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This chapter gives the basic formulas of Restricted Relativity and General Relativity.

Through the paragraph Definitions we will also show that many "mysteries" of space-time are due either to a rigor lack in the words or concepts use or to the formulas application outside of their applicability field. In particular :

- Confusion between Time and Duration.

- Confusion between proper Time and apparent Time.

- Different (but not contradictory) uses of the Time concept. For many mathematicians, time is reduced to a **purely operative concept** intervening for example in the Lorentz-Poincare Equations of Restricted Relativity. For many physicists, time on the contrary is a **physical real time**, measured by mean of spatially located clocks capable of synchronizing by exchanging light signals.

- Strong standardization of some constants (for example, light speed (c) or universal gravitational constant (G) set to 1) which makes the formulas non-homogeneous at the units level and favors numerical application errors.

- Wrong application of the Lorentz-Poincare equations to reference frames which are not in uniform translation movement relative to each other.

"On a human scale, the light speed is prodigiously high (about 300 000 km/s). When a light source sends us a signal, the light provides us with almost instantaneous information. We believe we see space at a given moment. Time seems absolute, separated from space." [AND Théorie - Partie 1]

Imagine two observers O and O' in relative movement relative to each other, who wish to set their watches by the exchange of optical signals. Suppose that the two watches moreover are synchronized, by any means, so that they indicate the same time at the same initial instant t = t '= 0. At this initial instant, each observer then sends a signal to the other. What time does each watch indicate when each observer receives the signal from the other ? It is obvious that this is not the same time.

"The transmission duration in fact is not the same in both directions since the observer O, for example, advances in front of the optical perturbation emanating from O', while the observer O' flies before the perturbation emanating from O. The watches will indicate the local time of each observer, so that one of them will appear to delay on the other." [POI L'Etat]

The indicated time would be the same for both observers only in the case of observers fixed relative to each other or in the thought hypothesis of a light having an infinite speed.

"The instant universe is unobservable. It appears as a Space-time where each observed object is seen at a space point and at a time point that is not the same for all space points." [AND Théorie - Partie 1]

"Until the end of the 19th century, classical mechanics founded by **Galileo and Newton** constituted an undisputed basis of physics.

In 1887 an American physicist **Albert Michelson** and his colleague **Edward Morley** showed that the light speed did not verify the Galilean law of addition of velocities. On the contrary, the light speed in the vacuum was independent of the motion of the emitting source.

At the end of the 19th century, a second enigma disrupted the certainties of the scientists. The famous **equations of British James Maxwell** which describe all the phenomena of electromagnetism no longer have the same form when they are transposed from one reference system into another by a uniform translation.

Should not the Galilean principle be, if not abandoned, at least rehabilitated ?

In 1905 **Henri Poincare** laid the fundamental foundations of Restricted Relativity which erased at once all the anxieties of physicists about these two enigmas.

In 1915 **Albert Einstein** elaborated General Relativity with the help of various mathematicians to take into account relativistic gravitation in particular." [HLA Pour_comprendre]

Today there remains one final challenge : the unification of General Relativity and **Quantum theory** in order to make coherent gravitation on a macroscopic scale and gravitational interaction on a microscopic scale where the quantum character of the elementary particles intervenes.

We consider a reference frame R' in **uniform translation** at the speed V relative to a reference frame R (see Figure above).

The two reference frames have their origin O and O' which coincide at time t = 0.

Let an arbitrary point M of abscissae x' in R' and x in R.

**The transformation of Galileo** from R to R' can be written as follows :

(G1) x' = x - V t

(G2) t' = t

**The transformation of Lorentz-Poincare** introduces a new entity to describe the physical phenomena : Space-time. This can be written as follows :

(L1) **x' = gamma (x - V t)**

(L2) **t' = gamma (t - B x)**

(L3) **gamma = 1 / (1 - V ^{2}/c^{2})^{1/2}**, named "

(L4)

where c is a constant (space-time structure constant) which is similar to a limiting speed and which appears during the presentation of the equations (L). The constant c is taken equal to the highest speed currently measured which is that of electromagnetic phenomena in vacuum, in this case the light speed in vacuum.

