Extra references added 2005.
Original by Steve Carlip.
The fundamental laws of physics, as we presently understand them, depend on about 25 parameters, such as Planck's constant h, the gravitational constant G, and the mass and charge of the electron. It is natural to ask whether these parameters are really constants, or whether they vary in space or time.
Interest in this question was spurred by Dirac's large number hypothesis. The "large number" in question is the ratio of the electric and the gravitational force between two electrons, which is about 1040; there is no obvious explanation of why such a huge number should appear in physics. Dirac pointed out that this number is nearly the same as the age of the Universe in atomic units, and suggested in 1937 that this coincidence could be understood if fundamental constants—in particular, G—varied as the Universe aged. The ratio of electromagnetic and gravitational interactions would then be large simply because the Universe is old. Such a variation lies outside ordinary general relativity, but can be incorporated by a fairly simple modification of the theory. Other models, including the Brans-Dicke theory of gravity and some versions of superstring theory, also predict physical "constants" that vary.
Over the past few decades, there have been extensive searches for evidence of variation of fundamental "constants." Among the methods used have been astrophysical observations of the spectra of distant stars, searches for variations of planetary radii and moments of inertia, investigations of orbital evolution, searches for anomalous luminosities of faint stars, studies of abundance ratios of radioactive nuclides, and (for current variations) direct laboratory measurements.
One powerful approach has been to study the "Oklo phenomenon," a uranium deposit in Gabon that became a natural nuclear reactor about 1.8 thousand million years ago; the isotopic composition of fission products has permitted a detailed investigation of possible changes in nuclear interactions. Another has been to examine ratios of spectral lines of distant quasars coming from different types of atomic transitions (resonant, fine structure, and hyperfine). The resulting frequencies have different dependences on the electron charge and mass, the speed of light, and Planck's constant, and can be used to compare these parameters to their present values on Earth. Solar eclipses provide another sensitive test of variations of the gravitational constant. If G had varied, the eclipse track would have been different from the one we calculate today, so the mere fact that a total eclipse occurred at a particular location provides a powerful constraint, even if the date is poorly known.
So far, these investigations have found no evidence of variation of fundamental "constants." The current observational limits for most constants are on the order of one part in 1010 to one part in 1011 per year. So to the best of our current ability to observe, the fundamental constants really are constant.
References:
For a good short introduction to the large number hypothesis and the constancy of G, see:
C.M. Will, Was Einstein Right? (Basic Books, 1986)
For more technical analyses of a variety of measurements, see:
P. Sisterna and H. Vucetich, Physical Review D41 (1990) 1034 and Physical Review D44 (1991) 3096
E.R. Cohen, in Gravitational Measurements, Fundamental Metrology and Constants, V. De Sabbata and V.N. Melnikov, editors (Kluwer Academic Publishers, 1988)
"The Constants of Physics," Philosophical Transactions of the Royal Society of London A310 (1983) 209–363
"The Oklo bound on the time variation of the fine structure constant revisited" T. Damour and F. Dyson, Nucl. Phys. B480 (1996) 37–54, hep-ph/9606486
Michael Duff: "Comment on time-variation of fundamental constants", hep-th/0208093 (2004)
Duff, Okun, and Veneziano: "Trialogue on the number of fundamental constants", JHEP 203 23 (2002), physics/0110060