Curvature 24 Nov 2008

Peter Fisher

11/27/08

8.033

In the last post, we showed how to measure the curvature in space-time caused by the presence of a gravitational field. There is also the curvature k that comes into the FRW metric:  dr 2  dτ 2 = dt 2 − a (t )  + r 2 dθ 2 + r 2 sin 2 θ dφ  2  1 − kr 

To start with, k must have units of 1/m2. We can work out a value for k from the kc2 relationship in PS 9, Problem 3: (1 − Ω 0 ) = − where a0 H 02 Ωo = Ω m + ΩΛ + Ωγ = 0.24 + 0.73 + 0.0005 = 0.9705 ⇒ k = −0.0295H 02 / c 2 . The Hubble constant is measured to be 70 ks/s/Mpc and 1 Mpc is the distance at which the radius of the Earth’s orbit around the Sun subtends 1”, so 1AU 153 ×10 9 m 2.3 ×10 −18 1 . This gives 1pc = = = 3.1×1016 m = 3.4ly ⇒ H 0 = = π 1 1" s 14Gy 180 60 × 60 2 2 −54    1  −1.7 ×10 1 k= =  =  .  7.6 ×10 26 m   80Gly  m2

We can see a bit about how curvature works by substituting

k r = sinh χ 1. Then

k dr = cosh χ d χ = 1+ sinh2 χ d χ = 1 − kr 2 d χ and we can write the FRW metric as

 d χ 2 sinh 2 χ sinh2 χ sin 2 θ 2  2  dτ = dt − a (t )  + dθ + dφ  = dt 2 − ds 2 . We can view the k k  k  space part of the metric at a fixed time to by forming the following embedding: w = R0 cosh χ 2

2

2

x = R0 sinh χ sinθ cos φ y = R0 sinh χ sin θ cos φ

. It is easy to see that R02 = w 2 − x 2 − y 2 − z 2 , which is some kind

z = R0 sinh χ cosθ of hyperbola in four dimensions. Suppose at a fixed time t=t0, we start at the origin: r=0. Then, w=R0 and x=y=z=0. Next, we measure out along θ = φ = 0 a distance s0=10Mly, about to the edge of the Local Group of galaxies Then, s0 χ0 a (t 0 ) χ 10Mly s0 = ∫ ds = ∫ d χ = 0 ⇒ χ 0 = s0 k = = 0.000125 and w = R0 cosh χ 0 , 80Gly k k 0 0

z = R0 sinh χ 0 . Now, measure the circumference of a circle by letting θ go from 0 to 2π.

1

Recall: sinh x ≡ (e x − e−x ) 2, cosh x ≡ (e x + e− x ) 2

Peter Fisher

11/27/08 2π

This distance will be C =

∫ ds = ∫ 0

sinh χ 0 k

dθ = 2π

8.033

sinh χ 0 k

so the circumference of a

circle with measured radius s0 is not 2πs0. 2 π s0 2 π s0 χ0 = 1 , in this case = = 0.999999997 . The C C sinh χ 0 curvature k causes the radius measured from the origin to differ from the radius measured from the circumference by about three parts per billion. This is what is meant by spacetime being curved.

In flat space-time, the ratio

We have worked this out for the case k<0. If k>0, we would use the substitution kr = sin χ and found 2π s0 C0 = 1.000000003 . This is also a good example of replacing a bad coordinate (r) with a good coordinate (χ). In the original metric, Eq. 1, r = 1 k appeared problematic. However, replacing r with χ gives the metric in Eq. 2, which clearly does not have any problems: r = 1 k ⇒ sinh χ = 1 ⇒ χ = 0.882 . Finally, this highlights the relation between book keeper coordinates (r,χ, θ and φ) and the coordinates measured in the observer frame (s0 and C). The latter are actually measurements and the former are used to make comparisons between different points in space-time.