The first important thing about spacetime are the transformations mixing space and time. They are called Lorentz transformations because Lorentz figured them out and made them famous before Einstein. Everyone agrees to that. Lorentz even got a Nobel prize in 1902 for related work. See History of Lorentz transformations for details.
Second is combining space and time into a 4-dimension object and giving it a non-Euclidean geometry. Poincare did this in 1905, defining the geometry by the metric and symmetry group. Minkowski elaborated on this with world lines and diagrams, and popularized it in 1908. As a result, spacetime is often called Minkowski space. In the words of a Harvard historian of science:
In sum Minkowski still hoped for the completion of the Electromagnetic World Picture through relativity theory. Moreover, he saw his own work as completing the program of Lorentz, Einstein, Planck, and Poincare. Of these it was Poincare who most directly influenced the mathematics of Minkowski's space-time. As Minkowski acknowledges many times in "The Principle of Relativity," his concept of space-time owes a great deal to Poincare's work.35 [Galison,1979]Third is defining a relativistic theory on spacetime. This means that the observable physical variables and equations must be covariant under the geometric structure. That is, observing from a rotated or moving frame must induce a symmetry in the laws of physics that is a mathematical consequence of the underlying geometry. Poincare proved this for Maxwell's theory of electromagnetism, and constructed a relativistic theory of gravity, in his 1905 paper. Minkowski appears to be the only one who understood Poincare's paper.
While Einstein understood Lorentz's work in step 1, and gave his own presentation of it, he completely missed steps 2 and 3. Even after Minkowski spelled them out clearly in a paper that was accepted all over Europe, Einstein said, "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."
While Poincare and Minkowski explicitly advocated geometric interpretations different from Lorentz's, Einstein denied that he had any differences from what Lorentz published.
While Einstein, in collaboration with Grossmann, Hilbert, and others, eventually built on Minkowski's work for general relativity, he denied the geometric interpretations of special and general relativity that are in textbooks today.
In short, Einstein had almost nothing to do with the development or acceptance of spacetime among physicists.
He started to become a cult figure with the general public when the NY Times published this 1919 story:
LIGHTS ALL ASKEW IN THE HEAVENS;And today, hardly anyone mentions spacetime without also mentioning Einstein's name.
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