Local and global gravitational aspects of domain wall space-times

TL;DR

The paper addresses how eternal vacuum domain walls between vacua with non-positive cosmological constants influence local and global gravity. It develops a comoving-wall metric framework, applies Israel’s thin-wall formalism, and classifies walls into extreme, non-extreme, and ultra-extreme by the surface density σ, linking each class to distinct interior/exterior geometries (M_4, AdS_4, and dS_4) and to their global causal structures. Tolman mass analysis shows that extreme walls have zero total gravitational mass per area when balanced by AdS_4 regions, supporting a positive-mass protection conjecture that forbids exterior observers from encountering negative-mass objects. The global space-times exhibit causal features akin to black holes, including Cauchy horizons and lattice extensions that avoid singularities, with extensions across horizons allowing well-posed Cauchy problems in AdS_4/M_4 mosaics. The work connects these idealized solutions to physical domain walls in cosmology, highlighting implications for vacuum decay, supersymmetry, and gravitational shielding, while underscoring their value as a didactic model for GR concepts.

Abstract

Local and global gravitational effects induced by eternal vacuum domain walls are studied. We concentrate on thin walls between non-equal and non-positive cosmological constants on each side of the wall. These vacuum domain walls fall in three classes depending on the value of their energy density $σ$: (1)\ extreme walls with $σ= σ_{\text{ext}}$ are planar, static walls corresponding to supersymmetric configurations, (2)\ non-extreme walls with $σ= σ_{\text{non}} > σ_{\text{ext}}$ correspond to expanding bubbles with observers on either side of the wall being {\em inside\/} the bubble, and (3)\ ultra-extreme walls with $σ= σ_{\text{ultra}} < σ_{\text{ext}}$ represent the bubbles of false vacuum decay. On the sides with less negative cosmological constant, the extreme, non-extreme, and ultra-extreme walls exhibit no, repulsive, and attractive effective ``gravitational forces,'' respectively. These ``gravitational forces'' are global effects not caused by local curvature. Since the non-extreme wall encloses observers on both sides, the supersymmetric system has the lowest gravitational mass accessable to outside observers. It is conjectured that similar positive mass protection occurs in all physical systems and that no finite negative mass object can exist inside the universe. We also discuss the global space-time structure of these singularity free space-times and point out intriguing analogies with the causal structure of black holes.

Figures (10)

• Figure 1: Conformal diagram of the extreme Type I domain wall, which separates AdS$_{4}$ from M$_{4}$. The $x$- and $y$-directions are suppressed; therefore, each point represents an infinite plane with distances in the plane conformally compressed by $A(z)$. The compact null coordinates are $u',v' = 2\tan^{-1}[\alpha(t \mp z)]$. The domain wall is the double time-like arc splitting the diamonds. The complete extension consists of an infinite lattice of diamonds. The vertices are infinitely conformally compressed points; i.e., they are an infinite affine distance away from points interior. Cauchy horizons for data placed on the constant time slices in one diamond are the dashed nulls separating the AdS$_{4}$ patches. The walls smooth out the singularities at the time-like boundaries of pure AdS$_{4}$ seen in Fig. 10. The removal of the time-like boundary allows for a formulation of the Cauchy problem on the covering space-time which prescribes initial data on one slice across an AdS$_{4}$ region and freely chooses boundary data on the past null infinities of the countably infinite number of M$_{4}$ regions. Note the similarity of the extension taken here to that of the extreme Kerr black hole along its symmetry axis [30,32] (Diagram taken after Ref. [18]).
• Figure 2: Conformal diagram of the extreme Type II domain wall. Conventions follow Fig. 1. AdS$_{4}$ regions are on both sides of the wall. As there are Cauchy horizons on both sides of the wall, the geodesically complete extension covers the whole plane with an infinite lattice of domain wall diamonds.
• Figure 3: Conformal diagram of the extreme Type III domain wall. Conventions follow Fig. 1. AdS$_{4}$ regions are on both sides of the domain wall. The irremovable singularity at $z=z^{*}=z'_{-}$ is represented by the time-like affine boundaries. This diagram has the same causal structure as pure AdS$_{4}$ seen in Fig. 10 (see also Ref. [20]). In this way, this system can be thought of as a generalized AdS$_{4}$.
• Figure 4: Conformal diagram for the M$_{4}$ side of the non-extreme bubble or the M$_{4}$ side inside an ultra-extreme bubble. This diagram is part of (3+1)-dimensional Minkowski space as seen in the compactified $(\underline{t},\underline{r})$ plane. Angular coordinates $(\theta,\phi)$ are suppressed. The axis of symmetry represents the world line of the center of the bubble at $\underline{r} = 0$. Opposite points on the right and left sides of $\underline{r} = 0$ represent antipodal points $\theta \rightarrow \pi - \theta$ and $\phi \rightarrow \phi + \pi$. The time direction increases upward. The solid curved lines asymptoting to the dotted nulls are the world-lines of anti-podal points of the non-extreme bubble wall at $z=0$, or equivalently $\underline{r}^{2} - \underline{t}^{2} = -\tan(u'/2) \tan(v'/2) = \beta^{-2}$. This diagram is a cross-section of the hyperboloid of dS$_{3}$ as embedded in M$_{4}$ (see Ref. [30] for the analogous case of dS$_{4}$). The rest frame of the wall is a Rindler frame whose acceleration has magnitude $\beta$. The dotted nulls are the Rindler horizons on which the comoving coordinates $(t,z)$ degenerate ($z = \infty$). In order to remain with the wall, observers must accelerate towards it. In this sense, the non-extreme bubble exhibits "repulsinve gravity." The unique extension of the comoving coordinates across the Rindler horizons is onto pure Minkowski space-time.
• Figure 5: Conformal diagram for the M$_{4}$ region outside of the AdS$_{4}$--M$_{4}$ ultra-extreme bubble. This diagram is the complement of the non-extreme bubble of Fig. 4. The two sides represent spatially anti-podal pieces of the spherically symmetric space-time. In this case, the M$_{4}$ side corresponds to the outside of the de Sitter hyperboloid of Fig. 4. These wedges are covered by the comoving coordinates $(t,z)$. The solid curved line is the hyperbolic trajectory of the wall at $z=0$. The solid nulls are the affine boundaries. Time-like observers with insufficient acceleration eventually encounter the wall. In this sense, the ultra-extreme bubble exhibits "attractive gravity."