Since the light speed is slowed down in various media according to their refractive index n, note that it is possible to accelerate particles that go faster than light in the same medium.

Note also that if two luminous particles move away from each other, their relative speed is equal to c and not 2c (law of speeds composition, see below).

In 1975 **Jean-Marc Levy-Leblong** published an article on Restricted Relativity presented in a modern form deduced only from the properties of space and time (Poincare's postulates), without need for reference to electromagnetism [LEV One_more]. Einstein's postulate on the invariance of the light speed in all reference frames then appears as a simple consequence of the Lorentz-Poincare transformation describing the Restricted Relativity.

In 2001 **Jean Hladik** published, with one of his colleagues **Michel Chrysos**, the first book on Restricted Relativity presented in this modern form [HLA Pour_comprendre].

Inspired by the works listed below in the Bibliography we present here an elegant and rigorous presentation of the Lorentz-Poincare transformation only based on the four Poincare's postulates.

**Postulat 1 : Space is homogeneous and isotropic**

Space has the same properties at every point and in every direction. In other words space is invariant by translation and rotation.

**Postulate 2 : Time is homogeneous**

The time is identical in every point of the same reference frame. All fixed clocks in a given reference frame must be strictly set at the same time. In other words time is invariant by translation.

**Postulate 3 (Principle of Relativity) : The laws of physical phenomena must be the same either for a fixed observer or for an observer entrained in a uniform translation movement.**

The form of the equations which describe the mechanical phenomena is invariant by changing the reference frame by uniform translation.

**Postulate 4 : Causality must be respected**

When a phenomenon A is the cause of a phenomenon B, then A must occur before B in any reference frame.

The postulates of space and time homogeneity induce that the desired transformation is linear of the following form :

(Ha) **x' = C(V) x + D(V) t**

(Hb) **t' = E(V) t + F(V) x**

where the four functions C, D, E and F are to be determined.

The particular point M = O' correspond to : x' = 0 and x = V t

Equations (H) can be rewritten as follows :

(C1a) **x' = gamma (x - V t)**

(C1b) **t' = gamma (A t - B x)**

The unknowns become gamma, A and B which are three functions dependent only of V. Namely : gamma = gamma(V) ; A = A(V) ; B = B(V).

When V = 0 we must have : x' = x and t' = t corresponding to the identity transformation and it can be deduced that :

(C2) gamma(0) = 1

The postulate of **space isotropy** induces that the form of the equations is invariant by reflection (x -> -x ; x' -> -x' ; V -> -V) corresponding to the passage of the " -R " reference frame to the " -R' " reference frame. From this it can be deduced that :

(C3a) gamma(V) = gamma(-V)

(C3b) A(V) = A(-V)

(C3c) B(V) = - B(-V)

The postulate of **form invariance** induces that the form of the equations is invariant by inverse transformation (x' <-> x ; t' <-> t ; V <-> -V) corresponding to the exchange of the reference frames R and R'. From this it can be deduced that :

(C4a) x = gamma(-V) (x' + V t')

(C4b) t = gamma(-V) (A(-V) t' - B(-V) x')

From relations (C1)(C3) it can be deduced that :

(C5a) **A = 1**

(C5b) gamma^{2} (1 - V B) = 1

It remains to determine the unknown B.

The postulate of **form invariance** induces that the form of the equations is invariant by composition of the transformations (R -> R') and (R' -> R"). From relation (C5a) it can be deduced that :

(C6a) x" = gamma(U) (x' - U t')

(C6b) t" = gamma(U) (t' - B(U) x')

where U is the uniform translation speed of R" relative to R'.

Let W the uniform translation speed of R" relative to R.

