Some Problems of Logic in Mathematics – Errors of even top mathematicians

In mathematics, it is commonly known that an assertion or theorem can be right or wrong.¹⁾ However, in logic, there is actually a third case that the condition is OK but some implicit assumption is invalid. Thus, the theorem is not right or wrong, but misleading. Such an error can be made by top mathematicians such as M. Atiyah.²⁾

For instance, 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. However, since the Einstein equation has no dynamic solution, which is bounded [3], the assumption of asymptotically flat implies that the solution is a static solution such as the Schwarzschild solution, the harmonic solution and etc.

Therefore, Schoen and Yau actually prove a trivial result that the total mass of a static solution is positive. Note that since the dynamic case is actually excluded from the consideration in the positive energy theorem, this explains why it was found from such a theorem that Einstein's theory is consistent and stable. This is, of course, misleading.

The condition of asymptotically flat is a common condition in physics and thus, for a valid theory in physics, it should not exclude the case of a dynamic solution. The problem rises form the fact that the Einstein equation does not have a bounded dynamic solution as Gullstrand, the Chairman (1922-1929) of Nobel Committee for Physics suspected [4]. 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.

In fact, Yau [1], Christodoulou [5]. Wald, and Penrose & Hawking [6] make essentially the same error of defining a set of solutions that actually includes no dynamic solutions [7-9]. Their fatal error is that they neglected to find explicit examples to support their claims. Had they tried, they should have discovered their errors. This shows that a top mathematician can made an elementary mistake.

Endnotes:

1)     This problem was raised by “xyz12345” in May 26, 2013.

2)     Micheal Francis Atiyah has been president of the Royal Society (1990-1995), master of Trinity College, Cambridge (1990-1997), chancellor of the University of Leicester (1995-2005), and president of the Royal Society of Edinburgh (2005-2008). Since 1997, he has been an honorary professor at the University of Edinburgh (Wikipedia).

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, 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.

4.       A. Pais, 'Subtle is the Lord ...' (Oxford University Press, New York, 1996).

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

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

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

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

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