# General Relativity from Causality

## TL;DR

This paper demonstrates that theories with multiple interacting spin 2 particles generally exhibit superluminal propagation and acausality, highlighting the uniqueness of general relativity as the consistent effective theory of a single massless spin 2 field.

## Contribution

It systematically analyzes causality constraints in theories with multiple spin 2 particles, showing most such theories are acausal, thus supporting the uniqueness of general relativity.

## Key findings

- Most multi-spin 2 theories lead to superluminal propagation.
- General relativity with a single spin 2 is causally consistent.
- Higher derivative interactions in spin 1 theories can be causal.

## Abstract

We study large families of theories of interacting spin 2 particles from the point of view of causality. Although it is often stated that there is a unique Lorentz invariant effective theory of massless spin 2, namely general relativity, other theories that utilize higher derivative interactions do in fact exist. These theories are distinct from general relativity, as they permit any number of species of spin 2 particles, are described by a much larger set of parameters, and are not constrained to satisfy the equivalence principle. We consider the leading spin 2 couplings to scalars, fermions, and vectors, and systematically study signal propagation in all these other families of theories. We find that most interactions directly lead to superluminal propagation of either a spin 2 particle or a matter particle, and interactions that are subluminal generate other interactions that are superluminal. Hence, such theories of interacting multiple spin 2 species have superluminality, and by extension, acausality. This is radically different to the special case of general relativity with a single species of minimally coupled spin 2, which leads to subluminal propagation from sources satisfying the null energy condition. This pathology persists even if the spin 2 field is massive. We compare these findings to the analogous case of spin 1 theories, where higher derivative interactions can be causal. This makes the spin 2 case very special, and suggests that multiple species of spin 2 is forbidden, leading us to general relativity as essentially the unique internally consistent effective theory of spin 2.