Foundations 3: Black Holes - Greg Egan

"The previous article in this series began building the framework of ideas needed for general relativity by describing the geometry of manifolds — mathematical spaces without any notion of distance or angle — and then showing how it was possible to add a metric that defined these things in a very general way. The idea of parallel transport of a vector was introduced: moving along any path, you can carry a kind of “reference copy” of a vector from your starting point with you. A path is called a geodesic if it continues to follow the parallel-transported copy of its initial direction, never swerving away from its original bearing. Parallel transport of a vector around a closed loop can produce a reference copy back at the starting point that fails to match the original vector, and this effect is used to quantify the curvature of space (or spacetime), via the Riemann curvature tensor.

Einstein's equation links the curvature of spacetime with the presence of matter and energy. We haven't quite said all that we need to about curvature, but this article will begin by attacking the other side of the equation. This will give us some insight into why the equation takes the form it does, before we reach the final goal: examining one solution of the equation, the Schwarzschild solution, which describes a black hole.


Mass

If we want to quantify the amount of matter and energy in a region of spacetime, a good place to start is the idea of mass. According to Newtonian physics, when we weigh an object we're measuring the gravitational force that the Earth exerts upon it, and this force is taken to be proportional to the object's mass. Mass is usually defined quite differently, though, through the property of inertia: in the absence of complications like friction, when you apply a certain force to an object its rate of acceleration will be inversely proportional to its mass. Imagine pushing two items of furniture on frictionless pallets across a level surface; even though you're not opposing gravity, the same push will accelerate a 100-kilogram sofa half as much as a 50-kilogram bookcase."


5 out of 5

"The previous article in this series began building the framework of ideas needed for general relativity by describing the geometry of manifolds — mathematical spaces without any notion of distance or angle — and then showing how it was possible to add a metric that defined these things in a very general way. The idea of parallel transport of a vector was introduced: moving along any path, you can carry a kind of “reference copy” of a vector from your starting point with you. A path is called a geodesic if it continues to follow the parallel-transported copy of its initial direction, never swerving away from its original bearing. Parallel transport of a vector around a closed loop can produce a reference copy back at the starting point that fails to match the original vector, and this effect is used to quantify the curvature of space (or spacetime), via the Riemann curvature tensor.

Einstein's equation links the curvature of spacetime with the presence of matter and energy. We haven't quite said all that we need to about curvature, but this article will begin by attacking the other side of the equation. This will give us some insight into why the equation takes the form it does, before we reach the final goal: examining one solution of the equation, the Schwarzschild solution, which describes a black hole.


Mass

If we want to quantify the amount of matter and energy in a region of spacetime, a good place to start is the idea of mass. According to Newtonian physics, when we weigh an object we're measuring the gravitational force that the Earth exerts upon it, and this force is taken to be proportional to the object's mass. Mass is usually defined quite differently, though, through the property of inertia: in the absence of complications like friction, when you apply a certain force to an object its rate of acceleration will be inversely proportional to its mass. Imagine pushing two items of furniture on frictionless pallets across a level surface; even though you're not opposing gravity, the same push will accelerate a 100-kilogram sofa half as much as a 50-kilogram bookcase."


5 out of 5

http://www.gregegan.net/FOUNDATIONS/03/found03.html