[Paper Review] Emergent general relativity in the tensor models possessing Gaussian classical solutions

Naoki Sasakura|arXiv (Cornell University)|Nov 6, 2009

Cosmology and Gravitation Theories20 references3 citations

TL;DR

This paper demonstrates that in tensor models with Gaussian classical solutions, low-energy long-wavelength fluctuations of the tensor fields precisely reproduce the spectrum of gravitational fluctuations in general relativity across D=1 to D=4 spacetime dimensions. The emergent geometry arises from spontaneous symmetry breaking of the tensor model's orthogonal invariance, which corresponds to the local translation symmetry of general relativity, indicating a unified emergence of space, gravity, and gauge symmetry from a single non-perturbative framework.

ABSTRACT

This paper gives a summary of the author's works concerning the emergent general relativity in a particular class of tensor models, which possess Gaussian classical solutions. In general, a classical solution in a tensor model may be physically regarded as a background space, and small fluctuations about the solution as emergent fields on the space. The numerical analyses of the tensor models possessing Gaussian classical background solutions have shown that the low-lying long-wavelength fluctuations around the backgrounds are in one-to-one correspondence with the geometric fluctuations on flat spaces in the general relativity. It has also been shown that part of the orthogonal symmetry of the tensor model spontaneously broken by the backgrounds can be identified with the local translation symmetry of the general relativity. Thus the tensor model provides an interesting model of simultaneous emergence of space, the general relativity, and its local gauge symmetry of translation.

Motivation & Objective

• To investigate whether general relativity can emerge as an effective theory from tensor models with Gaussian classical solutions.
• To identify the correspondence between fluctuations in the tensor model and geometric degrees of freedom in general relativity.
• To explore the origin of local gauge symmetry (local translations) in general relativity as arising from spontaneously broken orthogonal symmetry in the tensor model.
• To establish a numerical framework for comparing tensor model spectra with those of general relativity in various dimensions.
• To assess the generality and robustness of the emergence mechanism beyond specific Gaussian solutions.

Proposed method

• The tensor model is formulated with a real symmetric rank-three tensor $ C_{abc} $ invariant under orthogonal group $ O(N) $, with action and measure defined via the invariant metric $ ds_C^2 = dC_{abc} \, dC_{abc} $.
• Classical solutions are constructed as Gaussian backgrounds $ C^0_{abc} $, representing commutative nonassociative fuzzy flat spaces of arbitrary dimension $ D $.
• Small fluctuations $ \delta C_{abc} $ around these backgrounds are analyzed using a path integral approach with gauge fixing via the Faddeev-Popov method and BRST quantization.
• The fluctuation spectra are computed numerically and compared with the linearized general relativity spectrum on flat space, including metric, ghost, and gauge field modes.
• The gauge-fixing procedure introduces ghost fields and kinetic terms, whose spectra are matched to those from the tensor model via the DeWitt supermetric and inner products.
• The correspondence is tested across dimensions $ D=1,2,3,4 $, with momentum-space mode profiles and spectral ratios compared between the two theories.

Experimental results

Research questions

• RQ1 Can the low-energy dynamics of tensor models with Gaussian backgrounds reproduce the spectrum of linearized gravity in general relativity?
• RQ2 How does the local gauge symmetry of general relativity emerge from the global orthogonal symmetry of the tensor model?
• RQ3 To what extent is the correspondence between tensor model fluctuations and gravitational modes robust across different spacetime dimensions?
• RQ4 What is the role of the Faddeev-Popov ghosts in the tensor model, and how do their spectra match those in general relativity?
• RQ5 Is the emergence of general relativity in this framework limited to Gaussian backgrounds, or does it extend to more general classical solutions?

Key findings

• The long-wavelength, low-lying fluctuation spectra in the tensor model for $ D=1,2,3,4 $ match exactly with those of linearized general relativity on flat space.
• The spectral ratio of the two lowest-lying ghost modes in the tensor model is $ 2.5 $, matching the general relativity prediction of $ 20/8 = 2.5 $.
• The mode profiles of the ghost fields in the tensor model are in excellent agreement with the normal and longitudinal modes of general relativity's Faddeev-Popov ghosts.
• The part of the orthogonal symmetry spontaneously broken by the Gaussian background corresponds precisely to the local translation symmetry of general relativity.
• The framework is dimensionally independent, suggesting that the emergence of general relativity should hold in any spacetime dimension.
• The results are robust across multiple dimensions and are consistent with the expectation that general relativity is the unique effective theory of symmetric rank-two tensor fields with local gauge symmetry.