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Balance celestial and collapse of high dimensional Lovelock gravity

Author: ZhouKang
Tutor: YangZhanYing
School: Northwestern University
Course: Theoretical Physics
Keywords: Lovelock gravity TOV equation red shift naked singularity horizon photonsphere
CLC: O412.1
Type: Master's thesis
Year: 2012
Downloads: 23
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Abstract

In this paper, we have explored static stars in Lovelock gravity in high dimensional spacetime, and investigated the final fate of stars in gravitational collapse. With regard to stars under hydrostatic equilibrium, we have discussed the spherically symmetric perfect fluid whose internal pressure keeps balance with gravity in second order Lovelock theory with dimensions above5. We present an analytical solution of the metric, point out such solution will reproduce the vacuum solution outside the star, and deduce the extended TOV equation which can impose restriction on the distribution of pressure and density in stars. It turns out that the presence of the second order Lovelock term changes thermal and gravitational energies, and the upper bound of the red shift of spectral lines from the surface of stars. Unlike in general relativity, the upper bound of the red shift depends on the density of stars in our case. Moreover, we have proven that two theorems for judging the stability of equilibrium of stars in4-dimensional general relativity still hold in second order Lovelock gravity. A star will collapse while its gravity is so powerful that could not reach the hydrostatic equilibrium. We have simplified the collapsing star to a spherically symmetry and incoherent perfect fluid which is consisted with freely falling dust, and chosen the commoving coordinate to express the metric. Cases in second and third order Lovelock gravity have been discussed respectively. We have found the rigid solutions, and proven these solutions attach to the external vacuum solutions analytically by a coordinate transformation. Investigation on such solutions indicates that the contribution of high order Lovelock corrections have obvious influences on the nature of singularities, and alter the global structure of spacetime. For instance, the positive coupling constant of second order Lovelock term delays the formation of singularities; in5-dimensional second order and7-dimensional third order Lovelock gravity, the outgoing property of singularities depends on mass of stars, and the ingoing property in maximally symmetric spacetime is null rather than spacelike. Interestingly, a massive, timelike, forever observable naked singularity can be formed in gravitational collapse in5-dimensional second order and7-dimensional third order Lovelock gravity, which is forbidden in general relativity. Hence, a counterexample to Penrose’s cosmic censorship hypothesis has occurred, which does not exist in general relativity. We have explained these properties one by one, and portrayed the Penrose diagram to reflect the global structure of spacetime. Recently, Virbhadra and his cooperators proposed a new cosmic censorship hypothesis based on their investigation on gravitational lensing effects. We have proven the new cosmic censorship hypothesis also has been violated in5-dimensional second order and7-dimensional third order Lovelock gravity. However, our analyses about the strength of singularities manifest that naked singularities formed in5-dimensional second order and7-dimensional third order Lovelock gravity may be gravitational weak therefore may not be a serious threat to cosmic censorship hypothesis despite old one or new one. Moreover, based on conclusions in second and third order Lovelock gravity, we present two conjectures for gravitational collapse in general Lovelock gravity with arbitrarily order.

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