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


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 whose movement followed the universal expansion, since the big bang).
The derivative of each variable with respect to time will be denoted by the prime symbol ( ')


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.

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. This 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.

It contributes to the intrinsic curvature of S by its square: h = − H2/c2.

The coefficient −1/c2 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 constant 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 .


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

Olber's paradox and its explanation

The question of Olber's paradox is, if the universe is uniformly populated with stars then why is the night sky still dark ?
The correct explanation is that : Namely it would take about 100 billion light years distance to just fill the sky with galaxies. Now, after just a few billion light years we have red shift which makes things even fainter, and what is the background of the sky behind the galaxies is not exactly total darkness (there is no such a thing as a totally dark background in the universe !) but the cosmic microwave background, which came from a period where the temperature of the universe was about 3000K. This temperature is among that of the surface of red stars (so, quite less bright than the Sun which has a surface temperature of almost 6000K) but, of course, it was very much red shifted so that it is now of only 3K, that can be qualified as total darkness for our concerns of visible light.
To compute things more precisely : by the Friedmann equations, during the expansion of a flat universe, the energy density remains proportional to the square of the Hubble parameter (expansion speed per distance). This Hubble parameter is inversely proportional to the "size of the visible universe" (distance beyond which luminosity is significantly red shifted to induce a darkness effect from bright sources, and which also represents the age of the universe). Thus if we assume galaxies with fixed sizes, masses and absolute luminosities which concentrate a fixed proportion of the energy of the universe (which is actually not the case !), a doubling of this size divides the energy density by 4, which makes the universe twice more transparent through that doubled size.
We must point out that the universe was initially very opaque (thus bright), so that such a big expansion (to a universe with very low density) was needed to get us away from that to a transparent universe (thus darkened by redshift), because the above relation between energy density and expansion involves the gravitational constant which has a very small value compared to the rest of physical laws. To that we must moreover add the roles of dark matter and dark energy, and the fact that galaxies are absolutely fainter now than they were a few billion years ago.

French version : Expansion de l’Univers en Relativité Générale
Back to :
List of physics theories
Set Theory and foundations of mathematics