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

[binbagsss]

Homework Statement

Question attached

Homework Equations

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...

 

[binbagsss]

Homework Statement

Question attached View attachment 203944

Homework Equations

3. The Attempt at a Solution [/B]

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