The universe’s very structure of space and time is more mysterious than we ever thought. Scientists have long been intrigued by the nature of dark energy and the peculiar behavior of stars that seem to defy the conventional rules of physics. A new study offers a fresh perspective on these cosmic enigmas, suggesting that an underlying principle of minimum speed might be the key to unlocking their secrets. This intriguing concept challenges our understanding of space and time, potentially reshaping our view of the universe itself.

A study published in Physics of the Dark Universe has unveiled a novel approach to understanding dark energy stars, proposing that Lorentz violation with an invariant minimum speed forms the foundation of Gravitational Bose-Einstein Condensate (GBEC) of a dark energy star. This innovative research, conducted by Dr. Cláudio Nassif Cruz from the Centro de Pesquisas em Física Teórica, offers a profound shift in cosmological models, shedding light on the complexities of the quantum vacuum and spacetime.

Dr. Cruz proposes that the introduction of an invariant minimum speed, linked to the concept of Symmetrical Special Relativity (SSR), fundamentally alters the structure of spacetime. This minimum speed, denoted as V, is directly associated with the Planck length and changes the causal structure of spacetime, thus influencing the formation and behavior of GBEC within dark energy stars. The research posits that this new understanding could solve the puzzle of the event horizon singularity traditionally predicted in gravitational collapse scenarios.

“Our study shows that the presence of an invariant minimum speed in the fabric of spacetime leads to a phase transition between gravity and anti-gravity, which prevents the formation of a singularity at the event horizon. This transition occurs before reaching the Schwarzschild radius, thus eliminating the problematic divergence and allowing for a more stable structure,” explained Dr. Cruz.

The study outlines the significant implications of this phase transition, where a repulsive core, described by a GBEC metric, replaces the traditional concept of a black hole’s event horizon. This repulsive core is enveloped by a phase coexistence region that bridges gravity and anti-gravity, preventing the divergence of spacetime metrics and allowing signal propagation, which is impossible in classical black hole models.

By mapping the SSR metric, which accounts for the minimum speed, into the GBEC metric, Dr. Cruz was able to link the cosmological constant, which represents the vacuum energy density, to this invariant minimum speed. This innovative approach provides a quantum interpretation of the GBEC, where the minimum speed induces a strong anisotropy that leads to the observed phase transition during a star’s collapse.

“The SSR metric we developed is similar to the de Sitter metric, which is known for representing the spacetime of a universe with a positive cosmological constant. However, our metric includes a minimum speed, providing a more comprehensive understanding of the vacuum energy and its role in the cosmos,” elaborated Dr. Cruz.

The implications of this study are vast, as it challenges the traditional view of black holes and provides a new model for understanding the universe’s dark energy components. The elimination of the event horizon singularity and the introduction of a phase transition region offer new possibilities for exploring the behavior of extreme gravitational systems and the nature of dark energy.

“This research not only provides a solution to the longstanding issue of singularity in black holes but also opens up new avenues for exploring the interplay between quantum mechanics and general relativity, namely the quantum gravity theory. The idea that a minimum speed could be as fundamental as the speed of light in the fabric of spacetime is truly revolutionary shedding light on Astrophysics, Cosmology and various areas of Physics,” said Dr. Cruz.

In summary, the study by Dr. Cruz presents a transformative perspective on dark energy stars, offering a robust theoretical framework that could reshape our understanding of the universe’s most enigmatic phenomena. The incorporation of Lorentz violation due to an invariant minimum speed into the model of GBEC provides a comprehensive solution to the issues posed by classical black hole theories and paves the way for future explorations into the mysteries of dark energy.

Journal Reference

Nassif Cruz, C., dos Santos, R. F., & Amaro de Faria Jr., A. C. “Lorentz violation with an invariant minimum speed as foundation of the Gravitational Bose Einstein Condensate of a Dark Energy Star.” Physics of the Dark Universe (2020). DOI:

About the Author

Cláudio Nassif Cruz is retired Professor of Physics, Federal University of Ouro Preto (UFOP) , Ouro Preto, Minas Gerais, Brazil. He was born in Além Paraíba, Minas Gerais, in August 1967.

He got his bachelor’s degree of Physics (1992) in Federal University of Juiz de Fora (UFJF), Minas Gerais, Brazil.

His master’s degree (1992) and Ph.D degree (2002) of Physics was in Federal University of Minas Gerais (UFMG), Belo Horizonte, Minas Gerais, Brazil.

He has long experience in the area of Condensed Matter Physics, with emphasis on State Equation, Phase Equilibrium and Phase Transitions, focusing on the following topics as a line of work, namely Thompsons approach, Renormalization Group, dynamical and stationary critical exponents for various systems, diffusion-limited chemical reactions, polymers, surface growth, N-vector model without random field. Also some topics in field theories, such as Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD) are treated on Thompsons approach.

In another line of original research that he himself introduced, he works on the exploration of another possibility of Lorentz symmetry breaking for a Deformed Special Relativity with an invariant minimum speed (Symmetrical Special Relativity), where a background field is generated by a non-Lorentzian dynamics at low energies, thus explaining the tiny positive value of the cosmological constant and also explaining the principle of quantum uncertainty, allowing us to make a connection between quantum physics and cosmology.