Uncovering the gauge symmetries of general relativity via Noether’s theorem.

By Merced Montesinos, Diego Gonzalez, and Mariano Celada 

Symmetries are the cornerstone of modern physics. They are present in almost all its subfields and have become the language in which the underlying laws of the universe are expressed. Indeed, in the standard model of particle physics, our best understanding of nature down to the subatomic world, the interactions among fundamental particles are dictated by internal gauge symmetries.

Although the four fundamental interactions can be fitted within the framework of gauge theories, gravity still remains as the weird family member. While gravity can be conceived as a gauge theory on its own, it seems to be one that differs from those describing the non-gravitational interactions. Indeed, the latter are embedded within the so-called Yang-Mills theories, but gravity is something else.

Merced Montesinos (centre) is a theoretical physicist at Departamento de Física, Cinvestav, Mexico.
Diego Gonzalez (left) is a postdoctoral researcher at Instituto de Ciencias Nucleares, UNAM, Mexico.
Mariano Celada (right) is a postdoctoral researcher at Departamento de Física, UAM-I, Mexico.

Humanity’s best description of gravity is provided by Einstein’s general relativity, which has been around for more than a century. In our approach, we follow its first-order formalism, where both the local frame and the connection are treated as independent degrees of freedom. Current wisdom teaches us that within this approach there are two fundamental symmetries that define general relativity, namely, local Lorentz and diffeomorphism invariances. The latter is considered the holy grail of general relativity since it embodies Einstein’s principle of relativity, and at the same time it is this symmetry that sets general relativity apart from the “ordinary” gauge theories describing the non-gravitational interactions. What if we could supersede diffeomorphism invariance with another symmetry, one that makes general relativity closer to an ordinary gauge theory? This question has been in the air for several decades, and no definitive answer has been given yet.

On the other side, from the mathematical viewpoint the use of symmetries entails a powerful result due to Emmy Noether, who showed that there is an interplay between symmetries of an action principle and conservation laws. Her result has proved to be essential in the developments of modern theoretical physics and is cherished not only by its beauty but also by its deep consequences and wide range of applicability. Noether’s theorem is usually divided into two theorems; the first one deals with symmetries parametrized by a countable number of constant parameters (global symmetries), whereas the second one applies to symmetries depending on a given number of arbitrary functions (local gauge symmetries).  For the latter the conservation laws take the form of “Noether identities”, which consist of nontrivial relations off-shell among the variational derivatives of the action (these quantities dictate, on-shell, the classical dynamics of the system under consideration). Once these identities are identified, we can resort to the converse of Noether’s second theorem to unveil the gauge symmetry behind each identity.

In our work we apply the converse of Noether’s second theorem to the n-dimensional Palatini and Holst actions for general relativity with a cosmological constant coupled to several matter fields (scalar, Yang-Mills and fermions), and unveil the gauge symmetries involved in these formulations. As expected, we find that one of the gauge symmetries corresponds to local Lorentz symmetry, but instead of the symmetry under diffeomorphisms, we find an alternative internal gauge symmetry that can be regarded as a nontrivial higher-dimensional generalisation of three-dimensional local translations. This symmetry turns out to depend on the explicit form of the action principle, the spacetime dimension, and the matter fields themselves through their respective energy-momentum tensors. Outstandingly, infinitesimal diffeomorphisms can be written in terms of the field transformations associated to both the alternative symmetry and local Lorentz symmetry, and therefore, in this approach, they are no longer considered fundamental, but rather a derived symmetry. Thus, by challenging current wisdom, we can replace diffeomorphism invariance with the alternative gauge symmetry and describe the fundamental gauge freedom of general relativity (in the first-order order formalism) coupled to matter fields with an internal set of gauge symmetries made up of the alternative and local Lorentz symmetries.  In brief, our framework provides a way of thinking the full gauge symmetry of general relativity as being comprised of symmetries acting in an internal domain, something that might not only be a step forward towards speaking of general relativity in a language closer to that reserved for ordinary gauge theories, but also spark deep insights into the elusive quantum nature of gravity as a result of the traces left in the alternative symmetry by the action principle underlying the theory.

Given the potential usage of our method based on Noether’s theorem for reading off the gauge symmetries not only in general relativity but also in other alternative theories of gravity, we encourage the audience to stay tuned and read our paper in Classical and Quantum Gravity.

Read the full article in Classical and Quantum Gravity:
The gauge symmetries of first-order general relativity with matter fields
Merced Montesinos et al 2018 Class. Quantum Grav. 35 205005

About the authors:
Merced Montesinos is a theoretical physicist at Departamento de Física, Cinvestav, Mexico. Webpage: http://www.fis.cinvestav.mx/~merced/
Diego Gonzalez is a postdoctoral researcher at Instituto de Ciencias Nucleares, UNAM, Mexico.
Mariano Celada is a postdoctoral researcher at Departamento de Física, UAM-I, Mexico.

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