<PERSON> saw the universe, as a 17th century man would, as having a fixed time speed everywhere and lengths that appeared the same to all observers. Within this model, he created a really excellent set of rules for motion under the influence of gravity.
<PERSON> allowed for the speed of time and length measurements to vary based on the observer's reference frame. The resulting complications can be addressed using 19th century mathematics (<PERSON> famously said to a mathematician friend, "<PERSON>, you must help me or I will go mad!"). However, I think people go too far when they turn around and "see" the mathematics as defining a sort of higher reality. I think it is just a model; a very wonderful model that works, but still just a model to address the differences in viewpoints of time and length measurements.
If one takes a more prosaic view, and simply says "<PERSON> was pretty much right other than adjusting for different times speeds and measured lengths," then one can develop a parallel model for motion under the influence of gravity, that works every bit as well at matching observations including gravitational lensing, the <PERSON> time delay, planetary precession, satellite precession, relativistic rocket travel, photon sphere, gravitational waves, and the dynamics of falling into a non-roating black holes. The key is to retain all of <PERSON>'s laws, include conservation of energy, but allow the local traveler to experience motion according to his reference frame (he feels he is closer to the gravitational source, and that path lengths are shorter, and that the distant viewer has a clock that runs slower due to velocity and faster due to less gravity). For example, remove Mercury and put a spaceship in its place with a physicist and perfect measuring equipment - at no point in the orbit would he ever measure an error in <PERSON>'s laws nor see a day's motion that was inconsistent with <PERSON>'s laws, but sure enough, after 1 Mercury year, he'd find that he ended up with a precessed perihelion.
Conversely, there are things that the geometric curved spacetime models struggle to explain. For example, as people note, it is vital that there not be an absolute maximum time speed (which would imply an absolute zero velocity). Yet I can populate the universe with clocks created at the big bang and set in a no-dipole-in-the-cosmic-microwave-background-radiation reference frame, allowing them to separate with universal expansion. 562
It is impossible to create a clock in any other reference frame that could be beside one of my clocks and record more time having passed since creation. Hence it appears that I have defined a maximum time rate and an absolute zero velocity. (In fact, it is necessary for us to define an absolute zero velocity in order to define an age of the universe.)
There are other problems with the geometric mathematical solution, in that the complexity leads to false conclusions or the reliance on adages that may not be true in all cases. For example, one response says that "Objects moving through space-time curve not because they are "pulled" by the force of gravity, but because they are taking the shortest distance through curved space-time." But consider the path of two clocks starting at a point on the moon and returning to the origin. One clock is thrown vertically upward, while the other is thrown very fast to circle the whole moon at a very low altitude and return. The clock thrown vertically will record more time than a clock that sits at the origin, while the clock that flew all around the moon will record less time than the clock at the origin. The vertically thrown clock followed a "shorter distance" through spacetime than the still clock, but the low angle throw followed a longer distance through space time than the still clock, yet both paths are geodesics.
And there are more problems, like the conclusion that the astronaut twin ages less because she accelerated and later reversed direction while her brother stayed still. But have the astronaut accelerate to leave Earth on January 1 at noon, and have her stay in position fixed relative to the stars, and her brother will come around next January 1st to find that his sister has aged more than him by a fraction of a second (this is true even if the circling the sun twin is on board a spaceship with no gravity rather than on the planet Earth).
That is - <PERSON> is not so bad. We can adjust <PERSON> to account for changing time speeds and lengths, and do a fine job at dealing with motion through the universe under the impact of gravity using his laws. The conventional curved space model is useful, but it is not proven to have any more "truth" than <PERSON>'s model with adjustments, and it can lead to incorrect conclusions. 562