On Einstein∨s Principle of Covariance and the Royal Society* by C. Y. Lo Based on Einstein∨s ¨principle of covariance〃 [1, 2], some argued that diffeomorphic solutions (i.e., mathemati-cally they can be transformed from each other) would be physically equivalent. In Einstein∨s ¨principle of covariance〃, what has been missing is the relation between coordinates and measurements. A difference between physics and mathematics is that a coordinate system in physics is related to measurement and its method, whereas a coordinate system in mathematics need not be related to these [3]. For the covariance principle, Einstein∨s supporting arguments [1] are as follows: "That this requirement of general covariance, which takes away from space and time the last remnant of physical objectivity, is a natural one, will be seen from the following reflexion. All our space-time veri-fications invariably amount to a determination of space-time coincidences. If, for example, events con-sisted merely in the motion of material points, then ultimately nothing would be observable but the meetings of two or more of these points. Moreover, the results of our measurings are nothing but verifi-cations of such meetings of the material points of our measuring instruments with other material points, coincidences between the hands of a clock and points on the clock dial, and observed point-events hap-pening at the same place at the same time. The introduction of a system of reference serves no other purpose than to facilitate the description of the totality of such coincidences." Note that the meaning of measurements is crucially omitted. Moreover, in order to predict events, one must be able to relate events of different locations in a definite manner [3]. This means a valid method of measurement is necessary as Eddington [4] commented.^1) Of course, the outcome of a real experiment cannot depend on a choice of physical coordinates because their physical meaning is based on measurements. However, in general relativity the physical meaning of the coordinates would implicitly depend on the gauge, which may or may not be physically realizable [5]. As shown by the diffeo-morphic relation between the Schwarzschild and the isotropic solutions [6], they cannot be both physically realizable. This relation shows also that the gauges are not arbitrary. A basic error is assuming that physical results can be ob-tained from any gauge. Einstein∨s theory of measurement, which is invalid in physics [7], is based on invalid applications of special relativity [1] such that the notion of local distance in Riemannian geometry could be justified. Whitehead [8, p.83], strongly objected, ¨By identifying the potential mass impetus of a kinematic element with a spatio-temporal measurement Einstein, in my opinion, leaves the whole antecedent theory of measurement in confusion, when it is confronted with the actual conditions of our perceptual knowledge. The potential impetus shares in the contingency of appearances. It therefore follows that measurement on his theory lacks systematic uni-formity and requires a knowledge of the actual contingent physical field before it is possible.〃 Unfortunately, Whitehead also rejected Einstein∨s equivalence principle since Einstein falsely claimed that his theory of measurement, which is inconsistent with the observed light bending [9, 10, 11], was based on his equivalence prin-ciple [1]. In a way, Einstein∨s equivalence principle is an ingenious answer to the question of measurement for a physical Riemannian space. For a physical space, from the invariant ds, his principle provides a measurement of the space con-tractions and the time dilation [7, 9, 12]. Einstein∨s error is that he overlooked that his equivalence principle provides only a dynamic method of measurement for space contractions since the measurements are done in a movable local space resting but under the influence of gravitational acceleration, but are not done in the frame of reference with a static method of attachment. Moreover, a physically realizable Euclidean-like structure^2) is operationally defined in terms of spatial meas-urements essentially the same as Einstein defined the frame of reference for special relativity [13]. If the measuring rods are attached to the frame of reference, since the measuring rods and the coordinates being measured are under the same influence of