A Guide to Einstein's Cosmology calculation in his book “The Meaning of Relativity”

Einstein “The Meaning of Relativity” Princeton University Press:

“Appendix for the 2^nd Edition” (5^th ed: soft cover)

We’ll use x₄ as time coordinate: avoid confusion: we often use 0 for time coordinate, Einstein uses it specifically for spatial part of a metric, see below.

We assume that we can split the metric into time and space parts, and so we write the metric as:

ds² = dx₄² - G²(t) A²(x,y,z)(dx₁²+ dx₂²+ dx₃²)

Note that previously we used the letter R instead of G, ie in the metric above we had R(t) rather than G(t).

We call the whole space part ds²:

ds² = G²(t) A²(x,y,z)(dx₁²+ dx₂²+ dx₃²)

so we can write:

ds² = dx₄² - ds²

We can define:

g(x,y,z,t) = G²(t) A²(x,y,z)

and so we can write:

ds² = g(x,y,z,t) (dx₁²+ dx₂²+ dx₃²)

and therefore we can write:

ds² = dx₄² - g(x,y,z,t) (dx₁²+ dx₂²+ dx₃²)

If use ⁰g = g(x,y,z) to signify the spatial part of g(x,y,z,t) , we can write:

g(x,y,z,t) = G²(t) A²(x,y,z) = ⁰g G²(t)

We can take G(t) out of ds² and write the purely spatial part of ds² (which itself is the spatial part of the metric) as: ⁰ds² : ie:

⁰ds² = A²(x,y,z)(dx₁²+ dx₂²+ dx₃²)

ds² = ⁰ds² G²(t)

We’ll want to find the equation for the function A.

Since G is a common term we can take it out of the sum and we can write the spatial metric as:

ds² = G²(t) ⁰ds²

where the spatial metric is:

⁰ds² = A(x,y,z)(dx₁²+ dx₂²+ dx₃²)

and the entire spacetime metric is:

ds² = dx₄²- ds² = dx₄² - G²(t) ⁰ds²

If we start with a more general metric form, non-orthogonal, we’d need cross terms like dxdz etc, and each term would have a different function as coefficient, ie we’d have as the spatial part of the metric:

ds² = g_ik(x,y,z,t) dx_idx_{k i,k = 1,2,3}

or in short:

ds² = g_ik dx_idx_k

This spatial part of the spacetime metric is composed of mixed spatial dx’s with the metric functions g_ik as coefficient.

Therefore the metric for the spacetime would be written as:

ds² = dx₄² - g_ik dx_idx_k

ie to describe the metric we could write the set of equations:

• ds² = dx₄² - ds²

• s² = g_ik dx_idx_{k i,k = 1,2,3}

The above are equation (2) on page 114.

Einstein then assumes that the g_ik can be decomposed into a product of a spatial and time functions:

g_ik (x,y,z,t) = ⁰g_ik(x,y,z)G²(t) [This is eq 2a on p114.]

(Note: He has the 0 below the g).

Therefore he writes:

⁰ds² = ⁰g_ik dx_idx_{k i,j,k = 1,2,3} [This is eq 2b on p114.]

He also has an equation for the constant B (see eq 2c on page 115), we’ll ignore that. [eq 2c on p115.]

He then writes the equation we wrote above:

⁰ds² = A(x,y,z)(dx₁²+ dx₂²+ dx₃²) [eq 2d on p115.]

He finds equations for A, eqs 3, and finds the solution as eq 3a:

A = c₁/[c₂+c₃r²],

And in eq 3c he sets the constants to 1:

A = 1/[1 + cr²] .

thereby obtaining the same solution we found. We used k rather than c so as not to confuse readers into thinking it is the speed of light. He later (p117) uses the letter z for this constant.

Note that we used spherical polar coordinates rather than Cartesian coordinates, and placed the curvature into the r coordinate, ie we wrote the spatial part of the spatial metric as:

⁰ds² = g_rr(r)dr² + r²dq²+ r²sin²q df²

rather than Einstein’s

⁰ds² = A(x,y,z)(dx₁²+ dx₂²+ dx₃²)

Thus the equations Einstein finds for A (eq 3 on p115) are not the ones we found for g_rr, nevertheless the solution is the same, ie: g_rr = A = 1/[1 + cr²] .

Exercise: write the metric as Einstein did and solve to find the Riemann components for A, and set them equal to the constant B. Compare your result with eqs 3,3a,3b,3c [p115-116].

The Field Equations. P117-118.

Eq 4 on p117 is the Einstein field eq:

G _ik = -k T _ik,

where the components G _ik of the Einstein tensor are given by:

G _ik = R _ik – ½ g _ik R,

Ie

R _ik – ½ g _ik R = -k T _ik,

With the rhs, ie -kT _ik, moved over to the lhs, we can write it as Einstein does:

[R _ik – ½ g _ik R] + k T _ik = 0. [Eq 4 on p117.]

He then computes the formulae for [R _ik – ½ g _ik R] . There are three types of such terms: where one of the two indices i,k are t, where both are t, and where neither is t.

He finds for these the same formulae as we found, just that he uses z rather than k, and G rather than R, and he doesn’t give them a name, whereas e did, we called these the G’s. Beware that we called the Einstein tensor G whereas he uses G for the time function in the metric.

He then assumes that there’s no pressure, ie the universe is a collection of ‘dust’ (each particle of ‘dust being eg a cluster of galaxies), and therefore the source term P is zero, and only T ₄₄ = r remains.

He then writes the field equations with the lhs being the formulae we found earlier, and the rhs being 0 for the spatial component, and r for the time component.(Eqs 5 on p118.)

He then obtains from these eq 5a. We were able to obtain it directly via the geodesic deviation eq.

On p119 he solves the field eqs (our way was similar enough for this solution to be easily recognizable).

……………

Einstein's “The Meaning of Relativity”

Princeton University Press: (5^th ed: soft cover)

Most of the following will be comprehensible to readers of my GR book:

• p107-108: Einstein discusses the relative merits of a closed universe to a non-closed one;

• p109 – 128: the Friedman solution

• p129 - p132: He discusses “the beginning of the world” and the age of the universe.

• p127, 129 Re ‘creation’ etc (singularity?)]

[AR: I underlined pages, and/or marked on the inside front cover]