General relativity is a modern theory of gravity that is consistent with special relativity. When Albert Einstein completed the classical equations of electromagnetism with his equations of special relativity, he damaged the Newtonian theory of gravity. While an object can move at any speed in Newton's theory of gravity, it cannot travel faster than the speed of light under special relativity. Albert Einstein's general relativity is a theory of gravity that preserves the speed-of-light constraint of special relativity. But beyond this new constraint, general relativity adds a new idea, the idea that the motion in a gravitational field can be thought of as travel along a straight line in a curved space.
In Newtonian gravity, an object couples to a gravitational field through the object's mass, which makes the mass drop out of an object's equation of motion. All objects are accelerated in the same way by gravity, a property that inspires the broader idea of the equivalence principle. According to this principle, someone in free-fall cannot know through any local experiment whether he is falling in a gravitational field. For instance, if I measure the spectrum of light from a hot gas while free falling in a gravitational field, I would find the same result as someone floating in space away from any gravitational field. If I were accelerating at a constant rate, the physics would again be the same for acceleration created by a rocket as for acceleration from standing on the surface of Earth. The gravitational field does not change the physics we see locally. If we are to detect a gravitational field, we must look more globally, so that we see the variation of the gravitational field over distance through its tidal force. The equivalence principle demands that over short distances all equations of physics—the equations of motion, the equations of electromagnetism—have the same form in a gravitational field as in open space, whether in free fall or in a laboratory under constant acceleration. Over longer distances, the effects of gravity appear in our equations as tidal forces.
Pondering an object's motion under the equivalence principle, you may realize that it is similar to drawing a line on a curved surface. The area immediately surrounding a point on a curved surface appears to be a plane, and if I draw several straight lines in this small area, they will appear to obey the rules of Euclidean geometry. But if I then make a series of short, straight extensions to each of these lines, I eventually would find that the lines deviate from their expected Euclidean behavior. For instance, let's start with a pair of short lines that are parallel—a perpendicular line drawn through one is perpendicular as it passes through the other. Extending these lines, I may find that they come closer together or farther apart, contradicting our Euclidean expectations.
This behavior is precisely what we are looking for in a description of motion in a gravitational field, where the tidal force of a large gravitating body can draw together two small objects that are in free fall and initially at rest relative to each other. An example would be two objects dropped far above Earth's surface. Watching from a distance, we would see them at first maintain a constant separation as they travel on parallel paths, but slowly they would begin to accelerate towards each other, making their paths deviate from parallel lines. This is the very image of lines moving through a curved space.
But what if we were free falling with the two objects? At first we would see the two objects floating motionless relative to us and to each other. As time passed, these objects would start to move towards each other. How do we fit this into a curved-space framework? The answer is clear if we plot the positions of the objects against time: in this space-time plot we recover a pair of lines that are initially parallel, but at latter times curve towards each other. This is precisely the previous case, because in that example we used the free fall motion the objects as a proxy for time. From this we learn that if we want to use the mathematics of curved surfaces to describe gravity, we must apply this mathematics to the three spatial dimensions and to time. This is why we talk of gravity as curving space and time.
This is the key idea behind general relativity: rather than add a gravitational force term to our equations of motion, we incorporate the gravitational field into our variables as a curvature of space-time. This approach to gravity immediately produces a theory that preserves the equivalence principle.
The great advantage of incorporating gravity into the coordinates of space-time is that it automatically incorporates the force of gravity into all of our other equations of physics. We don't add a gravitational force to the equations of electromagnetism; instead, the effects of gravity appear through the definitions of the space and time variables. The same is true for the equations of motion for an object: for the equation of motion we have mass times acceleration equals the force, but the force term contains no gravitational components; instead the effects of gravity appear through the definition of acceleration.
The great disadvantage of incorporating gravity into the definition of our coordinates is that each time we solve a problem in general relativity, we must interpret what our coordinates represent. Generally this means that we must not only solve how objects move in general relativity, but how the light from those objects reach us. This is generally a much more involved process than encountered in special relativity, where the light moves in a straight line with a fixed frequency; in general relativity, the light from an object moves on a curved path, with both it's frequency and brightness changing.
By treating gravity as a curvature of space and time, we automatically preserve the characteristics of acceleration in special relativity. This means that under acceleration, we expect light to be Doppler shifted to the blue when moving counter to the acceleration, and to be Doppler shifted to the red when moving in the direction of acceleration. We expect the path of light to travel a curved path past us as we accelerate. Finally, we expect the formation of an event horizon below us when we accelerate.
Our decision to incorporate gravity into our physics as a curvature of space and time only gets us half way to a theory of gravity. The other half of the problem—the link between matter and the curvature of space-time—must yet to be specified. And here we run into a problem. While we have the equivalence principle to guide us in how we describe the influence of gravity on the motion of matter and energy, we have no broad principle that tells us how matter curves space-time. The only guide we have is the requirement that when the gravitational field is weak, the equations of general relativity must revert to the Newtonian equations of gravity. Ideally we would have measurements of strong gravitational fields to guide us, but such fields are only found in objects too distant to observe accurately. This lack of guidance allows an infinite number of solutions. The solution that is used within the scientific community is that given by Einstein. He chose the mathematically simplest equation that relates matter and energy to the curvature of space-time. But in truth, there is no reason why reality should conform to our desire for mathematical simplicity. The greatest uncertainty in the science of gravity is whether matter and energy couple to the curvature of space-time in the was hypothesized by Einstein.