General Relativity and Differential Geometry textbook problem

[shooride]

I'm studying General Relativity and Differential Geometry. In my textbook, the author has written $x^2=d(x,.)$ where d(x,y) is distance between two points $x,y\in M$. I couldn't understand what d(x,.) means. Moreover, I am not sure if this is generally true to write $x^2=g_{\mu\nu} x^\mu x^\nu=d(x,.)$. Under what conditions can one write $x^2=d(x,.)$?

 

[shooride]

I'm studying General Relativity and Differential Geometry. In my textbook, the author has written $x^2=d(x,.)$ where d(x,y) is distance between two points $x,y\in M$. I couldn't understand what d(x,.) means. Moreover, I am not sure if this is generally true to write $x^2=g_{\mu\nu} x^\mu x^\nu=d(x,.)$. Under what conditions can one write $x^2=d(x,.)$?

Oops! I should write $x^2=d(x,.)^2$

 

[Nugatory]

In my textbook...

Which textbook?

 

[shooride]

Which textbook?

Unfortunately, that book has been written in my native language! By this you mean it is meaningless to write x^2=d(x,.)?! I think that one can write x^2=d(x,.)^2 at least in Riemann normal coordinates...

 

[fzero]

Writing $x^2 = g_{\mu\nu} x^\mu x^\nu$ is common shorthand. I'm not sure why the author writes $d(x,.)$ because $d(x,x)$ would make more sense there. When you write something like $d(x,.)$ sometimes that means an operator that takes an object $v^\mu$ and acts to form $d(x,v)$. It doesn't appear to be the case here, but perhaps more context would help.

 

[shooride]

Writing $x^2 = g_{\mu\nu} x^\mu x^\nu$ is common shorthand. I'm not sure why the author writes $d(x,.)$ because $d(x,x)$ would make more sense there. When you write something like $d(x,.)$ sometimes that means an operator that takes an object $v^\mu$ and acts to form $d(x,v)$. It doesn't appear to be the case here, but perhaps more context would help.

After that I posted this question, I realized $d(x,.)$ means distance function. BTW, I've tried to evaluate the distance between two point $x^\mu$ and $y^\mu$ in Riemann normal coordinates, what I get is
$d(x,y)^2=g_{\mu\nu} (x^\mu-y^\mu)(x^\nu-y^\nu) +1/3 R_{\mu\nu\rho\sigma}x^\rho x^\sigma(x^\mu-y^\mu)(x^\mu- y^\nu) + O((x^\mu-y^\mu)^3)$
So
$d(x,0)^2=g_{\mu\nu} x^\mu x^\nu +1/3 R_{\mu\nu\rho\sigma}x^\rho x^\sigma x^\mu x^\mu + O(x^3)$
in this way, I think what author really means is
$d(x,0)^2=x^2+O(x^2)$
with an abuse of notation!

 

[bcrowell]

After that I posted this question, I realized $d(x,.)$ means distance function.

That doesn't make sense. I think fzero's interpretation is much more likely.