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GR conditions conserved quantities AdS s-t; t-l geodesic

  • Thread starter binbagsss
  • Start date
1,153
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1. The problem statement, all variables and given/known data

Question attached
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2. Relevant equations

3. The attempt at a solution

part a) ##ds^2=\frac{R^2}{z^2}(-dt^2+dy^2+dx^2+dz^2)##
part b) it is clear there is a conserved quantity associated with ##t,y,x##

From Euler-Lagrange equations ## \dot{t}=k ## , k a constant ; similar for ## \dot{y}=c ## and ## \dot{x}=b ## , ##b,c## constants

I get the Lagrangian as ## L=\frac{R^2}{z^2}( \dot{x^2} + \dot{y^2} + \dot{z^2} - \dot{t^2} )##

Let me combine all the constants as ##\kappa## then I can write this as:

##\frac{R^2}{z^2}( \kappa + \dot{z^2} )<0## ; since ##L## must be ##<0## for a time-like geodesic. I'm not sure what to do now...
 
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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 
1,153
6
Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
my pockets are all empty, ta bot bruv' though
 
1,153
6
1. The problem statement, all variables and given/known data

Question attached View attachment 203944

2. Relevant equations

3. The attempt at a solution


part a) ##ds^2=\frac{R^2}{z^2}(-dt^2+dy^2+dx^2+dz^2)##
part b) it is clear there is a conserved quantity associated with ##t,y,x##

From Euler-Lagrange equations ## \dot{t}=k ## , k a constant ; similar for ## \dot{y}=c ## and ## \dot{x}=b ## , ##b,c## constants

I get the Lagrangian as ## L=\frac{R^2}{z^2}( \dot{x^2} + \dot{y^2} + \dot{z^2} - \dot{t^2} )##

Let me combine all the constants as ##\kappa <0 ## then I can write this as:

##\frac{R^2}{z^2}( \kappa + \dot{z^2} )<0## ; since ##L## must be ##<0## for a time-like geodesic. I'm not sure what to do now...
So is it simply ## \kappa <0## ? Seems too trivial / simple ..

Many thanks
 

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