## A Laymen's Guide To General Relativity: Part IV

Posted by on Friday, November 27, 2015

In the last article we saw how special relativity is really nothing more than a function that gives the distance between two points in a four-dimensional spacetime, and the requirement that this distance be the same for all coordinate systems we might use. We also saw how, in the most general metric function this distance could vary depending on both time and on position within space. And that leads us to a new problem.

If the distance between two points can vary depending on where in spacetime the measurement is taken, then it becomes difficult to compare distances or lengths. For example if you are measuring the speed of two cars on the highway, then you can measure the speed of one of them, and then go to another place and time and measure the speed of the other. And then compare the two numbers to find the relative speeds.

But what happens if space is compressed in front of one of the cars? They might be traveling at the same speed, but because space is compressed one travels further in the same amount of time. Or a more realistic situation, what happens if you need to compare the colour (and hence the wavelength) of light coming from two different stars? Even if the same wavelength is emitted by both, the effects of compression or expansion of space itself will make them appear to be different.

And so we introduce the concept of a

Usually it is written out either using several indices to represent which components of the vector are being mapped to which components of the output vector, or using notation from differential geometry which require more mathematical background. For this simple review I will instead use the notation

where D is called the covariant derivative, A is the vector being moved along another vector B, where B*dA is the rate of change of A moving along B in ordinary Minkowski space, and W is the connection which maps A at one end of vector B to a new vector located at the other end of vector B. (If this last line is confusing, feel free to ignore it. The notation will not be used overly much in the coming articles anyway).

That is the entirety of the second level of general relativity. We now have an operator that can move vectors from one place and one time to another place and time, and as a result a method of comparing two vectors located in different parts of spacetime.

But that creates a new problem, which we will introduce in the next article...

If the distance between two points can vary depending on where in spacetime the measurement is taken, then it becomes difficult to compare distances or lengths. For example if you are measuring the speed of two cars on the highway, then you can measure the speed of one of them, and then go to another place and time and measure the speed of the other. And then compare the two numbers to find the relative speeds.

But what happens if space is compressed in front of one of the cars? They might be traveling at the same speed, but because space is compressed one travels further in the same amount of time. Or a more realistic situation, what happens if you need to compare the colour (and hence the wavelength) of light coming from two different stars? Even if the same wavelength is emitted by both, the effects of compression or expansion of space itself will make them appear to be different.

And so we introduce the concept of a

*connection (*also referred to as covariant derivatives or Christoffel symbols*)*. The formulas for the spacetime connection are two complicated to include in this simple review, but underneath all of the mathematics is a simple operator that takes a vector (or the distance between two points) in one time and location, and moves it to another time and location without changing the vector in anyway other than to correct for effects of different coordinates and of the curvature of spacetime itself.Usually it is written out either using several indices to represent which components of the vector are being mapped to which components of the output vector, or using notation from differential geometry which require more mathematical background. For this simple review I will instead use the notation

D(A,B) = B*dA + B*W(A)

where D is called the covariant derivative, A is the vector being moved along another vector B, where B*dA is the rate of change of A moving along B in ordinary Minkowski space, and W is the connection which maps A at one end of vector B to a new vector located at the other end of vector B. (If this last line is confusing, feel free to ignore it. The notation will not be used overly much in the coming articles anyway).

That is the entirety of the second level of general relativity. We now have an operator that can move vectors from one place and one time to another place and time, and as a result a method of comparing two vectors located in different parts of spacetime.

But that creates a new problem, which we will introduce in the next article...