in the tangent space) in the gauging of the Poincaré group already in Ref. [27]. The frame field can be obtained by the nonlinear realisation of the translation gauge potential [28]. We refer the reader to Fig. 1 for the relations between the eight types of affinely connected spacetimes.

Beltrán et al. [22] introduced the Palatini formalism for teleparallel and symmetric teleparallel gravity theories. In these notes, however, we adopt the inertial variation

^{1}There are 10 possible terms:

*M*, so as to remain dimensionless. The special case with the five non-vanishing constants \(a_1\), \(b_1\), \(c_1\), \(d_1\) and \(b_3\) has been considered previously [22, 24, 25].

Having obtained commonality with known results for the most general, parity-even action that is quadratic in curvature within Riemannian geometry, we now turn our attention to GR. From (2), we know that the equivalent of the Einstein-Hilbert Lagrangian is given by \(\mathcal {L}_{G} = Q^2\). Upon reflection, we observe that the Lagrangian (5) reduces to that of GR for the non-vanishing parameters \(a_1=-b_1=c_1=-d_1=1\), which obey (10), as required.

Concerning higher derivatives, we note that the connection equation of motion^{2} is generically third order in metric derivatives unless parameters are chosen in such a way that (10) holds. Investigating whether this would pose a problem for the Cauchy formulation is beyond the scope of this study.

*minimally coupled*in that the field \(u^a\) does not enter into the matter lagrangian \(\mathcal {L}_M\). This is identically true in a vacuum and for canonical scalar and vector fields. To couple spinor matter one may assume the equivalent of the standard Levi-Civita connection recast in non-metric geometry [25]. The standard metric coupling is a consequence of the Hermitian property of the Dirac action [30].

We have now isolated the propagator for the metric field but this is not the full story. We arrived at the result (22) by integrating out the connection, using its own equation of motion (8). In general, the reverse can not be similarly achieved, i.e. we cannot integrate out the metric to isolate the propagator for the connection due to the non-trivial coupling of the two fields. We can however arrive at the propagator for the combined field content of the theory.

As expected, the field \(u^a\) can carry a spin-1 d.o.f. It was explained and confirmed by various considerations in Refs. [31, 32] that pathologies cannot be avoided unless \(a+b=0\) so that \(P^{(1)}\)-part of the propagator disappears.^{3} As such, this is a constraint also upon the parameters of our theory. Furthermore, we require \(a>0\) so as to avoid a ghost-like pole for the graviton. While the signs of the two remaining scalar pieces in the propagator (23) should be chosen so as to avoid negative residues, we cannot exclude the possibility of these pieces remaining in a viable theory. To start with, the Newtonian limit [33] quite strictly constrains the additional scalar modes in the cases that do not satisfy (10). To categorically exclude such models, perhaps based on the possible difficulties with Cauchy formulation noted from (8), or by the potential issues such as acausal propagation that may be activated at a nonlinear order, see e.g. [34, 35], presents a considerable technical challenge.

It is clear that the equivalents of all Riemannian metric theories are contained within the symmetric teleparallel geometry, since the propagators of the latter are included in (22). We note also that some of the potentially viable scalar-tensor theories contained in the action (5) would, in the Riemannian context, require resorting to non-analytic functions with problematical integral operators such as \(1/\Box \) [33, 36], which is due to the higher derivatives of the Riemannian invariants. However, it is impossible to give a mass to the graviton when restricting to analytic functions, which is easy to see since a massive graviton would require a propagator of the form \(P^{\bar{2}}/(k^2+m^2)\). Massive scalar fluctuations of the form \(P^0/(k^2+m^2)\) are permitted, but the mass should be sufficiently large so as not to run into conflict with local constraints on long-range forces.

^{4}function, such as \(A(\Box )=C(\Box )= e^{-\frac{\Box }{M^2}}\), we can improve the scaling \(\sim k^{-2}\) of the GR propagator that leads to divergences in the ultra-violet. This is realised by the following infinite-derivative generalisation of the symmetric teleparallel equivalent:

The elegance of the new formulations gives rise to optimism for technical progress in the investigations into infinite-derivative gravity [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], but also hint at a possible shortcut towards a finite quantum theory. Non-locality has been recognised as a key to reconcile unitarity with renormalisability [3, 4, 5, 6]. In the newly *purified gravity* [22], we further avoid the conceptual difficulty of reconciling the local character of the equivalence principle and the the non-local character of the quantum uncertainty principle [37, 38]. In teleparallel gravity [19], in contrast to GR, it is possible to separate the inertial effects from gravitation and to consider its quantisation. This separation is in-built to our geometry.

## Footnotes

- 1.
For the purpose of deriving the propagator for an arbitrary theory, it suffices to expand the action to the quadratic order in the field strength. Furthermore, the action is parity-even if it is invariant under parity-transformations whereby the sign of the spatial coordinates is flipped.

- 2.
It can be seen from Eq. (5.8.21) of Ref. [21] that in the frame formulation, the case with non-metricity and vanishing torsion leads to higher-derivatives in the first (i.e. the frame) field equation, whereas in our Palatini formulation they appear in the second (i.e. connection) field equation.

- 3.
In Ref. [32], the 4-vector was separated into two transverse 3-vectors in an explicit expansion of the action, and it was demonstrated that one of the two 3-vectors will acquire the wrong sign for its kinetic term. Setting \(a+b=0\) decouples the piece of the vector \(u^a\) for which \(\nabla _a u^a=0\) from the theory, essentially rendering the metric fluctuation \(h_{ab}\) independently invariant under transverse diffeomorphisms.

- 4.
For a suitable function: 1) \(A(-k^2\rightarrow 0) \rightarrow 1\) so that GR is recovered in the infra-red, 2) \(A(\Box )=e^{\gamma (\Box )}\), where \(\gamma (\Box )\) is an entire function, so there are no additional poles, and 3) falls off sufficiently fast as \(A(-k^2 \rightarrow -\infty ) \rightarrow \infty \) so as to tame ultra-violet singularities. In the following examples, \(\gamma ({\Box })=-\Box /M^2\).

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