Some Scientifically Inaccurate Claims Concerning Cosmology and Relativity

Original by Chris Hillman (Last modified by Chris Hillman, 31 Jan 2001)

This page is provided for the benefit of three kinds of people:

Before explaining the contents of this page, I'd like to immediately mention an excellent popular article by John Farrell (Multimedia Specialist, Shapiro Institute for Education & Research, Harvard Medical School), Did Einstein Cheat?, which appeared in the July 5, 2000 issue of Salon Magazine .

In this page, I offer links to rebuttals by qualified scientists of some of the more common scientifically inaccurate claims made in

Since many of these claims "aren't even wrong", to use the caustic phrase of Wolfgang Pauli--- by which he meant an "argument" too vague to be susceptible to a reasoned counterargument--- it is generally difficult to give a point by point rebuttal of every such claim. Moreover, since there are so many scientifically inaccurate mass market books, Usenet posts, and personal web pages, it would be impossible to even compile a "crank bibliography". Hence, this page is neccessarily incomplete.

The Scientifically Inaccurate Claims Discussed on This Page

In roughly increasing order of sophistication of the conceptual errors involved:
Note well! On this page, I have made no attempt to provide a catalog of erroneous claims concerning relativity; I simply discuss some of the mistaken claims which have appeared in preprints posted to the LANL preprint server or in the popular press. If you know about a claim I haven't listed--- I really don't think I want to know!

"The Big Bang Theory Is Wrong."

Most people who say this don't even understand what cosmologists really mean when they talk about "the standard hot Big Bang theory". Thus, before reading about some erroneous claims and how scientists refute them, it is well worthwhile to spend some time learning what scientists really are saying about the origin and early history of the universe.

What the Standard Hot Big Bang Theory Really Says

Here are some short and fairly nontechnical overviews of the standard hot Big Bang Theory of cosmology:

Tutorials on the Standard Model of particle physics are harder to come by, which is unfortunate, because you need to know something about this to understand the standard hot Big Bang model in its full glory. Here are the best I could find:

I suspect that the preceding blurbs on the Standard Model of particle physics will confuse many readers. As an antidote, I offer this brief post by John Baez, who has a real talent for explaining complicated scientific issues in a simple way.

The essential lesson to be gained from the above sources is this:

Now that you have a better idea of what the standard hot Big Bang theory really says, I hope your appetite has been whetted for more! If so, I recommend that you study the fascinating textbook by Peacock. Readers with limited time may prefer to simply look at some more tutorials before proceeding.

Next, let's press on and look at some of the "alternative theories" which have been put forth at various times by people with a huge range of degrees of scientific credibility.

Once Credible Alternatives to the Hot Big Bang Theory

Not only has the hot Big Bang Theory been the reigning Queen of Cosmology for forty years, there really are no other contenders for the throne. It was not always so; the current queen was once a gamin! Here are two theories which were taken very seriously by scientists until the discovery of the cosmic microwave background radiation suddenly and completely unexpectedly elevated the unknown maiden to the throne.

Alternative Theories Offered by Authors With Scientific Credentials

Here are some theories whose authors at least have some scientific credentials.

Pseudoscientific "Theories"

A visit to yahoo shows that more crank sites are listed than scientifically accurate sites under "origins of the universe". Indeed, a little poking around the web soon shows that there are hundreds of web pages put up by nonscientists, all claiming to cast doubt upon the hot Big Bang Theory (which as I have already noted, does not say what most of these deluded souls imagine). What is the reason for this?

I [Chris Hillman] think it is perfectly clear that resistance to hot Big Bang Theory by various religious and political groups is not about science or reason. It is about two things: power and fear. Let me explain.


On the one hand, some people fear that science is somehow "contradicting" the validity of their religious convictions, and this has led some religious groups, particularly some "fundamentalist churches" in the U.S., to condemn such cornerstones of modern science as the theory of evolution and the theory of the Big Bang, prefering to stick to the literal truth of the creation story told in the Bible. (Speaking of which, you may want to read the official statement on "Creationism" of the American Physical Society; see also this book published by the National Academy of Sciences and debunkings offered by Tom Schneider of NIH.) For example, here are two sites which obviously fall under this category of total condemnation of scientific knowledge (under the feeble guise of claiming to use logical counterarguments):


Science is universally respected, even by those who fear it, and with good reason. Everyone knows that we live the way we do (in developed countries) because of the benefits of technology which grew out of basic scientific research. Scientists (and mathematicians) have almost become secular demigods, and it should be no surprise that those religious and political groups which are hostile to science but which do not condem scientific activity altogether, have tried to coopt scientific theories by claiming that these theories somehow prove the validity of their religious or political beliefs. For example, here are some sites which obviously fall under this category:

As you can see, religious and political extremism can make some pretty odd bedfellows.

In fact, to the hilarity of a bemused onlooker, a closer look at many "Creation-Science" or "Fundamentalist" websites, in particular, soon reveals that there are many warring factions within this pseudoscientific subculture.

Since I am not a student of sociology or psychopathology, I won't attempt to pursue this bizzaire situation any further.