From relation (C1) it can be deduced that :

(C7a) W = (V + U) / (1 + U B)

(C7b) B(U) / U = B / V

The relation (C7a) is the **law of speeds composition**.

The relation (C7b) shows that B is of the form :

(C8) **B(V) = b V**

where b is any constant (negative, zero or positive).

From particular relation (C2) the relation (C5b) can be written :

(C9) **gamma ^{2} = 1 / ( 1 - b V^{2})** avec

From relations (C8)(C9) the equations (C1) can be written :

(C10a)

(C10b)

(C10c)

It remains to determine the unknown b.

Let M1 and M2 two any points of the reference frame R.

From relation (C10b) it can be deduced that :

(t2' - t1')/(t2 - t1) = ( 1 - b V ((x2 - x1)/(t2 - t1)) ) / (1 - b V

The postulate of

(C11) b V (x2 - x1)/(t2 - t1) < 1

If b is negative this relation is not satisfied for any values of V, (x2 - x1) and (t2 - t1). The causality assumption is not respected for the case b < 0.

If b is positive or zero it can be written in the following form :

(C12)

From relation (C12) the relation (C10c) can be written :

(C13)

(C14) ((x2 - x1) / (t2 - t1)) / u < 1

From relations (C12) (C13) (C14) the relation (C11) is verified. The causality assumption is respected for the case b > 0 or b = 0.

Note that some authors such J. HLADIK arrive at the same conclusion (b > 0 or b = 0) without using the postulate of causality.

In practice the mathematical limit u is taken appropriately equal to the light speed c in the vacuum.

The Restricted Relativity applies only to reference frames in uniform translation and in a space-time where the gravitational effects are completely neglected as if the matter did not exist.

Einstein will rethink the notion of Newtonian gravitation which being propagated instantaneously is no longer compatible with the existence of a limiting speed.

He will also postulate that all laws of Nature must have the same form in all reference frames whatever their state of motion (uniform or accelerated).

General Relativity was born.

The fundamental equations of General Relativity, called **Einstein equations** or **equations of the gravitational field**, connect a local deformation of the geometry of space-time with the presence of local tensions (see Figure above).

These equations can be seen as a generalization of the **law of elasticity of Hooke** in a weakly deformed continuous medium for which the deformation of an elastic structure is proportional to the tension exerted on this structure.

Einstein equations are written :

**Eab** = KHI **Tab**

with : **Eab** = **Rab** - (1/2) **gab** R + LAMBDA **gab**

**Eab** is the Einstein tensor which measures the local deformation of the space-time geometry and represents its curvature at a given point. There is no gravitational force in General Relativity since this curvature of space-time takes its place. This tensor has remarkable property of having a zero Divergence.

**Tab** is the Energy-impulse tensor which describes at a point of space-time the energy and the impulse associated with matter or any other form of non-gravitational field such as the electromagnetic field. This tensor depends on the pressure p and the density rho of the physical environment that fills the space. This tensor is constructed so that its zero Divergence expresses the local conservation of impulse and energy.

a and b are the indices of the different tensors with a and b ranging from 0 to 3

KHI is the coefficient of proportionality : KHI = 8 Pi G / c^{4} (in m^{-1}.kg^{-1}.s^{2}). This coefficient was chosen so as to verify the Poisson equation of the Newtonian gravitation as a particular case of Einstein equations.

G is the universal gravitational constant : G = 6,6726 10^{-11} m^{3}.kg^{-1}.s^{-2}

c is the light speed in the vacuum : c = 2,99792458 10^{8} m.s^{-1}

**gab** is the Metric tensor solution of Einstein equations. The 16 gab components of this tensor are called **gravitational potentials**.

**Rab** est le Ricci tensor producted by Contraction of the Curvature tensor.

R is the Ricci curvature (or scalar curvature) producted by Contraction of the Ricci tensor.

