Pure double-layer bubbles in quadratic $F(R)$ gravity

Pure double-layer bubbles in quadratic $F (R)$ gravity

Ernesto F. Eiroa, Griselda Figueroa Aguirre, and José M. M. Senovilla

Phys. Rev. D 95, 124021 – Published 14 June 2017

Abstract

We present a class of spherically symmetric spacetimes corresponding to bubbles separating two regions with constant values of the scalar curvature, or equivalently with two different cosmological constants, in quadratic $F (R)$ theory. The bubbles are obtained by means of the junction formalism, and the matching hypersurface supports in general a thin shell and a gravitational double layer. In particular, we find that pure double layers are possible for appropriate values of the parameters of the model whenever the quadratic coefficient is negative. This is the first example of a pure double layer in a gravitational theory.

Received 4 April 2017

DOI:https://doi.org/10.1103/PhysRevD.95.124021

Physics Subject Headings (PhySH)

Alternative gravity theories Gravitation

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Ernesto F. Eiroa^1,2,*, Griselda Figueroa Aguirre^1,†, and José M. M. Senovilla^3,‡

¹Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), Casilla de Correo 67, Sucursal 28, 1428 Buenos Aires, Argentina
²Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pabellón I, 1428 Buenos Aires, Argentina
³Física Teórica, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain

^*eiroa@iafe.uba.ar
^†gfigueroa@iafe.uba.ar
^‡josemm.senovilla@ehu.eus

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Vol. 95, Iss. 12 — 15 June 2017

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Images

Figure 1
Pure double layer bubbles in quadratic $F (R)$ . Plot of $α R_{2}$ as a function of $α R_{1}$ , with $R_{1}$ and $R_{2}$ the scalar curvatures, and $α$ the quadratic coefficient of the theory. The values of $R_{1}$ and $R_{2}$ are positive while $α$ is negative (see text). The allowed values for $R_{1}$ and $R_{2}$ compatible with $M > 0$ and a positive quasilocal energy are shown by the solid line. The continuation shown with the dashed line would be feasible if $M < 0$ were permitted, but this leads to negative energies as proven in Sec. 4a.
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Figure 2
Pure double layer bubbles in quadratic $F (R)$ . The dipole distribution strength has components $κ P_{\hat{𝚤} \hat{j}} = Ω h_{\hat{𝚤} \hat{j}}$ , with $Ω = 2 α [R]$ shown as a function of $α R_{1}$ .
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Figure 3
Conformal diagrams of the two regions $M_{1}$ and $M_{2}$ composing the total spacetime. As usual null radial lines are at 45°. On the left side we show the portion of static de Sitter spacetime describing the inner part of the bubble. The origin of coordinates $r = 0$ is represented by the dotted line, while the pure double layer is placed at $r = a$ . Only the shaded zone describes the inner region $M_{1}$ , and this has to be joined to the exterior $M_{2}$ depicted on the right picture, which represents the conformal diagram of the portion of Schwarzschild–de Sitter spacetime describing the outer part $M_{2}$ of the bubble. This spacetime “starts” at the timelike hypersurface $r = a$ —representing the pure double layer—within a static region. Then, the metric can be extended, to the future and to the past, across the cosmological horizon labeled $r = r_{c}$ , leading to past and future null infinities $𝒥^{\pm}$ , and also to a new event horizon $r = r_{h}$ that encloses the singularities shown by double lines and marked as $r = 0$ . The metric can then be extended toward “the right” indefinitely, by alternating an infinite number of cosmological and event horizons. As we see, such a bubble only removes some of the many singularities of the Kottler spacetime, and therefore it might be better to cut the outer part by placing a second, symmetric, bubble, before the first event horizon appears. This is represented in Fig. 4.
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Figure 4
The conformal diagram of the total bubble, avoiding any curvature singularities but allowing for distributional curvature tensors. This metric is obtained by joining the two portions shown in Fig. 3 across the hypersurfaces labeled $r = a$ as explained in the main text, and by then doing a similar procedure on the symmetric “right” part of the cosmological horizon. The two shaded parts are equal copies of the portion $r < a$ of static de Sitter spacetime with $Λ_{1} > 0$ , and the nonshaded part is a portion of Kottler spacetime with $Λ_{2} \in (Λ_{1} / 2, Λ_{1})$ , with mass parameter $M = (Λ_{1} - Λ_{2}) a^{3} / 3$ , and without any event horizons—so that the curvature singularities have been removed. There are pure double layers on the hypersurfaces $r = a$ drawn with thick lines, having total strength $Ω = - 8 α (Λ_{1} - Λ_{2}) = - 24 α M / a^{3}$ which is positive—as $α$ is required to be negative. In the limit $Λ_{2} \to Λ_{1}$ the pure double layer disappears, $M$ vanishes, and one ends up with the entire de Sitter spacetime.
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