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.
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
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.
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
Moreover, as K is constant along time, its first derivative is 0 :
R'.r2 + 2 R.r.r'= 2r'.r"/c2
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
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)
Finally, eliminating R between both equations gives the
following equation for the evolution of the expansion rate
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.
Back to :
List of physics theories
Set Theory and foundations of mathematics