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..
- 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
Necessary preliminary : the
different notions of curvature in geometry.
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.
- 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
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
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
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
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
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
But to give this intrinsic curvature the same type as space Gaussian
curvature by proper conversion
between space and time quantities, we define
Rt = 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.
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
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 = (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 − 2Rt) = 0
The equation from mechanics
Introduce the notations
The work of P on this expanding volume, "consumes" its
internal energy at the rhythm
- ρ = mass density,
- U = c2 ρ = energy density
- E = U r3 = energy in an expanding
region of space with volume r3
- P = pressure.
−E' = P(r3)'
= 3 P r'r2
E' = (U r3)' = U'r3
+ 3 Ur'r2
U' + 3 H (U + P)
Putting all together to get the Friedmann's equations
equation of General relativity relates the 10-dimensional
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 Rt.
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'
Integrating the first equation we get
G* (U + P) = 2R − 2Rt
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
||= 3Λ − 3R − 3G* P
||= 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).
- 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 Px, Py,
Pz 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 Px 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 Rt in the (y,t) and (z,t)
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
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 .
Then the last term of the "potential energy" is
m = (4/3)πr3ρ the mass inside a
ball with radius r, so that E= c2ρr3
G = c4G*/8π
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.
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
It contributes to the potential as − (c2G*/6)
U r2, that is proportional to r−(1+α)
(ln U)' = U'/U = −(3+α)r'/r =
U is proportional to r−(3+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
Particular cases are
1/β = (3+α)/2
β = 2/(3+α)
- 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 t2/3.
- 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
Thus the "potential" is now proportional to - r-2,
and an expansion dominated by this contribution has r
proportional to √t.
French version : Expansion
de l’Univers en Relativité Générale
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