Minkowski spacetime is the four-dimensional geometrical realisation of the relationships of special relativity. To mathematicians, Minkowski spacetime tends to be considered as the more fundamental entity, with the special theory being defined as the associated physics.
Minkowski spacetime necessarily makes the same assumptions as special relativity regarding light - it assumes that when physical bodies exchange light-signals, the presence and relative motion of those bodies has zero effect on the propagation of light in the region. Under Minkowski spacetime, the physics of interacting bodies is dictated by the preexisting properties of spacetime, derived from the assumption of flat empty space.
Special relativity removed the idea of an absolute reference for inertial motion. This "absolute space" could be criticised for being a prior cause – "absolute space" dictated rather than negotiated the resulting physics, suggesting a form of "action without reaction" – a thing that could influence without itself being influence in turn. This was consixdered by some to be against basic philosophical principles and to "bad design".
Although Minkowski spacetiem has no preferred frame, Einstein criticised its concept as sufferring for the same basic weakness – Minkowski spacetime operated as a "prior cause" that dicated how bodies ought to move, without in any way being affected by them. Minkowski spacetime did not seem to support the idea of there being a "back-reaction".
General relativity's modified spacetime behaviour
The spacetime of general relativity addressed this problem. GR's spacetime was "democratic" rather than "dictatorial", and interactive rather than prescriptive (Wheeler:
Spacetime tells matter how to move, matter tells spacetime how to bend". According to the GPoR, the forcible acceleration of a body changed the geometry – applying a force to a body physically distorted the shape of the surrounding region of spacetime.
Criticism of GR1960's spacetime
Although the "new" spacetime behaviour appeared to eliminate the "prior cause" problem (and eliminated the "artificial" distinction between inertial and noninertial physics), the fact that Einstein's general theory was designed to reduce exactly to the physics of special relativity meant that Minkowski spacetime, with its perceived defects, lived on as a part of the larger system.
This became even more problematic after the 1960 crisis, when the community's need to defend SR from being invalidated by the GPoR meant that statements started to be made within the context of the revised (1960) general theory, suggesting that perhaps physical acceleration was NOT now associated with spacetime curvature.