LAMBDA is the **cosmological constant** of dimension m^{-2} and may be negative, zero or positive. LAMBDA was introduced by Einstein only later in applications to cosmology. The problem of the planets motion, considered as particles in an empty space around the sun (Schwarzschild metric), is solved by taking LAMBDA = 0 and **Tab** = **0**. In cosmology, the universe model (Friedmann-Lemaitre-Robertson-Walker metric) is determined by a priori non-zero LAMBDA value and the universal space is considered as filled with a real gas of galaxies with density rho and pressure p = 0 (Standard cosmological model).

For any tensor equation of type **A** = **0** it can be proved that its components retain the same form in any change of coordinates. The use of the tensorial formalism allows any physical law expressed with Einstein equations to remain invariant in any change of reference frame (principle of General Relativity). This is the extraordinary power of tensorial calculation.

About the Einstein equations themselves they are not demonstrated on the basis of more fundamental principles. This is the whole genius of Einstein to have postulated them.

The components of the **Eab** Einstein tensor are function only of the gravitational potentials gab and their first and second derivatives. These components are linear relative to the second derivatives and involve the Christoffel symbols which are function of these gab.

The resolution of these coupled differential equations of the second order is extremely difficult.
The symmetry of the tensors **Rab**, **gab** and **Tab** reduces to 10 the number of distinct equations and the 4 conditions of zero Divergence reduce them to 6 independent equations.

On their side, by symmetry, only 10 of gab are distinct. In a four-space frame the values of 4 of them can be chosen arbitrarily which also reduces to 6 the number of functions gab to be determined.

Several Relativistic Metrics are then available in General Relativity (see Figure above).

The Friedmann-Lemaitre-Robertson-Walker metric (F) is used in cosmology to describe the universe evolution at large scales. It is the main tool leading to the construction of the standard cosmological model : the **Big Bang theory**.

The Schwarzschild metric (S1, S2...) describes the geometry around the masses (M1, M2...).

The Minkowski Metric (K) describes the geometry away from the large masses, on the asymptotically flat part of the previous metrics, according to a tangent Euclidean space-time of the Restricted Relativity.

Under the hypothesis that the gravitational field is **static and centrally symmetrical** (Schwarzschild metric) as the case of Sun and many stars, the gravitational potentials gab are expressed in spherical coordinates (r, theta, phi) relative to two parameters nu and alpha only functions of r.

These gab allow to calculate the components of the Ricci tensor (**Rab**) and then, by Contraction, the Ricci curvature (R). See calculations detailed below.

In the particular case of a **zero cosmological constant** (LAMBDA = 0) and a gravitational field **in vacuum** (when the Energie-impulse tensor (**Tab**) is zero), Einstein equations then are reduced to a system of two differential equations of the functions nu and alpha. Their integration gives the expressions nu and alpha. See calculation detailed below.

The Schwarzschild metric ds^{2} is finally completely determined as follows :

g00 = -(1 - psi/r)

g11 = 1 / (1 - psi/r)

g22 = r^{2}

g33 = r^{2} sin^{2}[theta]

gij = 0 for i and j taken different between 0 and 3

where psi is a constant called **Schwarzschild radius** or **gravitational radius**.

In the particular case of a gravitational field created by a **symmetrical central mass M**, we have : psi = 2 G M / c^{2}, producted by comparing the Schwarzschild g00 with the g00 of the Newtonian approximation.

The particular values r = 0 and r = psi, which make the coefficients g00 and g11 infinite, delimit a singular region which is in practice located deep inside the mass M, which is not inconvenient for **planets, ordinary stars and neutron stars**.

For **black holes** the singularity r = psi can be eliminated by a suitable choice of the coordinate system. On the other hand, the singularity r = 0 is a singularity of the Metric tensor **g** which shows the limit of the black holes description by the General Relativity and probably requires the use of a **quantum theory of gravitation** which does not really exist to date.

When r tends to infinity, the coefficients gab are reduced to the components of the Minkowski metric expressed in spherical coordinates. The space-time described by the Schwarzschild metric is thus **asymptotically flat**.

