G11 Metric Tensor: What is it & How Does it Work?

[Mathematicsresear]

What is g₁₁? I am very curious, can someone briefly describe what the metric tensor is, please?

 

[phyzguy]

https://en.wikipedia.org/wiki/Metric_tensor

 

[Nugatory]

Tell us more. Where did you hear about the metric tensor and what did it say that prompted you to ask about specifically $g_{11}$? You'll get better answers if we know more about where you're coming from.

But with said... There may not be a good answer to your question unless you already are comfortable with the notion of the dot-product of vectors, but we can try.

The metric tensor is a mathematical tool that tells us the distance between two points. Of course that's trivial in ordinary three-dimensional space using Cartesian coordinates like we learned in our first year of algebra: The Pythagorean theorem says that the distance $s$ is $s=\sqrt{\Delta{x}^2+\Delta{y}^2+\Delta{z}^2}$. However, it gets much trickier if you aren't using Cartesian coordinates (for example, what's the formula for the distance between two points in a plane given their $r,\theta$ polar coordinates?) and even trickier if your points are on a curved surface (for example, the surface of the earth) where Euclidian geometry doesn't apply and the Pythagorean theorem doesn't work.

And as for why we should care about it? It is vitally important to general relativity, because GR is based on the idea that spacetime is curved. The simplest reasonable explanation still demands much more math than belongs in a B-level thread, but if you're curious you could take a look at https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf.