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In this section we will construct the Robertson-Walker interval in various forms from an expanding universe (big bang) model, in particular

ds² = dct² - a(t)²[dr² + r²( dq² + sin²qdf²)]/(1 + (1/4)kr²/R₀²)²

(8.1.1)

Recall the discussion of Gaussian curvature in section 6.1. Though it is popular to associate curvature with the Riemann tensor for the reason discussed there, Gaussian curvature is really what cosmologists mean when they talk about the universe being curved. Lets say we describe the motion of the bug on the sphere with the following coordinates.

R = the radius of the sphere.

y = the polar angle coordinate.

W = the azimuthal angle

Its easy to show that when the bug traces out a small circle around the pole it will find that "the curvature" of its space as defined by the equation in that section is

K = +1/R²

(Hint: r = Ry, C = 2pRsiny)

We next let the size of the sphere vary in time and note that the rigid ruler length for the path the bug travels is given by

dL² = R(t)²[dy² + sin²ydW²]

(8.1.2)

90 Chapter 8 Robertson-Walker and the Big Bang

Since this type of space-like hyper-surface is totally uniform(it looks like the same sphere from any place on it) so we know that modeling a 3d+1 space-time by

ds² = dct² - R(t)²[dy² + sin²ydW²]

(8.1.3)

where we have

dW² = dq² + sin²qdf²

(8.1.4)

will result in a solution of Einstein's field equations for a totally uniform matter distribution for the universe.

Doing a coordinate transformation

s = siny

(8.1.5)

results in

ds² = dct² - R(t)²[ds²/(1 - s²) + s²( dq² + sin²qdf²)]

(8.1.6)

Now we see that R(t) is now the meaning of the radius of the universe as a function of time. This is called the Robertson-Walker interval. Though one can imagine positively curved and yet open universes with non-uniform matter distributions, according to this model a positively curved universe implies a closed curved universe. Any frame in which the interval is expressed in this form is called a comoving frame. The time according to a comoving frame given the initial condition of t = 0 at R(t) = 0 is called cosmological time. It is this time that is being referred to when a physicist is talking about the age of the universe. The r coordinate of a comoving frame is related to the Radius of the universe and the coordinate s By the relation

r = sR₀.

(8.1.7)

(Note the subtle difference between this r and the bug's ruler distance based coordinate r)

We could also use the relation to the curvature

K = + 1/R₀²

(8.1.8)

8.1 Gaussian Curvature and the Robertson-Walker Interval 91

to express the interval as:

ds² = dct² - R(t)²[ds²/(1 - Kr²) + s²( dq² + sin²qdf²)]

(8.1.9)

Doing this its obvious that should the universe be negatively curved

K = -1/R₀²

(8.1.10)

the interval would become

ds² = dct² - R(t)²[ds²/(1 + s²) + s²( dq² + sin²qdf²)]

(8.1.11)

In which case a coordinate transformation also expresses it:

ds² = dct² - R(t)²[dy² + sinh²ydW²].

(8.1.12)

A positively curved conceptual model can be either open or closed curved, though a Robertson-Walker positively curved model is always also closed. However, there are no conceptual closed curved universe models that are also everywhere negatively curved. Mathematically the big bang creating an expanding universe can be modeled whether positively or negatively curved just fine. Conceptually one can imagine a negatively curved universe using 2d cross sections just as we have done for the positively curved universe. Conceptually one can imagine a positively and closed curved universe collapsing all the way back to the moment of the bang. However, there is no conceptualization for an negatively curved universe extrapolating back to a big bang. As a result many physicists believe that the universe must be closed curved and positively curved. In order for this to be the case, there must be much more matter in the universe than we have yet observed. This is called the missing mass. Experimentally the universe is found to be too nearly to tell for sure. In other words flat K in Eqn.8.1.9 is to small to tell from zero or determine the sign. If we were to model the universe with a stress-energy tensor given by Eqn. 6.3.8, then in order to be positivley curved and closed which will also make it collapse back on itself it would have to have a minimum

r₀ = (3H₀² - c²l)/(8pG)

where H₀ is Eqn. 8.1.21, and l is the cosmological constant.

We know that there is much more matter here than luminous matter. The nonluminous matter is called dark matter. This constitutes some of the difference. A positive cosmological constant would account for more. Neutrino mass constitutes some as well, but even considering these, there would still be missing matter left to find.

