Expansion of the Universe in General Relativity

A simple derivation of the Friedmann equations


This text can be taken as an introduction to General Relativity, by its way to independently explain, justify and use the Einstein's equation of General Relativity, in a simplified way thanks to the simplicity of this particular problem.
However it assumes the concept of Riemannian curvature in 2 dimensions.
An introduction to general relativity will be developed later to explain this concept as well as the more general form of Riemannian curvature in any dimension and the Einstein's equation.

Cosmological principle

We start with the Cosmological Principle, which says that in the large scale the universe is uniform (there is no center, everthing happens the same from the viewpoint of any galaxy that follows the expansion) and there is also no priviledged direction: each galaxy sees each other galaxy as going away radially, at a speed only depending on the distance, independently of the direction where they are seen.
We shall here take the approximation of neglecting the local inhomogeneities, assuming a perfect homogeneity of everything.

Introducing the variables

The variables here will describe properties of space-time and matter as implicit function of the time t defined as the age of the universe, that is the age of a galaxy present at that place, whose movement followed the universal expansion.
Their derivative with respect to time will be denoted by the prime notation

Let us fix a choice of 2 galaxies, and let the variable r denote their distance. More precisely, r is defined by summing up the distances in a chain of aligned intermediate galaxies between both, all taken at the same age. In other words, we may just take 2 nearby galaxies so that r will be small and we shall only consider properties at the first order of approximation relatively to r.

Thus the Hubble "constant" (which varies in time) is H = r'/r.

The cosmological principle gives symmetries to space-time so that among the 20 components of space-time curvature, only 6 are nonzero, those of the "diagonal" of the 6*6 matrices of curvature in a natural coordinate system around a point, describing the internal curvature of small surfaces in each of the 6 directions of planes defined by pairs of coordinates : (x,y), (x,z), (y,z), (x,t), (y,t), (z,t).

Moreover the symmetries of the problem make the first 3 equal to each other, and also the last 3, so that only 2 variables will describe the components of the space-time curvature in an expanding homogeneous universe.

The space curvature (that of a "flat surface" in space), will be denoted R (like "Riemann curvature"). This is the usual notation for the Gaussian curvature of a surface.

Next we introduce the mass density ρ, the energy density U = c2 ρ, and the pressure P.

The equation from geometry

Take a third galaxy such that the area of the triangle formed by the 3 galaxies is equal to r2. This property is conserved along time, as the area of any expanding surface keeps the same proportionality to r2.

Let K = sum of angles - π. As each galaxy sees each other galaxy fleeing radially, each angle of the triangle remains constant. Therefore K is also constant in time.

The equation of 2-dimensional curved geometry says that K= k.r2 where k is the Gaussian curvature of the surface.

But this surface is curved in space-time, towards the time direction. Therefore the value of k is a sum of 2 terms : k = R+h where h is the term due to the external curvature of an instantaneity surface (t=constant) in the time direction. This term is due to universal expansion.

If space-time was flat, we would have r proportional to the time t, the galaxies of the same age t would form a sphere with time radius t, and we would have H=1/t.

Changing the geometry of space-time into Euclidean geometry, the external curvature of an instantaneity surface would also be 1/t = H, and the Gaussian curvature would be h= t-2 = H2 taking the same units for time and space. Distinguishing time and space units and converting this result to Minkowskian geometry, gives h = - H2/c2
This describes how things go in a slice of space-time between two nearby ages, disregarding the rest of space-time : the expansion H describes the difference of direction of nearby time vectors (orthogonal to the instantaneity surface) in proportion to their distance, and thus the external curvature of the surface. It gives a relativistic small negative ( -1/c2) contribution to the intrinsic curvature, corresponding to the intrinsic curvature of a sphere with time radius (that is a hyperboloid in the flat Minkowskian space-time).

In conclusion, we get the geometric equation:

K = (R - H2/c2).r2 = R.r2 - r'2/c2

(which gives back K= - r'2/c2 in a flat space-time).

Moreover, as K is constant along time, its first derivative is 0 :

R'.r2 + 2 R.r.r'= 2r'.r"/c2

which can also be written as
 
R'+ 2R.H=2H r"/rc2


The equation from mechanics

Consider an expanding region of space with volume r3 and internal energy E = mc2 = U r3 (so U is the energy density). The internal pressure P in this expanding volume induces a variation of energy E'= - P(r3)'= - 3 P.r'.r2. Dividing both terms by r3 gives

U'  + 3 U.H + 3 P.H = 0

The Einstein's equation of General Relativity

This equation relates the 10-dimensional components of the stress-energy tensor with the 20-dimensional curvature tensor.
Here the study is restricted to a simple case where only two variables appear on each side:
On one side are the two variables of space curvature R and space-time curvature r"/r.
On the other side are the energy density U and the pressure P.
The relation must not involve the variable H, as this does not describe a physical quantity at the considered point but only a way in which physical quantities vary around it. 

It will be expressed by making both equations proportional to each other. (Sorry this is not absolutely rigorous proof, but already suggestive, and the formulas are accurate expressions of General Relativity in this case, so...)
Let us denote the proportionality coefficient 3/G*. In fact we will have G*=8πG/c4, where G is the gravitational constant. So we have

G*U' = 3R'
G*(U+P) = 2(R − r"/rc2)

The first equation gives G*U+Λ = 3R where Λ is the Cosmological Constant.
From this, the latter gives G*P = 2(R− r"/rc2)+Λ−3R = Λ+R−2r"/r.
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.
This means that the Cosmological Constant can be interpreted as a hidden combination of constant density of energy and constant negative pressure, which do not interact with ordinary matter but is only manifested in a gravitational way : such a mechanical system remains invariant during expansion, just like a film of soap keeps a constant surface density of energy with a constant surface tension during the expansion of a bubble.

Finally we have the following system (which may be simplified by letting Λ=0)

G*U = 3R − Λ
G*P = Λ − R − 2r"/rc2.

We can admire this system for its property of showing that the space and time dimensions obey the same laws:

Finally, eliminating R between both equations gives the following equation for the evolution of the expansion rate

6 r" =  rc2(2Λ − G*(U + 3 P))

Resolving the equation

Let us make the equation of the universal expansion look like the equation of movement of a particle in a field of potential energy. From above we have  r'2= c2 (R.r2 - K).
Since K is constant, let us interpret -Kc2 as the "total energy" (though it has nothing to do with an energy anymore), r'2 as the "kinetic energy". So the "potential energy" is
V= -c2 Rr2 = -c2 r2 (G*U+Λ)/3 = -8πG m/3r - c2 r2Λ/3

The term -c2 r2Λ/3 coming from the cosmological constant, will be the dominating term for large values of r. Such a potential, proportional to -r2, makes r diverge at exponential speed; while it can be neglected for small values of r.

In the case if P=0 all along, the parameter m of total mass inside the volume r3 is a constant, so that the remaining term of the potential -8πG m/3r behaves as a Newtonian gravitational potential, in -r-1.

But it is generally variable otherwise.

Near the Big Bang we have periods dominated by hot matter and radiative energy, so that most of the energy consists in particles going at or near the speed of light.
In this case we have P=U/3. Let us resolve this by the equation of mechanics: U'  + 3 U.H + 3 P.H = 0 where H = r'/r.
So, U' + 4U.r'/r =0.
(ln U)'=-4(ln r)'
ln U = -4 ln r + constant
U is proportional to r-4.
Thus the "potential" is now proportional to - r-2.
If we have a matter at rest not interacting with a radiative background then both separately contribute to the "potential" with respective terms in -r-1and - r-2.


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