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Home , De Sitter space, Hyperboloid, Pseudosphere

of (Anti-)De Sitter space-time

Author: Ricard Monge Calvo. Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.

Advisor: Dr. Jaume Garriga

Abstract: This work is an introduction to the De Sitter and Anti-de Sitter , as the maximally symmetric constant spacetimes with positive and negative scalar R. We discuss their causal properties and the characterization of their geodesics, and look at p,q the spaces embedded in flat R spacetimes with an additional . We conclude that the geodesics in these spaces can be regarded as intersections with planes going through the origin of the embedding space, and comment on the consequences.

I. INTRODUCTION In the case of dS4, introducing the coordinates (T, χ, θ, φ) given by: Einstein’s postulates that T  T  is a differential (Lorentzian) manifold of dimension 4, X0 = a sinh X~ = a cosh ~n (4) a a whose Ricci curvature tensor is determined by its mass- energy contents, according to the equations: where X~ = X1,X2,X3,X4 and ~n = ( cos χ, sin χ cos θ, sin χ sin θ cos φ, sin χ sin θ sin φ) with T ∈ (−∞, ∞), 0 ≤ 1 8πG χ ≤ π, 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π, then the element Rµλ − Rgµλ + Λgµλ = 4 Tµλ (1) 2 c is: where Rµλ is the Ricci curvature tensor, R te Ricci scalar T  ds2 = −dT 2 + a2 cosh2 [dχ2 + sin2 χ dΩ2] (5) curvature, gµλ the , Λ the cosmological con- a 2 stant, G the universal gravitational constant, c the speed of light in vacuum and Tµλ the energy-momentum ten- where the surfaces of constant time dT = 0 have metric 2 2 2 2 sor. dl = dχ + sin χ dΩ2, which represents a 3-dimensional After the formulation of general relativity, physicists and S3 in spherical coordinates. Thus, we can con- mathematicians have tried to find exact solutions to Ein- clude that dS4 is spatially closed. steins equations, often relying on the assumption of cer- Disregarding the coordinate singularities at χ = 0, π, tain symmetries. θ = 0, π and φ = 0, 2π, we have a complete map of The simplest solutions to these equations are those with dS4. This map, suppressing two , allows us to the highest degree of symmetry, called maximally sym- give a visual representation of the space as a one-sheeted metric spaces. These can be shown to have constant Ricci 2-dimensional inside a 3-dimensional space scalar R and are uniquely determined by its value [1]. (see figure 1). The most common is the well known , with R = 0. The aim of this work is the study of the maximally sym- metric space with positive curvature R > 0 called de Sit- ter Space (abbreviated dS), and the space with negative curvature R < 0 called Anti-de Sitter space (abbreviated AdS).

II. (ANTI-)DE SITTER SPACE-TIMES

A simple way to understand the n-dimensional De Sitter space (dS ) is by its embedding in a (n+1)- n FIG. 1: De Sitter space with coordinates (T, χ, θ, φ) dimensional flat space Rn,1 with Cartesian coordinates (X0,X1, ..., Xn) and metric: with sections of constant T and χ. Reduced model suppressing two dimensions θ and φ. ds2 = −(dX0)2 + (dX1)2 + ... + (dXn)2. (2) In the reduced model we see that the spatial sections De Sitter space is then defined as the hyperboloid of ra- contract to a minimum spatial volume (for T = 0) and dius a > 0, the with equation: then re-expand to infinity, given the congruence associ- ated with this frame. Therefore, we say De Sitter space µ 0 2 1 2 n 2 2 XµX = −(X ) + (X ) + ... + (X ) = a . (3) is an expanding space-time for T > 0. Geometry of (Anti-)De Sitter space-time Ricard Monge Calvo

