As far as I could understand, it seems that you want to know whether timelike geodesics can reach the conformal boundary of AdS. If that's the case (please do confirm), the answer is no - no timelike geodesic can reach conformal infinity, it rather gets constantly refocused back into the bulk in a periodic fashion. You need timelike curves which have some acceleration in order to avoid this. Maximally extended null geodesics (i.e. light rays), on the other hand, always reach conformal infinity, both in the past and in the future. An illustration of these facts using Penrose diagrams can be found, for instance, in Section 5.2, pp. 131-134 of the book by S. W. Hawking and G. F. R. Ellis, "The Large Scale Structure of Space-Time" (Cambridge, 1973).
The detailed reasoning behind the above paragraph can be seen in a global, geometric way. In what follows, I'll largely follow the argument presented in the book by B. O'Neill, "Semi-Riemannian Geometry - With Applications to Relativity" (Academic Press, 1983), specially Proposition 4.28 and subsequent remarks, pp. 112-113. For the benefit of those with no access to O'Neill's book, I'll present the self-contained argument in full detail. I'll make use of the fact that AdS_4 is the universal covering of the embedded hyperboloid H_m (m>0) in \mathbb{R}^{2,3}=(\mathbb{R}^5,\eta)
H_m=\{x\in\mathbb{R}^5\ |\ \eta(x,x)\doteq -x_0^2+x_1^2+x_2^2+x_3^2-x_4^2=-m\}\ .
The covering map \Phi:AdS_4\ni(t,r,\theta,\phi)\mapsto (x_0,x_1,x_2,x_3,x_4)\in H_m\subset\mathbb{R}^{2,3} through the global coordinates (t\in\mathbb{R},r\geq 0,0\leq\theta\leq\pi,0\leq\phi<2\pi) is given by
x_0=\sqrt{m(1+r^2)}\sin t\ ; x_1=\sqrt{m}r\sin\theta\cos\phi\ ; x_2=\sqrt{m}r\sin\theta\sin\phi\ ; x_3=\sqrt{m}r\cos\theta\ ; x_4=\sqrt{m(1+r^2)}\cos t\ .
The pullback of the ambient, flat pseudo-Riemannian metric \eta defined above (with signature (-+++-)) by \Phi after restriction to H_m yields the AdS_4 metric in the form appearing in the question and in Pedro Figueroa's nice answer up to a constant, positive factor:
ds^2= m\left[-(m+r^2)dt^2+(m+r^2)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)\right]\ .
The conformal completion of AdS_4, on its turn, is obtained by means of the change of radial variable u=\sqrt{m+r^2}-r, so that r=\frac{m-u^2}{2u}, dr=-\frac{1}{u}(\frac{m+u^2}{2u})du and m+r^2=(\frac{m+u^2}{2u})^2, yielding
ds^2=\frac{m}{u^2}\left[-\left(\frac{m+u^2}{2}\right)^2dt^2+du^2+\left(\frac{m-u^2}{2}\right)^2(d\theta^2+\sin^2\theta d\phi^2)\right]\ .
Conformal infinity is reached by taking r\rightarrow+\infty, which is the same as u\searrow 0. The rescaled metric \Omega^2 ds^2, \Omega=m^{-\frac{1}{2}}u yields the three-dimensional Einstein static universe as the conformal boundary (i.e. u=0).
It's clear that H_m is a level set of the function f:\mathbb{R}^5\rightarrow\mathbb{R} given by f(x)=\eta(x,x). Therefore, the vector field X_x=\frac{1}{2}\mathrm{grad}_\eta f(x)=x (where \mathrm{grad}_\eta is the gradient operator defined with respect to \eta) is everywhere normal to H_m - that is, any tangent vector X_x\in T_x H_m satisfies \eta(X_x,T_x)=0. Given two vector fields T,S tangent to H_m, the intrinsic covariant derivative \nabla_T S on H_m is simply given by the tangential component of the ambient (flat) covariant derivative (\partial_T S)^a=T^b\partial_b S^a:
\nabla_T S=\partial_T S-\frac{\eta(X,\partial_T S)}{\eta(X,X)}X=\partial_T S+\frac{\eta(X,\partial_T S)}{m}X\ .
