The Schwarzschild Solution

Karl Schwarzschild

Karl Schwarzschild, image courtesy of Wikipedia.

The name of this page comes from the man that was the first to find an exact solution to Einstein’s equation: Karl Schwarzchild. Einstein had came up with an approximate solution, but was shocked when someone came up with an exact one.

There is a great Wikipedia page about Karl, here!


 

Remember the equation we used in the section on Tensors, and the section on Space-time curvature? It involved tensors, and more tensors. To refresh our memories, here it is:

Image courtesy of scienceblogs.com

In many different forms (substituting the left side for the Einstein Tensor) this is Einstein’s Equation. Something I have not mentioned, and if you delve into the study of mathematics (especially differential equations) you will see that some equations have multiple solutions. Why? How? It’s all dependent on different assumptions, conditions, and constraints that are put on the equations.

The website WolframMathWorld tells us: “The Einstein equation contains 16 separate, nonlinear partial differential equations” (check it out!). Yikes! But remember from the tensors section that symmetry can sometimes help us out. Thanks to symmetry, it reduces to 10 equations. So 10 equations, and the Schwarzschild solution is just one solution to them. What does it look like?

 

Time-like form of the solution
Time-like form of the solution, courtesy of “Exploring Black Holes”.

According to the text “Exploring Black Holes” by Edwin F. Taylor and John Archibald Wheeler, the Schwarzschild solution describes space-time external to any isolated spherically symmetric body in the universe”. We can use this solution to calculate the space-time curvature around any body (and, thanks to gravitation, massive objects tend to be spherical). Quite the powerful tool!

The Schwarzschild solution deals with a spherically symmetric, non-rotating mass. Let’s see where the r’s, Φ’s, and other variables come from. The “Φ” deals with the the angle the sphere subtends (in this case, we are considering events near the sphere, so it may not be a 360 degree angle). The “r” deals with the reduced circumference of the shell (a term arising from space-time curvature, meaning the circumference is not what one would normally calculate). The “t” in the equation deals with the time as measured by clocks far away from the center of attraction.

All of the terms with a “d” in front of them are differentials. A differential, simply put, deals with an infinitesimally small amount. Remember the analogy of the Klingon canoe, and all the strings we had to measure it’s shape? We could do the same with differentials: just make the strings smaller and smaller and smaller, and we would get a much more accurate shape of the canoe’s geometry.


WHAT IS ONE WAY IN WHICH ALL THE PREVIOUS CONTENT COMES TOGETHER? WELL, ANY DISCUSSION OF THE GLOBAL POSITIONING SYSTEM WOULD BE INCOMPLETE WITHOUT IT!

Earth: Not a perfect sphere