Regarding the n-dimensional Anti-de Sitter space the expressions: (AdSn), we can embed it in a (n+1)-dimensional flat n(n − 1) n(n − 1) space n−1,2 with metric: R = and R = − (10) R dSn a2 AdSn a2 ds2 = −(dX0)2 − (dX1)2 + (dX2)2 + ... + (dXn)2 (6) which can be computed from the given metrics (5) and (9). Furthermore, these spaces can be regarded as solu- and then AdSn is defined as the hyperboloid with equa- tions for the Einsteins equations of empty spaces (Tµλ = tion: 0) with cosmological constant Λ = (n − 2)R/2n with the corresponding Ricci scalar (see [1] and [3]). µ 0 2 1 2 2 2 n 2 2 XµX = −(X ) − (X ) + (X ) + ... + (X ) = −a . (7) In the case of AdS4, introducing the coordinates III. CAUSAL DIAGRAMS (t, r, θ, φ) given by: To study the causal properties of the (Anti-)De Sitter  t   r   t   r  X0 = a sin cosh X1 = a cos cosh space, it is useful to bring infinite coordinate values into a a a a finite range. By doing this, we will be able represent  r  X~ = a sinh ~n (8) the causal structure using the so-called Carter-Penrose a diagrams. For dS4 with coordinates given in (5) we perform the 2 3 4 where X~ = X ,X ,X and ~n = ( sin θ cos φ, sin θ sin φ, change of variable cosh(T/a) = 1/ cos η to get the con- cos θ) with coordinates 0 ≤ t ≤ 2πa, r ≥ 0, 0 ≤ θ ≤ π formal metric: and 0 ≤ φ ≤ 2π, then the line element is: 2 2 a 2 2 2 2 ds = 2 −dη + dχ + sin χ dΩ2 (11) 2 2  2  r  2 2 2  r  2 cos η ds = a − cosh dt + dr + sinh dΩ2 (9) a a for −π/2 ≤ η ≤ π/2. To draw the diagram we use the where the surfaces of constant time dt = 0 have metric reduced model with only η and χ coordinates. Note that 2 2 2 r  2 the points χ = 0 and χ = π represent antipodal points in dl = dr +sinh a dΩ2, which represents an hyperbolic 3 the hyperboloid and opposite poles in the . space H . We conclude that AdS4 is spatially open. Disregarding the coordinate singularities at r = 0, we In this conformal metric null geodesics are represented as straight lines with a slope of 45o(see figure 3). have a complete map of AdS4. Suppressing two dimen- sions we can give a visual representation like in dS4 case (see figure 2).

FIG. 3: Penrose diagram for the De Sitter space with FIG. 2: Anti-de Sitter space with coordinates (t, r, θ, φ). conformal metric and null geodesics as 45o lines. Every Reduced model suppressing two dimensions φ and θ. point inside the diagram corresponds to a S2 spatial section in the full model.

Contrary to the De Sitter case, in AdS4 we have a pe- riodic time coordinate, with the points (0, r, θ, φ) and Due to the relation of the conformal coordinates in dS4 (2πa, r, θ, φ) representing the same point in the space, we have an interesting property of the space. Given a which leads to physically unrealistic closed timelike static observer at χ = 0, a photon sent from a timelike curves. As the metric described does not explicitly in- observer in its antipodal point χ = π (that follows a clude the time periodicity in its expression, we can un- null geodesic) will not reach the first observer (or it will wrap the time coordinate by extending t ∈ (−∞, +∞), reach him for η → π/2 that is T → +∞). This means working with the so called Anti-de Sitter universal cov- that there are pairs of timelike observers in the space that ering space. We will work with this covering space in the are not causally connected, in the sense that they are not following sections. able to exchange signals. Intrinsically, these are analogous to a For AdS4 with coordinates given in (9) we perform µ 2 the change of variable sinh(r/a) = tan(ρ/a) to get the sphere in Euclidean space (XµX = a ) and are there- fore usually called . conformal metric: Both in the de Sitter and anti-de Sitter case, the hyper- a2 ds2 = −dt2 + dρ2 + sin2 ρ dΩ2 (12) boloid’s radius is related to the curvature Ricci scalar by cos2 ρ 2

