Conformal gravity is a generic name for gravity theories which are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations Failed to parse (Missing texvc executable; please see math/README to configure.): g_{ab}\rightarrow\Omega^2(x)g_{ab}

The simplest theory in this category has the square of the Weyl tensor as the Lagrangian

where Failed to parse (Missing texvc executable; please see math/README to configure.): C_{abcd}

where Failed to parse (Missing texvc executable; please see math/README to configure.): R_{ab}

Since these theories lead to fourth order equations for the fluctuations around a fixed background, they are not manifestly unitary. It has therefore been generally believed that they could not be consistently quantized. This is now disputed .

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:

Sir Roger Penrose OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. He has received a number of prizes and awards, including the 1988 Wolf Prize for physics which he shared with Stephen Hawking for their contribution to our understanding of the universe. He is renowned for his work in mathematical physics, in particular his contributions to general relativity and cosmology. He is also a recreational mathematician and philosopher.

He was born in Colchester, Essex, England, Roger Penrose is a son of Lionel S. Penrose and Margaret Leathes. Penrose is the brother of mathematician Oliver Penrose and of chess Grandmaster Jonathan Penrose. Penrose attended University College School and University College, London, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalized matrix inverse, also known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar (1951). Penrose earned his Ph.D. at Cambridge (St John's College) in 1958, writing a thesis on "tensor methods in algebraic geometry" under algebraist and geometer John A. Todd. He devised and popularised the Penrose triangle in the 1950s, describing it as "impossibility in its purest form" and exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it. Escher's Waterfall, and Ascending and Descending were in turn inspired by Penrose. As reviewer Manjit Kumar puts it:

Lee Smolin (born 1955) is an American theoretical physicist, a researcher at the Perimeter Institute for Theoretical Physics, and an adjunct professor of physics at the University of Waterloo. He is married to Dina Graser, a communications lawyer in Toronto. His brother is David M. Smolin.

Smolin is best known for devising several different approaches to quantum gravity, in particular loop quantum gravity. He advocates that the two primary approaches to quantum gravity, loop quantum gravity and string theory, can be reconciled as different aspects of the same underlying theory. His research interests also include cosmology, elementary particle theory, the foundations of quantum mechanics, and theoretical biology.

Smolin was born in New York City to Michael Smolin, an engineer, and Pauline Smolin, a playwright. His brother, David M. Smolin, became a professor in the Cumberland School of Law in Birmingham, Alabama. He was educated at Hampshire College. He received his Ph.D in theoretical physics from Harvard University in 1979.

Erik Peter Verlinde (born 21 January 1962, Woudenberg) is a Dutch theoretical physicist and string theorist. He is the identical twin brother of physicist Herman Verlinde. The Verlinde formula, which is important in conformal field theory and topological field theory, is named after him. His research deals with string theory, gravity, black holes and cosmology. Currently he works at the Institute for Theoretical Physics at the University of Amsterdam.

At a symposium at the Dutch Spinoza-institute on 8 December 2009 he introduced a theory of Entropic gravity. In this theory gravity exists because of a difference in concentration of information in the empty space between two masses and its surroundings; he also extrapolates this to General Relativity and quantum mechanics. He said in an interview with the newspaper de Volkskrant, "On the smallest level Newton's laws don't apply, but they do for apples and planets. You can compare this to pressure of gas. Molecules themselves don't have any pressure, but a barrel of gas has." It appears that Verlinde's approach to explaining gravity leads naturally to the correct observed strength of dark energy. In June 2011, he was awarded the prestigious Spinoza Prize with a 2.5 million euro grant for this work, including his paper.