The normal component of \partial_T S, on its turn, has a special form due to the nature of H_m (notice that \partial_a X^b=\partial_a x^b=\delta^b_a):
\eta(X,\partial_T S)=\underbrace{\partial_T(\eta(X,S))}_{=0\ ;}-\eta(S,\partial_T X)=-\eta(S,T)\ \Rightarrow\ \frac{\eta(X,\partial_T S)}{\eta(X,X)}X=\frac{\eta(S,T)}{m}X\ .
As such, we conclude that a curve \gamma:I\ni\lambda\mapsto\gamma(\lambda)\in H_m (I\subset\mathbb{R} is an interval with nonvoid interior) is a geodesic of H_m if and only if \frac{d^2\gamma(\lambda)}{d\lambda^2}(\lambda)\doteq\ddot{\gamma}(\lambda) is everywhere normal to H_m, that is,
\ddot{\gamma}(\lambda)=-\frac{1}{m}\eta(\ddot{\gamma}(\lambda),X_{\gamma(\lambda)})X_{\gamma(\lambda)}=\frac{1}{m}\eta(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))X_{\gamma(\lambda)}=\frac{1}{m}\eta(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))\gamma(\lambda)\ .
In particular, if \eta(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))=0, then \gamma is also a (null) geodesic in the ambient space \mathbb{R}^{2,3}.
Given x\in H_m, the linear span of X_x=x and any tangent vector T_x\neq 0 to H_m at x defines a 2-plane P(T_x) through the origin of \mathbb{R}^5 and containing x. In other words,
P(T_x)=\{\alpha X_x +\beta T_x\ |\ \alpha,\beta\in\mathbb{R}\}\ ,
and therefore
P(T_x)\cap H_m=\{y=\alpha X_x+\beta T_x\ |\ \eta(y,y)=-\alpha^2 m+\beta^2\eta(T_x,T_x)=-m\}\ .
This allows us already to classify P(T_x)\cap H_m according to the causal character of T_x:
- T_x timelike (i.e. -k=\eta(T_x,T_x)<0): we have that m\alpha^2+k\beta^2=m with k,m>0, hence P(T_m)\cap H_m is an ellipse;
- T_x spacelike (i.e. k=\eta(T_x,T_x)>0): we have that m\alpha^2-k\beta^2=m with k,m>0, hence P(T_m)\cap H_m is a pair of hyperbolae, one with \alpha>0 and the other with \alpha<0. The point x=X_x belongs to the first hyperbola;
- T_x lightlike (i.e. \eta(T_x,T_x)=0): we have that \alpha^2=1 with \beta arbitrary, hence P(T_m)\cap H_m is a pair of straight lines, one given by \alpha=1 and the other by \alpha=-1. The point x=X_x belongs to the first line. Notice that each of these lines is a null geodesic both in H_m and in \mathbb{R}^{2,3}!
Moreover, x=\gamma(0) and T_x=\dot{\gamma}(0) define a general initial condition for a geodesic \gamma starting at x. It remains to show that any curve that stays in P(T_x)\cap H_m is a geodesic in H_m. This is clearly true for T_x lightlike, since in this case we have already concluded that \gamma(\lambda)=x+\lambda T_x for all \lambda\in\mathbb{R}. For the remaining cases (i.e. \eta(T_x,T_x)\neq 0), consider a \mathscr{C}^2 curve \gamma in P(T_x)\cap H_m beginning at \gamma(0)=x with \dot{\gamma}(0)=\dot{\beta}(0)T_x (we assume that \dot{\gamma}(\lambda)\neq 0 for all \lambda). Writing \gamma(\lambda)=\alpha(\lambda)X_x+\beta(\lambda)T_x, we conclude from the above classification of P(T_x)\cap H_m that we can choose the parameter \lambda so that
- T_x timelike: \alpha(\lambda)=\cos\lambda, \beta(\lambda)=\sqrt{-\frac{m}{\eta(T_x,T_x)}}\sin\lambda, so that \eta(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))=-m with \dot{\beta}(0)=\sqrt{-\frac{m}{\eta(T_x,T_x)}};
- T_x spacelike: \alpha(\lambda)=\cosh\lambda, \beta(\lambda)=\sqrt{\frac{m}{\eta(T_x,T_x)}}\sinh\lambda, so that \eta(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))=+m with \dot{\beta}(0)=\sqrt{\frac{m}{\eta(T_x,T_x)}}.