Treball de Fi de Grau 2 Barcelona, June 2018 Geometry of (Anti-)De Sitter space-time Ricard Monge Calvo for 0 ≤ ρ ≤ πa/2. To draw the diagram we take the the following geodesic equations: reduced model with only t and ρ coordinates. Moreover, ¨ 2ρ ˙ due to the unwrapping of the time coordinate the tem- t + 2 2 ρ˙t = 0 poral axis extends for (−∞, +∞). In our diagram, once a + ρ   the null geodesic reaches the spatial infinity (ρ = π/2 or  2  ρ ρ 2 1 2 r → +∞) we assume reflecting boundary conditions and ρ¨ +  1 + t˙ − ρ˙  − a2 a2  ρ2  the geodesic bounces off and returns (see figure 4). In 1 + a2 addition, points at t = 0 and t = πa represent antipodal  ρ2  points while at t = 0 and t = 2πa represent the same ˙ 2 ˙2 −ρ 1 + 2 [θ + sin θ φ ] = 0 point in the hyperboloid. a 2 θ¨ + ρ˙θ˙ − cos θ sin θ φ˙2 = 0 ρ 2 1 φ¨ + ρ˙φ˙ − 2 θ˙φ˙ = 0 (14) ρ tan θ where the dot indicates the derivative with respect to the affine parameter τ. Furthermore, as the Lagrangian used to get the geodesic equations is explicitly independent of the time and angular coordinates t and φ, we have that the conjugate momenta pt = E and pφ = L are conserved, that is: FIG. 4: Partial and 2πa−periodic Penrose diagrams for the Anti-de Sitter space with universal covering and  ρ2  1 + t˙ = E = const. and ρ2φ˙ = L = const. (15) conformal metric, with null geodesics as 45o lines. a2 Every point inside the diagram corresponds to a S2 spatial section in the full model. Without loss of generality and due to the symmetries of the space, we can restrict to the equatorial θ = π/2. We first take a look at circular timelike geodesics when Given the relations between the conformal coordinates, ρ = const. → ρ˙ = 0. In this case, besides the previous we get an interesting property of AdS4. For a static ob- constants the geodesic equations are: server at ρ = 0 a photon emitted from its position will ρ reach spatial infinity ρ = πa/2 and come back (assuming t¨= φ¨ = 0 and t˙2 − ρφ˙2 = 0. (16) a2 reflecting boundary) in a finite amount of time. ˙ ˙ Aside from the properties deduced by looking at null From the second equation we get the relation t/a = φ. geodesics, we can wonder how the spaces behave in re- We see the circular geodesics are closed orbits of the form lation to other geodesics. For flat spacetimes, all points φ ∝ t/a with the same angular velocity 1/a regardless of inside the future (past) light of an observer can be the radial distance. We conclude that, unlike in the case accessed through a timelike geodesic. In the case of Anti- of flat spaces, we can build a rigidly-rotating frame of de Sitter space, we will see that this property is not true reference in AdS. Given one such frame with constant angular coordinate φ0 = φ − t/a = const. then dφ2 = by answering the question: can any two points which 2 2 are connected through a timelike curve also be connected dt /a and from the metric in (13) we get: through a timelike geodesic? A related question can be  2  2 ρ 2 2 asked for De Sitter spacelike curves and geodesics. ds = − 1 + cos θ dt . (17) a2 We see that observers at rest in the rotating frame have ds2 < 0 and are thus timelike. On the other hand, we have the following constants of IV. GEODESICS IN ANTI-DE SITTER SPACE motion for the geodesics (considering c = 1): ( ∓1, timelike/spacelike g x˙µx˙µ = (18) By performing the change of variables ρ = a sinh(r/a) µν 0, null in (9) we get the static metric: which in our case, and restricting to the radial motion with L = 0 → φ˙ = 0, translates to the following condi-  2   2 −1 2 ρ 2 ρ 2 2 2 tions for our geodesics: ds = − 1 + dt + 1 + dρ + ρ dΩ2 (13) a2 a2 (  ρ2  ρ˙2 ∓1, timelike/spacelike − 1 + t˙2 + = a2  ρ2  0, null with ρ ≥ 0. Using this metric and the Euler-Lagrange 1 + a2 1 ˙µ ˙ν equations with the Lagrangian L = 2 gµν x x , we get (19)

