5.5 The characteristic initial value 5 Further results5.3 The Newtonian limit

5.4 Newtonian cosmology 

Apart from the interest of the Newtonian limit, Newtonian gravitational theory itself may provide interesting lessons for general relativity. This is no less true for existence theorems than for other issues. In this context it is also interesting to consider a slight generalization of Newtonian theory, the Newton-Cartan theory. This allows a nice treatment of cosmological models, which are in conflict with the (sometimes implicit) assumption in Newtonian gravitational theory that only isolated systems are considered. It is also unproblematic to introduce a cosmological constant into the Newton-Cartan theory.

Three global existence theorems have been proved in Newtonian cosmology. The first[9] is an analogue of the cosmic no hair theorem (cf. Section 4.1) and concerns models with a positive cosmological constant. It asserts that homogeneous and isotropic models are nonlinearly stable if the matter is described by dust or a polytropic fluid with pressure. Thus it gives information about global existence and asymptotic behaviour for models arising from small (but finite) perturbations of homogeneous and isotropic data. The second and third results concern collisionless matter and the case of vanishing cosmological constant. The second[65] says that data which constitute a periodic (but not necessarily small) perturbation of a homogeneous and isotropic model which expands indefinitely give rise to solutions which exist globally in the future. The third[63] says that the homogeneous and isotropic models in Newtonian cosmology which correspond to a k =-1 Friedmann-Robertson-Walker model in general relativity are non-linearly stable.



5.5 The characteristic initial value 5 Further results5.3 The Newtonian limit

image Local and global existence theorems for the Einstein equations
Alan D. Rendall
http://www.livingreviews.org/lrr-1998-4
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de