But let me come back for a moment to "power and fear". In my view, the irony is that both those groups which fear science, and those which envy its power, are operating under the same basic mistaken assumption: that scientific facts "mean" something more, much more, than they really do. It should not really be surprising that the same people who see conspiracies everywhere are often the same people who are attracted to extremist religious or politial causes, or to psuedo-scientific quackery. To the mystic, a fact is never "just" a fact. I look at a geological rock formation and see rocks. By using my imagination, I can "see" atoms which make up the grains of spinel and other minerals apparent upon closer examination. A mystic might look at a rock formation and see the likeness of the Virgin Mary. This is perhaps the fundamental difference between the scientific and the mystical viewpoint. It's not that scientists are any less imaginative than mystics (the theory of general relativity is a perfect counterexample!); the difference is that scientific imagination is highly constrained by the demanding criteria of experimental evidence and consistency with all other scientific knowledge. Scientific imagination is a highly disciplined imagination, and that is ultimately what separates the scientist from the crank.

It might be worth adding a few comments about the nature of "scientific credentials".

Nonscientists may not realize that engineers (in particular) have not earned scientific degrees and usually do not have any actual scientific training, although they have probably taken some undergraduate math courses. Speaking as someone who has worked with many engineering students and also with professional engineers, I must caution against assuming that any given engineer must have some scientific knowledge or even the common sense God gave Little Green Dumplings. For what it is worth, I have known quite a few engineers and my experience suggests that those who take genuine pride in being engineers, not scientists, are in general not merely competent engineers but are also perfectly sane (and I like and respect several such people whom I happen to know well); on the other hand, engineers who (seemingly) really wanted to become mathematicians or physicists frequently turn out to be borderline lunatics. I might add that I have learned to become wary when I sense that someone trained as a philosopher really wanted to become a mathematician-- or vice versa. My experience suggests that people who are frustrated with their station in life are much more likely to become cranks or to go quietly nuts than people who are perfectly happy with who they are personally and professionally.

The common sense lesson here is that one should never assume that someone who appears to be intelligent and who claims to have some sort of professional degree, actually knows very much about anything outside his or her narrow field of professional specialization. In other words: a tort laywer probably can be trusted to know about torts, but you wouldn't want to have a tort laywer representing you if you were on trial for your life. If you have cancer of the colon, you'd be foolish to consult a cardiologist. If you are the coach of a football team and you need to kick in a field goal, you'd be an idiot to send out your neighbor who happens to be a professional golfer. If your car is making a funny noise, consulting that fellow who says he is the chief mechanic for the most recent winner of the Tour de France is probably a waste of time.

"Black Holes Do Not Exist."

Every physicist instinctively abhors physical quantities which diverge to infinity, but there is no question that general relativity does predict the existence of curvature singularities. (Maxwell's theory of electromagnetism is plagued by analogous singularities in the EM field of a point charge.) Indeed, it is a rare exact solution in general relativity (model spacetime) which does not have any curvature singularities. In particular, there is no question that the exact solutions in gtr which describe collapsed objects (black holes) contain curvature singularities.

Nonetheless, various people who ought to know better continue to claim otherwise. In the past few years, for example, three physicists have posted preprints to the Los Alamos preprint server in which they appear to make one or more of the following claims:

These apparent claims are all mathematically incorrect as statements about the predictions of gtr. The proofs can be found in any modern gtr textbook, except for the fourth claim, for which the interested reader can start with these tutorial papers I say apparent claims because (is there any need to say this?) the papers in question tend to be rather muddled, so it is frequently hard to tell what the author intends to say, probably because he himself is plainly confused.

Another person (trained as an engineer, not a physicist--- see the above comment!), who works in some capacity at a French observatory, also maintains that the Schwarzschild and Kerr solutions have been "misinterpreted". Unfortunately, his claims rest upon numerous and quite elementary mathematical misconceptions. This gentleman is badly confused about the role played by coordinate charts, spacelike hyperslices, topology, and geometric singularities in gtr. For example, he claims that

the Schwarzschild solution can be interpreted as a "space bridge", linking two universes, two space-times, working as a one way bridge. We show that the transit time of a test particle is finite and short, which immediately makes the classical black hole model questionable."
But it is an elementary exercise (frequently set in modern graduate level gtr courses) to show that the so-called "Einstein-Rosen bridge" or "wormhole throat" in the "eternal Schwarzschild vacuum" (or "maximal extension of the Schwarzschild exterior chart"), which has the Penrose diagram

         888888888888 i^+ ("future timelike infinity")
        /\          /\
       /  \ future /  \
      /    \ int. /    \
     /      \    /      \ scri^+ ("future null infinity")
    / second \  / first  \ 
   / exterior \/ exterior \
   \  region  /\  region  / i^0 ("spatial infinity")
    \        /  \        /
     \      /    \      /
      \    / past \    / scri^- ("past null infinity")
       \  /  int.  \  /
        \/          \/
         888888888888 i^- ("past timelike infinity")

"pinches shut" before any particle can pass through the wormhole. Indeed, this fact is evident from the Penrose diagram above! For more information about how to interpret this diagram, and for a brief description of more realistic models of gravitational collapse (note the the "eternal Schwarzschild vacuum" is not a model of a black hole formed by gravitational collapse!) see this archived post or this tutorial paper by Jiri Bicak. For a fairly detailed analysis of the Schwarzschild vacuum, including the behavior of geodesics, see this archived post. For more information about the theory of black holes, see these lectures by Paul Townsend, or this classic monograph by Chandrasekhar.