Treball de Fi de Grau 3 Barcelona, June 2018 Geometry of (Anti-)De Sitter space-time Ricard Monge Calvo

For null radial geodesics, using the constant of motion in To remove the multiplier we take the second derivative (18) we have that the radial equation is: of our constrain and get:

¨A B ˙A ˙B ρ¨ = 0 (20) F¨ = 2(ηABX X + ηABX X ) = 0 (25) which gives a linear dependence on proper time. where η X˙AX˙B is equal to the constant of motion from For timelike radial geodesics, aside from the relations AB (18) under the restriction. in (15), the equation for the radial coordinate is: ¨A B For null geodesics we get ηABX X = 0 and for time- 1 ¨A B ρ¨ + ρ = 0 (21) like/spacelike geodesics ηABX X = ±1. By multiply- 2 B a ing the geodesic equation by ηABX and using these 2 which is the equation of a simple harmonic oscillator with relations with the constrain F we get λ = ±1/2a for frequency ω = 1/a. Therefore, radial timelike geodesics timelike/spacelike geodesics and λ = 0 for null, and thus will be sinusoidal curves which end up returning to ρ = 0 the equations are respectively: for proper time τ = πa (which will be the antipodal 1 point) and again at τ = 2πa, that have a maximum finite X¨A ± XA = 0 and X¨A = 0. (26) a2 value for ρ(τ) depending in the initial conditions (initial position ρ(0) and velocity ρ0(0)). Like in the previous section, we have for timelike In the spacelike case, we have the same equations with geodesics the equation of a simple harmonic oscillator ω = −1/a so that the geodesics follow hiperbolic func- with frequency ω = 1/a, although in this case for all coor- tions from a point to infinity. dinates XA and general timelike geodesics. Likewise, the It is also worth noting that for L 6= 0 the constant in expressions for the coordinates XA are all 2πa−periodic. (18) includes the term φ˙, so that using the relation in (15) For spacelike and null geodesics we also have the same we get the following second-order nonlinear differential result than before but for all coordinates XA. equation: From this expression we get that the points in timelike geodesics of AdSn have a general form given by: 1 L2 ρ¨ + ρ − = 0. (22) τ  τ  a2 ρ3 XA = pA sin + qA cos (27) a a It is easy to check numerically that the solutions of this A A equation are still 2πa−periodic. The reason for this will where p and q are constant vectors that due to the be apparent in the following section. constrain F and the orthogonality of the solutions satisfy To summarize the previous results, we have seen that the conditions: null geodesics are linear on proper time and that both pAp = qAq = a2 and pAq = 0. (28) radial and circular timelike geodesics are 2πa−periodic. A A A This fact can be further generalized by looking at the Due to the maximal symmetry of the AdSn space, given a geodesics from the perspective of the embedding flat timelike geodesic G such that XA(τ = 0) = P is an arbi- space. trary point of the geodesic, we can find a transformation which makes the coordinates of P equal to X0(0) = a and Xi(0) = 0 for i = 1, ..., n. In addition, by per- V. GEODESICS IN THE EMBEDDING FLAT forming an appropriate rotation around the X0 axis we SPACE can make the tangent vector of the geodesic at point P (given by X˙A(0)) tangent to the X1 axis, so that we Let us recall the definition of the n−dimensional get X˙ 0(0) = X˙ j(0) = 0 for j = 2, ..., n and X˙ 1(0) = 1. (Anti-)de Sitter space as the hyperboloid inside a (n + (p,q) With these transformations we find the geodesic in the 1)−dimensional flat R space. In the case of AdSn (n−1,2) resulting frame of reference has expression: we have the embedding space R with metric ηAB given by (6). The geodesics in this space can be calcu- τ  τ  X0 = a cos ,X1 = a sin 1 ˙A ˙B a a lated using the Lagrangian L = 2 ηABX X . We shall apply the Lagrange multipliers method with the restric- Xj = 0 for j = 2, ...n (29) tion given by the hyperboloid equation (7) to get the Lagrangian for the hyperboloid’s geodesics, which is: which represents a circle of radius a in the embedding space. 1 ˙A ˙B A Consequently, we have found that all timelike geodesics L = ηABX X − λ(τ)F (X ) (23) 2 of AdSn can be represented as circles of radius a on an A A 2 A B 2 Euclidean plane through the origin (given an appropriate where F (X ) = XAX + a = ηABX X + a = 0 and λ is the Lagrange multiplier. The geodesic equations are coordinate transformation) which means that they are then: the intersection of the hyperboloid with planes through the origin of the embedding space. In general, two differ- X¨A + 2λXA = 0, for A = 0, ..., n. (24) ent timelike geodesics starting from the same point will

