Mannheim's Conformal Gravity
These days I am reading about alternative gravity theories and try to classify with Tegmark. I'll try to write brief summaries of some interesting ones in the coming days. I will start with Philip Mannheim's conformal gravity. I call it “Mannheim's” not because I am sure that he is the first to write this Lagrangian ('cos I am not) but he seems the only one writing about it for thirty years and review papers do not give it enough attention.
Idea is to write a fully conformal but generally covariant theory without introducing any new structure (like a gauge field as Weyl did or torsion). It is not possible at the first order in R. But in the second order there is a beautiful Lagrangian: C^2. It is the Weyl tensor properly contracted.
You get the field equation by varying with respect to the metric as usual. (Is it also meaningful to do variation of the conformal scaling field?) Of course the field equation is a mess with fourth derivative of the metric (gravitational potential in the Newtonian limit). But Mannheim managed to find a Schwarzschild like solution. As you can imagine it has three independent parameters instead of one (mass) in Schwarzschild. First thing you can do with this solution is to fit your galaxy curves. They seem to fit very good if you accept those two new parameters as new universal constants. I can't help remembering von Neumann's wise words: "With four parameters I can fit an elephant and with five I can make him wiggle his trunk".
Mannheim also proposes some accelerating cosmological solutions, though it doesn't surprise me in the vast probabilities of a fourth order equation. Physicists like second order equations :)
Like Weyl, he also needs to face with fact that particles are massive, so world is not conformal. He offers a kind of Higgs mechanism to generate mass. Another interesting property is G appears not in the fundamental theory but as an effective coupling constant.
If you are interested, start reading with gr-qc/9306025 for his criticism of GR. In conclusion, I don't see it as the strongest candidate before seeing the uniqueness (Birkhoff's) theorem of solutions and some testable predictions (which may already be in the literature that I did not notice). However it certainly deserves more attention in review papers.
Comments are welcome.
IMPORTANT UPDATE:
See my recent post for Mannheim's response.
Idea is to write a fully conformal but generally covariant theory without introducing any new structure (like a gauge field as Weyl did or torsion). It is not possible at the first order in R. But in the second order there is a beautiful Lagrangian: C^2. It is the Weyl tensor properly contracted.
You get the field equation by varying with respect to the metric as usual. (Is it also meaningful to do variation of the conformal scaling field?) Of course the field equation is a mess with fourth derivative of the metric (gravitational potential in the Newtonian limit). But Mannheim managed to find a Schwarzschild like solution. As you can imagine it has three independent parameters instead of one (mass) in Schwarzschild. First thing you can do with this solution is to fit your galaxy curves. They seem to fit very good if you accept those two new parameters as new universal constants. I can't help remembering von Neumann's wise words: "With four parameters I can fit an elephant and with five I can make him wiggle his trunk".
Mannheim also proposes some accelerating cosmological solutions, though it doesn't surprise me in the vast probabilities of a fourth order equation. Physicists like second order equations :)
Like Weyl, he also needs to face with fact that particles are massive, so world is not conformal. He offers a kind of Higgs mechanism to generate mass. Another interesting property is G appears not in the fundamental theory but as an effective coupling constant.
If you are interested, start reading with gr-qc/9306025 for his criticism of GR. In conclusion, I don't see it as the strongest candidate before seeing the uniqueness (Birkhoff's) theorem of solutions and some testable predictions (which may already be in the literature that I did not notice). However it certainly deserves more attention in review papers.
Comments are welcome.
IMPORTANT UPDATE:
See my recent post for Mannheim's response.
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