This person also claims that

This person also tries to study the "t = t0" hyperslices in the Schwarzschild vacuum, but is clearly unaware of two elementary but fundamental points: first, these slices all pass through a single two-sphere S^2 (see the "X" in the Penrose diagram above), and never enter either interior region, and second, the geodesics of these spatial hyperslices are not the "trajectories" of either test particles or laser pulses obtained by projected timelike or null world lines onto such a slice!

Again, for all of the above points, interested readers are referred to either the notes by Townsend or the monograph by Chandrasekhar.

"Gravitational Waves Do Not Exist."

The question of the existence and physical significance of gravitational radiation was once a serious issue in gtr, although now it is very hard to see what all the fuss is about: it is quite obvious from modern treatments that they exist and that they carry energy. At one point Einstein himself was convinced (following an elementary confusion of coordinate with curvature singularities) that his prediction in 1914 of gravitational waves was incorrect! Fortunately, a referee spotted the error, and while the paper was being held up, another physicist was able to convince Einstein he had goofed, and the published paper did not contain the erroneous retraction of the prediction of gravitational waves. A simple argument of Richard Feynman eventually convinced all the doubters (after Einstein's death): imagine some beads on a sticky rod. If the rod is oriented correctly, as a linearly polarized gravitional wave passes by (exact solutions in gtr describe such things), the beads slide back and forth on the rod (if it is corrrectly oriented), and heat the rod by friction. This finally convinced everyone that gravitiational waves exist and carry energy. (Daniel Kennefick has written an interesting history of the concept of gravitational waves and gravitational radiation reaction.)

For more about the theory of astrophysical sources of gravitational waves, see these papers. For a summary of the principles behind the laser-interferometric-gravitational wave detectors like LIGO/VIRGO now nearing completion, which will be the first instruments sensitive enough to be capable of detecting the very weak strains (on the order of 10^(-21)) characterizing the strongest gravitational waves which astronomers estimate reach our vicinity several times per year, look here.

As I say, in retrospect it is hard to believe this point was ever controversial; modern coordinate-free methods such as orthonormal bases and the invariant decomposition of congruence of timelike or null geodesics have rendered the detailed study of simple exact solutions very easy and greatly clarify the geometry. Unfortunately, not all physicists have studied the differential geometry, and this can get quite subtle, so newcomers often make conceptual errors. Recently, one of the confused physicists mentioned in the previous section has posted to LANL the incorrect claim that gravitational waves are not in fact a prediction of gtr. This is mathematically incorrect as a statement about the predictions of gtr, and the proof can be found in any modern gtr textbook (Wald's textbook is a particularly good choice for this purpose).

"General Relativity Is Not Physically Meaningful."

In a series of papers and preprints posted to the LANL server, Hussein Yilmaz (Electro-Optics Technology Center, Tufts University) and, in some cases, his occasional coauthor, Carroll Alley (Physics, University of Maryland), have made a number of specific claims, including the following:

Yilmaz and Alley, incidentally, futher claim that they have presented an alternative to gtr, which they call "the Yilmaz theory of gravitation". For example, they make much of the fact that the static, spherically symmmetric chart

ds^2 = -exp(-2m/R) dt^2 + exp(2m/R) [ dR^2 + R^2 (du^2 + sin(u)^2 dv^2 ] -infinity < t < infinity, 0 < R < infinity, 0 < u < Pi, -Pi < v < Pi

which they claim is a vacuum solution in their "theory", contains no curvature singularities. For comparison, in the weak field limit, the well known Schwarzschild vacuum solution to the EFE, has the static, spherically symmetric exterior chart

ds^2 = -(1-2m/R) dt^2 + (1+2m/R) [ dR^2 + R^2 (du^2 + sin(u)^2 dv^2 ] -infinity < t < infinity, R "large", 0 < u < Pi, -Pi < v < Pi

Ironically, Yilmaz and Alley seem to be completely unaware of the fact that the Chazy-Curzon vacuum

ds^2 = -exp(-2m/R) dt^2 + exp(2m/R) [ exp(-m^2 sin(u)^2/R^2) (dR^2 + R^2 du^2) + R^2 sin(u)^2 dv^2 ] -infinity < t < infinity, 0 < R < infinity, 0 < u < Pi, -Pi < v < Pi,

is an exact static solution to the EFE which is approximately spherically symmetric for large R, and which also has no curvature singularities. As a matter of fact, all three charts can be used as models for the vacuum exterior to the Sun, for example, without making much practical difference. (This is not true, however, for black holes!) Be this as it may, unfortunately for Yilmaz and Alley, everyone, including myself, who has examined their "field equation" seems to agree that while the LHS is well-defined, the RHS (their replacement for the stress-momentum-energy-tensor of gtr) is not. Moreover, the failure of Yilmaz and Alley to realize this seems to be due ultimately to the same fallacy discussed above, concerning the nature of divergence in curved spacetime. See for example this preprint by Misner (Physics, University of Maryland; Misner is one coauthor of this classic gtr textbook.)