92 Chapter 8 Robertson-Walker and the Big Bang

If we define a curvature parameter lowercase k by k = -1, 0, 1 for negatively curved, zero curved, and Positively curved universes we can write the Robertson-Walker interval as

ds² = dct² - R(t)²[ds²/(1 - ks²) + s²( dq² + sin²qdf²)]

(8.1.13)

in terms of r and making the definition a(t) = R(t)/R₀ we can write it

ds² = dct² - a(t)²[dr²/(1 - kr²/R₀²) + r²( dq² + sin²qdf²)]

(8.1.14)

or by doing the transformation

s = (r/R₀)/[1 + (1/4)kr²/R₀²]

(8.1.15)

we arive at the most simple general form

ds² = dct² - a(t)²[dr² + r²( dq² + sin²qdf²)]/(1 + (1/4)kr²/R₀²)²

(8.1.16)

Problem 8.1.1

Draw an example of a 2d cross section otherwise called a spacelike hypersurface or a spatial manifold for

a. A closed positive curved universe - this should be easy now.

b. An open positive curved universe.

c. A open negative curved universe.

Problem 8.1.2

Does the answer for b look the same or different from a point of view of a bug at different locations on the surface?

Problem 8.1.3

Given l = 0, what is the minimum value for r₀ to close the universe.

___________________________________________________________________________________________ 

8.2 The Robertson-Walker Interval and Hubble's "Constant" 93

In this section we will use Eqn. 8.1.16

ds² = dct² - a(t)²[dr² + r²( dq² + sin²qdf²)]/(1 + (1/4)kr²/R₀²)²

to derive Hubble's Law

v = HL

(8.1.17)

And show that wavelengths of light from other nearby galaxies in comoving frames undergos a shift in wavelength given by

(l_f - l₀)/l₀ = HL/c

 (8.1.18)

From the above form of the Robertson-Walker interval we see that the rigid ruler distance between here any a far distant galaxy is given by

L = a(t)ò₀^rdr/(1 + (1/4)kr²/R₀²)

(8.1.19)

Galaxies associated with comoving frames will have a zero coordinate velocity. dr/dt = 0.But since a(t) depends on time their ruler velocity v = dL/dt will be nonzero.

v = (da(t)/dt)ò₀^rdr/(1 + (1/4)kr²/R₀²)

v = (da(t)/dt)(L/a(t))

At this point we make a definition for Hubble's "constant"

H(t) º (da(t)/dt)/a(t)

(8.1.20)

Note Hubble's "constant" is not really a constant, but is a function of time. Inserting this definition we have Hubble's law Eqn. 8.1.17

v = HL

The current value of Hubble's constant is not very well known but observations indicate that it is

H₀ = (65 ± 8) km/(s× Mpc).

(8.1.21)

where 1 parsec = 3.26 ly

94 Chapter 8 Robertson-Walker and the Big Bang

Also note that for large enough distances v is greater than c. This is allowed because there are no local faster than c motions and as long as this is the case, faster than c motions are allowed in general relativity.

Next consider light waves comming to us from another galaxy in a comoving frame. ds = 0 for light so we have

0 = dct² - a(t)²dr²/(1 + (1/4)kr²/R₀²)²

which results in

dct/a(t) = -dr/(1 + (1/4)kr²/R₀²)

Integrating this over the time that it takes a wave crest to reach us results in

ò_ti^tfdct/a(t) = ò₀^rdr/(1 + (1/4)kr²/R₀²)

If the period of the wave is Ti where it is emitted and the period is Tf here where it is recieved then the equation for the following wave crest will also be

ò_ti+Ti^tf+Tfdct/a(t) = ò₀^rdr/(1 + (1/4)kr²/R₀²)

because a galaxy of a comoving frame doesn't change in its coordinate position. Therefor

ò_ti^tfdct/a(t) = ò_ti+Ti^tf+Tfdct/a(t)

A little manipulation results in

ò_ti^ti+Tidct/a(t) = ò_tf^tf+Tfdct/a(t)

For small periods this becomes

Ti/a(ti) = Tf/a(tf)

In terms of wavelength this can be rewritten

(l_f - l₀)/l₀ = [a(tf) - a(ti)]/a(ti)

(8.1.22)

If the cosmological transit time for the wave is written Dt = L/c so that the initial ruler distance to the galaxy is L, then this becomes

(l_f - l₀)/l₀ = (L/c)(Da/Dt)/a(ti)

8.2 The Robertson-Walker Interval and Hubble's "Constant" 95

If the galaxies are nearby then the transit time is small so Da/Dt = da/dt and recalling the definition for H(t) we have

(l_f - l₀)/l₀ = HL/c

(8.1.23) 

 At this point we can make a relation between the rigid ruler velocity and the wavelength shift for nearby comoving galaxies.

