# Toward a Mechanism for the Emergence of Gravity

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## Abstract

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## 1. Introduction

## 2. A Mechanism for the Emergence of Gauge Symmetries

**Non-emergence of gauge symmetry:**This corresponds to the standard situation in which the charges ${\mathcal{Q}}_{\mathsf{\Psi}}^{\Vert}$ parametrize the space $\mathcal{U}$ without redundancies. In this case, we will have that the projection onto the subspace $\mathcal{U}$ simply eliminates the degrees of freedom directly encoded in the conditions $\mathsf{\Psi}=0$.

**Emergence of gauge symmetry:**This case corresponds to the situation in which some of the charges ${\mathcal{Q}}_{\mathsf{\Psi}}^{\Vert}$ vanish when we restrict to $\mathcal{U}$. In this case, not all the solutions in $\mathcal{U}$ can be distinguished by just using operations within $\mathcal{U}$. Hence, the solutions related by a finite transformation associated with the charges ${\mathcal{Q}}_{\mathsf{\Psi}}^{\Vert}$ can be interpreted as describing a single physical configuration. A pictorical representation of this situation can be found in Figure 1.

## 3. Two Examples: Yang–Mills Theories and Linearized Gravity

#### 3.1. Yang–Mills Theories

**Massive case ${M}_{ab}\ne 0$.**In this case no gauge symmetries emerge, since the putative emergent gauge symmetries from Equations (5) and (6) do not leave $\mathcal{U}$ invariant. Actually, they translate into the following map for ${\phi}^{a}$

**Massless case ${M}_{ab}=0$.**In this case gauge symmetries emerge naturally, since the transformations from Equations (5) and (6) become gauge symmetries by leaving the scalar fields ${\phi}^{a}$ invariant (note that the last term of (9) now vanishes).

#### 3.2. Linear Graviton Physics

- 1
- The first term, ${W}^{\mu}$, vanishes on shell within the subspace $\mathcal{U}$;
- 2
- The second term, given by the divergence of ${N}^{\mu \nu}$, is a superpotential;
- 3
- The third term, ${M}^{\mu}$, vanishes identically if we take the transformations such that ${\nabla}_{\alpha}\left(\right)open="("\; close=")">{\nabla}_{\mu}{\chi}^{\mu}$;
- 4
- The fourth term, ${D}^{\mu}$, is such that it also identically vanishes within the subspace $\mathcal{U}$.

**First case:**${B}_{\mu}\ne 0$. If we take ${B}_{\mu}\ne 0$ we need to restrict ourselves to the subspace $h=0$ in order to have a trivial current. However, notice that these transformations do not preserve such subspace and hence we would be in the first situation described in Section 2, namely the non-emergence of gauge symmetry.

**Second case:**${B}_{\mu}=0$. In this second case we can relax the condition on h to be ${\nabla}_{\mu}h=0$ in order to have a trivial current. Notice that these transformations preserve the subspace ${\nabla}_{\mu}h=0$ and hence this second option corresponds to the second situation described in Section 2, i.e., the emergence of gauge symmetry. More explicitly, to have WTDiff emergent transformations, we need to restrict ourselves to the same subspace $\mathcal{U}$ introduced above. We recall that such space is defined by the conditions (42) and that we need to choose it because the emergent Weyl gauge symmetries preserve them and the Noether currents restricted to such subspace give identically vanishing charges. Hence, we conclude that it is possible to also find emergent Weyl symmetries with the mechanism introduced in Section 2.

- 1
- It describes massless excitations, namely the masses from the Lagrangian in Equation (20) are equal to zero (${m}_{1}={m}_{2}=0$).
- 2
- It couples to a two-index symmetric source ${T}^{\mu \nu}$ that is conserved, i.e., divergenceless ${\nabla}_{\mu}{T}^{\mu \nu}=0$.
- 3
- The source has constant trace, namely ${\nabla}_{\mu}T=0$.

## 4. No-Go Theorems for Emergent Gravity Reconsidered

#### 4.1. Weinberg–Witten Theorem

#### 4.2. Marolf’s Theorem

**Marolf’s**

**theorem:**

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Special Cases of Parameters for Linear Graviton Fields

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**Figure 1.**We represent the space of solutions $\mathcal{S}$ and the set of surfaces characterized by constant values of ${\mathcal{Q}}_{\mathsf{\Psi}}^{\perp}$. Among them, there is a privileged one represented in darker blue characterized by ${\mathcal{Q}}_{\mathsf{\Psi}}^{\perp}=0$, i.e., it represents the subspace $\mathcal{U}$ from the discussion above. The solid blue and brown curves represent curves of constant ${Q}_{\mathsf{\Psi}}^{\Vert}$ within the surfaces of ${\mathcal{Q}}_{\mathsf{\Psi}}^{\perp}=\mathrm{constant}$. The dashed brown curves within the subspace $\mathcal{U}$ represent the emergent gauge orbits, i.e., they connect points that cannot be distinguished by performing operations within $\mathcal{U}$ and are characterized by the same vanishing value of some of the charges ${Q}_{\mathsf{\Psi}}^{\Vert}$ in $\mathcal{U}$.

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Barceló, C.; Carballo-Rubio, R.; Garay, L.J.; García-Moreno, G.
Toward a Mechanism for the Emergence of Gravity. *Appl. Sci.* **2021**, *11*, 8763.
https://doi.org/10.3390/app11188763

**AMA Style**

Barceló C, Carballo-Rubio R, Garay LJ, García-Moreno G.
Toward a Mechanism for the Emergence of Gravity. *Applied Sciences*. 2021; 11(18):8763.
https://doi.org/10.3390/app11188763

**Chicago/Turabian Style**

Barceló, Carlos, Raúl Carballo-Rubio, Luis J. Garay, and Gerardo García-Moreno.
2021. "Toward a Mechanism for the Emergence of Gravity" *Applied Sciences* 11, no. 18: 8763.
https://doi.org/10.3390/app11188763