The motion of material systems and photons in the space considered is finally found by writing the equations of Geodesics. When their mass m is very small relative to the mass M of the central body of the Schwarzschild metric, it can be proved that their trajectories (orbits) are plane and become ellipses when r tends to infinity.

*
Detailed calculation of components g^{ab}, Rab, R, Eab, alpha and nu [GOUR Relativité Générale] :
In the case of a gravitational field with static and centrally symmetry (Schwarzschild metric), the gravitational potentials gij of the Metric tensor are the following :
g00 = -e^{2 nu}
g11 = e^{2 alpha}
g22 = r^{2}
g33 = (r^{2}) sin^{2}[theta]
gij = 0 for i and j taken different between 0 and 3
where nu and alpha are only functions of r.
The gravitational potentials g^{ij} of the inverse Metric tensor are then the following such that : g^{ij} gjk = DELTA^{i}k
where DELTA is the Kronecker symbol.
g^{00} = -e^{-2 nu}
g^{11} = e^{-2 alpha}
g^{22} = 1/r^{2}
g^{33} = (1/r^{2}) sin^{-2}[theta]
g^{ij} = 0 for i and j taken different between 0 and 3
The Christoffel symbols GAMMA^{i}jk are then written by the relations : GAMMA^{i}jk = (1/2) g^{il} (glk,j + glj,k - gjk,l)
GAMMA^{0}01 = GAMMA^{0}10 = nu'
GAMMA^{1}00 = e^{2 (nu - alpha)} nu' ; GAMMA^{1}11 = alpha' ; GAMMA^{1}22 = -r e^{-2 alpha} ; GAMMA^{1}33 = -r sin^{2}[theta] e^{-2 alpha}
GAMMA^{2}12 = GAMMA^{2}21 = 1/r ; GAMMA^{2}33 = -cos[theta] sin[theta]
GAMMA^{3}13 = GAMMA^{3}31 = 1/r ; GAMMA^{3}23 = GAMMA^{3}32 = 1/ tan[theta]
where nu' = d(nu)/dr and alpha' = d(alpha)/dr
The other Christoffel symbols are all zero.
The Rij components of Ricci tenseur are then written by the relations : Rij = R^{k}ikj = GAMMA^{k}ij,k - GAMMA^{k}ik,j + GAMMA^{k}kl GAMMA^{l}ij - GAMMA^{k}jl GAMMA^{l}ik
R00 = e^{2 (nu - alpha)} ( nu" + (nu')^{2} - nu' alpha' + 2 nu'/r )
R11 = -nu" - (nu')^{2} + nu' alpha' + 2 alpha'/r
R22 = e^{-2 alpha} ( r (alpha' - nu') - 1 ) + 1
R33 = sin^{2}[theta] R22
The other components Rij are all zero.
The Ricci curvature is then written by the relation : R = g^{ij} Rij
R = 2 e^{-2 alpha} ( -nu" - (nu')^{2} + nu' alpha' + 2 (alpha' - nu')/r + (e^{2 alpha} - 1)/r^{2} )
In the case of LAMBDA = 0, the Einstein tensor is then producted by the relation : Eab = Rab - (1/2) gab R
E00 = (1/r^{2}) e^{2 (nu - alpha)} (2 r alpha' + e^{2 alpha} - 1 )
E11 = (1/r^{2}) (2 r nu' - e^{2 alpha} + 1 )
E22 = r^{2} e^{-2 alpha} ( nu" + (nu')^{2} - nu' alpha' + (nu'- alpha')/r )
E33 = sin^{2}[theta] E22
The other components Eij are all zero.
The Einstein equations are then written by the relation : Eab = KHI Tab
E00 = KHI T00
E11 = KHI T11
E22 = KHI T22
E33 = KHI T33
0 = KHI Tij for i and j taken different between 0 and 3
In the case Tab = 0, the Einstein equations then reduce to the 3 following equations :
2 r alpha' + e^{2 alpha} - 1 = 0
2 r nu' - e^{2 alpha} + 1 = 0
nu" + (nu')^{2} - nu' alpha' + (nu'- alpha')/r = 0
The first equation is integrated into :
alpha = -(1/2) ln[ 1 - psi/r]
where psi is a constant.
By replacing this alpha value into the second equation, this one is integrated into :
nu = (1/2) ln[ 1 - psi/r] + b0
where b0 is a constant.
The zero of the gravitational field at infinity (so as to ensure an asymptotically flat metric with nu = 0 when r tends to infinity) requires that : b0 = 0.
By replacing these alpha and nu values in the third equation, this one is always satisfied.
We finally find :
g00 = -(1 - psi/r)
g11 = 1/(1 - psi/r)
*

