The Errors in the Fields Medals, 1982 to S. T. Yau and 1990 to E. Witten

C. Y. Lo

Applied and Pure Research Institute

7 Taggart Drive, Unit E, Nashua, NH 03060

Abstract

Due to inadequate understanding in physics, mathematicians have made misleading erroneous claims in general relativity three times that result in awarding the Fields Medal in 1982 to S. T. Yau, and 1990 to E. Witten, and the 2011 Shaw Prize to D. Christodoulou. As a result, a major error in general relativity was not recognized and thus this error is currently still a major obstacle to the progress of physics. Also, similar errors have been made by D. Hilbert and M. Atiyah.

Key Words: stable solution; dynamic solution. 

04.20.-q, 04.20.Cv

                In mathematics, it is commonly known that an assertion can be either right or wrong. However, in logic, there is actually a third case that the conditions in a theorem are valid in mathematics but some implicit assumption is not generally valid. Thus, the theorem is not simply right or wrong, but misleading. In fact, such an error can be made by top mathematicians such as M. Atiyah ¹⁾ and consequently such misleading errors in mathematics were cited as a main reason to award the 1982 and the 1990 Fields Medal to Yau and Witten ²⁾ and to award the 2011 Shaw Prize in mathematics to Christodoulou. ³⁾

To this end, the Positive Energy Theorem of Yau and Schoen [1, 2] for general relativity is an example. Briefly, the positive mass conjecture says that if a three-dimensional manifold has positive scalar curvature and is asymptotically flat, then a constant that appears in the asymptotic expansion of the metric is positive (Wikipedia). A crucial assumption in the theorem of Schoen and Yau is that the solution is asymptotically flat. Such an assumption can be considered as common in physics since this condition is satisfied in stable solutions such as the Schwarzschild solution the harmonic solution, the Kerr solution, and etc. ⁴⁾ Thus, it could be natural to assert (as in Wikipedia) that their proof of the positive energy theorem in general relativity demonstrated—sixty years after its discovery—that Einstein's theory is consistent and stable.

However, if one understands the physics in general relativity as well as Gullstrand [6], the Chairman (1922-1929) of the Nobel Prize Committee does, the above statement is clearly incorrect. Since the Einstein equation has no dynamic solution, which is bounded [3-5], the assumption of asymptotically flat implies the exclusion of the most important class of solutions, the dynamic solutions. Therefore, Schoen and Yau actually prove a trivial result that the total mass of a static (or stable) solution is positive. In other words, the conclusions drawn from the positive theorem are grossly misleading. This illustrates that, an inadequate understanding in physics can lead to an obstacle for the progress in physics.

The condition of asymptotically flat is a normal condition in physics and thus, for a valid theory, it should not exclude the case of a dynamic solution. The problem rises from the Einstein equation that does not have a bounded dynamic solution as Gullstrand, the Chairman (1922-1929) of Nobel Committee for Physics suspected [6]. In other words, the problem is due to that Yau and Schoen used an implicit assumption which is false but was not stated in their theorem. Nevertheless, Atiyah, being a pure mathematician, was not aware of the problem in physics. Thus, one should not be surprised that such an error was made twice over eight years (1982-1990) by the Fields medal.

It should be noted that D. Hilbert also made a mistake on approving Einstein’s calculation of the perihelion of Mercury because he was not aware that this calculation requires a bounded solution of the many-body problem [6]. However, Hilbert was lucky because he understood that the Einstein’s calculation is not valid, but Atiyah was not as lucky. Nevertheless, because of Atiyah’s reputation as an outstanding mathematician, some journals such as Nature would not dare to criticize him.

In fact theorists such as Yau [1], Christodoulou [7], Wald, and Penrose & Hawking [8] make essentially the same error of defining a set of solutions that actually includes no dynamic solutions [9-12]. Thus, this is a common mistake among theorists. The fatal error is that they neglected to find explicit examples to support their claims. Had they tried, they should have discovered their errors. Moreover, the same error was cited in awarding to Christodoulou the 2011 Shaw Prize.^{3), 5)} Subsequently, Christodoulou was elected to the Member of U.S. National Academy of Sciences (2012). It would be interesting to see how this special case would end up.

A crucial result of the errors of mathematician is that the non-existence of a dynamic solution for the Einstein equation was not recognized. Thus the need of modifying the Einstein equation with an additional gravitational energy-stress tensor with the anti-gravitational coupling as the source was overlooked [3]. Then, the energy-mass formula E = mc² was still incorrectly considered as unconditionally valid [11]. Consequently, the charge-mass interaction was not only overlooked, but also explicitly denied by Einstein and his colleagues. Hence, the need of unification between gravitation and electromagnetism is missed [13].

Currently, a main problem is that Misner, Thorne. & wheeler [14] used the errors of Yau [1] and Witten [2] to strengthen their incorrect claim on the existence bounded dynamic solutions. For instance, they incorrectly claim that for their eq. (35. 31), L’’ + (β’)²L = 0, there are dynamic solutions without giving a proof (see Appendix A). If such an error was overlooked, one could easily fall into agreeing with the other errors [15].

     After P. Morrison passed away, at MIT general relativity is dominated by the Wheeler School with their errors to mislead others in the open courses Phys 8.033 and Phys 8.962. E. Bertschinger and Scott A. Hughes studied the linearized equation of the Einstein equation. However, they do not understand that for the dynamic case, the non-linear Einstein equation and its linearized equation do not have any compatible solutions [3, 4]. Moreover, Max Tegmark even failed to tell the different between mathematics and physics [15]. In other words, in the Physics Department of MIT, nobody understands the basic essence of general relativity.

      However, MIT is not the only victim among universities because of the influences of the Wheeler School. Their errors were not recognized because they were supported everywhere under such influences [5, 13], though without the necessary evidence. Thus, it is necessary to enlighten them and their like with a paper. ⁶⁾

 Currently, mathematicians are often being considered in terms of a hierarchy system.⁷⁾ However, such a practice would result in errors in mathematics not being corrected. This article attempts to break such a vicious cycle and to point out the absurdity of such a practice by showing that the top mathematicians can also make an elementary mistake (see also Appendices A, B, C, D).

Acknowledgments

    This paper is dedicated to Professor S. Weinberg, Nobel Laureate, University of Texas at Austin, who taught us that general relativity must be understood in terms of physics. The author gratefully acknowledges stimulating discussions with Professors L. Ford, J. E. Hogarth, P. Morrison, A. Napier, H. C. Ohanian, R. M. Wald, and J. A. Wheeler, H. Yilmaz. Special thanks are to S. Holcombe for valuable suggestions. This publication is supported by Innotec Design, Inc., U.S.A. and the Chan Foundation.