gravity, a Euclidean-like structure emerges as if gravity did not exist [9, 12]. Hence, independent of the space-time metric, a physical space must have a frame of reference with a realizable Euclidean-like structure. A good example is metric of the rotating disk [7]. This is consistent with fact that a Euclidean-like structure appears in all Einstein∨s calculations of verified predictions. Moreover, since a gauge may or may not be physically realizable, the issue is really whether a gauge is valid in physics. Some theorists argued,¨ deflection of light bending is described by solving equations of motion in some background curved spacetime. The result must be independent of any coordinates since the light curve is a geometric object. The physical observation should also be possible to measure the angles which are independent of coordinates. Coordinates are just representations of geometric objects. I think that any physical argument can be expressed in a very clear way especially through mathematics.〃 However, this argument actually involves some implicit assumptions. Physical coordinates are representations of geometric objects since such coordinates are based on real measure-ments. However, the coordinates for a gauge is based on assumed measurements, which may or may not be realizable. Thus, the coordinates for a gauge need not be representing geometric objects. A crucial criterion is that physics in-volves actual measurement, whereas mathematics involved only assumed measurements or no measurements at all. To justify the ¨covariance principle〃, it was also argued [14], ¨If one makes the mistake of attributing incorrect physical properties to one's coordinates -- for example, assuming that because two objects in different coordinate systems are both labeled r, they must measure the same distance, or assuming that any coordinate labeled r must measure proper radial distance -- then one can mistakenly conclude that computations in different coordinate systems disagree. But as long as one identifies genuinely measurable quantities, their values cannot depend on coordinates.〃 However, it has been shown, for the case of the precession formulas, such ¨genuinely measurable〃 quantities do not exist at all [3]. Moreover, such a quantity is only a mathematical illusion since its invariance is not supported by measurements. Some theorists simply cannot tell the difference between mathematics and physics, which is based on measure-ments. They discuss about physics without addressing the issue of measurement, and make errors [3, 5, 7, 15-18].^3) An interesting example is a ¨Board Member Comments〃 of the Royal Society [19] on a paper of Lo [5] on the deflec-tion of light as follows: ¨This paper exhibits a fundamental misunderstanding of the meaning of coordinates and invariants. In section 2, the author considers several different coordinate systems that all have a coordinate called "r". He finds that the value of "r" that labels a point outside the Sun depends on which meaning of "r" he chooses. This is certainly true. The conclusion that "the shortest distance r0 from the center of the sun is not gauge invariant," on the other hand, is silly. The value of a coordinate is, of course, not "gauge in-variant" if one chooses to use the same letter to denote two different coordinates. But the value of a co-ordinate is not a "distance," either; it is an arbitrary, human-made label, and nothing more.^4) The author states, correctly, that "it is clear that b and r0 cannot be both gauge invariant." Indeed, the coordinate value r0 is not gauge invariant -- the value of a coordinate depends on what coordinate sys-tem one uses. The impact parameter b, on the other hand, is a physical observable, a proper distance,^4) and does not depend on one's arbitrary choice of labels. This is the object one actually measures when one measures a distance. Note that the author's complaint has nothing in particular to do with general relativity. One could equally well write down ordinary polar coordinates on the plane, redefine r' = r + a, and complain that "distance to the origin" is not gauge invariant. Ultimately, it is no different from saying, "You say that John Smith is 6' tall, but I know someone named John Smith who is only 5'8" tall, so the height of your John Smith must be ill-defined." The fundamental error here is what philosophers call reification -- the treatment of a hypothetical con-struct as if it were a concrete physical entity. The space around the sun does not come equipped with lit-tle labels giving the coordinates of points. A coordinate system is, rather, an arbitrary human-made set of labels, that can be chosen in any way one desires.