A Laymen's Guide To General Relativity: Part III
Posted by on Friday, November 27, 2015
To me it has always seemed natural to divide the equations of general relativity into five distinct levels, each of which has its own laws and constraints. In my opinion each level must be studied and understood separately before they can be combined into one complete theory.
The foundation of the theory is the spacetime metric, which is the subject of today's article.
And while the name sounds impressive, it is actually a very simple object. A metric is just a fancy mathematical name for any function that takes two points in some surface or space or spacetime, and outputs a number representing the 'distance' between those points. The equation I gave before for Pythagoras' theorem qualifies, as it gives the distance between two points in three-dimensional space. And the Minkowski metric qualifies, as it gives the distance between two points in space and time.
But we can do even better!
The most general metric for the sort of spaces and spacetimes we are considering could measure distances differently in different directions. It could also vary, giving different distances depending on where and when the measurement is made. A ruler could be measured as one meter on Earth, and ten meters in another galaxy or another time.
However for technical reasons, and for reasons of simplicity, we will restrict our considerations to four-dimensional Lorentzian metrics. The equations of general relativity apply equally well to any number of dimensions, and in fact some of my own research has been focused on applying to five, six, seven or even eleven dimensional spacetimes. And some people have also considered what would happen with multiple time dimensions, or even with no time dimensions. However the Universe we see has three space dimensions and a single time dimension, and so that is what we will use today.
In general, the distance between two points whose coordinates differ by (dt,dx,dy,dz) will be given by the metric:
where u and v represent the numbers 0,1,2,3 and xu is shorthand for the coordinates, (t,x,y,z) and where the g(t,x,y,z) are all arbitrary functions of space and time. At this point in the calculations we do not know what these functions are.
And as with the Minkowski metric, this distance cannot depend on the specific choice of coordinates. That also means that once we know the functions, we can calculate time dilation and energy momentum relations just as we did for special relativity. It also is our first hint of an important prediction of general relativity - namely that time moves at different speeds in different parts of spacetime. We will see later in these articles that the presence of matter and energy causes spacetime to distort, which in turn causes time to slow down, and effect which has been measured experimentally.
For now though, we do not know what the metric functions might be. Now on to the next level...
The foundation of the theory is the spacetime metric, which is the subject of today's article.
And while the name sounds impressive, it is actually a very simple object. A metric is just a fancy mathematical name for any function that takes two points in some surface or space or spacetime, and outputs a number representing the 'distance' between those points. The equation I gave before for Pythagoras' theorem qualifies, as it gives the distance between two points in three-dimensional space. And the Minkowski metric qualifies, as it gives the distance between two points in space and time.
But we can do even better!
The most general metric for the sort of spaces and spacetimes we are considering could measure distances differently in different directions. It could also vary, giving different distances depending on where and when the measurement is made. A ruler could be measured as one meter on Earth, and ten meters in another galaxy or another time.
However for technical reasons, and for reasons of simplicity, we will restrict our considerations to four-dimensional Lorentzian metrics. The equations of general relativity apply equally well to any number of dimensions, and in fact some of my own research has been focused on applying to five, six, seven or even eleven dimensional spacetimes. And some people have also considered what would happen with multiple time dimensions, or even with no time dimensions. However the Universe we see has three space dimensions and a single time dimension, and so that is what we will use today.
In general, the distance between two points whose coordinates differ by (dt,dx,dy,dz) will be given by the metric:
ds2 = guvdxudxv = g00dt2 - g11 dx2 - g12dx dy - ...
where u and v represent the numbers 0,1,2,3 and xu is shorthand for the coordinates, (t,x,y,z) and where the g(t,x,y,z) are all arbitrary functions of space and time. At this point in the calculations we do not know what these functions are.
And as with the Minkowski metric, this distance cannot depend on the specific choice of coordinates. That also means that once we know the functions, we can calculate time dilation and energy momentum relations just as we did for special relativity. It also is our first hint of an important prediction of general relativity - namely that time moves at different speeds in different parts of spacetime. We will see later in these articles that the presence of matter and energy causes spacetime to distort, which in turn causes time to slow down, and effect which has been measured experimentally.
For now though, we do not know what the metric functions might be. Now on to the next level...
I am a theoretical physicist & mathematician, with training in electronics, programming, robotics, and a number of other related fields.
