Expansion of the Universe in General Relativity

A simple derivation of the Friedmann equations
and the meaning of Einstein's equation


The mathematical structure of the expansion of the Universe according to General Relativity, is remarkable for its way of being both

Thus it can be used as a fine introduction to General Relativity..
Necessary preliminary : the different notions of curvature in geometry.

Cosmological principle

This principle says that the universe is
Of course it is not exactly true in our universe (there are local inhomogeneities), but it is a relatively good approximation in the large scale, so we shall assume it as true here, for simply getting a first approximation of how the universe expands.

The geometric equation

Introducing the variables

Time

The properties of space-time and matter will be described by variables that are implicit functions of the time t defined as the age of the universe (the age of a galaxy since the big bang, whose movement followed the universal expansion).
The derivative of each variable with respect to time will be denoted by the prime symbol ( ')

Distance

As t is a function (field) in space-time, each value of t cuts space-time by a 3-dimensional subspace S, made of all galaxies at age t, that is space-like (i.e. with no time-like extension).
All spaces S correspond to each other by a mere change of scale. So they can be represented by the same 3-dimensional map of galaxies in the universe, that is locally faithful (showing how galaxies relate to their neighbors, as seen by a local observer following the expansion), with a scale that depends on time.
This map has its own 3-dimensional space-like geometry (locally Euclidean in first approximation, and with a constant curvature thanks to the Cosmological principle, thus either Euclidean, spherical or hyperbolic depending on the sign of the curvature).
Inside this map, we can consider surfaces and triangles.
Let us fix a choice of 2 galaxies, and let the variable r denote their distance in S. So, r is defined by summing up the small distances in a chain of aligned intermediate galaxies between both, all taken at the same age. Or to justify the same formalism another way, we may just take 2 nearby galaxies so that r will be "small" (they can exchange light in a "small time" compared to the age of the universe) and properties will be considered following this approximation.

The Hubble coefficient

The Hubble "constant" (which also varies in time) is

H = r'/r

and is independent of the conventional choice of galaxies whose distance define r (another choice multiplies r by a same number along time, thus lets H be the same function of time).

Space-time curvature

Thanks to the space-time symmetries given by the cosmological principle, among the 20 components of the Riemann curvature of space-time around a point in a local coordinates system, only 6 will be nonzero. They are the diagonal elements of the 6×6 matrix of Riemann curvature, and correspond to the effects of space-time curvature, on the Gaussian curvatures of small surfaces in each of the 6 directions of surfaces following the pairs of coordinate axis.

Moreover, for the same reason, these 6 components can be summed up as 2 variables:

Writing the equation

Take a third galaxy such that, in S at every time, the area of the triangle formed by the 3 galaxies is r2. This property is conserved along time (keeping the same galaxies), as the area of any expanding surface stays proportional to r2.

Let
K = (∑ angles) − π = k.r2

where k is the intrinsic (Gaussian) curvature of the triangle, and the angles are defined in S, thus are faithful to the way each galaxy naturally sees both other galaxies in its night sky.

The Gaussian curvature of that triangle is a sum of 2 contributions :

k = R + h

where R is the intrinsic curvature of space-time in space directions (explained above), and h is the contribution from the extrinsic curvature of S in the time dimension.

The extrinsic curvature is given by the Hubble coefficient H = r'/r.
Indeed, H is the ratio between the angle r' between time vectors (orthogonal to S) at nearby points of S, and their distance r. This describes how things go in a thick slice of space-time between successive S, disregarding the rest of space-time.

The extrinsic curvature contributes to the intrinsic curvature by its square. The precise formula is
h = − H2/c2

The coefficient in this formula is the one needed to make types of quantities fit, by the general conversion between space and time quantities :

In a flat Euclidean space-time, r would be proportional to t so that H=1/t. Then S would be a sphere with radius t in time units, or c't in space units, and extrinsic curvature 1/c't = H/c' that contributes to the Gaussian curvature by its square H2/c'2. Finally, the substitution rule c'2 = − c2 gives the result.
This contribution has a negative sign, as, in the geometry of Minkowski (in a flat Minkowski space-time), a "sphere" with time-like radius, is an hyperboloid of two sheets, whose geometry (given from the Minkowski geometry of surrounding space-time, to not be confused with an hyperboloid of two sheets in an Euclidean space) is not spherical but hyperbolic, with a negative Gaussian curvature − H2/c2.