^4) To compute a real physical quantity, one must re-late one coordinate-independent object to another. Coordinates are useful intermediate quantities, but they disappear at the end. By insisting that the quantity he calls "r0" must be a physical distance, even as he shows that it can be redefined arbitrarily by a choice of relabeling, the author shows a basic misun-derstanding, not only of general relativity, but of all physics.〃 To justify the invalid invariance, this board member acts irrationally. He claimed the shortest distance r0, is just a label. Moreover, according to his understanding of coordinates, r0 should be considered as a ¨hypothetical construct〃. This is absurd since r0 is a physical quantity as shown in Einstein∨s calculations [1, 2]. In physics, physical quantities must have ways to be measured. If the coordinates were just arbitrary labels, where the physical meaning of the impact pa-rameter b comes from? It should be noted that, r' = r + a does not make sense in mathematics when r = 0, but in the related gauge transformation a = M [6] and r∨> 2M. This manifests that this board member does not understand the notion of gauge as well as related mathematics. Moreover, he assumed implicitly that two John Smith exist. However, physics requires that there is only one length for the shortest distance r0 from the sun center just as one length for the impact parameter b. This shows that he tried to justify logical errors with an incorrect assumption. Note also that b is independent of only some gauge choices, which are not arbitrary. Thus, the right question should be what the correct physical gauge is. The meaning of coordinates is implicitly included in all physical quantities. This board member claimed, ¨To compute a real physical quantity, one must relate one coordinate-independent object to another〃 after reciting Einstein∨s erroneous view that has never been put into practice [1-3, 7, 12, 20]. How-ever, this board member does not really know what he is talking about. If the deflection angle must be calculated in term of the coordinate-independent object such as the impact parameter b, what should be used to calculate b, and so on . As expected, this board member has a problem in logical thinking. A basic problem is however, that he implic-itly assumed that physical results can be obtained from any gauge. Thus, he absurdly believed that any gauge non-invariant quantity were not a concrete physical entity. Moreover, ¨genuinely measurable quantities〃 for the case of the precession formulas do not exist, but this board member did not address such a failure. Could one consider this just as his oversight? ^5) It seems, for this case he has difficulties in justifying his view that denounces physical quantities as ¨hypothetical constructs〃. Also, unlike Zhou [21] ^6) this board member failed to appreciate Einstein∨s equivalence principle ^7) that implies the covariance principle invalid [7]. In short, this board member cannot tell the difference between a result from calculation based on mathematical notion of r0 and an actual measurement of ¨b〃 that requires a choice between r0 + M and r0 + 2M. He also made inva-lid assumptions to justify his errors. In other words, he does not understand the difference between mathematics and physics. He probably believed that nobody know what the correct physical gauge is, and he felt save to claim r0 as an arbitrary ¨hypothetical construct〃. However, he overlooked that, to show r0 being not arbitrary, it is sufficient to know whether the first order approximation of a metric is valid. The gauge transformations [6] show the gauges are not arbi-trary and its first order is also the relation among the first order of the metrics. Apparently, this board member and his colleagues maintain narrow and out-dated knowledge based on misun-derstanding of physics. Consequently, they did not know that the Maxwell-Newton Approximation has been proven as the valid first order approximation [22] because of the need to explain observations of the binary pulsars [18, 23]. Concurrently, the Schwarzschild solution is proven invalid.