Abstract:  This article discusses a puzzling problem in the history of physics, that is, Einstein almost struggled to explore the gravitational field equation for nearly eight years, but Hilbert was able to reach the end of the race to establish the gravitational field equation almost simultaneously after listening to Einstein’s six relevant reports. Naturally, there is a difference between thinking about gravity in physics and thinking about gravity in mathematics, or in the way physicists and mathematicians approach and deal with gravity. Their deep thinking and analysis of Poisson's equation obtained important inspiration from it, and thus established the correct idea of establishing their own gravitational field equation. However, if one thinks deeply, it is natural to ask whether there are other academic viewpoints and lines of inquiry that prevent Einstein from thinking smoothly about gravity. Undoubtedly, this is a scientific community worthy of deep thought and discussion. In order to compare, this paper gives the whole process of deriving the equations of gravitational field from the physical concept of Einstein and from the variational theory of Hilbert. When it comes to tensor, a mathematical tool, the paper gives a simple, clear and conceptual discussion. In order to improve readability, the novel diagram from vector to tensor drawn by the author is attached. The author believes that readers who have read this article will certainly have a deeper understanding of the gravitational field equation of general relativity, the variational method and the relationship between tensor and vector; You will also gain insight into the different research styles and creative modes of two famous scientists, Einstein and Hilbert.

Key words: gravitational field, metric tensor, Riemann curved space, Minkowski space-time, principle of least action