Under the hypothesis that Space-time is **spatially homogeneous and isotropic** (Friedmann-Lemaitre-Robertson-Walker metric), the gravitational potentials gab are expressed in spherical coordinates (r, theta, phi) relative to two parameters k (constant) and a (function of t only).

These gab allow to calculate the components of the Ricci tensor (**Rab**) and then, by Contraction, the Ricci curvature (R).

By choosing a **Perfect Fluid model** for the Energy-Pulse Tensor (**Tab**), its components then can be calculated as a function of the pressure p and the density rho of the physical environment that fills the space.

The Einstein equations are then reduced to a system of two differential equations of the functions a(t), rho(t) and p(t), called **Friedmann equations** :

(F1) (a'/a)^{2} + k (c/a)^{2} = (1/3) rho KHI c^{4} + (1/3) LAMBDA c^{2}

(F2) a"/a = -(1/6) (rho + 3 p/c^{2}) KHI c^{4} + (1/3) LAMBDA c^{2}

The system is completed by giving a **cosmic fluid equation of state** as p = p(rho). An example of a frequently used equation of state is : p(t) = w rho(t) c^{2} where w is a constant that is equal to -1 (**quantum vacuum**), 0 (**zero pressure**) or 1/3 (**electromagnetic radiation**).

This equation of state, associated with the two equations (F1) and (F2), gives a remarkable relation linking rho(t) and a(t) :

**(Q0) rho(t) a(t) ^{3(1 + w)} = rho0 a0^{3(1 + w)} = constant**

where rho0 and a0 are two constants (index 0 generally corresponding to current data).

The system then reduces to a single differential equation of the function a(t) (see calculation detailed below) :

with A = (1/3) rho0 (a0)

and B = (1/3) LAMBDA c

This differential equation is analytically integrated for w = 0 or 1/3 (with any LAMBDA and k), which completely determines a(t) and the metric ds

g00 = -1

g11 = a(t)

g22 = a(t)

g33 = a(t)

gij = 0 for i and j taken different between 0 and 3

The first Friedmann equation (F1) is often presented in the condensed form :

k (c/a)

where :

H(t) =

OMEGA(t) =

OMEGAv(t) =

q(t) =

It would appear that the value to date of the deceleration parameter is negative (a" > 0), the slowing due to the matter attraction being totally compensated by the acceleration due to a hypothetical

The Friedmann second equation (F2) is also written in the form :

with F = (1/2) (1 + 3 w) A

Note the relation (Q2) is also obtained immediately by derivation of the relation (Q1).

In the standard case where

All these curves, except two, represent

- The curve C1 relative to case (LAMBDA < 0), or case (LAMBDA = 0) and (k > 0), corresponds to a closed model (decelerated expansion followed by an accelerated contraction occurring after the maximum point M1).

- The curve C2 relative to case (LAMBDA = 0) and (k = 0 or k < 0) correspond to an open model (decelerated expansion).