^8) Consequently, the core of his arguments, to claim the shortest distance r0 from the center of the sun as arbitrary, is now clearly only nonsense. Nevertheless, had he act care-fully according to logic demands, he could have avoided such errors. However, since the covariance principle itself is a product of inadequacy in logic, its followers inevitably have the same kind of problem. Note that the covariance principle is only an interim measure of Einstein [1]; he probably did not anticipate having faithful followers. Fundamental concepts in a great theory are often difficult to grasp [24]. To mention a few, this happened to New-ton, Maxwell, Planck, Schőrdinger, and C. N. Yang [25]. Einstein is simply not an exception of such a problem. Unlike Newton, Einstein did not have adequate background in mathematics, and this affects the logical structure of his theory. He believed the solutions with different gauges as equally valid [2], but did not see that his covariance princi-ple is inconsistent with his notion of weak gravity [26]. As Zhou [27] pointed out, the notion that gauges do not matter just does not make sense in physics. Nevertheless, Einstein is a great theorist since the implications of general relativity such as the need for unification have been discovered and verified [28, 29]. Einstein∨s accurate predictions created a faith on his theory. However, theoretical developments [29, 30] and NASA∨s discovery of the Pioneer anomaly imply that Einstein∨s theory is clear inadequate [31, 32]. Moreover, Einstein∨s previous conceptual errors are becoming obstacles to the progress of general relativity. To mention a few, theoretically anything related to the covariance principle is glossed over. The Stanford experiment Gravity Probe-B on precessions ignores the problem of covariance. Prominent theorists failed responding to the chal-lenge from the Royal society on inconsistence in general relativity [26]. Einstein∨s equivalence principle and its con-sequences are essentially ignored [18, 30, 33]. Thus unification being a natural consequence of general relativity was overlooked for a long time since 1916 [34, 35]. Zhou∨s experiment [21] on local light speeds has been practically abandoned. It is hope that this paper would facilitate making further progresses. ENDNOTES 1) A theory in physics must be supported by experiments, in addition to being logically self-consistent. 2) A Euclidean-like structure is a mathematical notion that the Pythagorean Theorem is satisfied. This notion is independent of the space-time metric, and thus can be different from the Euclidean space, which is based on physical measurements. 3) In order to defend an error, it would inevitably create more errors, logical errors in particular. 4) This board member considers, ¨the value of a coordinate is not a ˉdistance,∨ either; it is an arbitrary, human-made label, and nothing more〃. Therefore, he considered the shortest distance r0 from the center of the sun in Ein-stein∨s calculation not as a physical quantity but a ¨hypothetical construct〃. On the other hand, he considers b as ¨a proper distance〃, which is a distance in terms of a Euclidean-like structure, but not a distance that Einstein de-fined in terms of the metric. Thus, it is not clear what he meant as a distance. 5) This board member of the Royal Society has problems in at least three areas, namely: physics, mathematics, and logic. 6) Zhou [21, 27, 36] has proposed that the harmonic gauge condition to be necessary for an asymptotically flat met-ric, and thus rejected the Schwarzschild solution. He also correctly argued that the ¨covariance principle〃 is not valid in physics. 7) A satisfaction of Einstein∨s equivalence principle requires that a time-like geodesic must represent a physical free fall. The Einstein-Minkowski condition [2] requires that the co-moving local space must be locally Minkowski. Thus, the difference between Einstein∨s equivalence principle and Pauli∨s version is not just a matter of philoso-phy as some speculated [18]. 8) Invalidity of the covariance principle implies that general relativity is not yet a complete theory. Thus, the claim of Logunov and Mestvirishvili [37] that general relativity gives no definite predictions concerning gravitational effects could have been justified. Fortunately, it has been proven that the Maxwell-Newton Approximation is valid for the first order approximation of a physical gauge [22] such that the binary pulsar experiment can be ex-plained satisfactorily [18, 23]. REFERENCES 1. A. Einstein, H. A. Lorentz, H. Weyl, and H. Minkowski, The Principle of Relativity (Dover, New York, 1952); A. Einstein, Ann. Phys. (Leipig) 49, 769-822 (1916). 