View source for A Counter Example of Einstein's Covariance Principle

{{Infobox paper
| title = A Counter Example of Einstein's Covariance Principle
| url = [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_2368.pdf Link to paper]
| author = [[Chung Y Lo]]
| keywords = covariance, Einstein, General Relativity, Sun
| published = 2009
| num_pages = 11
}}

'''Read the full paper''' [http://www.naturalphilosophy.org/pdf/abstracts/abstracts_2368.pdf here]

==Abstract==

Recently, calculation of the deflection angle to the second order also shows gauge invariance in mathematics. Nevertheless, careful analysis shows that this calculation actually implies that the theory is intrinsically not gauge invariant since, for each gauge, the shortest distance r<sub>0</sub> from the sun center is different from that for another gauge. Some argued that r<sub>0</sub> is just a label, but not a physical quantity. This is directly in conflict with Einstein's calculation that the deflection angle is 4KM/r<sub>0</sub>, where M is the total mass of the sun and&nbsp;K&nbsp;= G/c<sup>2</sup> = 7.425x10<sup>-29</sup> cm/g. Thus, r<sub>0</sub> is not just a label. Hence, Einstein's covariance principle is intrinsically not valid in physics. Thus, logical maturity is currently a major problem in general relativity.

==Overview==

This is a conference paper (16th Annual Natural Philosophy Alliance meeting, University of Connecticut, 2009) from C. Y. Lo of the Applied and Pure Research Institute, and it is a narrow, technical attack on one specific doctrine: [[Albert Einstein|Einstein]]'s principle of general covariance, quoted in the paper as the requirement that "the general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates". Lo's opening move is to note that the principle would be unproblematic if every equation of the theory were an unrestricted tensor equation — but Einstein's [[Equivalence Principle|equivalence principle]] is not a tensor condition, and gauge conditions such as the harmonic gauge are not tensor conditions either. So, he argues, the principle is not a theorem but a claim requiring test.

The test he proposes uses the second-order [[Gravitational Lensing|deflection of light]] by the sun, following calculations by Bodenner and Will (2003) and G&eacute;rard and Piereaux (1999) which showed the deflection angle to be gauge invariant when expressed as a function of the impact parameter ''b''. Lo's counter-move is that this invariance is bought at a price: if &alpha;(''b'') is the same in every gauge, then the ''closest distance'' ''r''<sub>0</sub> from the sun's centre cannot be the same in every gauge. Since Einstein's own first-order result is written &alpha; = 4&kappa;''M''/''r''<sub>0</sub> and has been experimentally confirmed, ''r''<sub>0</sub> must be a physical quantity and not, as his critics maintain, "an arbitrary, human-made label". The conclusion is that [[General Relativity|general relativity]] is intrinsically not gauge invariant and that the covariance principle is invalid in physics.

==The argument==

===Setting up the three gauges===

Lo works with the static spherical form d''s''<sup>2</sup> = ''A''<sup>2</sup>(''r'')''c''<sup>2</sup>d''t''<sup>2</sup> &minus; ''B''<sup>2</sup>(''r'')d''r''<sup>2</sup> &minus; ''D''<sup>2</sup>(''r'')''r''<sup>2</sup>(d&theta;<sup>2</sup> + sin<sup>2</sup>&theta;&nbsp;d&phi;<sup>2</sup>) and writes out the harmonic, isotropic and Schwarzschild solutions explicitly. He emphasises at the outset that a spherical coordinate system is necessarily defined on what he calls the '''Euclidean-like structure''' of the frame of reference — a structure in which the Pythagorean theorem holds — and is therefore "certainly not an arbitrary coordinate system".

The diffeomorphisms relating the three are standard: ''r''&prime; = &rho; + ''M''&kappa; between Schwarzschild and harmonic, and ''r''&prime; = ''r''(1 + ''M''&kappa;/2''r'')<sup>2</sup> between Schwarzschild and isotropic. Lo's point is that these very relations show "the Euclidean-like structures are different for different gauges": the impact parameter ''b'' comes out gauge invariant precisely because the radial coordinate does not.

