,RμνRμν,RμνρσRμνρσ are finite. The SFS is different from the respect that the metric is nonsingular at sudden future singularity for t=ts, but all the curvature invariants R,RμνRμν,RμνρσRμνρσ have a blow-up due to the blow-up of the Riemann tensor as a consequence of the divergence of the second derivative of the scale factor. In fact, the geodesic deviation equation feel SFS due to the divergence of the Riemann tensor.
where T=1/2πα′ is the string tension, α′ the Regge slope, τ and σ are the (spacelike and timelike, respectively) string coordinates, ηab is a 2-dimensional world–sheet (flat) metric (a,b=0,1), Xμ(τ,σ) (μ,ν=0,1,2,3) are the coordinates of the string world–sheet in a 4-dimensional spacetime with metric gμν.
where: (...).≡∂∂τ, (...)′≡∂∂σ, and λ=1 for tensile strings, while λ=0 for tensionless/null strings null ; dablar ; iza02. The meaning of the constraints (3) and (4) for a null string is as follows. The first one shows that the string moves with the speed of light and the second says that the velocity vector of an element of string is perpendicular to this element.
= ∫2π0√−t′2+a2f2r′2+a2r2θ′2+a2r2sin2θφ′2dσ.
From (24) we see that a Big-Rip singularity a(η)→∞ appears for η→0. This singularity corresponds to the limit τ→∞ and the ‘+′ sign or τ→−∞ and the ‘−′ sign in (26). Then, it is clear from (27) that a string size is infinite in either of these limits. This means a string will be infinitely stretched at a Big-Rip singularity.
Then, one can see that at a SFS, η is finite which means that the scale factor (33) is finite and so the invariant string size at a SFS is finite, too. This means a string may cross smoothly a SFS singularity and eventually approach a Big Crunch singularity where it may collapse to a zero size.
Let us now come to the discussion of the problem for the null strings (λ=0 in (2)-(4)). We will briefly show that the null string can be considered as a collection of particles in which σ is assigned to a null particle in a collection, while τ is a parameter of a geodesic with a particular value of σ. Formally, Xμ(τ,σ) is a geodesic with an index σ.
= −ηρkν∇ρkν=−kνηρ∇ρkν.
In conclusion, the evolution of the null string can always be reduced to the evolution of the geodesics.
respectively, and D=B′sinθ0+P′3cosθ0.
Now let us come to the problem of the null strings at SFS. Since for t=ts all the quantities in the expression for the invariant string size (19) are finite, then the size of the string at a sudden future singularity is finite. In all three cases k=0,±1 the string is spanned on the surface of either of the two cones of an angle θ0 each. The cones are attached to each other with their apexes at r=0. The cones have a common symmetry axis given by θ=0. For k=+1 the string oscillates around r=0 and the frequency of its oscillations decreases while its invariant size (19) grows, though not to infinity. For k=0 the string escapes from the point r=0 – its coordinate velocities (40)-(41) decrease while its size grows. For k=−1 the string escapes from r=0, its coordinate velocities and size grow rapidly.
Now let us study the null strings at Big-Rip. We assume a simple model of evolution of the scale factor which admits such a singularity caldwell, i.e., a(t)=aR(tm)[∣γ∣+1−(∣γ∣+1)(ttm)]−2/3∣γ∣, where γ<0 and tm is the time of a Big-Rip. The integral ∫dx/a(x) is convergent in an arbitrary small interval before a BR singularity. For k=+1 the frequency of oscillations of the string size is decreasing. Besides, the coordinate velocities (40)-(41) asymptotically tend to zero at a Big-Rip. For k=0 the coordinate velocities (40)-(41) decrease rapidly in order to finally reach zero. Since a(t)→∞ at Big-Rip and all other quantities in the invariant string size (19) are finite, it means that the string will be infinitely stretched at a Big-Rip, i.e., its size S→∞. However, as concluded from the equations (36)-(39), in the limit a(t)→∞ all the right-hand sides of these geodesic equations are zero which means that the four-velocity dXμ/dτ tends to zero at a Big-Rip. This is not the case for the acceleration vector d2Xμ/dτ2 which, as seen from the geodesic equations (14)-(17), is not regular due to a blow-up of the scale factor a(t) and its derivatives at a Big-Rip.
Finally, we briefly discuss strings at some other types of future singularities described in details in Refs. typeIII ; typeIV. These singularities appear for the scale factor of the form a(t)=asexp{h0(t−ts)1−α} and h0= const. If 0<α<1, then a=as= const., ϱ→∞, ∣p∣→∞ at t=ts (type III singularity of Ref. typeIII ). On the other hand, if α<−1, then a=as, ϱ→0 (or finite), ∣p∣→0 (or finite) and the higher order than two derivatives of the scale factor diverge at t=ts (type IV singularity of Ref. typeIV ). In the former case and for k=+1 the frequency of oscillations of a string approaching the singularity grows though remains finite at t=ts. What is important from the point of view of the main task of our discussion is the fact that the invariant string size (19) remains finite at the singularity. For k=0,−1 the coordinate velocities (40)-(41) near t=ts grow rapidly. The integral ∫dx/a(x) is convergent on an arbitrary interval of time before a singularity so that the coordinate velocities and the radial coordinate r are finite which means that the invariant string size is also finite at t=ts. In the latter case for k=+1 one has almost homogeneous and finite frequency of oscillations of a string near ts. The invariant size remains finite, too. For k=0 the coordinate velocities remain almost constant near ts and the invariant size is finite. For k=−1 one has a rapid growth of the coordinate velocities - since r remains finite, then the invariant size also remains finite.
where ζ=arctanP′3/B′ and Z=B′/(C′sinζ). Since (45) restricts the value of r, then this θ=θ(τ) case does not change a general picture in which the strings are not infinitely stretched at these future singularities. Exact solutions for η(θ) can easily be given but since they do not contribute a new quality to the discussion we will not present them here.
In conclusion, we emphasize that bearing in mind the system of geodesic equations (36)-(39) one can see that the geodesics may be extended through a sudden type, type III and type IV singularities discussed in this paper. This is a consequence of the finiteness of the right-hand sides of these equations since in all the cases we have studied the radial coordinate r was finite which implied the finiteness o the function f(r). Then, strings may survive these singularities. On the other hand, at a Big-Rip, geodesic equations are singular (though the four-velocity is zero) and the strings will be infinitely stretched. Finally, it is worth mentioning that in Ref. tolley the propagation of strings through a Big-Crunch/Big-Bang singularity in the ekpyrotic/cyclic scenario turok was investigated, which is complementary to our work about string propagation at Big-Rip and sudden future singularities.
M.P.D. acknowledges the support of the Polish Ministry of Education and Science grant No 1 P03B 043 29 (years 2005-2007).
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