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

General enough for containing the full substance of the Einstein's equation of General Relativity and its proof;
Simple enough (by its symmetries) for all this to be expressed in a short, elementary and self-sufficient way, bypassing the full mathematical expression as tensors, of the objects involved.

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

Uniform (all places are similar): the distribution of mass is homogeneous, there is no center, all galaxies are assumed to obey the general expansion, everything happens the same as viewed from any galaxy as from any other galaxy.
Isotropic (at every place, all directions are similar) : each galaxy, at any time, sees each other galaxy as going away radially, at a speed that only depends on the distance between them (the speed is proportional to the distance in the approximation where distance and speeds are small, respectively compared to the visible universe and the speed of light; and also if they are large if we exactly define them as described below).

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 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 ( ')

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:

The 3 "pure space" components (with directions (x,y), (x,z), (y,z)) have the same value, that we shall here denote R. This is the Gaussian curvature around any point of a "small" space-like surface that is extrinsically "straight" or "flat" (its extrinsic curvature is zero). The cancellation of the extrinsic curvature towards the 3rd space dimension is natural here for symmetry reasons; but the extrinsic curvature towards the time direction, needs to be cared for, as it is proportional to the movement of expansion (such as the universal expansion) of instruments that measure this surface as being in a "pure space" direction, i.e. made of "simultaneous" events (occuring at the "same time"). But S is externally curved into the time dimension by universal expansion. So, the "pure space" component R of space-time curvature will differ from the intrinsic curvature of S. The correct measurement of R requires to take a "flat" surface (with zero extrinsic curvature) that is tangent to S at the considered point, and can be defined as instantaneous relatively to a non-expanding apparatus instead of using our time variable t.
The 3 curvatures in space-time directions (with directions (x,t), (y,t), (z,t)), correspond to an acceleration (or deceleration) of expansion. Indeed, consider just one space dimension and pretend time is a second space dimension, let them form a surface obeying spherical geometry (or more generally, circular symmetry), where the time coordinate t is the latitude (distance from a pole), the spaces S of teach time are the circles of latitude, and the roles of world lines of galaxies (with "time direction"), are played by meridians. Let us define the distance r between two given meridians by the length of arc of circle of latitude between them. Then, following a meridian (at a speed conventionally taken as 1), the Gaussian curvature of the sphere "accelerates" attractively (slows down) the growth of r, proportionally to r. So this Gaussian curvature is defined as −r"/r. Indeed, r' is the angle between time directions of both galaxies measured by parallel transport along S, thus −r" is the flow of angles received from parallel transport around a space-time surface (the segment between both galaxies, extended in time) whose area between times t₁ and t₂ increases at "speed" r (when t₁ is fixed and t₂ varies), that is the flow of area swept by the border t=t₂, parallel curve with length r, that moves at unit speed when t₂ varies. But to give this intrinsic curvature the same type as space Gaussian curvature by proper conversion between space and time quantities, we define

R_t = r"/rc²

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. 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 = − c²gives 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 = (R − H²/c²).r² = R.r² − r'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 − 2R_t) = 0

The equation from mechanics

Introduce the notations

ρ = mass density,
U = c² ρ = energy density
E = U r³ = energy in an expanding region of space with volume r³
P = pressure.

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

−E' = P(r³)' = 3 P r'r²E' = (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 stress-energy tensor is represented by the energy density U and the pressure P.
The Riemann curvature is represented by the space curvature R and the space-time curvature R_t.

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 − 2R_t

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 R_t
6 R_t	= 3Λ − 3R − 3G* P
6 R_t	= 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 R_t = 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 R_t 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 R_t) with the stress-energy tensor (symmetric 4×4 matrix, here reduced to its diagonal with coefficients U and three occurrences of P):

U is the time component of the stress-energy tensor. It represents the energy density, that we can describe as "inside a small space volume" or "flowing through this volume" in the time direction where it is conserved; it is related to the sum of the 3 components of Riemann curvature along the space-like surface directions in a given coordinates system, all equal to R.
All 3 space components P_x, P_y, P_z of the diagonal of the stress-energy tensor are equal to P. Each one measures the force over (thus the flow of momentum through) what appears as a surface inside usual space, but that is in fact 3-dimensional due to its time extension. For example P_x is related to the sum of the 3 components of the Riemann curvature in that space with coordinates y,z,t (orthogonal to x) : R in the (y,z) direction, and both components equal to R_t in the (y,t) and (z,t) directions.

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.r² − r'²/c² which was ignored while we only used the cancellation of its derivative:

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

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

− c²K/2 as the "total energy" (though it is no more an energy, or... it can somehow be understood as expressing here the geometrical quantity that plays the role of the conserved energy in General Relativity),
r'²/2 as the "kinetic energy".
So the "potential energy" is − c² R r²/2 = − (c²Λ r²/6) − (c²G*/6) E/r

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 t^β (no matter the coefficient, that is a pure convention for r) we have r' proportional to t^β−1 = r^1−1/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

The cosmological constant, also called dark energy, can be seen as the particular case α = − 3, giving its potential in r². As a main contribution to the potential, this case is the exceptional one : we have a division by 0, letting r be an "infinite power" of t, in fact an exponential function of time.
The inert (cold) matter (α= 0) gives the Newtonian potential in 1/r. As a main contribution to the potential, it makes r proportional to t^2/3.
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, P = U/3, i.e. α=1. Indeed, this way, E = U r³ is proportional to 1/r, which can be interpreted as the effect of redshift on this radiative energy, as the expansion dilates its wavelength (this is just another interpretation of the redshift by relative speed).
Thus the "potential" is now proportional to - r^-2, and an expansion dominated by this contribution has r proportional to √t.

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 :

galaxies are relatively thin and with low density of stars enough to be almost transparent, so that with or without dust they are anyway quite dark and the question is mainly reduced to the inter-galactic issue ;
the universe is old enough to leave us with a quite low density of something like 1 important galaxy (most of which are less than 100,000 light years of diameter) per cube of 10 million light years. Such a low density results in a very low occupation of the sky surface.

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
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