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**1. The problem statement, all variables and given/known data**

Question attached

**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...