Late-time tails and mass inflation for the spherically symmetric Einstein-Maxwell-scalar field system

Summary

The Einstein-Maxwell-scalar field system models gravity in the presence of an electromagnetic field and a scalar field. A solution to this system is a quadruple (M,g,φ,F) consisting of a 4-dimensional manifold M with Lorentzian metric g, a real-valued scalar field φ, and a 2-form F. The system is globally well-posed by [27] and the data considered is compactly supported.

Highlights

• The Einstein-Maxwell-scalar field system is a model of gravity in the presence of an electromagnetic field and a scalar field.
• The system is globally well-posed and the data considered is compactly supported.
• The characteristic rectangle Rchar is defined by [1,∞)×[1,∞) and is normalized so that 1∂−vrµ|{r=rH} = 1, where rH = supHr is the limiting value of the area radius function along the event horizon.
• The double null gauge (u,v) is used, which is adapted to the commutator vector fields V and S.
• The initial data norm Dk[φ] is defined, which controls the scalar field and its first k derivatives.
• The theorem states that for every ϵ>0 and integer k≥0, there is C=C(ϵ,k,ϖi,cH,rmin,Dk[φ])>0 such that |(v∂)kφ||H≤CDk[φ]v−1+ϵ.
• The scaling vector field S is used, which is equal to v∂ along the horizon and is tangent to the event horizon.

Key Insights

• The Einstein-Maxwell-scalar field system is a complex model that requires a deep understanding of the interplay between gravity, electromagnetism, and scalar fields. The use of a double null gauge and the definition of a scaling vector field S are crucial in the analysis of this system.
• The compact support of the data is a key assumption in the theorem, which allows for the use of a characteristic rectangle Rchar and the definition of an initial data norm Dk[φ].
• The theorem provides a bound on the scalar field and its derivatives, which is essential in understanding the behavior of the system near the event horizon. The use of the scaling vector field S and the double null gauge allows for a precise analysis of the system's behavior.
• The analysis of the system requires a combination of geometric and analytic techniques, including the use of commutator vector fields and the definition of a scaling vector field S. The use of these techniques allows for a precise understanding of the system's behavior near the event horizon.
• The theorem has implications for our understanding of black holes and the behavior of matter and energy near the event horizon. The use of a double null gauge and the definition of a scaling vector field S provide a new perspective on the analysis of black holes.
• The Einstein-Maxwell-scalar field system is a fundamental model in general relativity, and the analysis of this system provides insights into the behavior of gravity and electromagnetism in extreme environments. The use of a double null gauge and the definition of a scaling vector field S are essential tools in the analysis of this system.
• The theorem provides a new understanding of the behavior of scalar fields near the event horizon of a black hole. The use of the scaling vector field S and the double null gauge allows for a precise analysis of the system's behavior, which has implications for our understanding of black holes and the behavior of matter and energy in extreme environments.

Mindmap

Citation

Gautam, O. (2024). Late-time tails and mass inflation for the spherically symmetric Einstein-Maxwell-scalar field system (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.17927