According to what Tom Van Flandern told Tom Bethell, the author of this article, which appeared in the magazine American Spectator, an un-named person claimed in conversation with Van Flandern that Einstein's classic computation of the perihelion of Mercury was "fudged". Another poster in sci.physics.relativity reported having a similar conversation with Carroll Alley. However, Alley later told John Farrell, the author of this article, which appeared in Salon Magazine (and was far better researched, as casual inspection will show!), that while he is indeed the person mentioned in the article by Bethell, what he actually told Van Flandern was something rather different. If so, that is good for him, because the assertion is of course erroneous; it can only rest upon a profound misunderstanding of the role of approximation and linearization in mathematical physics in general, and in Einstein's classic computation in particular. (For a detailed sketch of how the precession of the perihelion of a "quasi-Keplerian nearly-elliptical orbit" in the Schwarzschild static vacuum, may be derived, by following Einstein's adaptation of a clever trick first introduced by the famous nineteenth century differential geometer Binet [in the context of solving the Newtonian two-body problem!] to carefully reduce the problem of finding the approximate (planar) motion of a (nonspinning) test particle to a linear differential equation, see for example this post; for the full details, see for example this classic monograph.)

"General Relativity Is An Aether Theory."

Albert Einstein, in his essay On the Aether (1924), made some injudicious comments to the effect that relativity theory could be said to ascribe physical properties to spacetime itself, and in that sense, to involve a kind of "aether". He clearly did not mean the kind of "aether" which had been envisioned by Maxwell and others in the nineteenth century, but his remarks have been seized upon ever since, by various cranks and other ill-informed persons, as evidence that "gtr is an aether theory". Here's a typical claim of this sort:

...the aether is restored in General Relativity see Einstein's 1924 essay "On the Aether". Einstein recanted on his 1905 rejection of the aether since the mutable curved space-geometry is a dynamical object (with shift and lapse fields in ADM formulation), hence an aether.
This claim is misleading, to say the least. What Einstein really meant was that the aether which had been overthrown by str (and thus was incompatible with gtr, which incorporates str) involved a a specific "preferred frame of reference" in the classical field theory, whereas the field equation of gtr involves no "prior geometry" (such as the euclidean geometry of "space" which has assumed by Maxwell and his contemporaries), much less any "preferred frame". Nonetheless, gtr does not quite say there is "nothing" in "empty space"; in general there will be gravitational waves running about, and these carry (very tiny) amounts of energy, which gravitate. So in this sense, a very different kind of "aether" in the very weak sense of there being "something there" in a vacuum (namely nonlocalizable gravitational field energy, metric properties of "space" in a 3+1 decomposition, etc.), could be said to enter into gtr. In modern quantum field theories, of course, there are still more "things which are there" in a vacuum, but again these do not constitute an "aether" in the nineteenth century sense in which this word was used as a technical term.

Einstein was criticizing people who claimed, in effect, that the classical notion of the aether was such nonsense that people like Maxwell should have known better. He was saying that the problem with the classical aether was not ontological, merely that it is inconsistent with observation and experiment; hence the need for str.

Many years ago, Andrei Sakharov (yes, that Sakharov!) proposed to interpret gtr in terms of something like "stresses" on spacetime as something like a material. This is discussed in Chapter 17 of MTW, but here too, ill-informed readers of that theory have badly misunderstood the meaning of Sakharov's work.

"Tensors Are Unnecessary in GTR."

In the web version of the paper by Steven C. Bell, A numerical solution of the relativistic Kepler problem, Computers in Physics 9 (1995) 281-285, the author (a programmer who claimed special expertise by virtue of having worked on commercial satellite launches using the Titan launch vehicle) erroneously claimed that there are stable circular orbits in the Schwarzschild solution inside r = 3m (a correct analysis of orbits in the Schwarzschild and Kerr geometries can be found in the textbook of Schutz.). In posts to sci.physics.relativity Bell even claimed that tensor calculus is unnecessary for doing computations in gtr, and claimed further that the interpretation of spacetime curvature is not needed (see the next section).

"The Speed of Gravity Is Faster Than Light."

This one seems to be the special province of Tom Van Flandern (you may recall that Van Flandern is also one of the Big Bang doubters). Van Flandern has published a notorious and flagrantly erroneous paper, T. Van Flandern, "The speed of gravity-what the experiments say", Phys. Lett. A 250 (1998): 1 -- 11, which is also available on-line here. Recently, Steve Carlip, one of the world's leading experts on gravitation physics, has written a short rebuttal of this paper, which is based upon a misconception concerning the role of abberation in gtr, among other errors. It is worth pointing out that Phys. Lett. A is not a journal whose editors are specialists in gtr; indeed, this journal is one of the few willing to publish "controversial" papers which would not pass refereeing by experts.

Tom Bethell, the Washington correspondent for the magazine The American Spectator, has written at least one article quoting approvingly from Van Flandern's paper and books, apparently in ignorance of the many rebuttals and corrections which reputable physicists have offered over the past five years of Van Flandern's many flagrantly erroneous statements. Bethell's description of the refereeing of this paper is certainly rather startling; the version I have heard from other sources is that the editors promised to publish the paper without any refereeing whatsoever. Bethell's description of a paper by Paul Gerber is historically and scientifically inaccurate, as is his description of the GPS system (see below).

