Only those theories which come very close to predicting the same results as General Relativity have survived the test of experiment, although it is still possible the the experimental results could, if refined, falisfy the predictions of both General Relativity and many of its proposed alternatives.
Final results are not expected until 2010 and are contingent upon additional funding (the project is about $4 million short and has spent about $800 million to date). Current preliminary results has quite a high margin of error due to various experimental issues.
Garth's post from April 20, 2007 stated (with links to papers regarding all but the first two theories):
the first results have verified the GR geodetic prediction to 1% but there is no handle on the frame-dragging prediction, basically because unexpected signals so far swamp it, except for 'glimpses'.
By the end of the year the correct removal of these effects will give a robust reading to both precessions.
The running now stands:
1. Einstein's General Relativity(GR)
2. Brans-Dicke theory (BD)
3. Barber's Self Creation Cosmology (SCC),
4. Moffat's Nonsymmetric Gravitational Theory (NGT),
5. Hai-Long Zhao's Mass Variance SR Theory (MVSR),
6. Stanley Robertson's Newtonian Gravity Theory (NG),
7. Junhao & Xiang's Flat Space-Time Theory (FST).
8. R. L. Collin's Mass-Metric Relativity (MMR) and
9. F. Henry-Couannier's Dark Gravity Theory (DG).
10. Alexander and Yunes' prediction for the Chern-Simons gravity theory (CS).
11. Kris Krogh's Wave Gravity Theory (WG)
12. Hongya Liu & J. M. Overduin prediction of the Kaluza-Klein gravity theory (KK).
13. Kerr's Planck Scale Gravity: now accepted for publication Predictions of Experimental Results from a Gravity Theory (PSG)
The following are still in the running:
GPB Geodetic precession (North-South)
1. GR = 6.6144 arcsec/yr.
2. BD = 6.6144 arcsec/yr. where now >60.
4. NGT = 6.6144 - a small correction arcsec/yr.
6. NG = 6.6144 arcsec/yr.
9. DG = 6.6144 arcsec/yr.
10. CS = 6.6144 arcsec/yr.
11. WG = 6.6144 arcsec/yr.
12. KK = (1 + b/6 - 3b2 + ...) 6.6144 arcsec/yr. where 0 < b < 0.07.
We await the GPB gravitomagnetic frame dragging precession (East-West) result.
1. GR = 0.0409 arcsec/yr.
2. BD = 0.0409 arcsec/yr.
4. NGT = 0.0409 arcsec/yr.
6. NG = 0.0102 arcsec/yr.
9. DG = 0.0000 arcsec/yr.
10. CS = 0.0409 arcsec/yr. + CS correction
11. WG = 0.0000 arcsec/yr.
12. KK = 0.0409 arcsec/yr.
Those that have fallen by the wayside:
3. SCC = 4.4096 arcsec/yr.
5. MVSR = 0.0 arcsec/yr.
7. FST = 4.4096 arcsec/yr.
8. MMR = -6.56124 arcsec/yr.
13. PSG = 0.0000 arcsec/yr/
He was the man behind SCC, and was brutally honest in accepting the falsification provided by GP-B (not be daunted he revised the theory to fit the data in short order).
As he more fully explains the data from GP-B on page 14 of that thread:
Just to make clear what the present situation is:
(mas = milliarcsec)
The geodetic N-S precession is predicted by GR to be - 6606 mas/yr, however there is a solar geodetic precession N-S component of + 7 mas/yr and the proper motion of IM Pegasi +28 mas/yr to take into account, resulting in a net expected N-S precession of -6571 1 mas/yr.
The frame-dragging E-W precession is predicted by GR to be -39 mas/yr, the solar geodetic precession E-W component of -16 mas/yr and the proper motion of IM Pegasi -20 mas/yr to include, resulting in a net expected N-S precession of -75 1 mas/yr.
From pages 20 and 21 of Francis Everitt's April APS talk, we find: A series of error ellipses on the N-S v E-W precession plot with centres respectively at (-6584 60, -83 22 mas/yr) June 2006, (-6597 17, -92 15 mas/yr) December 2006, (-6595 12, -98 7 mas/yr) March 2007 and (-6603 8, -98 7 mas/yr) March 2007.
It was this last reading for the geodetic precession that Francis Everitt reported at his April APS talk. If we also include that 'glimpse' of the E-W precession as well we have net values of:
(-6603 8, -98 7 mas/yr)
whereas GR predicts:
(-6571 1, -75 1 mas/yr).
In other words the actual readings are larger than GR predicts by 32 mas/yr in geodetic precession and 23 mas/yr in frame-dragging precession.
However, they reported an overall error, which is still being reduced, caused by residual gyro-to-gyro inconsistencies due to incomplete modeling of ~ 100 mas/yr.
This renders the present geodetic 'glimpse' as being consistent with GR to within about 1%, whereas the frame-dragging precession is at present swamped by noise.
Page 17 of that thread notes a December 2007 update of the GP-B data:
Einstein expectation: -6571 +- 1*
4-gyro result (1 sigma) for 85 days
(12 Dec 04 -- 4 Mar 05) -6632 +- 43
The December 2007 update puts experimental results 1.4 standard deviations greater than General Relativity predicts, with partial data. This is still reasonably close. Random chance calls for results within 1 standard deviation about 68% of the time and within 2 standard deviations about 95% of the time, so this isn't a horribly quirky result, although it also isn't as confirming as one might hope if one wanted an unequivocal GR confirmation.
Why would anyone even bother rethinking a well established theory like General Relativity? As a paper described in the thread notes:
[T]here are important reasons to question the validity of Einstein’s theory of gravity. Despite the beauty and simplicity of general relativity, our present understanding of the fundamental laws of physics has several shortcomings. The continued inability to merge gravity with quantum mechanics, and recent cosmological observations that lead to the unexpected discovery of the accelerated expansion of the universe (i.e., “dark energy”) indicate that the pure tensor gravity field of general relativity needs modification.
Some theorists believe that modifications to General Relativity could also explain some or all of the phenomena described as "dark matter" which require matter of a type fundamentally different from known "baryonic" matter which has never actually observed to fit the experimental data.
Dark matter and dark energy are both phenomena that are observed only at very long ranges where gravitational fields are very weak. Many theorists think that a quantum formulation of general relativity could produce subtle differences from General Relativity, which is a classical theory, rather than a quantum one, to produce subtle empirical differences from classical General Relativity.