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n-field approach to Black Holes Production in deSitter Spacetime

n-field approach to Black Holes Production in deSitter Spacetime. A.A. Bagrov Steklov Mathematical Institute (based on joint work with I.Ya. Aref’eva and E.A. Guseva). Problem definition. We consider non-expanding shock-waves in the 4-dimensional dS spacetime

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n-field approach to Black Holes Production in deSitter Spacetime

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  1. n-field approach toBlack Holes Productionin deSitter Spacetime A.A. Bagrov Steklov Mathematical Institute (based on joint work with I.Ya. Aref’eva and E.A. Guseva)

  2. Problem definition • We consider non-expanding shock-waves in the 4-dimensional dS spacetime • Structure of null-geodesics beams in such perturbed spacetime is being analyzed. • In the case of two colliding shock-waves we derive and study the equation for trapped surface.

  3. Motivation • Shock-waves seem like plausible approximation for gravitational fields of ultrarelativistic particles. • It has been shown by several authors (Banks, Fishler, Aref’eva) that in collisions of particles at TeV energies black holes can be formed. • dS background in our considerations is a model for dark energy. • In this context we analyze possible influence of dark energy on the process of black holes production in collisions.

  4. Metric of the spacetime • It is convinient to immerse 4-dimensional dS spacetime with shock wave as submanifold to 5-dimensional Minkowski space. So: • The potential of shock-wave could be obtained by boosting of the Schwarzschild solution in dS: M. Hotta, M. Tanaka – 1992

  5. n-field approach

  6. n-field approach • The null-geodesics could be derived from n-field-like Lagrangian: • Where:

  7. Null-geodesics • Based on this Lagrangian we can derive the equations of geodesics in following simple form:

  8. Null-geodesics • Without significant mathematical difficulties we can find non-singular solution of these equations:

  9. Behavior of geodesics X2 U X3 For each value of initial parameter X40 we have different focal length

  10. Independent coordinates • Let us transfrom geodesics and metric tensor to independent smooth coordinates. We can do it in two steps. • Projection: • Regularization:

  11. Independent coordinates • Metric in these coordinates become regular: • In these terms geodesics depends on affine parameter nonlinearly. But the general picture remains similar.

  12. Trapped surface • A trapped surface is a two dimensional spacelike surface whose two null normals have zero convergence (Neighbouring light rays, normal to the surface, must move towards one another) • Th. (Hawking-Penrose) A spacetime (M; g) with a complete future null infinity which contains a closed trapped surface must contain a future event horizon, the interior of which contains the trapped surface

  13. Trapped surface W Σ X

  14. Trapped surface (basic objects)‏

  15. The equation for trapped surface • On the plane of first shock-wave (W=0) after all calculation we obtain: • On the second wave (Σ=0) the equation is the same due to notation symmetry (W – Σ) • For the AdS case analogous equation and its’ solution have been obtained by S. Gubser, S. Pufu, A. Yarom (hep-th: 0902.4062)

  16. Conclusions • Geometrical structure of deSitter spacetime with non-expanding shock-waves has been studied. • Focusing of null-geodesics on shock-waves has been shown. • The equation for trapped surface in case of two head-on colliding shock waves has been derived. • In future we are going to analyze can the dark enegy serves as catalyst for balck holes production.

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