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Use precisely the stated convention for the Riemann curvature tensor, and let $D={\nabla }_T$. The geodesic equation says ${\nabla }_{T}T=0$ because proper time is an affine parameter. The commuting variation and the torsion-free connection giveThus the geodesic deviation equation isHere $(R(X,Y)Z{)}^a=R^a{}_{bcd}{Z}^b{X}^c{Y}^d$. Declaring this index convention fixes the sign: writing $R(S,T)T$ instead would require a minus sign. A deviation vector satisfying this equation is a Jacobi field.

The metric tensor of the planar warped spacetime depends only on $z$. The three translations therefore preserve it, givingThey are linearly independent Killing vector fields everywhere. The associated geodesic conserved quantities from Killing vectors are $-A^2\dot{t}$, $A^2\dot{x}$ and $A^2\dot{y}$, for an affine parameter; the conventional positive energy is $E=A^2\dot{t}$. No restriction on the positive function $A$ is needed for these three symmetries.