Treball de Fi de Grau 4 Barcelona, June 2018 Geometry of (Anti-)De Sitter space-time Ricard Monge Calvo only intersect in this point and its antipodal in the hy- perboloid, where their planes intersect. Moreover, with this discussion we see that the distinction between radial and circular timelike geodesics is coordinate dependent, as intrinsically they are all intersections with planes in the embedding space. Likewise, spacelike geodesics are branches that are also intersections with planes through the origin. We now go back to the causal diagram for AdS4 and represent timelike geodesics. From an origin point at ρ = 0, the timelike geodesics are curves that diverge and reconverge to the same point in πa−periods, and thus FIG. 6: Penrose diagram for the De Sitter space with the points of these paths are bounded inside the infinite conformal metric, null geodesics as 45o lines and diamonds formed by the null geodesics (see figure 5). spacelike geodesics as dashed lines.

VI. CONCLUSIONS

To summarize, we have seen that the geometric struc- ture of dS and AdS spaces introduces unusual causal properties that differ from those in flat spaces, like the existence of timelike observers that are not able to ex- change signals in dS or the finite time interval of a pho- ton reflected at spatial infinity in AdS. Furthermore, we have also seen some interesting proper- ties of the geodesics of the spaces, like the existence of rigidly-rotating frames for all radial distances in AdS or FIG. 5: Partial and 2πa−period Penrose diagrams for the periodic timelike or spacelike geodesics in AdS and the Anti-de Sitter space with universal covering and dS respectively. These properties become easily appar- conformal metric, with null geodesics as 45o lines and ent by studying the spaces through their embedding in timelike geodesics as dashed lines. flat R(p,q) spaces with an additional dimension, where we see that these geodesics are intersections with planes Therefore, the points outside this regions (but inside the through the origin of the embedding space. causal future) can not be accessed by timelike geodesics although we can construct timelike curves. Alternatively, the spacelike and timelike geodesics of De Sitter space can also be shown to be the intersec- Acknowledgments tions with planes through the origin of the embedding space ([2] and [5]) and thus spacelike geodesics will also This work was done with the help of my adviser Dr. be periodic starting in a point and ending in its antipo- Jaume Garriga, from whom i received all the necessary dal (see figure 6). In the same way that the timelike guides and references to study the subject at hand. geodesics for AdS, the points that can be accessed by In addition, I would like to thank my family and friends spacelike geodesics in dS are restricted to the diamond for their continuous support and companionship over shaped limited by the null geodesics. these years.

[1] S. Weinberg, Gravitation and Cosmology principles and [4] Leszek M. Sokolowski, The bizarre anti-de Sitter space- applications of the general theory of relativity, John Wiley time, International Journal of Geometric Methods in Mod- & Sons Inc. 13: 375-406 (1972). ern Physics. 13: 9 (2016). [2] E. Schr¨odinger, Expanding Universes, Cambdrige Univer- [5] L. M. Soko lowski, Z. A. Golda, Every timelike geodesic sity Press. 1: 1-40 (1956). in anti–de Sitter spacetime is a circle of the same radius, [3] S.W. Hawking and G.F.R. Ellis, Large-scale structure International Journal of Modern Physics. 25: 1 (2016). of space-time, Cambridge University Press. 5: 117-133 (1973).

Treball de Fi de Grau 5 Barcelona, June 2018