We finally get the geometric equation

K = (RH2/c2).r2 = R.r2r'2/c2

As each galaxy sees each other galaxy fleeing radially, each angle of the triangle remains constant. Thus K is also constant along time, its first derivative is 0 :

R'.r2 + 2 Rrr' − 2r'.r"/c2 = 0
R' + H (2R − 2Rt) = 0

The equation from mechanics

Introduce the notations

The work of P on this expanding volume, "consumes" its internal energy at the rhythm

E' = P(r3)' = 3 P r'r2
E' = (U r3)' = U'r3 + 3 Ur'r2

U' + 3 H (U + P) = 0

Putting all together to get the Friedmann's equations

The Einstein's equation of General relativity relates the 10-dimensional stress-energy tensor with the 20-dimensional Riemann curvature tensor, in a way that more precisely expresses the former in terms of the latter.
But in the simple case of the universal expansion, this is reduced to a relation between 2 pairs of variables:

The relation will not involve the variable H, which does not describe a direct, local physical quantity (but only how our coordinates system varies around).

We can obtain this relation from the above equations, by assuming that their terms are proportional, with a proportionality coefficient that we shall define as 3/G* (in fact G* = 8πG/c4, where G is Newton's gravitational constant) : 

G* U' = 3R'
G* (U + P) = 2R − 2Rt

Integrating the first equation we get

Λ + G* U = 3R

where Λ is the Cosmological Constant. This can be used to re-express the other equation in 2 possible simplified ways:

Λ− G* P R + 2 Rt
6 Rt = 3Λ − 3R − 3G* P
6 Rt = 2Λ G* (U +3P)

The Cosmological Constant can be formally cancelled by taking G* U +Λ as the new definition of G* U, and G* P − Λ as the new definition of G* P.
It can thus be interpreted as a hidden combination of a constant energy density and a constant negative pressure, which does not interact with ordinary matter apart from its gravitational effects : such a mechanical system remains invariant during expansion, just like a film of soap (a bubble) can expand keeping a constant surface density of energy with a constant surface tension.

Finally, by Rt = r"/rc2 we get the law of evolution of the expansion

r" = r ((Λc2/3) − (G*c2/6) (U +3P))

The meaning of Einstein's equation

We can admire the system of both equations Λ + G* U = 3R and Λ− G* P R + 2 Rt as showing the similarity between space and time dimensions, in how they relate the Riemann curvature (symmetric 6×6 matrix, here reduced to its diagonal whose coefficients are three times R and three times Rt) with the stress-energy tensor (symmetric 4×4 matrix, here reduced to its diagonal with coefficients U and three occurrences of P):
These are the 4 main (diagonal) components of Einstein's equation of General Relativity, that identifies, at each event, the (10-dimensional) stress-energy tensor, as a function of the (20-dimensional) Riemann curvature tensor. The remaining (6 non-diagonal) components of that equation, can be deduced from these 4 (or even just from one of them), just by re-applying the same relations to other coordinates systems (rotated in all possible ways around the considered point).

Integrated form of the equation

The evolution of the expansion can be expressed in an integrated form (involving r' instead of r'') by using the constant value of K = R.r2r'2/c2 which was ignored while we only used the cancellation of its derivative:

3R = Λ + G* E/r3
c2K = r'2c2 R r2 = r'2 − (c2Λ/3) r2 − (c2G*/3) E/r
r'2/2 = (c2Λ/6) r2 +  (c2G*/6) E/rc2K/2

It resembles the equation of movement of a particle in a field of potential energy, interpreting

If Λ > 0 then the term − (c2Λ r2/6), that can be neglected for small values of r, will be the dominating term for large values of r. Such a potential makes the movement of universal expansion behave like a mechanical system diverging away from a point of unstable equilibrium, thus at exponential speed with respect to time, with characteristic time 1 / c Λ/3 .

Let

m = (4/3)πr3ρ the mass inside a ball with radius r, so that E= c2ρr3 = (3c2/4π)m
G = c4G*/8π

Then the last term of the "potential energy" is

− (c2G*/6) E/r = − G m/r

that is the Newtonian gravitational potential around the mass m.
There is a difference with the classical Newtonian theory, though : here, m varies along time if P is nonzero.

Diverse solutions

Imagine a universe where matter (combination of energy and pressure) is a superposition of independent substances, each with its own proportionality between energy density U and pressure P :
3P = α U.
U' = − 3 H (U + P) = − H (3+α)U
(ln U)' = U'/U = −(3+α)r'/r = −(3+α)(ln r)'
U is proportional to r−(3+α)
It contributes to the potential as − (c2G*/6) U r2, that is proportional to r−(1+α)
If the "kinetic energy" r'2/2 was just the opposite of this "potential energy" (to be caused by it) then r' would be proportional to r−(1+α)/2:
Looking for expressions of r as proportional to tβ (no matter the coefficient, that is a pure convention for r) we have r' proportional to tβ−1 = r1−1/β. Thus α and β are related by
1−1/β = −(1+α)/2
1/β = (3+α)/2
β = 2/(3+α)
Particular cases are


French version : Expansion de l’Univers en Relativité Générale
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