A few of the problems...
Relativity assumes the failure of the MM experiment and amounts to an attempt to adapt the rest of physics to the supposed imperviousness of light speed. One problem is that Dayton Miller, one of the best physicists of the era immediately after Michelson and Morley, re-ran the experiment with better equipment and at higher altitude (to attenuate the effect of gravitational drag from the Earth itself) and, apparently, the experiment did not fail:
http://www.orgonelab.org/miller.htm
The book I'd normally recommend to somebody wishing a basic grasp of relativity would be Lewis Carroll Epstein's Relativity Visualized:
http://www.amazon.com/Relativity-Visualized-Lewis-Carroll-Epstein/dp/093521805X
Epstein uses the following analogy for what Einstein did or tried to do: Imagine that you have a house in which all windows and every door other than for one worked and opened and shut easily but that the one door was binding. Normally you'd simply plane material from the one door until it worked properly. But you COULD go to your local Walmart and buy a couple of hundred jacks and jack the foundation of the house until the one door worked, and then re-adjust every other door and window in the house and/or plane THEM or do whatever it took to ensure that they all worked again.... In the analogy of course, the house is modern physics, the one bad door is light, and the other doors and windows are all of the other things in the house of physics, time, distance, inertia, velocity... Epstein claims that relativity is the one case you would ever encounter in which this second approach was the right one but it seems sufficiently obvious to me that this is a gross violation of Occam's basic principle and that there could never be such a case.
There is the problem that even if you accept the proposition of light speed does not vary, there may be other explanations for that and, apparently, nobody investigated any of those other explanations. One version of such an explanation involves the sub-electron particles which Ralph Sansbury describes:
http://www.amazon.com/Faster-Than-Light-Relativity-Reconsidered/dp/1477584587
Sansbury describes light as an instantaneous force; there is another possible explanation for light and involving sub-electron particles, which would involve an analogy with rifle fire.
There is another problem in that Ron Hatch, the man who holds most of the basic patents for GPS as I read it, claims that relativity is not compatible with the actual research involved in GPS
https://www.google.com/search?client=opera&q=ron+hatch++relativity+gps&sourceid=opera&ie=UTF-8&oe=UTF-8
There is an obvious problem with gravity (which propagates instantaneously) and any sort of a claim that information cannot be transmitted faster than C.
And there is a gigantic problem with claiming that gravity amounts to some sort of a 4-dimensional differential geometry sort of thing. You cannot start with that and believe that gravity could have recently undergone any sort of a very large (3 - 1) change near the surface of our own planet, nonetheless it is an easy demonstration that it has. That demonstration goes like this:
You lose power/weight RATIO as you get larger no matter what you do. Weight is proportional to volume, which is a cubed figure; strength is proportional to cross section of bone and muscle, which is a squared figure. Double your dimensions, and that factor of two gets cubed for volume and weight and only squared for cross section and strength, you'll be eight times heavier and only four times stronger; you'll have cut your power/weight ratio exactly in half.
Obviously, you can only halve your power/weight ratio so many times and still stand up and walk and since muscle tissue is basically the same for vertebrate animals, we can compute a rough limit for the world from what we know about weight lifting sports.
The strongest human athletes are the top unlimited weight category power-lifters who compete in the World's Strongest Man competitions which you've seen. Take Benedikt Magnusson for example, who holds the world's record for the dead-lift.
Magnusson weighs around 380 and the record lift was 1015 lbs. The dead-lift pretty much uses every muscle in the athlete's body to a maximal extent or at least comes closer to that than any other exercise.
For all lifting events, you compensate for the effect of the square/cube problem by dividing through by 2/3 power of the athletes' weights, i.e. that lets you compare the lifts of the champions of the various weight divisions. When you divide the championship numbers for a particular event by 2/3 power of the competitors' weights, the numbers almost line up and become the same number; one will stand out a bit from the rest and that guy has basically done the best pound-for-pound lift overall. This works, at least up to the point of the super heavyweight division, because the athletes are all roughly built along the same lines.
