The Genesis of General Relativity
(6) How did Einstein overcome the "Hole" Argument?
Appendix: Howard & Norton's Reconstruction of Einstein's Solution
"Out of the Labyrinth? Einstein, Hertz, and the Goettingen Answer to the Hole Argument", in Earman et al 1993, pp. 30-62.
As I mentioned in (6), Howard & Norton (1993) is quite good for explaining Einstein's difficulty in 1914 and his own solution of it in 1915. So I would like to reproduce their exposition with my own figures, which are adapted from their figures.
First, the following figures show Einstein's construction of Hole Argument in 1914. The figures are adapted from Howard & Norton (1993), Figure 1 and 2. The grey circle is the "Hole", and the coordinate transformation (within the hole) takes the original point <2, 2> to <1, 1> of the new coordinate system K'. Follow the order (1), (2), and (3) in the Figure. Also, recall that metric tensor gmn in this case are, g11, g12, g21, and g22, and they are coefficients of each combination of two variables.
Why does this pose a difficulty? The presupposition of the argument is as follows:
Thus it seems undesirable that the uniqueness of the solution is violated, since the two solutions can be obtained within the same coordinate system K; Einstein says that the "principle of causality" is violated. Then, how does Einstein solve the problem in 1915? Here is his "point-coincidence argument", and the figure is adapted from Howard & Norton (1993), Figure 3.
Einstein now argues that, although there are two mathematically distinct solutions, they express one and the same physical coincidence, and therefore the physical uniqueness is not violated.
Have you noticed that, in the preceding, a version of Leibnizian principle of identity of indiscernibles revived? We've got to distinguish between abstract ("ideal") spacetime and physical spacetime!
References
Earman, J., Janssen, M., and Norton, J.D., eds., (1993) The Attraction of Gravitation, Einstein Studies vol. 5, Birkhaeuser.
Howard, D. and Stachel, J., eds. (1989) Einstein and the History of General Relativity, Einstein Studies vol. 1, Birkhaeser.
Howard, D., and Norton, J.D. (1993) "Out of the Labyrinth? Einstein, Hertz, and the Goettingen Answer to the Hole Argument", in Earman et al 1993, pp. 30-62.
Stachel, J. (1989) "Einstein's Search for General Covariance, 1912-1915", in Howard &Stachel 1989, pp. 63-100.
