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Description
Isometric embedding of a pseudo-Riemannian spacetime is a description of this spacetime as a surface in an ambient spacetime of higher dimension. This procedure has been used for more than a century in the examination of solutions of Einstein equations, since the embedding class (i.e. the minimal codimension of a surface in a flat ambient spacetime) is an invariant characteristic of a spacetime.
Isometric embeddings are also used in modifying gravity within the so-called Regge-Teitelboim approach [1].
Embedding procedure for a spacetime with a given metric reduces to satisfying a condition of the metric inducibility:
\begin{equation}
\partial_\mu Y^a \partial_\nu Y^b \eta_{ab} = g_{\mu\nu}(x)
\end{equation}
where $Y$ is the embedding function, and $\eta$ and $g$ are the metrics of the ambient and embedded spaces, respectively.
In 2012, S. A. Paston proposed [2] a group-theoretic method for the separation of variables in this system. Using this method, all possible global minimal symmetric embeddings were found for the Friedmann model, non-rotating black holes (Schwarzschild-(anti) de Sitter and Reissner-Nordström-(anti) de Sitter), the BTZ black hole, and others.
Embeddings of gravitational wave metrics have been studied very poorly. Apart from mentions of a few specific cases of embeddings in the literature, a systematic investigation of this embedding, to our knowledge, has only been attempted once in [3]. However, the author of that work, employing the method of solving the Gauss-Codazzi equations, apparently did not aim to find the explicit form of the embeddings, and therefore only a few particular examples of the obtained surfaces are provided. Furthermore, some possible types of surfaces remained unstudied.
This work is devoted to a systematic study of embedded surfaces with the metric of gravitational pp-wave:
\begin{equation}
ds^2 = 2H(u,y,z)du^2-2dudv-dy^2-dz^2,
\end{equation}
where H is an arbitrary function whose transverse Laplacian is zero, and u and v are light-cone coordinates. The ambient space is chosen to be 2+4-dimensional because, according to [4] and [5], such an embedding can not exist in a 5-dimensional or globally hyperbolic space. We construct the surfaces using two ansatzes for the embedding function: the one used in [3] and the one given by method described in [2].
The obtained embeddings can be used for the analysis of gravitational wave solutions in the Regge-Teitelboim equations.
References
- Sheykin A. A., Paston S. A. The approach to gravity as a theory of embedded surface // AIP Conference Proceedings. 2014. V. 1606. P. 400.
- Paston S. A., Sheykin A. A. Embeddings for Schwarzschild metric: classification and new results // Class. Quant. Grav. 2012. V. 29. P.095022.
- Collinson, C. D. Embeddings of the Plane-Fronted Waves and Other Space-Times // J. Math. Phys. 1968. V. 9. P. 403.
- Kasner, E. Geometrical Theorems on Einstein's Cosmological Equations // Am.J.Math. 1921. V.43. N.4. P.217.
- Penrose, R. A Remarkable Property of Plane Waves in General Relativity // Rev. Mod. Phys. 1965. V. 37. P. 215.