However for the thought experiment of scaling the SAME athlete to different sizes, this isometric scaling is perfect since the symmetry at different sizes is perfect. The idea is to answer this question: at what point in size does it become the same effort for Magnusson to simply stand up and lift his OWN weight, as it is for that 1015-lb lift at his normal size of 380. On the left side of the equation you want the 1015 for the bar plus the 380 for Magnusson divided by the 2/3 power of 380, and on the right side of the equation you want x/(2/3 power of x), i.e. the guy just lifting his own weight:
1395/380^.67 = cube root of x
x = 17,718 lbs
I.e. at around 18,000 lbs, it would be everything in the world Magnusson could do just to stand up.
If you put a large sauropod dinosaur next to Magnusson, you're looking at one animal at the top of the food chain and the other near the bottom. Magnusson's body is mainly muscle and that terrifyingly well trained; the sauropod's body is mostly gut and digestive system for processing leaves and grass. If Magnusson couldn't stand and walk at 20,000 lbs, the sauropod sure as hell couldn't. Magnusson is very much stronger than any possible quadruped herbivore his size. The only thing any 400-lb quadruped herbivore could do with a 1000-lb weight is be crushed by it. The first quadruped herbivore which could do anything at all with such a weight other than be crushed would be an elephant.
They're finding dinosaurs now which were 150' long or thereabouts. Some (brachiosaurids) held their necks upwards, others (diplodocids) held their necks outwards. The seismosaur was one of the kinds which held his neck outwards, and that neck would have been 40' - 60' long and could easily have weighed 40,000 lbs or more. If the center of gravity of that neck was even 10' from the shoulders, you'd be looking at trying to hold 400,000 foot pounds of torque with flesh and blood on a 24/7/365 basis in our gravity. Move that COG five feet further out...
In real life, the only thing there is on this planet which figures in the ballpark of a half million to a million foot pounds of torque would be the combined max torque of all of the engines of one of our largest ships. A seismosaur in our gravity would be trying to hold that much torque with flesh and blood structures.
The problem for the other kind of sauropod (brachiosaurids) which held their necks upwards is that the heart it would take to get blood to their heads would not fit in their bodies, that problem is well known.
There are similar problems with the 500 - 1000-lb flying creatures of past ages; in our gravity, the largest birds which can (with great difficulty) take off or land in our present world are around 30 lbs.
Scientists aware of these problems keep trying to low-ball the weight estimates for sauropods. Christopher McGowen "Dinosaurs, Spitfires, and Sea Dragons", claimed a volumetrically derived weight of 180 tons for the ultrasaur and despite the grief he caught for that I suspect that number is reasonable. If you assume an equal level of effort to stand for the ultrasaur in his gravity and the largest possible elephant (16,000 lbs) to stand in our gravity and solve, you get a necessary attenuation in gravity of about 2.8 - 1 for the ultrasaur to function.
The thing about dinosaurs and gravity does not involve higher math or anything beyond grasping the difference between squared and cubed things. I don't see how anybody could look at that and want to claim that gravity was some sort of a geometrical thing, particularly when we're now getting radiocarbon dates of 20K - 40K years for dinosaur remains, finding soft tissue in dinosaur remains, and finding accurate depictions of known dinosaur types in native American petroglyphs, i.e. since there is no longer any respectable way to claim that the Earth has had 65M years to increase its mass:
http://www.newgeology.us/presentation32.html
The most recent findings involving dinosaur remains amount to one of the two slices of bread for what I term the basic evolutionist time sandwich, the other slice being the Haldane dilemma (they need quadrillions of years and only have a few tens of thousands). Again, there is no reasonable way to posit a very large change in our planet's mass in that space of time. The most reasonable claim I know of as to what gravity actually is involves Sansbury's sub-electron particles again and amounts to a claim that gravity is a sort of an electrostatic dipole effect:
http://www.holoscience.com/wp/electric-gravity-in-an-electric-universe/