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But if we're ''very'' careful, and give the golf ball ''just the right'' push, it will curve completely around the bowling ball and return to our hand.

But if we're ''very'' careful, and give the golf ball ''just the right'' push, it will curve completely around the bowling ball and return to our hand.

This is, in a nutshell, how spacetime and matter interact under the theory of general relativity. Massive objects — represented in our analogy by the bowling ball — curve spacetime. Less-massive objects also curve spacetime, but to a lesser extent. If the object is small enough, like our golf ball, the amount of curvature is so slight that we can't even measure it.

This is, in a nutshell, how spacetime and matter interact under the theory of general relativity. Massive objects — represented in our analogy by the bowling ball — curve spacetime. Less-massive objects also curve spacetime, but to a lesser extent. If the object is small enough, like our golf ball, the amount of curvature is so slight that we can't even measure it.

The way the golf ball moved in the three scenarios we imagined correspond to ''conic-section orbits,'' or ''Kepler orbits.'' When we just placed the golf ball and it rolled straight toward the bowling ball, that was a ''degenerate'' orbit: a straight line. When we gave it a push and it curved around the bowling ball and off the edge of the trampoline, that was a ''hyperbolic'' orbit. And when we gave it just the right push so that it curved around the bowling ball and back to our hand, that was an ''elliptical'' orbit.

The way the golf ball moved in the three scenarios we imagined correspond to ''conic-section orbits,'' or ''Kepler orbits.'' When we just placed the golf ball and it rolled straight toward the bowling ball, that was a ''degenerate'' orbit: a straight line. When we gave it a push and it curved around the bowling ball and off the edge of the trampoline, that was a ''hyperbolic'' orbit. And when we gave it just the right push so that it curved around the bowling ball and back to our hand, that was an ''elliptical'' orbit.

General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional [[manifold]], which looks locally like [[Minkowski space]], and which acquires [[curvature]] due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from [[Euclidean geometry]]: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along [[geodesic]]s in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.

General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional [[manifold]], which looks locally like [[Minkowski space]], and which acquires [[curvature]] due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from [[Euclidean geometry]]: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along [[geodesic]]s in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.

The GR field equations are  

The mathematical expression of the theory of general relativity takes the form of the ''Einstein field equations,'' a set of ten nonlinear partial differential equations. While solving these equations is quite difficult, examining them provides valuable insight into the structure and meaning of the theory.

:<math> G_{uv} = 8\pi\, T_{uv} </math>

 

where ''G_uv'' is the [[Einstein curvature tensor]], and ''T_uv'' is the [[stress-energy tensor]], ''G_uv'' and ''T_uv'' are both rank 2 symmetric tensors. The GR field equations is a system of [[partial differential equations]] that relates the curvature of space to the mass occupying the space.

In their general form, the Einstein field equations are written as a single [[tensor]] equation in [[abstract index notation]] relating the curvature of spacetime to sources of curvature such as energy density and momentum.

 

:<math> G_{\mu\nu} = 8\,\pi\,G\,T_{\mu\nu} </math>

 

In this form, <math>G_{\mu\nu}</math> represents the ''Einstein tensor,'' <math>G</math> is the same ''gravitational constant'' that appears in the [[Law of Universal Gravitation|law of universal gravitation]], and <math>T_{\mu\nu}</math> is the ''stress-energy tensor'' (sometimes referred to as the ''energy-momentum tensor).'' The indices <math>\mu</math> and <math>\nu</math> range from zero to three, representing the time coordinate and the three space coordinates in a manner consistent with [[Special theory of relativity|special relativity]].

 

The left side of the equation — the Einstein tensor — describes the curvature of spacetime in the region under examination. The right side of the equation describes everything in that region that affects the curvature of spacetime.

 

As we can clearly see even in this simplified form, the Einstein field equations can be solved "in either direction." Given a description of the gravitating matter, energy, momentum and fields in a region of spacetime, we can calculate the curvature of spacetime surrounding that region. On the other hand, given a description of the curvature of a region spacetime, we can calculate the motion of a test particle anywhere within that region.

 

Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it.

===The right side of the equation: the stress-energy tensor===

===The right side of the equation: the stress-energy tensor===