On the gravitational field of a plane plate in general relativity

Novotný, Jan; Kučera, Jaromír; Horský, Jan

doi:10.1007/bf00759098

Cited by 8 publications

(19 citation statements)

References 6 publications

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“…In this paper, we want to illustrate this and other curious features of relativistic gravitation by means of a simple exact solution: the gravitational field of a static plane symmetric relativistic perfect incompressible fluid with positive density located below z = 0 matched to vacuum solutions. In reference 3 , we analyze in detail the properties of this internal solution, originally found by A. H. Taub 4 (see also 5,6,7 ), and we find that it finishes up down below at an inner singularity at finite depth d, where 0 < d < π 24ρ . Depending on the value of a parameter κ, it turns out to be gravitational attractive (κ < κ crit ), neutral (κ = κ crit ) or repulsive (κ > κ crit ), where κ crit = 1.2143 .…”

Section: Introductionmentioning

confidence: 74%

The Gravitational Field of a Plane Slab

Saraví

Ricardo²

2009

Int. J. Mod. Phys. A

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We discuss the exact solution of Einstein's equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below z = 0 matched to vacuum solutions. The internal solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space-times finish down below at an inner singularity at finite depth d ≤ q π 24ρ . We show that for κ ≥ 0.3513 . . . , the dominant energy condition is satisfied all over the space-time.We match these singular solutions to the vacuum one and compute the external gravitational field in terms of slab's parameters. Depending on the value of κ, these slabs are either attractive, repulsive or neutral. The external solution turns out to be a Rindler's space-time. Repulsive slabs explicitly show how negative, but finite pressure can dominate the attraction of the matter. In this case, the presence of horizons in the vacuum shows that there are null geodesics which never reach the surface of the slab.We also consider a static and plane symmetric non-singular distribution of matter with constant positive density ρ and thickness d (0 < d < q π 24ρ ) surrounded by two external vacuums. We explicitly write down the pressure and the external gravitational fields in terms of ρ and d. The solution turns out to be attractive and remarkably asymmetric: the "upper" solution is Rindler's vacuum, whereas the "lower" one is the singular part of Taub's plane symmetric solution. Inside the slab, the pressure is positive and bounded, presenting a maximum at an asymmetrical position between the boundaries. We show that if 0 < √ 6πρ d < 1.527 . . . , the dominant energy condition is satisfied all over the space-time. We also show how the mirror symmetry is restored at the Newtonian limit.We also find thinner repulsive slabs by matching a singular slice of the inner solution to the vacuum.We also discuss solutions in which an attractive slab and a repulsive one, and two neutral ones are joined. We also discuss how to assemble a "gravitational capacitor" by inserting a slice of vacuum between two such slabs.

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Section: Introductionmentioning

confidence: 74%

The Gravitational Field of a Plane Slab

Saraví

Ricardo²

2009

Int. J. Mod. Phys. A

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“…The main reason for doing so is because in this case the Newtonian theory has an unique solution. Moreover, as showed in [4,5], its Newtonian limit also exists. To construct such sources, we shall use the ansatz, z → |z| (7) in the solutions (5) and (6), respectively.…”

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confidence: 83%

“…On the other hand, from the work of Vilenkin [4] and Novotný et al [5] we can see that the Newtonian limit of such space-times exists. As a matter of fact, it takes the form…”

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confidence: 95%

On the sources of static plane symmetric vacuum space-times

1998

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The static vacuum plane spacetimes are considered, which have two non-trivial solutions: The Taub solution and the Rindler solution. Imposed reflection symmetry, we find that the source for the Taub solution does not satisfy any energy conditions, which is consistent with previous studies, while the source for the Rindler solution satisfies the weak and strong energy conditions (but not the dominant one). It is argued that the counterpart of the Einstein theory to the gravitational field of a massive Newtonian plane should be described by the Rindler solution, which represents also a uniform gravitational field

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“…[11], where some properties of the inner solution are studied. In particular, it is shown that the solution cannot have a "plane" of mirror symmetry in a region where p(z) ≥ 0.…”

Section: Discussionmentioning

confidence: 99%

Infinite slabs and other weird plane symmetric space–times with constant positive density

Saraví

2008

Gen Relativ Gravit

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We present the exact solution of Einstein's equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below z = 0. This solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space-times finish down below at an inner singularity at finite depth. We show that for κ ≥ 0.3513 . . . the dominant energy condition is satisfied all over the space-time. We match this solution to the vacuum one and compute the external gravitational field in terms of slab's parameters. Depending on the value of κ, these slabs can be attractive, repulsive or neutral. In the first case, the space-time also finishes up above at an empty repelling singular boundary. In the other cases, they turn out to be semi-infinite and asymptotically flat when z → ∞. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss how to assemble a "gravitational capacitor" by inserting a slice of vacuum between two such slabs.

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On the gravitational field of a plane plate in general relativity

Cited by 8 publications

References 6 publications

The Gravitational Field of a Plane Slab

The Gravitational Field of a Plane Slab

On the sources of static plane symmetric vacuum space-times

Infinite slabs and other weird plane symmetric space–times with constant positive density

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