General Relativity and Differential Geometry textbook problem

  • #1
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I'm studying General Relativity and Differential Geometry. In my text book, 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,.)##?
 

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  • #2
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I'm studying General Relativity and Differential Geometry. In my text book, 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##
 
  • #4
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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...
 
  • #5
fzero
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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.
 
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  • #6
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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
[itex]
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)
[/itex]
So
[itex]
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)
[/itex]
in this way, I think what author really means is
[itex]
d(x,0)^2=x^2+O(x^2)
[/itex]
with an abuse of notation!
 
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  • #7
bcrowell
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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.
 

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