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- Title
Bilgisayarlı hesaplama yöntemleri ile beş boyutlu uzayzamanlarda dalga denklemlerinin incelenmesi.
- Authors
Birkandan, Tolga; Hortaçsu, Mahmut
- Abstract
The interest in higher dimensional wave equations is driven by the usage of higher dimensional metrics in general relativity and string theory. Instanton solutions of general relativity are the counterparts of Yang-Mills instantons which are finite-action solutions of the Yang-Mills equations. They have an important contribution to the path-integral in the quantization of the Yang-Mills fields. The general relativistic instantons are also expected to play a similar role in the path-integral approach to quantum gravity. Weierstrass' general local solution of minimal surfaces yields to a general instanton metric and Nutku's helicoid metric is a special case which corresponds to the helicoid minimal surface of this general metric. Dirac and Laplace equations can be solved in terms of Mathieu functions in the four dimensional case. If a time coordinate is added trivially to the metric, the solutions become double confluent Heun functions which are known to arise in higher dimensional solutions in the literature. One can trade the irregular singularity at zero by two regular singularities at plus and minus one by a transformation, to reach at the same singularity structure of the Mathieu equation and give the solutions in this form. But the main difference between the two cases is that, although both the radial and the angular parts can be written in terms of Mathieu functions, the constants are different, modified by the presence of the new term coming from the timedependence, which makes the summation of these functions to form the propagator quite difficult. In four dimensions one can use the summation formula for the product of four Mathieu functions -two of them for the angular and the other two for the radial part- summing them to give us a Bessel type expression. Nutku helicoid solution has a curvature singularity at the origin. Therefore, in order to have a precise result we tried to apply the Atiyah-Patodi-Singer spectral boundary conditions which was necessitated by the dimension of the spacetime. One is free to choose the local boundary conditions in even dimensions. However, we applied the same type of boundary conditions to conserve γ5 and charge conjugation symmetries of the Dirac operator in the four dimensional case. The application of the spectral boundary conditions involves the solution of the so called the little Dirac equation, the Dirac equation written on the boundary of the manifold. The analytical solutions of the little Dirac equation could not be obtained. The singularity structure of the little Dirac equation can reveal why we could not obtain an analytical solution.…
- Subjects
WAVE equation; STRING models (Physics); THEORY of everything (Physics); RELATIVITY (Physics); GEOMETRIC quantization; QUANTUM theory; MATHIEU functions; BOUNDARY value problems; DIRAC equation
- Publication
ITU Journal Series C: Basic Sciences, 2009, Vol 7, Issue 1, p9
- ISSN
1303-7021
- Publication type
Article