Van Flandern has published several other erroneous papers repeating the same errors; see T. Van Flandern, "Possible new properties of gravity" Astrophysics and Space Science 244 (1996): 249 -- 61 and T. Van Flandern, "On the `speed of gravity' ", Galileian Electrodynamics 4 (1993) 35 -- 7. The latter journal was founded by the late Petr Beckmann, an electrical engineer who was on the faculty of the University of Colorado for many years, and who did not accept the special theory of relativity. Needless to say, this journal is not highly regarded by the vast majority of physicists. Van Flandern has recently published a book called Dark Matter, Missing Planets.

In the past year or so, Van Flandern (one must give him a point for self-consistency here) has gone further and now claims that not only do changed in gravitational "forces" propagate faster than light, but that changes in electromagnetic "forces" propagate faster than light! In other words, according to Van Flandern, not only is "the speed of gravity" infinite, but so is "the speed of light"! Needless to say, Van Flandern is simply very, very confused.

Van Flandern has been making incorrect claims about gravitation for a very long time, which have been corrected more or less patiently by experts; see this (very, very long--- currently 85,000 lines and growing fast) collection of posts by Steve Carlip, John Baez, and other leading gravitation physicists. See also this excellent article by John Farrell, which quotes numerous leading physicists who debunk various of Van Flandern's misstatements and misrepresentations.

"Spacetime Curvature Is Fictitious."

In a cosmology textbook which was widely used some years ago, Steven Weinberg (the Nobel Prize winning particle physicist) made the following comments:

...the passage of time has taught us not to expect that the strong, weak and electromagnetic interactions can be understood in geometrical terms, and too great an emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics...

...the non-vanishing of the the tensor R_{mnvk} is the true expression of the presence of a gravitational field... Riemann [had historically] introduced the curvature tensor R_{mnvk} to generalize [Gauss's] concept of curvature to three or more dimensions. It is therefore not surprising that Einstein and his successors have regarded the effects of a gravitational field as producing a change in the geometry of space and time. At one time it was even hoped that the rest of physics could be brought into a geometric formulation, but this hope has met with disappointment, and the geometric interpretation of the theory of gravitation has dwindled to a mere analogy, which lingers in our language in terms like "metric", "affine connection", and "curvature", but is not otherwise very useful.

Some people have apparently taken this to mean the Weinberg thought geometry has no role to play in physics (a glance at my
reading list will show that such a viewpoint is completely incorrect). In fact, Weinberg apparently meant that differential geometry alone was not likely to be capable of unifying gravity with the other fundamental interactions, and indeed this viewpoint has if anything gained support with the passage of time. Weinberg continued:
The important thing is to be able to make predictions about the images on the astronomer's photographic plates, frequencies of spectral lines, and so on, and it simply doesn't matter whether we ascribe these predictions to the physical effect of a gravitational field on the motion of planets and photons or to a curvature of space and time. (The reader should be warned that these views are heterodox and would meet with objections from many general relativists.)
John Baez (a researcher on quantum gravity) comments
I believe [Weinberg] later retracted this comment. But his attitude was quite common among particle physicists of his day, and this attitude still influences research in string theory, where until recently, most calculations were done on flat spacetime... Presumably his use of the word "connections" was a Freudian slip indicating that subconsciously he knew better: like gravity, the strong, weak and electromagnetic interactions can be understood geometrically using the theory of CONNECTIONS on fiber bundles.

Be this as it may, incorrect claims are often made that it is always possible to interpret any solution in gtr in terms of "funny fields" on Minkowski vacuum which mimic the effects of spacetime curvature. This is incorrect!, for several reasons:

In summary, upon close examinations, claims that spacetime curvature is merely a mathematical artifact of the conventional tensor formalism of gtr are all based upon various misunderstandings of some rather subtle technical and conceptual issues.

"Relativity Theory Isn't Working In The GPS."

In a paper remarkable chiefly for the extraordinary number of obvious errors it contained (see above), Tom Van Flandern, ("The speed of gravity-- what the experiments say" Phys.Lett.A 250 (1998) 1-11, also available here), stated:

the Global Positioning System (GPS) showed the remarkable fact that all atomic clocks on board orbiting satellites moving at high speeds in different directions could be simultaneously and continuously synchronized with each other and with all ground clocks. No "relativity of simultaneity" corrections, as required by SR, were needed. This too seemed initially to falsify SR. But on further inspection, continually changing synchronization corrections for each clock exist such that the predictions of SR are fulfilled for any local co-moving frame. To avoid the embarrassment of that complexity, GPS analysis is now done exclusively in the Earth-centered inertial frame (the local gravity field). And the pre-launch adjustment of clock rates to compensate for relativistic effects then hides the fact that all orbiting satellite clocks would be seen to tick slower than ground clocks if not rate-compensated for their orbital motion, and that no reciprocity would exist when satellites view ground clocks.

