The Newtonian limit of spacetimes for accelerated particles and black holes

Abstract

Solutions of vacuum Einstein’s field equations describing uniformly accelerated particles or black holes belong to the class of boost-rotation symmetric spacetimes. They are the only explicit solutions known which represent moving finite objects. Their Newtonian limit is analyzed using the Ehlers frame theory. Generic spacetimes with axial and boost symmetries are first studied from the Newtonian perspective. The results are then illustrated by specific examples such as C-metric, Bonnor–Swaminarayan solutions, self-accelerating “dipole particles”, and generalized boost-rotation symmetric solutions describing freely falling particles in an external field. In contrast to some previous discussions, our results are physically plausible in the sense that the Newtonian limit corresponds to the fields of classical point masses accelerated uniformly in classical mechanics. This corroborates the physical significance of the boost-rotation symmetric spacetimes. Dedicated to the memory of Jürgen Ehlers (29 December 1929 to 20 May 2008

Variations on Spacetimes with Boost-Rotation Symmetry

“…Our procedure then results into a classical point particle undergoing uniform acceleration, z = 1 2 gt 2 , which generates classical field described by the Newtonian gravitational potential Φ = m √ r 2 +(z− 1 2 gt 2 ) 2 . This follows from the limit of various examples of these spacetimes [2]. Our results thus strongly support the physical significance of the boostrotation symmetric spacetimes (in contrast to some previous conclusions [5] which made the limit in the regions II and IV ).…”

“…Our procedure then results into a classical point particle undergoing uniform acceleration, z = 1 2 gt 2 , which generates classical field described by the Newtonian gravitational potential Φ = m √ r 2 +(z− 1 2 gt 2 ) 2 . This follows from the limit of various examples of these spacetimes [2].…”

“…Newtonian limit of a relativistic spacetime greatly corroborates its physical interpretation. We perform the limit within the framework of the Ehlers frame theory (see [2] for more details and references). The key point is the causality constant λ = c −2 which becomes zero in the limit.…”

Rotating charged black holes accelerated by an electric field

“…For reviews of the boost-rotation symmetric spacetimes, including their radiative properties, see, e.g., [3], [4] and references therein. Recently, we analyzed the Newtonian limit of these spacetimes using the rigorous Ehlers frame theory [5]. The analysis corroborated their physical significance: the Newtonian limit describes the fields of classical point masses accelerated uniformly in classical mechanics.…”

Accelerating electromagnetic magic field from the C-metric

“…A number of papers were written on various specific aspects of the boostrotation symmetric solutions, in particular on the C-metric. In our very recent work [1] dedicated to the memory of Jürgen Ehlers we analyzed the generic spacetimes with axial and boost symmetries from the Newtonian perspective by employing the Ehlers frame theory [9], [10], to construct the Newtonian limit rigorously. This work corroborated the physical significance of the boost-rotation symmetric spacetimes by demonstrating that the Newtonian limit corresponds to the gravitational field of classical point masses accelerated uniformly in the classical mechanics.…”