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We examine the role of consistency with causality and quantum mechanics in determining the properties of gravitation. We begin by examining two different classes of interacting theories of massless spin 2 particles—gravitons. One involves coupling the graviton with the lowest number of derivatives to matter, the other involves coupling the graviton with higher derivatives to matter, making use of the linearized Riemann tensor. The first class requires an infinite tower of terms for consistency, which is known to lead uniquely to general relativity. The second class only requires a finite number of terms for consistency, which appears as another class of theories of massless spin 2. We recap the causal consistency of general relativity and show how this fails in the second class for the special case of coupling to photons, exploiting related calculations in the literature. In a companion paper Hertzberg and Sandora (2017), this result is generalized to a much broader set of theories. Then, as a causal modification of general relativity, we add light scalar particles and recap the generic violation of universal free-fall they introduce and its quantum resolution. This leads to a discussion of a special type of scalar-tensor theory: the

General relativity is consistent with observations over a vast range of length scales. The

The latter has provided some of the central motivations for considering alternatives to general relativity. It is difficult to understand why the vacuum energy is so small, despite there being known large contributions from massive particles running in loops, such as top quarks. Furthermore, the coincidence problem (why there is a comparable amount of matter and dark energy today), as well as the cosmological horizon, homogeneity, and flatness problems are also sometimes invoked as motivations. Also, there are a suite of difficulties in understanding general relativity as a quantum theory, including nonrenormalizability, trans-Planckian unitarity violation, black hole information paradox, and global issues associated with de Sitter space and eternal inflation.

This range of primarily conceptual challenges, leads one to enquire just how inevitable general relativity is; whether theoretically consistent alternatives exist. It is sometimes thought that indeed general relativity follows inevitably as the

In this letter we clarify some aspects of this basic idea. Firstly, we point out that in fact classes of theories of massless spin 2 particles exist, which are Lorentz invariant and do not propagate new degrees of freedom. The most basic one involving the least number of derivatives, so-called minimal coupling, and the others involving higher derivatives (the latter can be organized to not propagate any additional degrees of freedom). While the first leads to general relativity, the seconds appears as another class of theories of spin 2, and was earlier introduced in Ref. [

Secondly, we study conventional ways to modify general relativity by the addition of new light scalars. We emphasize that in the Standard Model of particle physics only the Lorentz symmetry is postulated and most couplings compatible with it are observed. Similarly, new scalars should generically come with many parameters leading to violation of the observed universality of free-fall. However, quantum effects typically remove the problem by making the scalars heavy. This leads to an examination of the so-called

We are interested in constructing theories of massless spin 2 particles from the ground up. As is well known, the massless spin 2 unitary representation of the Lorentz group involves two helicities in 3+1 dimensions. It is useful to embed these two degrees of freedom into a symmetric tensor field in order to build a local theory. We shall denote this

If one fixes the gauge, then under a Lorentz transformation

The free theory of these spin 2 particles is associated with terms of the form

The interaction that involves the least number of derivatives, and hence would be most relevant at large distances, is to attempt to couple

It is easy to see that under a gauge transformation and an integration by parts, the problem is fixed by taking

This ensures the theory is gauge invariant on-shell. To also be gauge invariant off-shell one must endow the matter fields with a gauge transformation rule, such as

Hence Type I coupling leads uniquely to general relativity and all of its successes. This theory can even be quantized, in the low energy regime, with concrete quantum gravity predictions such as [

Here we would like to describe a much bigger class of theories of massless spin 2 particles; this was introduced earlier in Ref. [

As a concrete example of the interaction, if the matter involves

If we considered the equivalence principle to be another fundamental postulate, then this would suffice to reject this entire Type II class in favor of the very special Type I. However, in this work we only take the Lorentz symmetry as a fundamental postulate, and the equivalence principle is to be derived rather than assumed.

In fact it is useful to put this point of view in a broader perspective. It is often suggested that modern particle physics is built out of various additional postulates, such as the “gauge principle” or “principle of minimal coupling”. However, if we examine the structure of the Standard Model, in particular its symmetries, a different picture emerges. (i) Exact symmetries: CPT derives from locality and unitarity, while

Furthermore, the global

There is reason to think that causality provides a possible answer to this question. We examine this in greater detail and for a much broader class of models in a companion paper [

It is well known that in general relativity with standard matter sources there is no problem with causality [

Let us focus on the case of the four index object

In the geometric optics limit, the leading deflection from null propagation on the Minkowski cone is [

In a related context of QED, minimally coupled to gravity, it is known that one can integrate out the electron and generate terms of the form (

There does exist a manifestly causal way to modify gravitation. This involves the introduction of additional degrees of freedom. Since fermions do not mediate long range forces, and vectors (with minimal coupling) have sources that tend to neutralize, we focus on the remaining case of adding light scalars.

Usually in the literature a scalar

So, to assume the form (

This poses a challenge to deriving the (weak) equivalence principle. However, if we take quantum effects into account, then a generic scalar

A popular framework that is both causal and enforces the universality of free-fall is the so-called

A popular example is

Here we would like to examine

Note that the effective Lagrangian in eq. (

By contrast, the

One might attempt to quantize

We note that in other formulations of gravity, such as Palatini, similar conclusions hold. Namely, in constructions in which there are no new degrees of freedom, then the theory can always be recast into the general relativity form with a collections of appropriate counter-terms, while, in constructions in which there is a new degree of freedom, it can only be self-consistently quantized by reorganization into the scalar-tensor form.

The above arguments help toward deriving general relativity as the only consistent theory involving massless spin 2 at low energies. (i) Type II theories that utilize higher derivative couplings (but can avoid extra degrees of freedom) can lead to problems with causality; see [

However, important puzzles remain, including understanding dark energy. The smallness of the vacuum energy within the framework of general relativity does have a candidate explanation by introducing many (heavy) scalars, leading to a potential with an exponentially large number of vacua. Though it is unclear how to formulate probabilities in this context. While the behavior of gravitation at the Planck scale requires further new physics.

The paper does not refer to any specific data set.

An earlier version of this manuscript was presented at the 13th International Symposium on Cosmology and Particle Astrophysics (CosPA 2016).

The author declares that they have no conflicts of interest.

I would like to thank Raphael Flauger, Jaume Garriga, Alan Guth, David Kaiser, Juan Maldacena, McCullen Sandora, and Mark Trodden for helpful conversations. I would like to thank the Tufts Institute of Cosmology for support.