Conformal Gravity - Four Derivative Theories

Four Derivative Theories

Conformal gravity is an example of a 4-derivative theory. This means that each term in the wave equation can contain up to 4 derivatives. There are pros and cons of 4-derivative theories. The pros are that the quantized version of the theory is more convergent and renormalisable. The cons are that there may be issues with causality. A simpler example of a 4-derivative wave equation is the scalar 4-derivative wave equation:

$\Box^2 \Phi =0$

The solution for this is in a central field of force is:

$\Phi(r)= 1 -2m/r +ar +br^2$

The first two terms are the same as a normal wave equation. Since this equation is a simpler approximation to conformal gravity then m corresponds to mass of the central source. The last two terms are unique to 4-derivative wave equations. It has been suggested to assign small values to them to account for the galactic acceleration constant (also known as dark matter) and the dark energy constant. The solution equivalent to the Schwarzschield solution in General Relativity for a spherical source for conformal gravity has a metric with:

$\phi(r) = g^{00} = (1-6bc)^\frac{1}{2} - \frac{2b}{r} + c r + \frac{d}{3} r^2$

to show the difference between General Relativity. 6mc is very small so can be ignored. The problem is that now c is the total mass-energy of the source, b is the integral of density times distance to source squared. So this is a completely different potential to General Relativity and not just a small modification.

The main issue with conformal gravity theories, as well as any theory with higher derivatives, is the typical presence of ghosts, which point to instabilities of the quantum version of the theory, although there might be a solution to the ghost problem.

Read more about this topic: Conformal Gravity

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