- The curve C4 relative to the case (LAMBDA > 0) and (k = 0 or k < 0), or to the case (LAMBDA > LAMBDA_F) and (k > 0) correspond to an open model with inflection point I (decelerated expansion followed by accelerated expansion). The sub-case (LAMBDA & gt; 0) and (k = 0) corresponds to the Standard cosmological model when the pressure is zero (w = 0).

- The curves C5, and again C1, are related to the case (0 < LAMBDA < LAMBDA_F) and (k > 0). They correspond to two possible behaviors : an open model of non-Big Bang type (decelerated contraction followed by accelerated expansion after the minimum point M2), and a closed model with a maximum point M1.

- The curves C3, and again C4, are related to the

Note these curves represent a subset of curves listed by Harrison [HAR "Classification"].

LAMBDA_F is the

LAMBDA_F = 3 (k/m)

n = 2/(1 + 3 w) > 0

m = n + 1

LAMBDA_F is linked to the singular scale factor aF as follows :

LAMBDA_F = 3 (k/m) aF

aF

By expressing the constant A at the singular point 0 = F, we obtain the expressions of

LAMBDA_E = (1/n) rhoE KHI c

aE

Below are some particularly simple solutions for a(t) (index 0 generally corresponding to data to date).

Apart from the first two solutions, the others are almost all

**1. Einstein static universe**

It is the **static cosmological model** with : a(t) = aE ; rho(t) = rhoE ; p(t) = pE

where aE, rhoE and pE are constants.

The second Friedmann equation (F2) then becomes : LAMBDA = LAMBDA_E

where : LAMBDA_E = (1/2)(rhoE + 3 pE / c^{2}) KHI c^{2}

LAMBDA_E is the **singular cosmological constant of Einstein** which characterizes a static universe.

Note that outside a vacuum (rhoE = pE = 0), a static solution can exist only with a non-zero cosmological constant.

By replacing this value of LAMBDA in the first Friedmann equation (F1), we find :

k / aE^{2} = (1/2)(rhoE + pE / c^{2}) KHI c^{2}

If the cosmic fluid satisfies the **strict low energy condition** then : rhoE + pE / c^{2} > 0 and therefore necessarily : k > 0, so : k = 1

The curve a(t) is thus a constant (see curve C3 in Figure 1 above) :

a(t) = aE = ( (1/2)(rhoE + pE/c^{2}) KHI c^{2} )^{-1/2}

**2. De Sitter Space-time**

It is the **cosmological model of the vacuum (rho = p = 0) with LAMBDA > 0 and k = 0 (flat curvature)**.

The first Friedmann equation (F1) then becomes : (a'/ a) ^{2} = (H0)^{2}

with H0 = B^{1/2} = c (LAMBDA / 3)^{1/2}

This equation is integrated into :

a(t) = a0 e^{H0 (t - t0)}

where a0 and t0 are constants.

The curve a(t) is of exponential type.

**3. Friedmann model with open curvature**

It is the **cosmological model without pressure (w = 0) with LAMBDA = 0 and k = -1 (open curvature)**

By replacing these values in differential equation (Q1), we find :

a'^{2} = A a^{-1} + c^{2}

where A = (1/3) rho0 (a0)^{3} KHI c^{4}

This equation is integrated in the form of a parametric equation :

a(t) = D (cosh[m] - 1)

t - ti = (D/c) (sinh[m] - m)

with D = (1/2) A c^{-2} and parameter m > 0

where a0, rho0 and ti are constants.

The term (t - ti) expressed more simply as a function of (a) in the form :

t - ti = (D/c) ( ((a/D)(2 + (a/D)))^{1/2} - ln[ (1 + (a/D)) + ((a/D)(2 + (a/D)))^{1/2} ] )

The curve a(t) is of hyperbolic type (see Figure 2 above for k = -1).

The constant ti is generally set to 0 by an original choice of the coordinate t.

