Was Galileo wrong?

The answer is no. Then what is special in Einstein's Special Theory of Relativity(STR)? The answer is that in STR , we know about the kinematics of high speed objects. In a certain limit the Lorentz Transformation or Einstein's Relativity merge with Galilean Transformation. The limit is " Object is moving very slowly, to be most precise the object is moving slowly with respect to speed of light".

In STR the master assumption is " In every inertial frame of reference the speed of light in vacuum is constant( which is known as c )." From this assumption there is a strong link of General Theory of Relativity (GTR) and what is that link? the link is that what we do if the frame of reference is non-inertial? This is the clue point of GTR.

GTR can be thought as the relativity theory in a frame of reference which is not moving in a constant velocity or it a classical field theory of Gravitation. If we consider the second aspect then it is natural to face a question: was Newton wrong? The answer is again no. Like Galilean Transformation, Newton's law of Gravity is a special case of GTR.

If we approximate that gravitational field is very weak and object is moving slowly the GTR merges with the Newtonian Gravity. Now in GTR there is a strong connection between the geometry and Physics. In one side of Einstein's field equation contains only the geometrical property of space-time and one the other side the term is energy momentum tensor which tells cause of this type of space-time geometry.

Obviously the geometry which involves in GTR is not simple...it is differential geometry. It is hard but very interesting.