In both cases, we conclude that
\ddot{\gamma}(\lambda)=\frac{\eta(\dot{\gamma}(\lambda),\dot{\gamma}(\lambda))}{m}\gamma(\lambda)\ ,
i.e. \gamma must satisfy the geodesic equation in H_m with the chosen parametrization, as wished. Since any pair of initial conditions for a geodesic determines a 2-plane through the origin in the above fashion, we conclude that the resulting geodesic in H_m will remain forever in that 2-plane. For later use, I remark that all geodesics of H_m cross at least once the 2-plane P_0=\{x\in\mathbb{R}^5\ |\ x_1=x_2=x_3=0\} - this can be easily seen from the classification of the sets P(T_x)\cap H_m. This allows us to prescribe initial conditions in P_0 for all geodesics in H_m.
Now we have complete knowledge of the geodesics in the fundamental domain H_m of AdS_4. What happens when we go back to the universal covering? What happens is that the lifts of spacelike and lightlike geodesics stay confined to a single copy of the fundamental domain, whereas the lifts of timelike geodesics do not. To see this, we exploit the fact that translations in the time coordinate t are isometries and the remark at the end of the previous paragraph to set \gamma(0)=X_x=x=(0,0,0,0,\sqrt{m}) in H_m (i.e. \gamma is made to start at P_0 with t=0), so that \dot{\gamma}(0)=T_x=(y_0,y_1,y_2,y_3,0)\ . We also normalize \eta(T_x,T_x) to -m, +m or zero depending on whether T_x is respectively timelike, spacelike or lightlike. Writing once more \gamma(\lambda)=\alpha(\lambda)X_x+\beta(\lambda)T_x, we use the classification of geodesics in H_m by their causal character to write explicit formulae for \gamma:
- T_x timelike \Rightarrow \gamma(\lambda)=(\cos\lambda)X_x+(\sin\lambda)T_x;
- T_x spacelike \Rightarrow \gamma(\lambda)=(\cosh\lambda)X_x+(\sinh\lambda)T_x;
- T_x lightlike \Rightarrow \gamma(\lambda)=X_x+\lambda T_x.
The above expressions show that, in the spacelike and lightlike cases, the last component \gamma(\lambda)_4 of \gamma(\lambda) never goes to zero, which implies by continuity that the time coordinate t stays within the interval (-\frac{\pi}{2},\frac{\pi}{2}), hence the lift of \gamma to AdS_4 stays within a single copy of its fundamental domain. One also sees that the spatial components (1,2,3) of \gamma(\lambda) go to infinity as \lambda\rightarrow\pm\infty, hence u\rightarrow 0 along these geodesics as \lambda\rightarrow\pm\infty. In the timelike case, the whole time interval [0,2\pi] is spanned by \gamma(\lambda) as \lambda spans the interval [0,2\pi]. Since the curve is closed, its lift to AdS_4 spans the whole time line \mathbb{R} as \lambda does so. On the other hand, it's clear that in this case the spatial components of \gamma(\lambda) just keep oscillating within a bounded interval of the coordinate r - hence, the coordinate u stays bounded away from zero. Therefore, a timelike geodesic \gamma never escapes to conformal infinity.
