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 AdS4 is the universal covering of the embedded hyperboloid Hm (m>0) in R2,3=(R5,η)
Hm={x∈R5 | η(x,x)≐−x20+x21+x22+x23−x24=−m} .
The covering map Φ:AdS4∋(t,r,θ,ϕ)↦(x0,x1,x2,x3,x4)∈Hm⊂R2,3 through the global coordinates (t∈R,r≥0,0≤θ≤π,0≤ϕ<2π) is given by
x0=√m(1+r2)sint ;
x1=√mrsinθcosϕ ;
x2=√mrsinθsinϕ ;
x3=√mrcosθ ;
x4=√m(1+r2)cost .
The pullback of the ambient, flat pseudo-Riemannian metric η defined above (with signature (−+++−)) by Φ after restriction to Hm yields the AdS4 metric in the form appearing in the question and in Pedro Figueroa's nice answer up to a constant, positive factor:
ds2=m[−(m+r2)dt2+(m+r2)−1dr2+r2(dθ2+sin2θdϕ2)] .
The conformal completion of AdS4, on its turn, is obtained by means of the change of radial variable u=√m+r2−r, so that r=m−u22u, dr=−1u(m+u22u)du and m+r2=(m+u22u)2, yielding
ds2=mu2[−(m+u22)2dt2+du2+(m−u22)2(dθ2+sin2θdϕ2)] .
Conformal infinity is reached by taking r→+∞, which is the same as u↘0. The rescaled metric Ω2ds2, Ω=m−12u yields the three-dimensional Einstein static universe as the conformal boundary (i.e. u=0).
It's clear that Hm is a level set of the function f:R5→R given by f(x)=η(x,x). Therefore, the vector field Xx=12gradηf(x)=x (where gradη is the gradient operator defined with respect to η) is everywhere normal to Hm - that is, any tangent vector Xx∈TxHm satisfies η(Xx,Tx)=0. Given two vector fields T,S tangent to Hm, the intrinsic covariant derivative ∇TS on Hm is simply given by the tangential component of the ambient (flat) covariant derivative (∂TS)a=Tb∂bSa:
∇TS=∂TS−η(X,∂TS)η(X,X)X=∂TS+η(X,∂TS)mX .
The normal component of ∂TS, on its turn, has a special form due to the nature of Hm (notice that ∂aXb=∂axb=δba):
η(X,∂TS)=∂T(η(X,S))⏟=0 ;−η(S,∂TX)=−η(S,T) ⇒ η(X,∂TS)η(X,X)X=η(S,T)mX .
As such, we conclude that a curve γ:I∋λ↦γ(λ)∈Hm (I⊂R is an interval with nonvoid interior) is a geodesic of Hm if and only if d2γ(λ)dλ2(λ)≐¨γ(λ) is everywhere normal to Hm, that is,
¨γ(λ)=−1mη(¨γ(λ),Xγ(λ))Xγ(λ)=1mη(˙γ(λ),˙γ(λ))Xγ(λ)=1mη(˙γ(λ),˙γ(λ))γ(λ) .
In particular, if η(˙γ(λ),˙γ(λ))=0, then γ is also a (null) geodesic in the ambient space R2,3.
Given x∈Hm, the linear span of Xx=x and any tangent vector Tx≠0 to Hm at x defines a 2-plane P(Tx) through the origin of R5 and containing x. In other words,
P(Tx)={αXx+βTx | α,β∈R} ,
and therefore
P(Tx)∩Hm={y=αXx+βTx | η(y,y)=−α2m+β2η(Tx,Tx)=−m} .
