Einstein Field Equations in Polynomial Form

Einstein Field Equations in Polynomial Form


Einstein's Field equations (and the action from which it is derived) are sometimes, I would say mistakenly, said to be `highly non-polynomial'. This little article explains one method of writing the equations so they are polynomial in a redefinition of the metric tensor to a suitable metric tensor-density. 


Another way to put the equations in polynomial form when there is a diliton field such as in Brans-Dicke gravity is to absorb the determinants of the metric into the definition of the diliton field. 

Comments

Post a Comment

Popular posts from this blog

Vortex Solution to Navier Stokes Equations

Image
This is a solution I found to the Navier Stokes equations . It uses the flow lines of the Hopf Fibration (which is basically just a way of covering the entire 3D space in circles). The flow lines can be seen in this image below: You can see my paper here. Solution to Navier-Stokes equation . Unfortunately I won't get a million pounds for winning the Clay Millennium prize because this is a compressible not incompressible solution. :( Oh, well. Nice try! I think it's pretty nice anyway. :) Also it uses the nice identity: which paramaterizes the Pythagorean quadruples!
Read more

Could the Universe be a Giant Knot?

Idea is that the state of the Universe is represented by a knot. The complex amplitude is given by a knot invariant. This is the sum of histories. A history is a series of slicings which return the knot to the unknot. Knot states related by Reidmeister moves are equivalent. The cosmological time of a knot is the 'unknotting' number. One possible flaw is that there are an infinite number of families of knots with 'unknotting number' 1. However, many of these may have very low amplitudes. The amplitudes might be given by the knot polynomials , e.g,for some constant,k,q: \[ q\psi(K_0) = e^{ik}\psi(K_+) - e^{-ik}\psi(K_-) \] and \( \psi( unknot ) = 1 \) . The question is, how would one relate a knot to a curved 3 dimensional manifold? What would the curvature be? Where would the points of space-time be? Does it satisfy: \[ \sum\limits_{K_2}\Delta(K_1,K_2)\Delta(K_2,K_3) = \Delta(K_1,K_3) \]
Read more

Tarski-Seidenberg theorem

Tarski-Seidenberg theorem States that a set of equalities and inequalities f(x1,x2,x3,....xn)>=0 can be reduced down to a set of equalities and inequalities f(x1,x2,x3,....xn-1)>=0 with one less variable. Thus all equations can be reduced to a set of equalities and inequalities on their coefficients. See also: cylindrical algebraic decomposition See:  Elimination These algorithms aren't very simple.
Read more