===The second-order deflection formula===

The deflection is written &alpha;(''b'') = 4''m''/''b'' + (15&pi;/16)(''m''/''b'')<sup>2</sup> + O(''m''<sup>3</sup>/''b''<sup>3</sup>), with ''m'' = ''GM''/''c''<sup>2</sup> and ''b'' = ''D''(''r''<sub>0</sub>)''r''<sub>0</sub>/''A''(''r''<sub>0</sub>). To first order ''A'' &asymp; ''D'' &asymp; 1, so &alpha; &asymp; 4''m''/''b'' &asymp; 4''m''/''r''<sub>0</sub> — Einstein's result. But when the same series is re-expressed in terms of each gauge's own closest distance ''d'', the second-order coefficient changes: it is (15&pi;/16 &minus; 1) for the Schwarzschild gauge and (15&pi;/16 &minus; 2) for the harmonic and isotropic gauges. Equivalently,

: ''b'' &asymp; ''m'' + ''r''<sub>0</sub>&nbsp;&nbsp;(Schwarzschild) &nbsp;&nbsp;or&nbsp;&nbsp; ''b'' &asymp; 2''m'' + ''r''<sub>0</sub>&nbsp;&nbsp;(harmonic, isotropic)

"From relation (7)," Lo writes, "it is even clearer that ''b'' and ''r''<sub>0</sub> cannot be both gauge invariant." Given a measured number for ''r''<sub>0</sub>, one must decide which of the three formulas to apply, so "such gauge invariance has no practical meaning unless the first order approximation of the metric is known."

===Coordinate light speeds===

A supporting calculation gives the coordinate radial and transverse light speeds (d''s''<sup>2</sup> = 0) in the three solutions and shows them to be explicitly different even though the solutions are diffeomorphic. Lo remarks that some theorists therefore "insist on the measured light speed in vacuum is always ''c''", though Einstein himself had written that the [[Speed of Light|light speed]] is no longer constant once a ray bends.

===The dispute over what counts as measurable===

Roughly half the paper is a rebuttal of the counter-arguments, quoted at length in an appendix. Against Einstein's own defence — that all space-time verifications reduce to coincidences, so a reference system "serves no other purpose than to facilitate the description of the totality of such coincidences" — Lo objects that "the meaning of measurements is crucially omitted", and that a coordinate system in physics is tied to a measurement method while a coordinate system in mathematics need not be. Against Bodenner and Will's identification of ''b'' with the ratio of the photon's angular momentum to its energy, he replies that measuring ''J'' raises the same problem. Against a Royal Society board member's charge that treating ''r''<sub>0</sub> as physical is "reification — the treatment of a hypothetical construct as if it were a concrete physical entity", he replies that a measurement of ''b'' necessarily involves ''r''<sub>0</sub>, so if coordinates were arbitrary labels, "where the physical meaning of the impact parameter ''b'' comes from?" He cites [[Alfred North Whitehead]]'s 1922 objection that Einstein's identification of measurement with the metric "leaves the whole antecedent theory of measurement in confusion", and Zhou Pei-Yuan's argument that since tensor components are expressed in coordinates, "the gauge does matter".

===Positive programme===

Lo's own resolution is that a physical gauge is unique for a given frame, fixed by the requirement that measuring rods be attached to the frame of reference so that a realisable Euclidean-like structure emerges. He claims the Maxwell-Newton Approximation has been proven to be the valid first-order approximation independently of the Einstein equation, and that this selects ''b'' &asymp; 2''m'' + ''r''<sub>0</sub>, making the second-order deflection obtainable from a measurement of ''r''<sub>0</sub>. He closes with a broad indictment — that Einstein "did not have adequate background in mathematics", that Gravity Probe-B "ignores the problem of covariance", that Straumann, Wald and Will failed to respond to an inconsistency identified by Bondi, Pirani and Robinson in 1959, and that NASA's Pioneer anomaly shows the theory to be inadequate.

