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Impearls: Battle-tested General Relativity

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Earthdate 2004-04-12

Battle-tested General Relativity

There's been an interesting conversation in National Review Online's blog The Corner between commentators Peter Robinson and John Derbyshire with regard to the testing of Albert Einstein's general theory of relativity.  (A satellite will soon launch carrying what is called a “Gravity B” experiment designed to further test general relativity.)  You can read the pieces of the discussion here: R1, R2, D3, R4, D5, R6, R7.

While quite enjoyable, I was a little concerned by a tendency to overlook, as I perceive it, the degree to which General Relativity has been tested and has prevailed against its competitors.  An excerpt from Misner, Thorne, and Wheeler's classic tome Gravitation (1973) on the subject of competing theories of gravity is illuminating in this regard: 1  (Ellipses in the text refer to omitted section cross-reference numbers where each topic is gone over in detail.)

§ 39.1.  Other theories

Among all bodies of physical law none has ever been found that is simpler or more beautiful than Einstein's geometric theory of gravity {…}; nor has any theory of gravity ever been discovered that is more compelling.

As experiment after experiment has been performed, and one theory after another has fallen by the wayside a victim of the observations, Einstein's theory has stood firm.  No purported inconsistency between experiment and Einstein's laws of gravity has ever surmounted the test of time.

Query:  Why then bother to examine alternative theories of gravity?  Reply:  To have “foils” against which to test Einstein's theory.

To say that Einstein's geometrodynamics is “battle-tested” is to say it has won every time it has been tried against a theory that makes a different prediction.  How then does one select new antagonists for decisive new trials by combat?

Not all theories of gravity are created equal.  Very few, among the multitude in the literature, are sufficiently viable to be worth comparison with general relativity or with future experiments.  The “worthy” theories are those which satisfy three criteria for viability: self-consistency, completeness, and agreement with past experiment.

Self-consistency is best illustrated by describing several theories that fail this test.  The classic example of an internally inconsistent theory is the spin-two field theory of gravity [Fierz and Pauli (1939) {…}], which is equivalent to linearized general relativity {…}.  The field equations of the spin-two theory imply that all gravitating bodies move along straight lines in global Lorentz reference frames, whereas the equations of motion of the theory insist that gravity deflects bodies away from straight-line motion.  (When one tries to remedy this inconsistency, one finds oneself being “bootstrapped” up to general relativity {…}.)  Another self-inconsistent theory is that of Kustaanheimo (1966).  It predicts zero gravitational redshift when the wave version of light (Maxwell theory) is used, and nonzero redshift when the particle version (photon) is used.

Completeness:  To be complete a theory of gravity must be capable of analyzing from “first principles” the outcome of every experiment of interest.  It must therefore mesh with and incorporate a consistent set of laws for electromagnetism, quantum mechanics, and all other physics.  No theory is complete if it postulates that atomic clocks measure the “interval”  = (− gαβ dxα dxβ)½  constructed from a particular metric.  Atomic clocks are complex systems whose behavior must be calculated from the fundamental laws of quantum theory and electromagnetism.  No theory is complete if it postulates that planets move on geodesics.  Planets are complex systems whose motion must be calculated from fundamental laws for the response of stressed matter to gravity.  {…}

Agreement with past experiment:  The necessity that a theory agree, to within several standard deviations, with the “four standard tests” (gravitational redshift, perihelion shift, electromagnetic-wave deflection, and radar time-delay) is obvious.  Equally obvious but often forgotten is the need to agree with the expansion of the universe (historically the ace among all aces of general relativity) and with observations at the more everyday, Newtonian level.  Example: Birkhoff's (1943) theory predicts the same redshift, perihelion shift, deflection, and time-delay as general relativity.  But it requires that the pressure inside gravitating bodies equal the total density of mass-energy, p = ρ; and, as a consequence, it demands that sound waves travel with the speed of light.  Of course, this prediction disagrees violently with experiment.  Therefore, Birkhoff's theory is not viable.  Another example:  Whitehead's (1922) theory of gravity was long considered a viable alternative to Einstein's theory, because it makes exactly the same prediction as Einstein for the “four standard tests.”  Not until the work of Will (1971b) was it realized that Whitehead's theory predicts a time-dependence for the ebb and flow of ocean tides that is completely contradicted by everyday experience {…}.

§ 39.2.  Metric theories of gravity

Two lines of argument narrow attention to a restricted class of gravitation theories, called metric theories.

The first line of argument constitutes the theme of the preceding chapter.  It examined experiment after experiment, and reached two conclusions:  (1) spacetime possesses a metric; and  (2) that metric satisfies the equivalence principle (the standard special relativistic laws of physics are valid in each local Lorentz frame).  Theories of gravity that incorporate these two principles are called metric theories.  In brief, Chapter 38 says, “For any adequate description of gravity, look to a metric theory.”  Exception:  Cartan's (1922b, 1923) theory [“general relativity plus torsion”; see Trautman (1972)] is nonmetric, but agrees with experiment and is experimentally indistinguishable from general relativity with the technology of the 1970's.

The second line of argument pointing to metric theories begins with the issue of completeness (preceding section).  To be complete, a theory must incorporate a self-consistent version of all the nongravitational laws of physics.  No one has found a way to incorporate the rest of physics with ease except to introduce a metric, and then invoke the principle of equivalence.  Other approaches lead to dismaying complexity, and usually to failure of the theory on one of the three counts of self-consistency, completeness, and agreement with past experiment.  All the theories known to be viable in 1973 are metric, except Cartan's.  [See Ni(1972b); Will (1972).]