(l_f - l₀)/l₀ = v/c

(8.1.24)

Consider for a moment the wavelength Doppler shift of special relativity. Eqn. 3.3.14.

l = l ₀' [(c + v)/(c - v)]^1/2

A little manipulation results in

(l - l₀' )/l₀' = [(c + v)/(c - v)]^1/2 - 1

Next look at the case for v << c. It becomes Eqn. 8.1.24.

(l - l₀' )/l₀' = v/c

This is in exact agreement. Thus on local scales it doesn't matter whether we say that the galaxies are "still" within an expanding space or if we say the galaxies are moving apart from each other within a static space. But beware, this is only a local equivalence. Since the universe is not a steady state universe, there is no frame for which the expanding universe, Robertson-Walker metric, is globally static. This means that globally speaking, of the two, only the expanding space paradigm is descriptive.

The time in any of the above expressions of the Robertson-Walker interval is called cosmological time. As we know by now, time is relative and so one might wonder how meaningful it is to discuss the age of the universe. It turns out that the typical average path of galaxies follow comoving frames and so cosmological time is meaningful for us traveling in a typical galaxy. The age of the universe is defined as the amount of time that has gone by since a(ct) was zero. To express time in a unitless way one can also define cosmic time in terms of cosmological time by t_cos = tH₀

96 Chapter 8 Robertson-Walker and the Big Bang

Problem 8.2.1

What is the rigid ruler distance L to a galaxy that travels faster than c according to Eqn. 8.1.17 ?

Problem 8.2.2

If the rigid ruler distance to a comoving galaxy is L = 10 Mpc, What is v and what is the wavelength we see of light that originated at l₀' = 650nm according to Eqn 8.1.23? What issues should be a concern related to discrepancy?

Problem 8.2.3

If a thin wall has a stress energy tensor whose only significant terms are

T⁰₀ = T^x_x = T^y_y = sd(z)c²

according to a local observer's frame, then it is called a domain wall and the vacuum field solution around the wall is

ds² = (1 - a|z|/c²)²dct² - [exp(-2act/c²)](1 - a|z|/c²)²(dx² + dy²) - dz²

Were we define a by

a = 2pGs

a. Use Einstein's field equations to show that the exact stress energy tensor for this exact spacetime interval is

T⁰⁰ = 4(c⁴/8pG)(a/c²){(a/c²)[Q(z)(1 - Q(z))] + d(z)(1 - a|z|/c²)}/(g₀₀)²

T^xx = 4(c⁴/8pG)(a/c²){(a/c²)[Q(z)(1 - Q(z))] + d(z)(1 - a|z|/c²)}/(g₀₀g_xx)

T^yy = 4(c⁴/8pG)(a/c²){(a/c²)[Q(z)(1 - Q(z))] + d(z)(1 - a|z|/c²)}/(g₀₀g_yy)

T^zz =12(c⁴/8pG)(a/c²)²[Q(z)(1 - Q(z))]/g₀₀₀₀

Q(z) = the step function. Q(z) is 0 for z less than 0 and is 1 For z greater than 0 and is 1/2 for z = 0.

b. Show that including the only significant terms at the origin that it does reduce to the first stress energy tensor and that even with all terms included it is a vacuum for any other z than 0.

[Hints: d|z|/dz = 2Q(z) - 1 and dQ(z)/dz = d(z) and at the origin any finite terms are insignificant compared to d(z)]

c. Calculate the Riemann tensor to show that is is zero off the wall.

d. Demonstrate that a test particle released from rest will accelerate away from the wall according to these coordinates which are for an observer who is stationary with respect to the wall.

Problem 8.2.4

I David Waite discovered the following exact solution of Einstein's field equations for a domain wall of a different pressure state

ds² =

(1 + a|z|/c²)²dct² - (1 + a|z|/c²)^-2ndx² - (1 + a|z|/c²)^-2n/(n-1)dy² - (1 + a|z|/c²)^{-2(n^2)/(n-1)}dz²

a. Verify that this spacetime corresponds to a stress-energy tensor for a domain wall whose only nonzero terms are

T⁰⁰ = 2(c⁴/8pG)(a/c²)(n²/(n-1))d (z)

T^xx = -2(c⁴/8pG)(a/c²)d (z)/(n-1)

T^yy = -2(c⁴/8pG)(a/c²)(n-1)d (z)

by using Einstein's field equations or a computor programed to calculate the Einstein tensor.

b. Find the Riemannian spacetime curvature for this spacetime.

c. Demonstrate that a test particle released from rest will accelerate toward the wall according to these coordinates which are for an observer who is stationary with respect to the wall.

d. What would happen to the system if walls of different pressure states such as these where released parallel to eachother?