## A Laymen's Guide to General Relativity: Part I

Posted by on Wednesday, November 25, 2015

This week is a very important anniversary in the scientific community, as it was 100 years ago this week that Albert Einstein submitted an academic paper for peer review, creating the General Theory of Relativity.

In the century that followed, the theory has developed from an obscure mathematical exercise that was famously claimed to be understood by only three people in the world, to a tool so useful that no GPS system would work properly without it. Only one hundred years after it was first proposed, every teenager and their smart phone and every lost tourist following their car's oddly spoken instructions is benefiting from the theory of relativity.

But what exactly is the general theory of relativity? Even now it is really only understood by serious professional physicists, and is rarely taught in any detail below the graduate student level. And yet aside from being incredibly useful to society, it is actually a very beautiful theory. It is true that the mathematics is so brutal that even Einstein enlisted help to work through it, but when you interpret it into geometrical concepts it is one of the simplest theories in modern physics.

A few weeks ago I was chatting on an online physics forum targeted towards undergraduate physics and mathematics students (and serious amateurs in science) and I made a similar comment about the underlying beauty of the theory of relativity. And of course, a few of the participants answered this comment with a challenge: Could I write an article (or series of articles) for the centennial of general relativity, which explained it in detail without using any advanced mathematics? Of course I can! (Which proves that I am either confident in presenting theoretical concepts in simple terms, or I am foolhardy enough to attempt the impossible)

The first step is to go back to everyone's favorite mathematical theorem, Pythagoras' Theorem. Suppose that you take two points in space, and some coordinate system. It could be North-South-East-West-Up-Down, or some XYZ system. If the distance between the points along one coordinate is dx, along a second coordinate is dy, and along a third coordinate system is dx, then the distance between those two points is:

In the century that followed, the theory has developed from an obscure mathematical exercise that was famously claimed to be understood by only three people in the world, to a tool so useful that no GPS system would work properly without it. Only one hundred years after it was first proposed, every teenager and their smart phone and every lost tourist following their car's oddly spoken instructions is benefiting from the theory of relativity.

But what exactly is the general theory of relativity? Even now it is really only understood by serious professional physicists, and is rarely taught in any detail below the graduate student level. And yet aside from being incredibly useful to society, it is actually a very beautiful theory. It is true that the mathematics is so brutal that even Einstein enlisted help to work through it, but when you interpret it into geometrical concepts it is one of the simplest theories in modern physics.

A few weeks ago I was chatting on an online physics forum targeted towards undergraduate physics and mathematics students (and serious amateurs in science) and I made a similar comment about the underlying beauty of the theory of relativity. And of course, a few of the participants answered this comment with a challenge: Could I write an article (or series of articles) for the centennial of general relativity, which explained it in detail without using any advanced mathematics? Of course I can! (Which proves that I am either confident in presenting theoretical concepts in simple terms, or I am foolhardy enough to attempt the impossible)

The first step is to go back to everyone's favorite mathematical theorem, Pythagoras' Theorem. Suppose that you take two points in space, and some coordinate system. It could be North-South-East-West-Up-Down, or some XYZ system. If the distance between the points along one coordinate is dx, along a second coordinate is dy, and along a third coordinate system is dx, then the distance between those two points is:

ds

^{2}= dx^{2}+ dy^{2}+ dz^{2}And that distance is the same regardless of the coordinates you use. If you have the same two points, but with different values of dx,dy,dz then the total distance between the two points must remain the same:

The fact that the distance does not depend on the coordinate system is one of the most important points in the theory of relativity, and unfortunately many people even at higher levels of physics do not grasp this concept.

The two points are not the same as their coordinates, and the distance between them is invariant under a change of coordinates.

So what would you expect to happen in four-dimensional spacetime? It is not what you think. In four-dimensional space we do indeed have

dx'

^{2}+dy'^{2}+ dz'^{2}= dx^{2}+ dy^{2}+ dz^{2}The fact that the distance does not depend on the coordinate system is one of the most important points in the theory of relativity, and unfortunately many people even at higher levels of physics do not grasp this concept.

The two points are not the same as their coordinates, and the distance between them is invariant under a change of coordinates.

So what would you expect to happen in four-dimensional spacetime? It is not what you think. In four-dimensional space we do indeed have

ds

However in order for the laws of physics to be the same for all observers, the temporal dimension must be slightly different. For reasons given in my Bondi k-calculus page, the distance between two points in spacetime is given by:

^{2}= dw^{2}+ dx^{2}+ dy^{2}+ dz^{2}However in order for the laws of physics to be the same for all observers, the temporal dimension must be slightly different. For reasons given in my Bondi k-calculus page, the distance between two points in spacetime is given by:

ds

^{2}= -c^{2}dt^{2}+ dx^{2}+ dy^{2}+ dz^{2}(It should be noted here that 'c' is just a conversion factor between units of space and units of time. It is also the speed of light. And in most serious research we just set it equal to 1, and so I won't continue to include it here either). This is called the "Minkowski metric", and this one equation together with the invariance of distance under a change of coordinates contains the entire theory of special relativity.

And so having reached special relativity, I will end the first part of this series. Tomorrow we start on general relativity!

And so having reached special relativity, I will end the first part of this series. Tomorrow we start on general relativity!