Philip D. Mannheim has developed a theory of

that is based on the conformal Weyl tensor, rather than ( as in I. E. Segal's theory ) based on the 15-dimensional conformal group Spin(2,4) = SU(2,2).

Penrose and Rindler, in volume 1 of their books Spinors and Space-Time (Cambridge 1986) say, at page 355:

"... A flat-space theory which is Poincare invariant and also conformally invariant in this [ Weyl-type ] sense, will be invariant under the 15-parameter conformal group [ SU(2,2) ]. ... But the [ Weyl ] type of conformal invariance ... is really more general ... since it applies to curved space-time also. ...".

In astro-ph/0104022, Mannheim says:

"... conformal gravity (viz. gravity based on the fully covariant, locally conformal invariant Weyl action ... ) ... possesses an explicit symmetry (conformal invariance) which when unbroken would require the cosmological constant to vanish.The cosmology associated with the conformal gravity theory was first presented in [ P. D. Mannheim, Astrophys. J. 391, 429 (1992).] where it was shown to possess no flatness problem, to thus release conformal cosmology from the need for the copious amounts of cosmological dark matter required of the standard theory. Subsequently [P. D. Mannheim, Conformal cosmology and the age of the universe, astro-ph/9601071. ... ], the cosmology was shown to also possess no horizon problem, no universe age problem, and, through negative spatial curvature, to naturally lead to cosmic repulsion. ... Finally, it was shown ... that even after the conformal symmetry is spontaneously broken by a /\ inducing cosmological phase transition, the cosmology is still able to control the contribution of the induced cosmological constant to cosmic evolution even in the event that /\ is in fact as big as particle physics suggests, to thereby provide a completely natural solution to the cosmological constant problem. In the present paper we show that this control actually enables us to provide for a complete and explicit accounting of the recent high z supernovae Hubble plot data without the need for any fine tuning at all. ...

... The non-relativistic terrestrial and solar system conformal gravity expectations are completely standard ... being controlled by a local G whose dynamical generation is totally decoupled ... from that of the cosmological Geff , with it being only in the continuation beyond the solar system that the standard and conformal gravity theories actually depart from each other. ...".

Some questions that occur to me are:

- Do the "local G" and "cosmological Geff" correspond to Segal's use of two energy operators, one a Minkowksi one used locally and the other a Conformal one used on larger astrophysical scales?
- Could the "decoupl[ing] ... in the continuation beyond
the solar system" be related to the anomalous
acceleration of the Pioneer spacecraft data from beyond our solar
system?

In astro-ph/9910093, Mannheim says:

"... within ... higher order derivative gravitational theories, one of them is immediately singled out, namely conformal gravity, a fully coordinate invariant gravitational theory which possesses an additional symmetry not enjoyed by the standard theory (viz. invariance under any and all local conformal stretchings ... of the geometry), a theory which consequently has as its uniquely allowed gravitational action the Weyl action ...... Conformal gravity is thus a gravitational theory which possess[es] no fundamental scale ( and thus no intrinsic G or fundamental /\ ) at all, and is thus a theory which can immediately lead to a cosmology which is free of intrinsic scales at sufficiently high enough temperatures.

As such, conformal gravity emerges as a potential gravitational analog of the Weinberg-Salam-Glashow electroweak theory, with it immediately being suggested ... that in it Newton's constant G might be generated as a low energy effective parameter in much the same manner as Fermi's constant GF is generated in the electroweak theory, with G as measured in a low energy Cavendish experiment then indeed nicely being decoupled from the hot early universe. However, it turns out that the low energy limit of conformal gravity need not emerge in precisely this fashion since ... it is not in fact necessary to spontaneously break the conformal gravity action down to the Einstein-Hilbert action ( i.e. down to the standard theory equations of motion ). Rather ... , it is only necessary to obtain the solutions to those equations in the kinematic region ( viz. solar system distance scales ) where those standard solutions have been tested. Thus, as had been noted by Eddington ... already in the very early days of relativity, the standard gravity vacuum Schwarzschild solution is just as equally a vacuum solution to higher derivative gravity theories also, since the vanishing of the Ricci tensor entails the vanishing of its derivatives as well. ...

... As a fundamental theory, conformal gravity has as a motivation the desire to give gravity a local invariance structure and a dimensionless coupling constant (and thus power counting renormalizability), to thereby make it analogous to the three other fundamental interactions. And indeed, ... the local conformal symmetry invoked to do this then not only excludes the existence of any fundamental mass scales such as a fundamental cosmological constant, even after mass scales are induced by spontaneous breakdown of the conformal symmetry, the (still) traceless energy-momentum tensor then constrains any induced (and necessarily negative) cosmological constant term to be of the same order of magnitude as all the other terms in T^(mu nu) , neither smaller nor larger. Thus, unlike standard gravity, precisely because of its additional symmetry, conformal gravity has a great deal of control over the cosmological constant (essentially, with all mass scales - of gravity and particle physics both - being jointly generated by spontaneous breakdown of the scale symmetry, conformal gravity knows exactly where the zero of energy is), and it is our purpose now to show that it is this very control which then provides for both a natural solution to the cosmological constant problem and for a complete accounting of the new high z data. ...

... Even though the cosmology expands far more slowly than the standard one, nonetheless, this gets compensated for in the conformal case because weak interactions are then found ... to remain in thermal equilibrium down to lower temperatures than in the standard case, with the conformal metallicity predictions then being found ... to actually outperform those of the standard model. ...

... the distinction between homogeneity and inhomogeneity which provides the demarcation between local and global gravity, to thus now enable us to consider repulsive cosmologies which are not incompatible with the attractive gravity observed on solar system distance scales. ...

... the conformal gravity extrapolation ... has been found to provide for a satisfactory explanation of galactic rotation curve systematics without the need to introduce any galactic dark matter at all. ...

... Despite the fact that conformal gravity has ... been found to be globally repulsive, nonetheless, it is important to note that in the conformal theory local solar system gravity can still be attractive ...".

Some further questions that occur to me are:

- Could the Segal Spin(2,4)=SU(2,2)
Conformal theory be related to the Mannheim Weyl Conformal
Gravity
in the same way that

- What is the relationship between Segal's
quadratic Hubble red-shift curve described,for example, in his
book Mathematical Cosmology and Extragalactic Astronomy (Academic
Press 1976)?
and

the red-shift curves described by Mannheim in astro-ph/0104022 and related papers?

Mannheim, in astro-ph/0302362, says:

"... We show that detailed exploration of the 1 < z < 2 redshift region can provide for definitive testing not only of the standard inflationary cosmological paradigm with its fine-tuned cosmological constant and its mysteriously late (z < 1) onset of cosmic acceleration, but also for the non finetuned, alternate conformal cosmological model, a cosmology which accelerates both above and below z = 1. In particular we confront both of these models with the currently available type Ia supernovae standard candle and extended FRII radio source standard yardstick data, with these latter data being particularly pertinent as they already include a sizeable number of points in the 1 < z < 2 region. We find that both models are able to account for all available 0 < z < 2 data equally well; and with the conformal model explicitly being able to fit the data while being an accelerating one in the z > 1 region, one is thus currently unable to ascertain whether the universe is accelerating or decelerating between z = 1 and z = 2. To be able to visualize the supernovae and radio galaxy data simultaneously, we present a representation of the radio galaxy data in terms of an equivalent apparent magnitude Hubble diagram.We discuss briefly some implications of the anisotropies in the cosmic microwave background for the conformal theory, and show that in that theory fluctuations which set in at around nucleosynthesis can readily generate the first peak in the anisotropy data. ...".

......