7/30/2019 A Problem of Logic in Mathematics

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A Problem of Logic in Mathematics & physicsErrors made by even top mathematicians

C. Y. Lo

In mathematics, it is commonly known that an theorem can be either right or wrong.1)

However, in logic, there is actually a third case that the conditions in a theorem are valid but some

implicit assumption is invalid. Thus, the theorem is not right or wrong, but misleading. Such anerror 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 isan example. Briefly, the positive mass conjecture says that if a three-dimensional manifold has positivescalar curvature and is asymptotically flat, then a constant that appears in the asymptotic expansion ofthe metric is positive (Wikipedia). A crucial assumption in the theorem of Schoen and Yau is that thesolution is asymptotically flat. However, since the Einstein equation has no dynamic solution, which isbounded [3], the assumption of asymptotically flat implies that the solution is a static solution such asthe Schwarzschild solution, the harmonic solution and etc.

Therefore, Schoen and Yau actually prove a trivial result that the total mass of a static solution ispositive. However, many incorrectly claimed that their proof of the positive energy theorem in generalrelativity demonstratedsixty years after its discoverythat Einstein's theory is consistent and stable

(Wikipedia). Note that since the dynamic case is actually excluded from the consideration in thepositive energy theorem, this explains why it was found from such a theorem that Einstein's theory isconsistent and stable. This is, of course, misleading.

The condition of asymptotically flat is a normal condition in physics and thus, for a valid theory inphysics, it should not exclude the case of a dynamic solution. The problem rises from the fact that theEinstein 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 andSchoen 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 (Wikipedia). Thus, oneshould not be surprised by the error made twice over eight years (1982-1990) by the Fields medal.

In fact, Yau [1], Christodoulou [5]. Wald, and Penrose & Hawking [6] make essentially the sameerror of defining a set of solutions that actually includes no dynamic solutions [7-9]. Thus, this is a

common mistake among theorists. Their fatal error is that they neglected to find explicit examples tosupport their claims. Had they tried, they should have discovered their errors. This shows that the topmathematicians can also made an elementary mistake.

Endnotes:

1) This problem was raised by xyz12345 bbs.creaders.netMay 26, 2013.

2) Micheal Francis Atiyah has been president of the Royal Society (1990-1995), master of TrinityCollege, Cambridge (1990-1997), chancellor of the University of Leicester (1995-2005), andpresident of the Royal Society of Edinburgh (2005-2008). Since 1997, he has been an honoraryprofessor 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, suggestsand approves the Appendix: The Gravitational Energy-Stress Tensor for the necessity of modifyingEinstein 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.

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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).