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3 results

  • 9 - Gravitational collapse and black holes

  • Stephen W. Hawking, University of Cambridge, George F. R. Ellis, University of Cape Town
  • Book:
    The Large Scale Structure of Space-Time
    Published online:
    17 February 2023
    Print publication:
    16 February 2023, pp 299-347

  • Summary

    In this chapter, we show that stars of more than about 1½ times the solar mass should collapse when they have exhausted their nuclear fuel. If the initial conditions are not too asymmetric, the conditions of theorem 2 should be satisfied and so there should be a singularity. This singularity is however probably hidden from the view of an external observer who sees only a ‘black hole’ where the star once was. We derive a number of properties of such black holes, and show that they probably settle down finally to a Kerr solution.

    In §9.1 we discuss stellar collapse, showing how one would expect a closed trapped surface to form around any sufficiently large spherical star at a late stage in its evolution. In §9.2 we discuss the event horizon which seems likely to form around such a collapsing body. In §9.3 we consider the final stationary state to which the solution outside the horizon settles down. This seems to be likely to be one of the Kerr family of solutions. Assuming that this is the case, one can place certain limits on the amount of energy which can be extracted from such solutions.

  • DEFORMATIONS OF THE VERONESE EMBEDDING AND FINSLER $2$-SPHERES OF CONSTANT CURVATURE

  • Part of
  • Christian Lange, Thomas Mettler
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 6 / November 2022
    Published online by Cambridge University Press:
    20 April 2021, pp. 2103-2134
    Print publication:
    November 2022
    • Article
      • You have access
      • Open access
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  • We establish a one-to-one correspondence between, on the one hand, Finsler structures on the $2$-sphere with constant curvature $1$ and all geodesics closed, and on the other hand, Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and whose geodesics are all closed. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb {CP}(a_1,a_2)\rightarrow \mathbb {CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide examples of Finsler $2$-spheres of constant curvature whose geodesics are all closed.

  • Constant curved minimal CR 3-spheres in CPn

  • Part of
  • Zhen-Qi Li, An-Min Huang
  • Journal:
    Journal of the Australian Mathematical Society / Volume 79 / Issue 1 / August 2005
    Published online by Cambridge University Press:
    09 April 2009, pp. 1-10
    Print publication:
    August 2005
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  • In this paper we prove that minimal 3-spheres of CR type with constant sectional curvature c in the complex projective space CPn are all equivariant and therefore the immersion is rigid. The curvature c of the sphere should be c = 1/(m2-1) for some integer m≥ 2, and the full dimension is n = 2m2-3. An explicit analytic expression for such an immersion is given.