Review on the Contributions of S. T. Yau

      From the free encyclopedia Wikipedia, the contributions of professor Yau was summarized as follows:

 ”Yau's contributions have had a significant impact on both physics and mathematics. Calabi–Yau manifolds are among the ‘standard toolkit’ for string theorists today. He has been active at the interface between geometry and theoretical physics. His proof of the positive energy theorem in general relativity demonstrated—sixty years after its discovery—that Einstein's theory is consistent and stable. His proof of the Calabi conjecture allowed physicists—using Calabi-Yau compactification—to show that string theory is a viable candidate for a unified theory of nature.”

It was claimed that his “proof “ of the positive energy theorem [1, 2] in general relativity has profound influence that leads to even the large research efforts on string theory. Based on his proof, it was also claimed that Einstein’s theory is consistent and stable. Thus, it is clear that there should be problems in Yau’s proof since Einstein’s theory has been proven clearly not consistent without the necessary rectifications [3], and thus not yet being a theory.

I. Some Known Problems.

To mention a few, they are as follows:

1.   It is known that Einstein’s covariance principle is not valid as pointed out by Zhou Pei-Yuen of Peking University [4, 5] and its invalidity is proven more recently with counter examples [6].

2.   A related problem is Einstein’s theory of measurement since the covariance principle was created to remedy the short comings of his theory of measurement. Note that such a theory was criticized by Whitehead as clearly invalid in physics [7] and is inconsistent with the observed light bending [8]. However, these criticisms were rejected although the light bending is calculated alternatively to avoid the obvious conflict [9].

3.   A major reason that theorists held of the Einstein’s equivalence principle is due to that his theory of measurement was justified with special relativity. Recently, it is proven that Einstein’s justifications are based on invalid applications of special relativity. Thus, the invalidity of Einstein’s covariance principle is finally settled [3]. 

4. In fact, Einstein’s covariance principle and Einstein’s equivalence principle [10, 11] actually are directly in conflict [12]. However, many misidentified Einstein’s equivalence principle of 1916 wrongly as the equivalent assumption of 1911 [13], which has been proven invalid by observations [8]. Moreover, due to an inadequate background in mathematics, the Wheeler School [7] misinterpreted Einstein’s equivalence principle, and created the invalid notion local Lorentz invariance [14] (as a distortion of the Einstein-Minkowski condition [10, 11]).

5.The famous formula E= mc^^2 is only conditionally valid [15]. In fact, Einstein failed to prove the formula for other cases in spite of his efforts of several years [16]. Moreover, he or others actually has not completed the proof for the case of photons [17].

6.It has been proved in 1993 [18] and published in 1995 [19, 20] that there is no dynamic solution or gravitational wave for the Einstein equation.

7.It was further proved that the crucial book [21] that claimed the existence of dynamic solution, published by Princeton University as mathematical classic is actually invalid [22-24] due to elementary errors in mathematics.

Nevertheless, after the necessary rectifications, Einstein’s general relativity can prove the need of unification, a dream of Einstein that he himself failed to complete.  Thus, one may wonder what went wrong in Yau’s proof.

   The problem started at least from 1981 because Yau did not understand the non-linear equation of Einstein. Note that Yau has wisely avoided committing himself to the errors of Christodoulou & S. Klainerman, by claiming that his earlier interest has been changed [21]. However, he was unable to see that the binary pulsar experiment of Hulse & Taylor not only confirms that there is no dynamic solution but also that the signs of  coupling constants is not unique. In fact, Yau has made the same errors of Penrose and Hawking [25], and implicitly uses the invalid assumption of unique sign in his positive energy theorem of 1981. Nevertheless, Prof. Yau is a very good mathematician as shown by his other works.

     Unfortunately, he made the crucial mistake of assuming that famous theorists such as Hawking and Penrose would understand physics correctly. Moreover, even Einstein and Princeton University can make a crucial error in spite of the warning of Gullstrand [26], a member of the 1921 Nobel Committee. Who could have discovered that the Wheeler School actually does not understand Einstein’s equivalence principle if you do not read carefully and question them [27]?

II. Important Omissions.

     An important omission of Wikipedia is Yau’s crucial contribution to solving the Poincare conjecture. It is well known that he leads Cao and Chu to finish the final stage of solving the Poincare conjecture. Although Yau did the pioneer work that lays the foundation before even Hamilton, who laid the subsequent basis for solving the conjecture, some throw mud at Yau by giving such a credit unfairly to Perelman, who has a very considerable contribution, but with questionable honesty.

