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Register a new userExact gravitational plane waves and two-dimensional gravity
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Authors
Jorge G. Russo
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We discuss dynamical aspects of gravitational plane waves in Einstein theory with massless scalar fields. The general analytic solution describes colliding gravitational waves with constant polarization, which interact with scalar waves and, for generic initial data, produce a spacetime singularity at the focusing hypersurface. There is, in addition, an infinite family of regular solutions and an intriguing static geometry supported by scalar fields. Upon dimensional reduction, the theory can be viewed as an exactly solvable two-dimensional gravity model. This provides a new viewpoint on the gravitational dynamics. Finally, we comment on a simple mechanism by which short-distance corrections in the two-dimensional model can remove the singularity.
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We give a higher even dimensional extension of vacuum colliding gravitational plane waves with the combinations of collinear and non-collinear polarized four-dimensional metric. The singularity structure of space-time depends on the parameters of the solution.
The behaviour of a test electromagnetic field in the background of an exact gravitational plane wave is investigated in the framework of Einsteins general relativity. We have expressed the general solution to the de Rham equations as a Fourier-like integral. In the general case we have reduced the problem to a set of ordinary differential equations and have explicitly written the solution in the case of linear polarization of the gravitational wave. We have expressed our results by means of Fermi Normal Coordinates (FNC), which define the proper reference frame of the laboratory. Moreover we have provided some gedanken experiments, showing that an external gravitational wave induces measurable effects of non tidal nature via electromagnetic interaction. Consequently it is not possible to eliminate gravitational effects on electromagnetic field, even in an arbitrarily small spatial region around an observer freely falling in the field of a gravitational wave. This is opposite to the case of mechanical interaction involving measurements of geodesic deviation effects. This behaviour is not in contrast with the principle of equivalence, which applies to arbitrarily small region of both space and time.
The geometry of twisted null geodesic congruences in gravitational plane wave spacetimes is explored, with special focus on homogeneous plane waves. The role of twist in the relation of the Rosen coordinates adapted to a null congruence with the fundamental Brinkmann coordinates is explained and a generalised form of the Rosen metric describing a gravitational plane wave is derived. The Killing vectors and isometry algebra of homogeneous plane waves (HPWs) are described in both Brinkmann and twisted Rosen form and used to demonstrate the coset space structure of HPWs. The van Vleck-Morette determinant for twisted congruences is evaluated in both Brinkmann and Rosen descriptions. The twisted null congruences of the Ozsvath-Schucking,`anti-Mach plane wave are investigated in detail. These developments provide the necessary geometric toolkit for future investigations of the role of twist in loop effects in quantum field theory in curved spacetime, where gravitational plane waves arise generically as Penrose limits; in string theory, where they are important as string backgrounds; and potentially in the detection of gravitational waves in astronomy.
Understanding the role of higher derivatives is probably one of the most relevant questions in quantum gravity theory. Already at the semiclassical level, when gravity is a classical background for quantum matter fields, the action of gravity should include fourth derivative terms to provide renormalizability in the vacuum sector. The same situation holds in the quantum theory of metric. At the same time, including the fourth derivative terms means the presence of massive ghosts, which are gauge-independent massive states with negative kinetic energy. At both classical and quantum level such ghosts violate stability and hence the theory becomes inconsistent. Several approaches to solve this contradiction were invented and we are proposing one more, which looks simpler than those what were considered before. We explore the dynamics of the gravitational waves on the background of classical solutions and give certain arguments that massive ghosts produce instability only when they are present as physical particles. At least on the cosmological background one can observe that if the initial frequency of the metric perturbations is much smaller than the mass of the ghost, no instabilities are present.
Junction conditions for vacuum solutions in five-dimensional Einstein-Gauss-Bonnet gravity are studied. We focus on those cases where two spherically symmetric regions of space-time are joined in such a way that the induced stress tensor on the junction surface vanishes. So a spherical vacuum shell, containing no matter, arises as a boundary between two regions of the space-time. A general analysis is given of solutions that can be constructed by this method of geometric surgery. Such solutions are a generalized kind of spherically symmetric empty space solutions, described by metric functions of the class $C^0$. New global structures arise with surprising features. In particular, we show that vacuum spherically symmetric wormholes do exist in this theory. These can be regarded as gravitational solitons, which connect two asymptotically (Anti) de-Sitter spaces with different masses and/or different effective cosmological constants. We prove the existence of both static and dynamical solutions and discuss their (in)stability under perturbations that preserve the symmetry. This leads us to discuss a new type of instability that arises in five-dimensional Lovelock theory of gravity for certain values of the coupling of the Gauss-Bonnet term. The issues of existence and uniqueness of solutions and determinism in the dynamical evolution are also discussed.
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