At first glance, Van Flandern here appears to be claiming that the fact that the GPS continues to operate with great accuracy has in fact disproven the predictions of str concerning moving clocks (Van Flandern doesn't mention the gtr effects, but they are also significant). On careful reading, in this paper he actually appears to be saying in effect that anything that can be explained using str can be explained just as well using the Lorentz ether theory (let), a theory which he has never specified but which is usually taken to be mathematically equivalent to str, but with a different interpretation of Lorentz transformations, one which most physicists since Lorentz's day have found implausible. However, more recently, in postings to sci.physics.relativity, Van Flandern has clearly stated that he believes that changes in electrostatic and gravititostatic potentials are transmitted instantly (literally!), just as if electromagnetism and gravity were truly governed by the Poisson equation, a viewpoint which is mathematically utterly inconsistent with both str and gtr, contrary to his claims in an earlier (and also wildly erroneous) paper, "Possible new properties of gravity", Astrophysics and Space Science 244 (1996), also available here.

Since Van Flandern also claims special expertise in the GPS system, by virtue of having worked as a "consultant" in its design, some nonphysicists might take his claims seriously. However, his name does not appear anywhere in the official GPS bibliography kept by NOAA, and according to the article by Farrell, Van Flandern "left the U.S. Naval Observatory under something of a cloud."

(On the other hand, Neil Ashby (Physics, University of Colorado) has written extensively in journals such as GPS World, IEEE Spectrum, and has written some of the official documentation for the GPS system; at this point, some readers may want to skip directly to General relativity in the global positioning system, a short paper by Ashby which quickly debunks Van Flandern's claims.)

Before we can understand why, contrary to Van Flandern's assertions, relativity theory is actually working just fine in the GPS, we need to understand the basic principles behind its design and daily operation, so I'll begin by explaining (in an oversimplified way) how the GPS works, what the most important non-relativistic sources errors are, and how they are overcome. Once this is out of the way, I'll discuss the relativistic sources of error, and how they are overcome.

The GPS has revolutionized the transportion industry, as well as offering unprecedented position and chronometer accuracy to field researchers involved in biology, botany, ecology, geology, and petroleum exploration, among others. While the system was not designed as a test of gtr, it turns out that in addition to numerous extremely complex Newtonian physical issues which must be taken account of, there are also about a dozen distinct str and gtr effects which must be taken into account in the design and operation of the system. This takes quite a bit of explaining, since the actual system is quite complex, and I certainly won't attempt to explain all the engineering details here (although I'll provide links to sites where you can obtain more detailed information).

The purpose of the GPS is, of course, to allow users with a GPS receiver to determine his/her location on the Earth, including altitude, latitude, longitude, and to inform the user of the precise Universal Coordinated Time (UTC) maintained by a reference atomic clock at the U.S. Naval Observatory in Bethesda, MD, as well as velocity and heading (if the user is in a moving vehicle, aircraft, or vessel), using coded signals transmitted by a constellation of 24 Earth orbiting satellites.

A good way to start thinking about the general principle behind GPS is as follows. Suppose you know your precise range r1 to an object S1 with precisely known position x1 (a point in E^3, ordinary Euclidean space). Then you know you are located somewhere on a sphere of radius r1 and center x1. Next, suppose you also know your precise range r2 to a second object S2, with precisely known position x2. Then you know you are located somewhere on the circle which is the intersection of the first sphere with the sphere of radius r2 and center x2. Now suppose you also know your precise range r3 to a third object S3, with precisely known position x3. If you know all three things at the same time, then you know you are located on one of the two points in which three circles intersect! This process is called trilateration by geographers.

The basic idea behind GPS is to adapt this idea by providing users with a "constellation" of satellites as "orbiting landmarks", which always know their precise position with respect to the Earth's surface, as well as the precise UTC at their location, and which continually transmit this dual information at regular intervals. The current (Block 2) GPS constellation consists of (at least) 24 Earth orbiting satellites, called SV's, in circular orbits about 11,000 nm (20,200 km) above the Earth's surface (that is, 26,750 km above the center of the Earth), traveling at 4 km/sec, giving an orbital period of precisely twelve sidereal hours; that is, the satellites rotate once every twelve hours with respect to the fixed stars, not with respect to the Earth, which is of course itself rotating underneath the satellites. (The actual number of satellites in orbit varies from time to time because new ones are launched, with a Delta 2 rocket, as old ones begin to wear out. The design life of each satellite is 7.5 years.)

This means that each satellite comes over the same location along the same track over a fixed location on the surface of the Earth every 24 hours, or rather, four minutes earlier each day; the four minute discrepancy is due to the Earth's advance in its own orbit around the Sun. Incidentally, a common question is: why are the satellites not in geostationary orbits, like many communication satellites? The answer is that geostationary orbits are only possible over the equator, and as we've seen, trilateration won't work unless you can measure your distance to satellites in more than one plane. Since the designers of GPS were also looking ahead to future space-based applications, e.g. actively steering spacecraft in perfect formation (despite buffeting from the solar wind) by keeping their position using GPS, arranging the satellites so that their orbital period is precisely 12 sidereal hours turns out to be the simplest choice.

Each satellite is visible above the horizon of a stationary user for about five hours. The satellites each orbit in one of six equally spaced planes (sixty degrees apart), each inclined at about 55 degrees to the equatorial plane of the Earth, and with (at least) four SV's in each plane. This configuration ensures that at any given at time and any given place on the Earth, five to eight satellites are in a direct line-of-sight from the user.