This allows us already to classify P(Tx)∩Hm according to the causal character of Tx:
- Tx timelike (i.e. −k=η(Tx,Tx)<0): we have that mα2+kβ2=m with k,m>0, hence P(Tm)∩Hm is an ellipse;
- Tx spacelike (i.e. k=η(Tx,Tx)>0): we have that mα2−kβ2=m with k,m>0, hence P(Tm)∩Hm is a pair of hyperbolae, one with α>0
and the other with α<0. The point x=Xx belongs to the first hyperbola;
- Tx lightlike (i.e. η(Tx,Tx)=0): we have that α2=1 with β arbitrary, hence P(Tm)∩Hm is a pair of straight lines, one given by α=1 and the other by α=−1. The point x=Xx belongs to the first line. Notice that each of these lines is a null geodesic both in Hm and in R2,3!
Moreover, x=γ(0) and Tx=˙γ(0) define a general initial condition for a geodesic γ starting at x. It remains to show that any curve that stays in P(Tx)∩Hm is a geodesic in Hm. This is clearly true for Tx lightlike, since in this case we have already concluded that γ(λ)=x+λTx for all λ∈R. For the remaining cases (i.e. η(Tx,Tx)≠0), consider a C2 curve γ in P(Tx)∩Hm beginning at γ(0)=x with ˙γ(0)=˙β(0)Tx (we assume that ˙γ(λ)≠0 for all λ). Writing γ(λ)=α(λ)Xx+β(λ)Tx, we conclude from the above classification of P(Tx)∩Hm that we can choose the parameter λ so that
- Tx timelike: α(λ)=cosλ, β(λ)=√−mη(Tx,Tx)sinλ, so that η(˙γ(λ),˙γ(λ))=−m with ˙β(0)=√−mη(Tx,Tx);
- Tx spacelike: α(λ)=coshλ, β(λ)=√mη(Tx,Tx)sinhλ, so that η(˙γ(λ),˙γ(λ))=+m with ˙β(0)=√mη(Tx,Tx).
In both cases, we conclude that
¨γ(λ)=η(˙γ(λ),˙γ(λ))mγ(λ) ,
i.e. γ must satisfy the geodesic equation in Hm 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 Hm will remain forever in that 2-plane. For later use, I remark that all geodesics of Hm cross at least once the 2-plane P0={x∈R5 | x1=x2=x3=0} - this can be easily seen from the classification of the sets P(Tx)∩Hm. This allows us to prescribe initial conditions in P0 for all geodesics in Hm.
Now we have complete knowledge of the geodesics in the fundamental domain Hm of AdS4. 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 γ(0)=Xx=x=(0,0,0,0,√m)
in
Hm (i.e.
γ is made to start at
P0 with
t=0), so that
˙γ(0)=Tx=(y0,y1,y2,y3,0) .
We also normalize
η(Tx,Tx) to
−m,
+m or zero depending on whether
Tx is respectively timelike, spacelike or lightlike. Writing once more
γ(λ)=α(λ)Xx+β(λ)Tx, we use the classification of geodesics in
Hm by their causal character to write explicit formulae for
γ:
- Tx timelike ⇒ γ(λ)=(cosλ)Xx+(sinλ)Tx;
- Tx spacelike ⇒ γ(λ)=(coshλ)Xx+(sinhλ)Tx;
- Tx lightlike ⇒ γ(λ)=Xx+λTx.
The above expressions show that, in the spacelike and lightlike cases, the last component γ(λ)4 of γ(λ) never goes to zero, which implies by continuity that the time coordinate t stays within the interval (−π2,π2), hence the lift of γ to AdS4 stays within a single copy of its fundamental domain. One also sees that the spatial components (1,2,3) of γ(λ) go to infinity as λ→±∞, hence u→0 along these geodesics as λ→±∞. In the timelike case, the whole time interval [0,2π] is spanned by γ(λ) as λ spans the interval [0,2π]. Since the curve is closed, its lift to AdS4 spans the whole time line R as λ does so. On the other hand, it's clear that in this case the spatial components of γ(λ) just keep oscillating within a bounded interval of the coordinate r - hence, the coordinate u stays bounded away from zero. Therefore, a timelike geodesic γ never escapes to conformal infinity.
This post imported from StackExchange Physics at 2014-06-14 16:03 (UCT), posted by SE-user Pedro Lauridsen Ribeiro