Properties of the magnetic universe with positive cosmological constant

Abstract: The properties of the Melvin-type spacetime with a positive cosmological constant $\Lambda$ in $d$-dimensional Einstein--Maxwell gravity is studied. The solution is parametrised in terms of the `de Sitter radius' $\ell\propto\Lambda^{-1/2}$ and the magnetic field parameter $\beta$, and they are warped products of the form $\mathbb{R}^{1,d-3}\times S^2$, where $\mathbb{R}^{1,d-3}$ is the $(d-2)$-dimensional Minkowski spacetime and $S^2$ is topologically a two-sphere which contains a conical singularity, whose nature depends on the product $\beta\ell$. In the limit $\ell\rightarrow\infty$, the $S^2$ decompactifies and the $d$-dimensional Melvin universe is recovered. The Freund--Rubin-type flux compactification model is shown to be another particular limit of this solution. We also calculate the flux and geodesics in this spacetime.

• The paper presents a Melvin-type spacetime model with positive Λ by employing d-dimensional Einstein–Maxwell gravity and warped product geometry.
• A detailed examination shows that a positive cosmological constant disperses magnetic fields through manipulated conical singularities and parameterized curvature effects.
• Geodesic analyses confirm circular orbit stability and bounded particle trajectories, offering insights for extradimensional stabilization and astrophysical applications.

Properties of the Magnetic Universe with Positive Cosmological Constant

Introduction

The study explores the Melvin-type spacetime in a gravitational framework where a positive cosmological constant and electromagnetic fields are considered. This configuration is examined within the context of $d$-dimensional Einstein–Maxwell gravity. The spatial solution is presented as a warped product, $\mathbb{R}^{1,d-3}\times S^2$, with particular attention to the conical singularity's nature. As $\ell$ trends towards infinity, the Melvin universe is recovered, highlighting the decoupling with compactified extra dimensions when considering strong magnetic fields.

Geometrical Analysis

In the field of positive $\Lambda$ (cosmological constant), spacetime presents an unorthodox topology characterized by the warped product between $(d-2)$-dimensional Minkowski space and an $S^2$ distorted sphere containing the conical singularity. The conical singularity is manipulated through parameters $\beta$ and $\ell$. A Freund–Rubin-type flux compactification model emerges as a particular case, prominently in $q=2$ scenarios associated with Maxwell fields, suggesting extradimensional stabilization mechanisms in low-energy string theory.

Magnetic Flux Examination

The gauge potential is defined explicitly, enabling the computation of magnetic flux through the system. Flux values demonstrate dependencies on spacetime curvature, especially with a varying cosmological constant. There is notable divergence in total flux as $\beta$, the magnetic field parameter, approaches zero, suggesting an absence of confinement and subsequent spreading in contrast to the Anti-de Sitter scenario. The presence of a positive $\Lambda$ enhances the dispersal of the magnetic field, a phenomenon attributed to the essential 'anti-box' nature of the positive cosmological space as opposed to the confinement seen in negative cosmological counterparts.

Geodesic Motion

Geodesic analyses reveal insights into the physical intuition behind the Melvin spacetime with $\Lambda>0$. Time-like and null geodesics reveal stability in circular orbits and the presence of finite bound orbits. Only specific radii, as determined by the spacetime parameters, permit circular null orbits. A bounded spatial domain within which these geodesics operate underlines the role of the background magnetic field in stabilizing particle trajectories, echoing previous findings in minimalistic solenoid models.

Conclusion

Through exploring the geometrical configurations, the research delineates the properties of the Melvin universe when a positive cosmological constant is applied. The notion of twisted compactified dimensions illustrates sophisticated interactions between gravity and electromagnetism within the field of higher-dimensional theories. While extending to positive $\Lambda$ provides a model with pertinent astrophysical applications, the spacetime ‘anti-box’ aspect demands further investigation into potential cosmological impacts and stability considerations within broader gravitational models.