In only one significant way do metric theories of gravity differ from each other: their laws for the generation of the metric.  In general relativity theory, the metric is generated directly by the stress-energy of matter and of nongravitational fields.  In Dicke-Brans-Jordan theory {…}, matter and nongravitational fields generate a scalar field φ; then φ acts together with the matter and other fields to generate the metric.  Expressed in the language of {…}, φ is a “new long-range field” that couples indirectly to matter.  As another example, a theory devised by Ni (1970, 1972) {…} possesses a flat-space metric η and a universal time coordinate t (“prior geometry” {…}); η acts together with matter and nongravitational fields to generate a scalar field φ; and then η, t, and φ combine to create the physical metric g that enters into the equivalence principle.

All three of the above theories — Einstein, Dicke-Brans-Jordan, Ni — were viable in the summer of 1971, when this section was written.  But in autumn 1971 Ni's theory, and many other theories that had been regarded as viable, were proved by Nordtvedt and Will (1972) to disagree with experiment.  This is an example of the rapidity of current progress in experimental tests of gravitational theory!

Notice that in all of these “trials by combat” against the searing fires of experiment, it is Einstein's general relativity that has consistently withstood the tests, like Daniel walking through the fiery furnace!

Quoting further from Misner, Thorne, and Wheeler's Gravitation (§44.2): 2 

No theory more resembles Maxwell's electrodynamics in its simplicity, beauty, and scope than Einstein's geometrodynamics.  Few principles in physics are more firmly established than those on which it rests: the local validity of special relativity {…}, the equivalence principle {…}, the conservation of momentum and energy {…}, and the prevalence of second-order field equations throughout physics {…}.  Those principles and the demand for no “extraneous fields” (e.g., Dicke's scalar field) and “no prior geometry” {…} lead to the conclusion that the geometry of spacetime must be Riemannian and the geometrodynamic law must be Einstein's.

To say that the geometry is Riemannian is to say that the interval between any two nearby events C and D, anywhere in spacetime, stated in terms of the interval AB between two nearby fiducial events, at quite another point in spacetime, has a value CD/AB independent of the route of intercomparison {…}.  There are a thousand routes.  By this hydraheaded prediction, Einstein's theory thus exposes itself to destruction in a thousand ways {…}.

Geometrodynamics lends itself to being disproven in other ways as well.  The geometry has no option about the control it exerts on the dynamics of particles and fields {…}.  The theory makes predictions about the equilibrium configurations and pulsations of compact stars {…}.  It gives formulas {…} for the deceleration of the expansion of the universe, for the density of mass-energy, and for the magnifying power of the curvature of space, the tests of which are not far off.  It predicts gravitational collapse, and the existence of black holes, and a wealth of physics associated with these objects {…}.  It predicts gravitational waves {…}.  In the appropriate approximation, it encompasses all the well-tested predictions of the Newtonian theory of gravity for the dynamics of the solar system, and predicts testable post-Newtonian corrections besides, including several already verified effects {…}.

No inconsistency of principle has ever been found in Einstein's geometric theory of gravity.  No purported observational evidence against the theory has ever stood the test of time.  No other acceptable account of physics of comparable simplicity and scope has ever been put forward.


References

1 Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, 1973, W. H. Freeman and Co., San Francisco; pp. 1066-1068.

2 Ibid., p. 1199.


UPDATE:  2004-04-19 00:51 UT:  A follow-up on the nature of scientific theories has been posted.

UPDATE:  2004-04-29 23:50 UT:  A follow-up “Copernicus Dethroned” has been posted.

UPDATE:  2004-04-18 16:30 UT:  Fred Kiesche at the stimulating The Eternal Golden Braid blog has linked to this piece, noting, ”To follow up on my recently posted news item about the Gravity B probe (launch still on track for Monday), Michael McNeil's Impearls (a great site, by the way; I'm still in debt for his help with J.D. Bernal's The World, the Flesh and the Devil) talks about Battle-Tested General Relativity.”

Additional articles on the Gravity B probe may be found here in the New York Times, and here in the journal Science (requires subscription or pay-per-view).

UPDATE:  2009-09-08 14:20 UT:  The foregoing excerpt from Misner, Thorne, and Wheeler's Gravitation was written three and a half decades ago, and thus one might imagine that things might well have changed in the meantime.  A while back, however, in 2007, I had an exchange of personal correspondence with well-known general relativist Sean Carroll of Caltech — author of the graduate-level text Spacetime and Geometry: An Introduction to General Relativity (2003), who blogs at Cosmic Variance (now sponsored by Discover Magazine) — and I had occasion to ask him if the lofty status that Gravitation ascribes to Einstein's geometrodynamics has held up over all those decades.  Here's Carroll's reply, which he's granted me permission to publicly quote:

Hi Michael —

All of that is still completely true, yes.

These days we actually have better reasons to consider alternatives to GR — namely, the apparent existence of dark matter and dark energy.  They are only “detected” through their gravitational fields, so it's natural to wonder whether the evidence in their favor is actually evidence for a modification of gravity.  Sadly, the attempts so far to modify gravity in the right ways have fallen a bit short; see these posts:

http://preposterousuniverse.blogspot.com/2004/05/was-einstein-wrong.html
http://preposterousuniverse.blogspot.com/2004/05/was-friedmann-wrong.html
http://cosmicvariance.com/2006/11/08/out-einsteining-einstein/

So we still think that Einstein's theory has passed all of its tests, as the tests themselves keep getting better and better.  But who knows?  Tomorrow we might get a surprise.

Sean


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