What's with all the index notation in General Relativity? I am self-studying General Relativity with Leonard Susskind's lectures from Stanford. The thing that is bothering me is the notation of GR, specifically, the index notation.

In simple layman terms first, can someone explain these and then move on to the more precise and physics explanation of why:

*

*Why do you need all that index notation?

*What is it good for?

*Provide a concrete (not abstract) and a simple example of this notation in action

Answering these questions will help me understand more about GR and provide me with the basics of GR.
 A: *

*Because it provides a nice, easy way of dealing with tensors and the operations that exist between them. Along with the summation convention, the index notation massively condenses the equations used in general relativity.


*It makes manipulations in general relativity as simple as knowing a few rules on how indices can and can't interact with each other.


*Even someone new to general relativity will be able to see that: $$T^\mu g_{\mu\nu}A_\mu=X_{\nu} \tag{1}$$ is an invalid tensor equation, because we have used the same index three times in one term (and we have the same index down twice). If we wrote this out without the index notation and without the summation convention it would be pretty hard to decipher and would not be immediately obvious that it was invalid. The index/summation conventions are ways of visually decluttering equations in GR without sacrificing important information.
A: The simple reason for using index notation is that most mathematicians and physicists find it to be (eventually) a natural and intuitive way of describing tensor operations. It just takes a bit of work to get to the "natural and intuitive" stage - but then $\frac {d^2x}{dt^2}$ must have looked weird when we first encountered it too.
However, index notation is not the only game in town - Roger Penrose invented and championed an alternative graphical notation. Maybe if we had been taught this from day one then you would be asking "what's with all the weird spaghetti diagrams in GR ?" instead.
A: It's all about distinguishing $a-b$ from $b-a$.
