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.

Necessary preliminary : the different notions of curvature in geometry.

**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).

The derivative of each variable with respect to time will be denoted by the prime symbol (

All spaces

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.

*H* = *r'*/*r*

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*t*_{1}and*t*_{2}increases at "speed"*r*(when*t*_{1}is fixed and*t*_{2}varies), that is the flow of area swept by the border*t*=*t*_{2}, parallel curve with length*r*, that moves at unit speed when*t*_{2}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"*/*rc*^{2}

Take a third galaxy such that, in *S* at every time, the
area of the triangle formed by the 3 galaxies is *r*^{2}.
This property is conserved along time (keeping the same galaxies),
as the area of any expanding surface stays proportional to *r*^{2}.

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*

Indeed,

It contributes to the intrinsic curvature of *S* by its square:
*h* = − *H*^{2}/*c*^{2}.

In a flat Euclidean space-time,

rwould be proportional totso thatH=1/t. ThenSwould be a sphere with radiustin time units, orc'tin space units, and extrinsic curvature 1/c't=H/c'that contributes to the Gaussian curvature by its squareH^{2}/c'^{2}. Finally, the substitution rulec'^{2}= −c^{2 }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 −H^{2}/c^{2}.

We finally get the geometric equation

*K* = (*R* − *H*^{2}/*c*^{2}).*r*^{2}
= *R.r*^{2} − *r*'^{2}/*c*^{2}

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*'.*r*^{2} + 2 *Rrr'*
− 2*r'.r"*/*c*^{2} = 0

*R*' + *H* (2*R* − 2*R _{t}*) = 0

Introduce the notations

*ρ*= mass density,

*U*=*c*^{2}*ρ*= energy density-
*E*=*U r*^{3}= energy in an expanding region of space with volume*r*^{3} *P*= pressure.

−*E'* = *P*(*r*^{3})*'*
= 3 *P r'r*^{2
}*E'* = (*U r*^{3})' = *U'**r*^{3}
+ 3 *U**r'r*^{2}

*U'* + 3 *H* (*U* + *P*)
= 0

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**
= 8π*G*/*c*^{4}, where *G* is Newton's
gravitational constant) :

*G** *U'* = 3*R'*

*G** (*U* + *P*) = 2*R* − 2*R _{t}*

Λ + *G** * U* = 3*R*

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

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}* =

*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*of the diagonal of the stress-energy tensor are equal to_{z}*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*is related to the sum of the 3 components of the Riemann curvature in that space with coordinates_{x}*y,z,t*(orthogonal to*x*) :*R*in the (*y,z*) direction, and both components equal to*R*in the (_{t}*y,t*) and (*z,t*) directions.

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.r*^{2} − *r*'^{2}/*c*^{2}
which was ignored while we only used the cancellation of its
derivative:

3*R* = Λ + *G** *E*/*r*^{3}

−*c*^{2}*K* = *r*'^{2} − *c*^{2}
*R **r*^{2} = *r*'^{2} − (*c*^{2}Λ/3)
*r*^{2} − (*c*^{2}*G**/3) *E*/*r*

*r*'^{2}/2 = (*c*^{2}Λ/6) *r*^{2}
+ (*c*^{2}*G**/6) *E*/*r* − *c*^{2}*K*/2

−

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

- −
*c*^{2}*K*/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 −
*c*^{2}*R**r*^{2}/2 = − (*c*^{2}Λ*r*^{2}/6) − (*c*^{2}*G**/6)*E*/*r*

If Λ > 0 then the term − (*c*^{2}Λ
*r*^{2}/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

Then the last term of the "potential energy" is

m= (4/3)πr^{3}ρthe mass inside a ball with radiusr, so thatE=c^{2}ρr^{3}= (3c^{2}/4π)m

G=c^{4}G*/8π

− (*c*^{2}*G**/6) *E*/*r* = −* **G
m*/*r*

There is a difference with the classical Newtonian theory, though : here,

3*P* = α *U*.

(ln U)' =

If the "kinetic energy"

Looking for expressions of

1−1/β = −(1+α)/2

1/β = (3+α)/2

β = 2/(3+α)

Particular cases are1/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
*r*^{2}. 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 r*^{3}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*.

French version : Expansion
de l’Univers en Relativité Générale

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