==Assessment==

The technical core of this paper is correct, and it is worth separating from the polemic around it. The diffeomorphisms Lo writes down are the standard ones; the impact parameter really is ''b'' = ''D''(''r''<sub>0</sub>)''r''<sub>0</sub>/''A''(''r''<sub>0</sub>); and expanding it gives ''b'' &minus; ''r''<sub>0</sub> = ''m'' in Schwarzschild coordinates and 2''m'' in harmonic and isotropic coordinates, exactly as claimed, which is why the second-order coefficient shifts by 1 or 2 when the series is written in the radial coordinate. His &kappa; = ''G''/''c''<sup>2</sup> = 7.425&times;10<sup>&minus;29</sup> cm g<sup>&minus;1</sup> checks out, and 4&kappa;''M''<sub>&#9737;</sub>/''R''<sub>&#9737;</sub> gives 1.75 arcsec as it should. The observation that gauge invariance of &alpha;(''b'') and gauge invariance of ''r''<sub>0</sub> cannot both hold is not a mistake — it is a correct statement about the same fact that his opponents are asserting, read in the opposite direction. Lo also puts his finger on something real in the endnotes: a Euclidean-like structure defined by rods attached to the frame is an operational stipulation, and it is fair to ask which gauge such a procedure actually realises.

Where the argument fails is in the step from "''r''<sub>0</sub> is gauge dependent" to "''r''<sub>0</sub> is measured, therefore the theory is inconsistent". The load-bearing claim is that "&alpha;(''r''<sub>0</sub>) has been verified as accurate up to the first order", and this is where the numbers should be checked. For the sun, ''m'' = ''GM''/''c''<sup>2</sup> = 1.48 km against ''R''<sub>&#9737;</sub> = 6.96&times;10<sup>5</sup> km, so ''m''/''r''<sub>0</sub> = 2.1&times;10<sup>&minus;6</sup>. The disputed difference between ''b'' = ''r''<sub>0</sub> + ''m'' and ''b'' = ''r''<sub>0</sub> + 2''m'' is therefore a fractional difference of 2&times;10<sup>&minus;6</sup> in the deflection — that is, it lives entirely in the second-order term, roughly one microarcsecond at the solar limb. A first-order confirmation cannot possibly distinguish the two, because at first order ''b'' and ''r''<sub>0</sub> are the same number to within a part in half a million. And the second-order term has never been measured: the sharpest constraints on light bending are the Cassini radio-delay determination of &gamma; to 2.3&times;10<sup>&minus;5</sup> and VLBI at comparable precision, both one to two orders of magnitude short of even the full 2.7 microarcsecond second-order deflection, let alone the microarcsecond gauge difference within it. The "counter example" of the title is thus not an empirical counter example. It shows that a coordinate label is coordinate dependent — which everyone agrees on — and then asserts a measurement that has not been performed to convert that into a physical contradiction.

The light-speed calculation cuts the same way. That the coordinate speed d''r''/d''t'' differs between diffeomorphic solutions is exactly the standard reason for saying coordinate speed is not an observable; the locally measured speed is ''c'' in every one of them. Lo presents this as a violation, but it is a demonstration of the point he is disputing.

Two further difficulties should be recorded. First, a substantial part of the paper is not physics but complaint — that a Royal Society board member "acts irrationally" and "has problems in at least three areas, namely: physics, mathematics, and logic", that Einstein lacked mathematical background, that referees are "collectively ill-informed". None of this bears on whether ''r''<sub>0</sub> is observable, and it displaces argument that the case needs. Second, the empirical props have since given way: the Pioneer anomaly, invoked here as showing "Einstein's theory is clear inadequate", was accounted for by anisotropic thermal recoil from the spacecraft's own radioisotope generators in analyses published shortly after this paper; and Gravity Probe-B, dismissed here for ignoring covariance, reported geodetic and frame-dragging precessions consistent with general relativity in 2011. The paper's own preferred alternative, the Maxwell-Newton Approximation as an independently proven first-order gauge, is asserted by reference to Lo's earlier work rather than derived here, so a reader cannot assess it from this document.

The paper is best read as a precise and largely correct piece of technical bookkeeping about what is and is not gauge invariant in the second-order deflection, wrapped around an interpretive claim — that ''r''<sub>0</sub> is measured — that the measurements do not yet support.

==See also==

* [[Chung Y Lo]]
* [[General Relativity]]
* [[Gravitational Lensing]]
* [[Equivalence Principle]]
* [[Albert Einstein]]
* [[Alfred North Whitehead]]
* [[Speed of Light]]
* [[Perihelion Precession of Mercury]]
* [[Maxwell's Equations]]

[[Category:Scientific Paper|counter example einstein 's covariance principle]]

[[Category:Relativity|counter example einstein 's covariance principle]]
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