Inconsistency and Problems in Einstein’s General Relativity Currently, Einstein’s general relativity is generally regarded as a top scientific achievement, although it is very difficult to understand. It is well known that observations accurately confirm the three predictions of Einstein [1, 2], namely: 1) the gravitational redshifts, 2) the perihelion of Mercury, and 3) the deflection of light. However, the difficulties in its understanding actually came, at least in part, due to its being not a self-consistent theory [3]. Einstein’s three accurate predictions created a faith on general relativity. Because of such a faith, few of his peers took a critical look of his theory. Although problems were raised by Whitehead [3] and Eddington [4] on Ein-stein’s theory of measurements, they are soon forgotten since nobody was able to solve them. Currently, instead of trying to improve the theory, many theorists tried very hard to make physical sense out of just any solutions of Einstein’s equation [5, 6, 7]. And such efforts often made their works sound more like science frictions than a scientific theory [8]. Unsolved problems were still there after more than 90 years although all the problems seem to be rectifiable. In other words, general relativity actually has never been well understood. It should be noted that, in spite of the confirmations of Einstein’s predictions, there are problems in verifying Ein-stein’s theory as follows: 1) The gravitational redshifts were based on Einstein’s 1911 preliminary assumption equivalence between ac-celeration and Newtonian gravity. However, such an assumption is inconsistent with Einstein’s equivalence principle proposed later in 1916 [1, 2]. Fock [9] found that it is impossible to have a metric that is consistent with Newtonian uniform gravity. This shows that gravitational red shifts can be derived from an invalid the-ory although the gravitational redshifts can be derived from Einstein’s equivalence principle [1, 2]. 2) Although Einstein did derive the perihelion of Mercury, Gullstrand [10] pointed out in his report to the Nobel Committee that Einstein’s field equation may not be able to produce a solution for a two-body problem. In other words, Einstein’s derivation may not be valid. Because of this, Einstein was awarded a prize for his work in the photo-electric effects. Moreover, it has been proven that Einstein’s field equation indeed cannot produce a physical solution for a two-body problem [11, 12]. Recently, ‘t Hooft [7] tried to rebuttal this con-clusion with a “counter” example of his. However, this only exposed his inadequacy in some aspect of phys-ics such as that he does not understand Einstein’s equivalence principle as well as the principle of causality [7, 13]. So, the perihelion actually cannot be considered as a verification of Einstein’s theory although it does suggest that his theory would be in the right direction. 3) From both the Schwarzschild and the harmonic solution, Einstein obtained the same first order deflection of light in terms of the shortest distance r0 from the sun center [1, 2]. Then, in support of his covariance princi-ple, Einstein [2] remarked, “It should be noted that this result, also, of the theory is not influenced by our ar-bitrary choice of a system of coordinates.” Obviously, this gauge invariance should have been supported by all physical quantities in all orders of calculations. Recently, calculation of the deflection angle to the second order also shows gauge invariance in terms of the impact parameter “b” [14, 15]. However, careful analysis shows that this calculation actually implies that the theory is intrinsically not gauge invariant since, for each gauge, the shortest distance r0 is different from that for another gauge [16]. To defend this inconsistency, the editorial of the Royal Society claimed [17] only b is a true measurable physical quantity, but r0 is just an arbi-trary label, a hypothetical construct. However, this is inconsistent with Einstein’s result on the first order ap-proximation [1, 2]. Thus, the editorial of the Royal Society has not reached the maturity in logic. Because Einstein’s covariance principle is invalid, general relativity of Einstein was not a complete theory. Fortu-nately, the Maxwell-Newton approximation has been proven as the valid first order approximation for gravity due to massive sources [18] such that the binary pulsar experiments can be explained satisfactorily [11, 12]. According to this approximation, r0 is at least accurate to the first order. Moreover, validity of this approximation implies also that the coupling constants have different signs [11] and thus the physical assumption of unique sign in singularity theorems of Penrose and Hawking is invalid. This logical immaturity also led to supporting [19] Bondi, Pirani & Robin [5] who rejected Einstein’s requirement on weak gravity since it is inconsistent with Einstein’s covariance principle. Nevertheless, prominent theorists such as Straumann [20], Wald [21], and Will [22], who believe in both Einstein’s requirement on weak gravity and his covariance principle, failed responding to this inconsistence [5] discovered since 1959. Moreover, such logic immaturity is not just isolated incidents of this Royal Society as shown in Hawking’s book [23, 24]. Moreover, although the International Society on General Relativity and Gravitation was formed, founders of the society such as P. G. Bergmann [25], H. Bondi [5], V. A. Fock [8], J. L. Synge [26], J. A. Wheeler [27], and etc. have never reached a general consensus on general relativity. Under the auspices of this society, “General Relativ-ity and Gravitation” is published. Surprisingly, members of the Editorial Board actually do not sufficiently under-stand physical principles, such as Einstein’s equivalence principle and the principle of causality [22-30]. For in-stance, except in Einstein's original works, there are no textbooks or reference books [28] (including the British Encyclopedia [2006]) that explained Einstein's equivalence principle correctly although this principle is stated squarely in page 57 of Einstein's book, “The Meaning of Relativity'” [2]. They also failed to understand that Ein-stein has changed his position on E = mc2 to as only conditionally valid [31], and also the experiments of the bi-nary pulsars. In addition, some of such theorists criticized Einstein without getting the facts straight first [8, 26]. Einstein’s difficulties are due to incorrectly adapt the mathematical notion of local distance in Riemannian ge-ometry as if valid in physics [32]. Moreover, Einstein’s theory of measurement is actually based on invalid appli-cations of special relativity [1]. Whitehead [3, p.83], strongly objected, “By identifying the potential mass impetus of a kinematic element with a spatio-temporal measure-ment 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 system-atic uniformity and requires a knowledge of the actual contingent physical field before it is possible.” Unfortunately, Whitehead also rejected Einstein’s equivalence principle, which actually rectifies Einstein’s theory of measurement [33]. His theory of measurement is also inconsistent with the observed light bending [34, 35], and is the root for ambiguity of coordinates and ended up the need of his covariance principle as an interim measure. Fundamental concepts in a great theory are often difficult to grasp [36]. To mention a few, this happened to Newton, Maxwell, Planck, Schőrdinger, and C. N. Yang [37]. Einstein is simply not an exception. 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 principle is inconsistent with his notion of weak gravity [5]. Zhou Pei-Yuan [38, 39] of Peking University was the first who correctly rejected Einstein’s covariance principle but accepted Einstein’s equivalence principle. Nevertheless, Ein-stein is a great theorist since the implications of general relativity such as the need for unification have been dis-covered and verified [40, 41]. However, theoretical developments [7, 41] and NASA’s discovery of the Pioneer anomaly imply that Einstein’s theory is clear inadequate [42, 43]. References: 1. A. Einstein, H. A. Lorentz, H. Minkowski, H. Weyl, The Principle of Relativity (Dover, N. Y., 1952), pp 115-118, p.162; 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. A. N. Whitehead, The Principle of Relativity (Cambridge Univ. Press, Cambridge, 1922). 4. A. S. Eddington, The Mathematical Theory of Relativity (1923) (Chelsa, New York, 1975), p. 10. 5. H. Bondi, F. A. E. Pirani & I. Robinson, Proc. R. Soc. London A 251, 519-533 (1959). 6. Penrose R., Rev. Mod. Phys. 37 (1), 215-220 (1964). 7. C. Y. Lo, The Principle of Causality and the Cylindrically Symmetric Metrics of Einstein & Rosen, Bulletin of Pure and Applied Sciences, 27D (2), 149-170 (2008). 8. K. S. Thorne, Black Holes and Time Warps (Norton, New York, 1994), pp. 105, 456. 