Let's discuss the example of the carnival ride in more depth. While it was moving, you could measure the circumference of it from the outside by measuring the rim. To find the radius, you could climb underneath the ride and measure from the axle to the rim. You would find that the circumference was 2π times the radius. If you have studied geometry, you would know that this is what you would expect. But if you were to make these same measurements while the ride was spinning, you would discover something different. As you near the outside of the ride, the path you take is longer than near the center, so you are moving more quickly. Because of the Lorentz contraction, if you were to measure the circumference with a ruler (or some other tool), the ruler would be shortened along its length. This means it would take more lengths of the ruler to measure the circumference, effectively giving you a larger measurement. On the other hand, if you measured the radius the ruler would be shortened only along its width because Lorentz contraction occurs in the direction of motion. The shortening of the width would not effect your measurement of the radius and you would arrive at the same answer as you found from outside the ride.
As you can see from the diagram, the carnival ride is warped. This effect can be explained as the warping of actual space. In this way, Einstein showed that regular "flat" geometry does not work for someone in accelerated motion. If you'll remember, accelerated motion is just like gravity. Einstein realized that gravity is the warping of space.
Extending the relation between space and time, you might see that time is also warped. As you neared the outside of the carnival ride, you would speed up and your watch would slow down. The warping of time is defined as a difference in the rate of the passage of time from place to place.
A frequently used visual portrayal of this effect is the idea of a rubber sheet. If you place a bowling ball on a rubber sheet, the sheet curves. This curvature of space is caused by the presence of 
mass or energy. If there were no matter or energy, Einstein said, space would be flat. (The problem with the rubber sheet analogy is that a bowling ball is pulled toward the ground by gravity, the very thing we are trying to illustrate. It is just a visual representation, nothing more.) If you were to roll a marble on the rubber sheet and it neared the bowling ball, it would follow the path of least resistance toward the bowling ball. If you rolled it just right it would roll around the bowling ball, in effect going into orbit.
In reality, this occurs in all three space dimensions, not just two. The graphic below begins to convey the idea of curvature in the three spacial dimensions. Curvature happens in time certainly, too, but that is even more difficult to show.
Einstein found that gravity effects travel like ripples in a pond, but the ripples don't travel instantly to everywhere. Instead, Einstein calculated, the effects of gravity travel at - surprise! - the speed of light. So Newton's theory was incorrect about this. If the sun exploded, it would not only take light from the explosion eight minutes to reach us, it would also take the changes in gravity eight minutes to reach us too. We would fall out of orbit at the same time we saw the explosion.
