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| We believe that, the latest LIGO gravitational wave detection results is undoubtedly a major achievement of general relativity experimental studies and an important event in the history of human science development. But the direct discovery of gravitational waves does not mean that general relativity is already a complete theory of gravity. On the contrary, it demonstrates the inadequacy of Einstein gravitational theory. Because gravitational waves, as the rigorous solutions of the linear gravitational field equations, do not satisfy the nonlinear Einstein equations. Thus, gravitational waves can not be space-time waves, but only probability waves of the gravitons. In order to solve a series of problems including gravitational waves, based on the quantized mathematics and new equivalence principle，we improve Einstein gravitational theory to include curved space-time geometry and linear gravity, thus forming a gravitational theory with special and general relativity expansions. Thereafter, the graviton solutions of this theory are obtained, and the wave-particle duality of the gravitons in curved space-time background are explained. In particular, we illustrate the internal structure and the external behavior of the gravitons, define extended graviton and non-extended graviton, as well as positive momentum graviton and negative momentum graviton; deduce the quantization and uncertainty of background space-time geometry, and explain the quantization of space-time and energy. Correspondingly, we introduce classical gravitational strings and define quantum graviton states，reveal the probability nature of gravitational strings and describe the change of graviton states with respect to graviton energy and background space-time geometry; describe the interaction of gravitons with matter. This paper also predicts the possible developments of our gravitational theory and explains the significant impacts of our graviton solutions on the existing physical theories. Finally, we point out that, as the average approximation of gravitational strings, gravitational waves can transmit energy and information，but do not warp the space-time they pass through. As for the LIGO discovery, the interference effects are only the results of the laser transmission path length modulations caused by the gravitons with positive and negative momentums, which act on the mirrors in the detectors and make them displacements. This no doubt will change our gravitational wave observation methods and data analysis. |
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| Keywords:The quantized mathematics; New equivalence principle; Curved space-time geometry; Linear gravity; Graviton; Wave-particle duality |
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