    However, Perelman only created several sub-conjectures from the original one and claimed he has solved them all and published them in the NET (and thus avoided questioning).  Unfortunate for him, one of his sub-conjectures is still not yet proved not only after Cao and Chu have completed the proof of the Poincare conjecture but also unclear even now. It is well known that conjectures are much easier to make than to proven them. This is the reason why there are so many conjectures around. This is especially true for mathematical analysis. Thus, Perelman’s several unexplained acts of disappearance would strengthen the suspicion of avoiding questions form colleagues after having taken a deceptive short cut. In short, not only there is no evidence that Perelman has actually proved the conjecture, but also his honesty is questionable.  Nevertheless, some theorists still unfairly claimed that Perelman has solved the conjecture. They even claimed that this was clear for them in spite of the fact that they make such an approval months after the publication of the work of Cao and Chu.  

    It is a shame that not only nobody from the mathematical community of China, except Yau’s students, stands out for the group of Yau et al. for a fair credit, but also Tin (Yau’s student) of Peking University made the shameless statement of belittling their Chinese colleagues. Professor Tin also distinguished himself by telling Perelman privately being correct soon after he made the sub-conjectures – an act ignored by Perelman as probably fishing for information; but failed to claim Perelman’s being essentially correct publicly immediate after the publication of Cao and Chu.

References:

1.  R. Schoen and S.-T. Yau, “Proof of the Positive Mass Theorem. II,” Commun. Math. Phys. 79, 231-260 (1981).

2.    E. Witten, “A New Proof of the Positive Energy Theorem,“ Commun. Math. Phys., 80, 381-402 (1981).

3. C. Y. Lo, Rectifiable Inconsistencies and Related Problems in General Relativity, Phys. Essays, 23 (2), 258-267 (2010).

4.  Zhou, Pei-Yuan, in Proc. of the Third Marcel Grossmann Meetings on Gen. Relativ. ed. Hu Ning, Sci. Press/North Holland. (1983), 1-20.

5.   P. Y. Zhou, Proc. of the International Symposium on Experimental Gravitational Physics, Guang Zhou, China (1987).

6. C. Y. Lo, On Gauge Invariance in Physics & Einstein’s Covariance Principle, Phys. Essays, 23 (3), 491-499 (Sept. 2010).

7.     A. N. Whitehead, The Principle of Relativity (Cambridge Univ. Press, Cambridge, 1962).

8.     C. Y. Lo, On Criticisms of Einstein’s Equivalence Principle, Phys. Essays 16 (1), 84-100 (March 2003).

9.     C. W. Misner, K. S. Thorne, & J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).

10. A. Einstein, The foundation of the general theory of relativity (translated from), Annalen der Physik, 49, 769-822 (1916); A. Einstein, H. A. Lorentz, H. Minkowski, & H. Weyl, The Principle of Relativity (Dover, 1923).

11. A. Einstein, The Meaning of Relativity (Princeton Univ. Press 1954).

12. C. Y. Lo, Einstein’s Principle of Equivalence, and the Einstein-Minkowski Condition, Bulletin of Pure and Applied Sciences, 26D (2), 73-88 (2007d).

13. A. Einstein, On the influence of Gravitation on the propagation of light, Annalen der Physik, 35, 898-908 (1911).

14. K-Y. Chung, S-w. Chiow, S. Herrmann, S. Chu, and H. Müller, Phys. Rev. D 80, 016002 (2009).

15. C. Y. Lo, Comments on Misunderstandings of Relativity, and the Theoretical Interpretation of the Kreuzer Experiment, Astrophys. J. 477, 700-704 (March 10, 1997).

16. "Einstein's Miraculous Year" edited and introduced by John Stachel, Princeton Univ. Press (1998).

17. C. Y. Lo, Completing Einstein’s Proof of E = mc², Progress in Phys., Vol. 4, 14-18 (2006).

18. C. Y. Lo, Einstein's Radiation Formula and Modifications in General Relativity, The Second William Fairbank Conference, Hong Kong Polytechnic, Hong Kong Dec. 13-16 (1993).

19. C. Y. Lo, Astrophys. J. 455, 421-428 (1995); Editor S. Chandrasekhar, a Nobel Laureate, suggests and approves the Appendix: The Gravitational Energy-Stress Tensor for the necessity of modifying Einstein equation.

20. C. Y. Lo, Astrophys. Space Sci., 306: 205-215 (2006).

21. D. Christodoulou & S. Klainerman, The Global Nonlinear Stability of the Minkowski Space (Princeton. Univ. Press, 1993); No. 42 of the Princeton Mathematical Series.

22. C. Y. Lo, Phys. Essays 13 (1), 109-120 (March 2000).

23. Volker Perlick, Zentralbl. f. Math. (827) (1996) 323, entry Nr. 53055.

24. Volker Perlick (republished with an editorial note), Gen. Relat. Grav. 32 (2000).

25. R. M. Wald, General Relativity (The Univ. of Chicago Press, Chicago, 1984).

26. A. Gullstrand, Ark. Mat. Astr. Fys. 16, No. 8 (1921); ibid, Ark. Mat. Astr. Fys. 17, No. 3 (1922).

27. C. Y. Lo, “The Question of Lorentz Invariance and Einstein’s Equivalence principle in General Relativity”, in preparation.