Abstract
This paper deals with the generalization of usual round spheres in the flat Minkowski spacetime to the case of a generic four-dimensional spacetime manifold M. We consider geometric properties of sphere-like submanifolds in M and impose conditions on its extrinsic geometry, which lead to a definition of a rigid sphere. The main result is a local existence theorem concerning such spheres. For this purpose, we apply the surjective implicit function theorem. The proof is based on a detailed analysis of the linearized problem and leads to an eight-parameter family of solutions in case when the metric tensor g of M is from a certain neighborhood of the flat Minkowski metric. This contribution continues the study of rigid spheres in (Class. Quantum Grav. 30: 175010, 2013).
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Notes
Einstein summation convention over repeated lower and upper indices is always assumed.
Throughout the paper, capital letters A, B,…∈{2,3}, whereas small letters a, b,…∈{0,1}.
By 〈f〉 we denote the mean value of the function f over \(\mathcal {S}\).
Hence, Definition 3 coincides with [2, Definition 4].
Observe that [2, Theorem 2] on the existence of a unique equilibrated spherical system on a topological sphere S 2 is also valid within Sobolev framework (substitute C k, α(S 2) with H k+2, k≥1).
References
Chruściel, T.P., Jezierski, J., Kijowski, J.: Hamiltonian Field Theory in the Radiating Regime. Lecture Notes in Physics. Monographs, vol. 70. Springer, Berlin (2001)
Gittel, H.-P., Jezierski, J., Kijowski, J., Łęski, S.: Rigid spheres in Riemannian spaces. Class. Quantum Grav. 30, 175010 (2013). doi:10.1088/0264-9381/30/17/175010
Erdèlyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1953)
Jezierski, J.: Energy and angular momentum of the weak gravitational waves on the schwarzschild background—quasilocal Gauge-invariant formulation. Gen. Rel. Grav. 31, 1855–1890 (1999)
Jezierski, J.: “Peeling Property” for linearized gravity in null coordinates. Class. Quantum Grav. 19, 2463–2490 (2002)
Kijowski, J.: A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity. Gen. Rel. Grav. 29, 307–343 (1997)
Szabados, L.: Quasi-local energy-momentum and angular momentum in GR: a review article. Living Rev. Relativ. 7, 4 (2004). (available on the Web: http://www.livingreviews.org/lrr-2004-4)
Taylor, M.E.: Partial Differential Equations, vol. 1–3. Springer, New York (1996)
Triebel, H.: Höhere Analysis. Deutscher Verlag der Wissenschaften, Berlin (1972)
Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. 1–4. Springer, New York (1988)
Acknowledgments
We are deeply indebted to Eberhard Zeidler, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, for fruitful discussions and valuable help. Further we thank the referee for hints of improvements in the text.
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Dedicated to Professor Dr. Eberhard Zeidler on the occasion of his 75th birthday.
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Gittel, HP., Jezierski, J. & Kijowski, J. On the Existence of Rigid Spheres in Four-Dimensional Spacetime Manifolds. Vietnam J. Math. 44, 231–249 (2016). https://doi.org/10.1007/s10013-016-0185-z
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DOI: https://doi.org/10.1007/s10013-016-0185-z
Keywords
- Minkowski metric
- Lorentzian spacetimes
- Extrinsic geometry
- Rigid spheres
- Local existence
- Implicit funtion Theorem