ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof. A. Achterberg, Dr. S. Larsen and Dr. F. Saueressig

Exercise 1: Flat space in spherical coordinates

For constant time t the line element of special relativity reduces to the flat three-dimensional Euclidean metric

Euclidean 2 SR 2 2 2 2 (ds ) = (ds ) =const = dx + dy + dz . (1) − |t Use the coordinate transformation

x = r sin θ cos φ, y = r sin θ sin φ , z = r cos θ , (2) with θ [0,π] and φ [0, √2 π[ to express the line element in spherical coordinates. ∈ ∈ Exercise 2: Distances, areas and volumes

For constant time the metric describing the spatial part of a homogeneous closed Friedmann- Robertson Walker (FRW) universe is given by 2 2 dr 2 2 2 2 ds = + r dθ + sin θ dφ , r [ 0 , a ] . (3) 1 (r/a)2
∈ − The scale factor a(t) depends on time only so that, for t = const, it can be considered as a fixed number. Compute from this line element

a) The proper circumference of the sphere around the equator.

b) The proper distance from the center of the sphere up to the coordinate radius a.

c) The proper area of the two-surface at r = a.

d) The proper volume inside r = a. Compare your results to the ones expected from flat Euclidean space. Which of the quantities are modified by the property that the line element (3) is curved?

Exercise 3: The metric is covariantly constant

In the lecture we derived a formula for the Christoffel symbol

Γα 1 gαβ g + g g . (4) µν ≡ 2 h µβ,ν νβ,µ − µν,βi Here the “, α” indicates a derivative with respect to the coordinate xα, i.e., g ∂ g ∂gµν . µν,α ≡ α µν ≡ ∂xα Note that an “upper index” in the denominator corresponds to a “lower index” in the numerator. Figure 1: The embedding of the (r, φ)-slice of the wormhole geometry (7) into three-dimensional flat space.

Based on the Christoffel symbol, we can define a covariant derivative Dµ by its action on (the components of) a tensor by

D T β1...βm ∂ T β1...βm µ α1...αn ≡ µ α1...αn ith position n m (5) ν β1...βm βi β1... ν ...βm Γ αiµ Tα1... ν ...αn + Γ νµ Tα1...αn . − X X z}|{ i=1 ith position i=1 |{z} Use this definition to show that the covariant derivative is compatible with the metric, i.e.

Dµ gαβ = 0 . (6)

β β This implies that the covariant derivative commutes with the metric, e.g., Dµ gαβ u = gαβ Dµ u .

Exercise 4: Transversing a wormhole (hand-in exercise)

The wormhole geometry depicted in Figure 1 allows to connect either two different (potentially far away) points of the universe or even different universes. The line element of the geometry is

2 2 2 2 2 2 2 2 ds = dt dr (r + b )(dθ + sin θdφ ) . (7) − − The factor b is coordinate independent parameter that determines the size of the wormhole’s throat.

a) Use the fact that geodesics extremize proper time to derive the geodesic equation for this particular geometry from the variational principle. Hint: Write down the Lagrangian = 1 2 L (ds/dσ)2 / for the line element (7) and compute the Lagrange equation for the four   components of the position vector xα = t,r,θ,φ . { } b) Verify your result by explicitly computing the Christoffel Symbols with upper index t and t r 0 1 r, i.e., Γ αβ and Γ αβ. Write down the geodesic equation for x = t and x = r. Does your answer agree with part a)?

2 c) Consider a spaceship that starts at coordinate position r = R and falls freely and radially through the wormhole throat. For a given initial velocity ur = U, how much time does it take on the spaceships own clock to fall through the wormhole and reach the corresponding point r = R? −

Exercise 5: The warp-drive spacetime (optional)

The Alcubierre warp-drive solution uses coordinates (t,x,y,z) and a spaceship moving along the curve x = xs(t),y = 0, z = 0, lying in the t-x-plane and moving through the origin. The line element specifying the metric is

2 2 2 2 2 ds = dt [dx V (t) f(r ) dt] dy dz . (8) − − s s − − Here V (t) dx (t)/dt is the velocity of the spaceship and r (x x (t))2 + y2 + z2 marks s ≡ s s ≡ − s its position. The function f(rs) is any smooth function interpolating between f(0) = 1 and f(R) = 0. If necessary, you can use f(r ) = 1 (r /R)4 for concreteness. s − s To see what is interesting about this geometry, consider two stationary space stations at xA and xB, separated by the coordinate distance D along the x-axis, and the spaceship connecting the two space stations in an elapsed coordinate time T < D. This looks like the ship has traveled faster than the speed of light! Indeed the curve xs(t) must have Vs > 1 for some time. Establish that, owed to the special curvature of the warp-drive spacetime, the spaceship is always moving at less than the local velocity of light, even if the coordinate velocity Vs is greater than 1.

a) The light cones at a point in the t-x plane are the curves emerging from the point with ds2 = 0. Give a qualitative sketch of the light cone structure entailed by the geometry in

the t-x-plane. Distinguish between the regions where f(rs) is vanishing and non-vanishing. Compare the light cone structure to the one of flat space.

b) Show that at every point along the curve xs(t) the four-velocity of the ship lies insight the local light cone.

Remark: Unfortunately, spacetimes as (8) are excluded in known classical physics. Creating the “warped” spacetime requires matter fields with negative local energy density. All known classical fields have positive energy density, however. Quantum mechanics allows negative energy densities, but it is presently unclear whether this can be used to create a geometry as the one above.

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