Now, if one can provide a way for a ground receiver to synchronize its clock with the satellite clock, a simple measurement of the time delay in the signal received from a satellite in view of the Earthbound user, should result in a precise range to that satellite. Imagine for example a ground receiver which finds and locks onto one visible satellite, synchronizes its clock with the satellite clock, measures the time delay, computes the range, stores this number along with the reported position of the satellite, then locks onto a second visible satellite, and repeats this process until ranges and positions from three satellites have been obtained. This isn't how GPS works, but it's getting close!

Before delving into a more accurate explanation of how GPS works, let's examine some nonrelativistic sources of error in the simple procedure I have just outlined. As I said, in principle, determining the range to each satellite is a matter of a simple computation: the time delay of each signal from the satellite, multiplied by the speed of light, gives the range! However, the signals are not in fact transmitted in along the geometric line of sight, because they are diffracted at the upper boundary of the ionosphere (at about 1000 km above the Earth's surface) and then again at the boundary between the ionosphere and the troposphere (at about 70 km above the Earth's surface). These boundary layers are move up and down, depending upon the time of day, the latitude, and other factors. Moreover, the lower 8-13 kilometers to the atomosphere are active participants in local weather conditions, and the speed of light in air is different from the speed of light in vacuo (and the temperature of the air matters!). In addition, satellites which are closer to the horizon will transmit signals which are subject to more atmospheric disturbance than those near the zenith. In sum, ionospheric, tropospheric, and local weather conditions can result in errors of ten meters or more. Furthemore, it is possible for the same signal to arrive at the receiver at different times, having taken multiple paths to get there; this can account for another half meter or so of inaccuracy. All these sources of error must be taken account of and corrected by the GPS system.

Further "Newtonian" sources of error arise in determining the precise position of each satellite. In principle, a Keplerian orbit about the Earth is determined by six orbital parameters, usually taken to be the following:

However, there are numerous influences which perturb the orbits of the satellites in an extremely complicated way. First, the Earth is not a perfect oblate spheroid, but has numerous bumps and valleys and mountains, and the mountains, being closer to the satellite, "tug" a bit more firmly. In addition, the Earth has an inhomogeneous mass density, so that even if it were a perfect sphere, its "gravitational tug" on the satellite would still vary from place to place in the orbit. For this reason, spherical harmonics up to order eight are used to model the gravitational potential at each point on the orbit. This requires a detailed gravimetric (density inhomogeneity) and geodetic (shape inhomogeneity) surveys of the Earth! Just to make things more complicated, the shape of the Earth is dynamic, principally because the lunar tides pull the surface of the Earth closer to the satellite twice a day (and the Moon itself also tugs on the satellite more when it is closer). Furthermore, the varying radiation pressure from sunlight striking the satellite turns out to be a significant source of error. This effect is extremely complex because the different parts of the satellite have different albedos, diffusivities, etc., and of course present different aspects to the Sun as the satellite and the Sun and Earth move around. This radiation pressure effect turns out to be the most difficult kind of error to eliminate in the GPS system. And of course, orbital injection is never perfect, and none of the GPS satellites are in perfect circular orbits, or aligned precisely as designed. There is also slight frictional drag due to the (extremely diffuse) atmosphere at the location of the satellite orbits, which causes a steady decay of each orbit, but this effect is largely negligible. (If you are curious about the details, see this file for the names and orbital parameters as determined by NORAD of the currently operational GPS satellites.)

Now, as I said, I have not yet described how GPS really works. There are two fundamental issues I have not addressed:

There is also a further component to the GPS system which I have not yet discussed. In addition to the satellites themselves, the space segment of the system, and the millions of hand held GPS receivers operated by people around the world, the user segment, there are also four unmanned ground stations (one located in Hawaii, a second on Kwajalein, an atoll in the Pacific Ocean; a third on Diego Garcia, an island in the Indian Ocean, and a fourth on Ascension Island in the Atlantic Ocean), which are devoted to carefully tracking the position of each GPS satellite, taking readings twice every three seconds. Each station automatically uses local weather and ionospheric conditions to average the tracking data, and every quarter hour, reports its best estimate of the position of each visible satellite to the master ground control station, which is located at Schriever AFB, Colorado Springs, CO. Here, the orbits and on-board clocks of each satellite can be adjusted if neccessary, using more sophisticated computer models (and bigger computers!) than can be carried on the SV's. This is the third component of GPS, the control segment.

As always, the devil is in the details, and it is the facts reported in the previous two paragraphs which have permitted Van Flandern to make a plausible sounding, but nonetheless completely incorrect, allegation. Let's take a closer look at how the above two issues are met in the design of the GPS.

Each GPS satellite (SV) has a mass of 930 kg and a length of 5.1 m, and carries four atomic clocks (two Cesium and two Rubidium oscillitors). These clocks are highly stable and accurate to within about 3 nanoseconds per day. They are occasionally reset from the ground, as required, in order to maintain sufficient synchrony with GPS time (for the moment, we can consider this to be the same as UTC time). Each satellite transmits two types of signals:

(This description is oversimplified; in fact, there are three modulations superimposed on the two carrier frequencies, as shown in this figure, taken from Peter Dana's website. Incidentally, one reason for using two frequencies in the GPS system is that the ionospheric disturbances are frequency dependent, so PPS users can use both frequencies to partially compensate for this source of error, using a mathematical model of the ionosphere. [SPS users are stuck with the errors!] The tropospheric disturbance is partially compensated for by estimating it using another mathematical model. However, all these ramifications are irrelevant for our purposes.)