2. A. Einstein, The Meaning of Relativity (Princeton Univ. Press, 1954), p. 63, p. 87 & p. 93. 3. C. Y. Lo, Phys. Essays, 18 (1), 112-124 (March, 2005). 4. A. S. Eddington, The Mathematical Theory of Relativity (Chelsea, New York, 1975), p. 10. 5. C. Y. Lo, Bulletin of Pure and Applied Sciences, 27D (1), 1-15 (2008). 6. S. Weinberg, Gravitation and Cosmology: (John Wiley Inc., New York, 1972). 7. C. Y. Lo, Phys. Essays, 18 (4), 547-560 (December 2005). 8. A. N. Whitehead, The Principle of Relativity (Cambridge Univ. Press, Cambridge, 1922). 9. C. Y. Lo, Phys. Essays, 15 (3), 303-321 (September, 2002). 10. C. Y. Lo, Phys. Essays 16 (1), 84-100 (March 2003). 11. C. Y. Lo, Progress in Phys., Vol. 1, 10-13 (Jan., 2006). 12. C. Y. Lo, Chinese J. of Phys., Vol. 41, No. 4, 332-342 (August, 2003). 13. A. Einstein, ˉThe Problem of Space, Ether, and the Field in Physics' (1934) in Ideas and Opinions (Dover, 1982). 14. Louise Le Bas, Publishing Editor, the Royal Society, A Board Member∨s Comments (July 24, 2007). 15. C. Y. Lo, Bulletin of Pure and Applied Sciences, 26D (2), 73-88 (2007). 16. V. A. Fock, The Theory of Space Time and Gravitation, translated by N. Kemmer (Pergamon Press, 1964). 17. H. C. Ohanian & R. Ruffini, Gravitation and Spacetime (Norton, New York, 1994). 18. C. Y. Lo, Phys. Essays 13 (4), 527-539 (December, 2000). 19. Louise Gardner, Editorial Coordinator, the Royal Society, A Board Member∨s Comments (Feb. 25, 2009). 20. C. Y. Lo, Chin. Phys. (Beijing), 16 (3), 635-639 (March 2007). 21. Zhou Pei-yuan, ¨Further Experiments to Test Einstein∨s Theory of Gravitation〃, International Symposium on Experimental Gravitational Physics (Guangzhou, 3-8 August 1987), edited by P. F. Michelson, 110-116 (World Sci., Singapore). 22. C. Y. Lo, Phys. Essays 12 (3), 508-526 (Sept. 1999). 23. C. Y. Lo, Astrophys. J. 455: 421-428 (Dec. 20, 1995). 24. Motz L & Weaver J H, The Story of Physics (Avon, New York, 1989). 25. C. N. Yang, Phys. Rev. Lett. 33, 445 (1974). 26. H. Bondi, F. A. E. Pirani & I. Robinson, Proc. R. Soc. London A 251, 519-533 (1959). 27. Zhou (Chou) Pei-Yuan, ¨On Coordinates and Coordinate Transformation in Einstein∨s Theory of Gravitation〃 in Proc. of the Third Marcel Grossmann Meetings on Gen. Relativ., ed. Hu Ning, Science Press & North Hol-land. (1983), 1-20. 28. C. Y. Lo, Bulletin of Pure and Applied Sciences, 26D (1), 29 - 42 (2007). 29. C. Y. Lo, The Necessity of Unifying Gravitation and Electromagnetism and the Mass-Charge Repulsive Effects in Gravity, Physical Interpretation of Relativity Theory: Proceedings of International Meeting. Moscow, 2 5 July 2007/ Edited by M. C. Duffy, V. O. Gladyshev, A. N. Morozov, P. Rowlands Moscow: BMSTU, 2007. p. 82. 30. C. Y. Lo, Bulletin of Pure and Applied Sciences, 27D (2), 149-170 (2008). 31. S. G. Turgshev, V. Toth, L. R. Kellogy, E. L. Lau, and K. J. Lee, ¨Ther Study of the Pioneer Anomaly: New Data Objectives for New Investigation〃 arXIV: gr-gc/0512121v2, 6 Mar. 2006. 32. C. Y. Lo, The Mass-Charge Repulsive Force and Space-Probes Pioneer Anomaly, in preparation. 33. C. Y. Lo, Bulletin of Pure and Applied Sciences, 26D (2), 73-88 (2007). 34. H. Resissner, 1916. Ann. Phy. (Germany) 50:106-120. 35. G. Nordstrom, ¨On the energy of gravitational field in Einstein∨s theory,〃 Proc. Kon. Ned. Akad. Wet. 20, p. 1238 (1918). 36. Peng Huang-Wu, Commun. Theor. Phys. (Beijing, China), 31, 13-20 (1999). 37. A. Logunov & M. Mestvirishvili, The Relativistic Theory of Gravitation (Mir Publishers, Moscow, 1989). * This article is taken out from a full paper on the discussion of Einstein∨s covariance principle with minor editions to makes clear things without the rest of the full article. Although this article has no mathematics that is usually in a pa-per of physics, the arguments are clear. Readers interested in more details can read the references provided. The author believes that this article would be understood by the educated general public. In particular, this article shows the root why some theorists could make general relativity sounded more like a science friction than physics.