9. V. A. Fock, The Theory of Space Time and Gravitation, translated by N. Kemmer (Pergamon Press, 1964), pp 6, 111, 119, 228-233 10. A. Gullstrand, Ark. Mat. Astr. Fys. 16, No. 8 (1921); ibid, Ark. Mat. Astr. Fys. 17, No. 3 (1922). 11. C. Y. Lo, Einstein's Radiation Formula and Modifications to the Einstein Equation, Astrophysical Journal 455, 421-428 (Dec. 20, 1995). 12. C. Y. Lo, On Incompatibility of Gravitational Radiation with the 1915 Einstein Equation, Phys. Essays 13 (4), 527-539 (December, 2000). 13. C. Y. Lo, Special Relativity, Misinterpretation of E = Mc2, and Einstein’s Theory of General Relativity, in Proc. IX International Scientific Conference on ‘Space, Time, Gravitation,’ August 7-11, 2006, Saint-Petersburg, Russian Academy of Sciences. 14. J. Bodenner & C. M. Will, Am. J. Phys. 71 (8), 770 (August 2003). 15. J. M. Gérard & S. Piereaux, “The Observable Light Deflection Angle, arXiv:gr-qc/9907034 v1 8 Jul 1999. 16. C. Y. Lo, The Deflection of Light to Second Order and Invalidity of the “Principle of Covariance”, Bulletin of Pure and Applied Sciences, 27D (1), 1-15 (2008). 17. Louise Gardner, Editorial Coordinator, the Royal Society, A Board Member’s Comments (Feb. 25, 2009). 18. C. Y. Lo, Compatibility with Einstein's Notion of Weak Gravity: Einstein's Equivalence Principle and the Absence of Dynamic Solutions for the 1915 Einstein Equation, Phys. Essays 12 (3), 508-526 (Sept. 1999). 19. Pring F, The Royal Society, “Board Member's Comments” (Jan. 8, 2007). 20. N. Straumann, General Relativity and Relativistic Astrophysics (Springer, New York, 1984). 21. R. M. Wald, General Relativity (The Univ. of Chicago Press, Chicago, 1984). 22. C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge. Univ. 1981). 23. S. Hawking, A Brief History of Time (Bantam Books, New York, 1988). 24. 霍金教授是现代的爱因斯坦吗？(2006年6月19日) www5./MainNews/Opinion. 25. P. G. Bergman, Introduction to the Theory of Relativity (Dover, New York, 1976). 26. J. L. Synge, Relativity; The General Theory (North-Holland, Amsterdam, 1971). 27. C. W. Misner, K. S. Thorne, & J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973). 28. C. Y. Lo, Einstein’s Principle of Equivalence, and the Einstein-Minkowski Condition, Bulletin of Pure and Applied Sciences, 26D (2), 73-88 (2007d). 29. C. Y. Lo, The Gravitational “Plane Waves” of Liu and Zhou and the Nonexistence of Dynamic Solutions for Einstein’s Equation, Astrophys. Space Sci., 306: 205-215 (2006 (DOI 10.1007/s10509-006-9221-x). 30. C. Y. Lo, Einstein’s Equivalence Principle, the Principle of Causality, and Plane-Wave Solutions, Phys. Es-says 20 (3) (Sept. 2007). 31. A. Einstein, “E = mc2 (1946),” in Ideas and Opinions (Crown, New York, 1954), p. 337. 32. C. Y. Lo, Misunderstandings Related to Einstein’s Principle of Equivalence, and Einstein’s Theoretical Errors on Measurements, Phys. Essays 18 (4), 547-560 (December, 2005). 33. C. Y. Lo, Space Contractions, Local Light Speeds, and the Question of Gauge in General Relativity, Chinese J. of Phys. (Taipei), 41 (4), 233-343 (August 2003). 34. C. Y. Lo, On Criticisms of Einstein’s Equivalence Principle, Phys. Essays 16 (1), 84-100 (March 2003). 35. C. Y. Lo, On Interpretations of Hubble's Law and the Bending of Light, Progress in Phys., Vol. 1, 10 (2006). 36. L. Motz & J. H.. Weaver, The Story of Physics (Avon, New York, 1989). 37. C. N. Yang, Phys. Rev. Lett. 33, 445 (1974). 38. 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 Holland. (1983), 1-20. 39. Zhou Pei-Yuan, “Further Experiments to Test Einstein’s Theory of Gravitation”, International Symposium on Experimental Gravitational Physics (Guangzhou, 3-8 Aug. 1987), edited by P. F. Michelson, 110-116 (World Sci., Singapore). 40. C. Y. Lo, The Necessity of Unifying Gravitation and Electromagnetism and the Mass-Charge Repulsive Ef-fects in Gravity, Physical Interpretation of Relativity Theory: Proceedings of International Meeting. Moscow, 2 ˉ 5 July 2007/ Edited by M.C. Duffy, V.O. Gladyshev, A.N. Morozov, P. Rowlands. ˉ Moscow: BMSTU, 2007, p. 82. 41. C. Y. Lo, Limitations of Einstein’s Equivalence Principle and the Mass-Charge Repulsive Force, Phys. Essays 21 (1), 44-51 (March 2008). 42. S. G. Turgshev, V. Toth, L. R. Kellogy, E. L. Lau, and K. J. Lee, “The Study of the Pioneer Anomaly: New Data Objectives for New Investigation” arXIV: gr-gc/0512121v2, 6 Mar. 2006. 43. C. Y. Lo, The Mass-Charge Repulsive Force and Space-Probes Pioneer Anomaly, in preparation.