100 Years of General Relativity

When Einstein was a 25 year old unknown patent clerk, he stunned the physics community by publishing what is now known as the Special Theory of Relativity.  This introduced the ideas of length contraction, time dilation, spacetime, E=mc², and maybe more fundamentally, it did away with Newtonian absolute space and time.  But it was also limited in that it couldn't account for gravity or acceleration.  Ten years later Einstein finished the more complete General Theory of Relativity that could account for both gravity and acceleration.  But whereas Special Relativity only requires some fairly basic algebra, General Relativity involves some pretty advanced math: tensors, manifolds, covariant derivatives, and other esoteric terms that are part of a branch of mathematics known as differential geometry.  From books and science documentaries I've become familiar with many of the concepts of General Relativity such as gravity being the warping, stretching, or curving of spacetime, time slowing down in locations of greater gravity, the equivalence of gravity and acceleration, etc., but I've never had the chance to engage with the math in any meaningful way.  Until now, dun dun dun.

Ok, my understanding is still quite cursory, but that's an improvement nonetheless.  I'll attempt to share of little of the little I understand.  Here is the beauty that is Einstein's Equation (that's its actual name):

It looks pretty harmless at first glance, but then you might notice a few things.  For starters the third term has this capital lambda that looks a bit suspicious, but where the real nastiness lies is in those little subscripts.  That means that the R, g, and T are what are known as tensors which are sort of like multi-dimensional vectors manifest in the form of matrices (i.e. this equation is shorthand for a series of equations).

The R term with the subscripts (known as the Ricci Tensor) expands out to this guy:

So it's actually a tensor calculated from other tensors (as indicated by the multi-subscripted symbols).  These upside down L looking things are capital gammas and are called Christoffel symbols (the fraction looking terms are partial derivatives...something that I'd actually previously heard of).  They represent another layer of nesting:

Stay with me here.  The "g"s in the above equation are the same as the ones in the original Einstein Equation that I started with.  They represent the properties of the coordinate system (e.g. cartesian, polar, etc.) and is called the Metric tensor, or just the Metric for short.  It get's uglier in other coordinate systems, but in cartesian coordinates the Metric is just the matrix in the lower right with ones and negative ones.

So the Christoffel symbols (the gammas) contain information about how a coordinate system changes as you move around space and the Ricci tensor (the R with subscripts) contains the information about how spacetime is curved or contoured from one location to another.  The R without subscripts is known as the Ricci scalar which is similar to the tensor but without as much information.

So essentially the entire left hand side of Einstein's Equation (more on the lambda term later) describes the curvature or topology of spacetime.

On the right-hand side of the equation the 4, pi, G, and c are all just constants (hooray!).  The T is known as the stress-energy tensor:

The Metric shows up in there again as well as these F tensor things.  These take into account how mass, energy, and pressure are distributed in space.  All of these warp or stretch spacetime and thus contribute of gravity (not just mass as Newton thought).

There you have it.  A bunch of nasty looking math to tell us how spacetime is curved (the left side) and why (the right side).