In addition to transmitting its on-board clock time and estimated orbital parameters and its own estimated current location (the basis ephemeris information), its assessment of its own "health", its best estimate of the current location of the other GPS satellites (the basic almanac information), each satellite must also transmit identifying data so that the ground receivers can distinguish between signals coming from different visible satellites, and lock onto four or more in turn. In addition, the system is designed so that it takes only 30 seconds to obtain a first, rough position, and 630 seconds to obtain a full precision position. Moreover, it is desirable to build in some degree of error protection, so that if the signal is slightly disrupted, the receiver can compensate for some lost bits (or if neccessary wait for the next transmission).

To meet these requirements, the signals transmitted by the satellites are neccessarily rather complex. Each signal consists of nine packets, with an overall transmission rate of 50 bits per second (including parity check digits). Each packet consists of about 200 Bytes and is divided in turn into five 300 bit subpackets, with one packet transmitted every 30 seconds. The first subpacket contains the clock corrections; the next two contain the precise location (ephemeris data) of the transmitting satellite; the next contains the UTC time and some ionospheric correction data, and the last contains partial almanac data for all 24 satellites. (The complete almanac and ionospheric data is built up over the 7.5 minutes required to transmit nine packets, making up one complete signal.) There is some error-detection/correction built into the coding: the subpackets consist of 30 bit words, of which 24 bits contain data and the remaining 6 bits allow parity checks.

So, how can the ground receiver synchronize its clock with the satellite clock, in order to carry out the basic time delay computation outlined above? The answer is that it doesn't--- instead of sequentially locking onto and synchronizing clocks with four different satellites, as suggested above, the simplest (and cheapest) GPS receivers first lock onto the signal of one satellite, i.e. compare their internally generated C/A signal with the satellite signal until they get their own (cheap, inaccurate) clock in "roughly in synch" (modulo a still undetermined offset) with GPS time, and then records a reception time (by its own clock). It then locks onto the next signal, and repeats this process until four reception times (modulo undetermined offsets) have been recorded. Only then does it combine this data to determine the common offset of its clock from the satellite clocks, in effect resynchronizing its clock with three on-board atomic clocks. It then uses trilateralization and the declared position of each satellite as described above to determine the its own position. One way to think about this is that by locking onto four satellite signals, the receiver can determine its position and one more number, the offset of its clock from the highly stable, accurate, and mutually synchronized clocks carried by the satellites.

The grand result is that SPS users can obtain their position accurate to within 100 meters (latitude and longitude), their altitude to within 150 meters, and the universal time to within 350 nanoseconds. PPS users can obtain their position accurate to within 20 meters, their altitude to within 30 meters, and the universal time to within 200 nanoseconds. Actually, even this is merely the tip of the iceberg--- more sophisticated users can use two or more sequential ground receivers, or a parallel receiver capable of locking onto several satellite signals at once, and can use various least squares estimations to collate the data collected in order to considerably improve on the stated accuracies, typically to a few meters in the position accuracy. Furthermore, stationary receivers (e.g. used by geologists to study the motion of tectonic plates, or volcanic terrain) currently achieve position accurracy on the order of one mm!

So far I've said quite a lot about GPS, and have yet to mention relativity. It is when we begin to discuss how GPS time, a sort of "imaginary time" stitched together from the onboard clocks of the 24 satellites, as correlated by the master control station, is converted to UTC time, that we can see some explicit relativistic computations entering into the daily operation of the GPS; see for example this figure, also from Peter Dana's site. I won't try to explain how GPS time is constructed, other than to say that one basic idea is to compare time signals from a given satellite as received by two ground stations, using a ground link between the two stations, with suitable time delays for the transmission times. Because the sphere in which the SV orbits reside is quite large (more than 40 million meters), the light time travel delays (which are of course the key to making the system work in the first place) must be very carefully compensated for in correlating the onboard clocks.

The first 10 GPS satellites, comprising Block I, were used for testing and for military geolocation, and were launched beginning in 1978. The next 24 satellites, comprising Block II, were launched between 1989 and 1994; these are the SV's used in the operational GPS system. The way in which Van Flandern's claims quoted above are misleading is now easily summarized:

At this point, I can do no better than send readers who have not already been there to Neil Ashby's paper for a detailed accounting of str and gtr effects which are significant in the GPS system. For a slightly longer version of this paper, look at this issue of Matters of Gravity, a newsletter in the field of gravitation physics. See also this paper by Thomas Bahder and references therein. You can also try this paper by Charles W. Misner (Physics, University of Maryland), and this one by Clifford Will (Physics, Washington University). You can find additional references in the posts by Tom Roberts included in this collection.

By the way, there is a Russian system, GLOSNASS, which has fewer satellites but is similar to GPS. I have seen the claim that this system does not exhibit the relativistic effects mentioned in Ashby's paper. This is of course completely false.

To conclude, here are some websites offering information on the general design and operation of the GPS:

Acknowledgements and Disclaimer

I [Chris Hillman] am grateful for helpful comments from several people. All opinions expressed on this page and any errors or scientific inaccuracy are of course the sole responsibility of Chris Hillman.

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