sourena
Jun13-11, 03:09 AM
As we know GR cannot explain the flat rotation curves fo galaxies without introducing unseen dark matter and, of course, the fact that it cannot explain the current accelerated expansion of the universe without the introduction of both dark matter and dark energy. This does not sound appealing at all and has lead to several attempts to modify or extend GR in different ways, addressing the dark matter or dark energy problem.
An alternative route is to modify GR by abandoning the simplicity assumption that the action should be linear in the scalar curvature R. So, instead of using the Einstein-Hilbert Lagrangian one can use a more general one, which depends on a generic function f(R). Hence, such theories are called f(R) theories of gravity. Such theories are unfortunately not free of problem!
There is a further modification of gravity that one can consider which does not necessarily involve any modification of the action, but rather the use of a different variational principle, known as the Palatini formalism. It treats the metric and the affine connection as independent geometrical quantities. One has to vary the Lagrangian with respect to both of them fo derive the field equations.
When we vary the Lagrangian with respect to affine connection, it means it is a dynamic variable. What is the physical interpretation of this sentence? What are pros and cons of this method? Can we extend the Palatini formalism to other Lagrangian not just containing Ricci scalar R, but Reimann tensor or Weyl tensor and if so, how can we vary a Lagrangian containing these tensors with respect to connection?
I would really appreciate it if anybody can help me.
An alternative route is to modify GR by abandoning the simplicity assumption that the action should be linear in the scalar curvature R. So, instead of using the Einstein-Hilbert Lagrangian one can use a more general one, which depends on a generic function f(R). Hence, such theories are called f(R) theories of gravity. Such theories are unfortunately not free of problem!
There is a further modification of gravity that one can consider which does not necessarily involve any modification of the action, but rather the use of a different variational principle, known as the Palatini formalism. It treats the metric and the affine connection as independent geometrical quantities. One has to vary the Lagrangian with respect to both of them fo derive the field equations.
When we vary the Lagrangian with respect to affine connection, it means it is a dynamic variable. What is the physical interpretation of this sentence? What are pros and cons of this method? Can we extend the Palatini formalism to other Lagrangian not just containing Ricci scalar R, but Reimann tensor or Weyl tensor and if so, how can we vary a Lagrangian containing these tensors with respect to connection?
I would really appreciate it if anybody can help me.