**4. Friedmann model with flat curvature (or Einstein-De Sitter Space-time)**

It is the **cosmological model without pressure (w = 0) with LAMBDA = 0 and k = 0 (flat curvature)**

By replacing these values in differential equation (Q1), we find :

a'^{2} = A a^{-1}

This equation is integrated into :

a(t) = a0 ( (t - ti)/T0 )^{2/3}

with T0 = ( (9/4) A a0^{-3} )^{-1/2} = ( (3/4) rho0 KHI c^{4} )^{-1/2}

where a0, rho0 and ti are constants.

The curve a(t) is of parabolic type (see Figure 2 above for k = 0).

The constant ti is generally set to 0 by an original choice of the coordinate t.

**5. Friedmann model with closed curvature**

It is the **cosmological model without pressure (w = 0) with LAMBDA = 0 and k = 1 (closed curvature)**

By replacing these values in differential equation (Q1), we find :

a'^{2} = A a^{-1} - c^{2}

with A = (1/3) rho0 (a0)^{3} KHI c^{4}

This equation is integrated in the form of a parametric equation :

a(t) = D (1 - cos[m])

t - ti = (D/c) (m - sin[m])

with D = (1/2) A c^{-2} and parameter m varying from 0 to 2 Pi

where a0, rho0 and ti are constants.

The term (t - ti) expressed more simply as a function of (a) in the form :

For t - ti < Pi (D/c) : t - ti = (D/c) ( Arccos[1 - (a/D)] - ((a/D)(2 - (a/D)))^{1/2} )

For t - ti > Pi (D/c) : t - ti = 2 Pi (D/c) - (expression (t - ti) of the previous case)

The curve a(t) is a cycloïde (circle point rolling on a straight line). It is symmetrical relative to the value t - ti = Pi (D/c) (see Figure 2 above for k = 1).

Note that the curve goes from the "**Big Bang**" point (t - ti = 0) to the "**Big Crunch**" point (t - ti = 2 Pi (D/c)) through an expansion phase (a' > 0) and then a contraction phase (a' < 0).

The constant ti is generally set to 0 by an original choice of the coordinate t.

**6. Model without pressure (w = 0) with non-zero LAMBDA**

The exact solution of this model is given by [KHA "Some exact solutions"].

**7. Model without pressure (w = 0) with non-zero LAMBDA and k = 0 (flat curvature)**

By replacing these values in differential equation (Q1), we find :

a'^{2} = A a^{-1} + B a^{2}

with A = (1/3) rho0 (a0)^{3} KHI c^{4}

and B = (1/3) LAMBDA c^{2}

This equation is integrated into :

if LAMBDA < 0 : a(t) = (-A/B)^{1/3} sin^{2/3}[ (3/2) (-B)^{1/2} (t - ti) ]

if LAMBDA > 0 : a(t) = (A/B)^{1/3} sinh^{2/3}[ (3/2) B^{1/2} (t - ti) ]

where a0, rho0 and ti are constants.

If LAMBDA > 0, the curve a(t) have two successive expansion phases (a' > 0). The first phase is similar to the open curve of the Friedmann model (see Figure 2 above for k = -1) with deceleration (a" < 0) but leading to an inflection point I (a" = 0). The second phase is again an open curve but with acceleration (a" > 0) (see curve C4 in Figure 1 above).

If LAMBDA < 0, the curve a(t) is similar to the closed curve of the Friedmann model (see Figure 2 above for k = 1).

The constant ti is generally set to 0 by an original choice of the coordinate t.

**8. Model for electromagnetic radiation (w = 1/3) with non-zero LAMBDA**

The exact solution of this model is given by [KHA "Some exact solutions"].

**9. Model for electromagnetic radiation (w = 1/3) with LAMBDA = 0**

By replacing these values in differential equation (Q1), we find :

a'^{2} + k c^{2} = A* a^{-2}

with A* = (1/3) rho0 (a0)^{4} KHI c^{4}

This equation is integrated into :

For k = -1 : a(t) = E c ( (1 + (1/E)(t - ti))^{2} - 1 )^{1/2}

For k = 0 : a(t) = (4 A*)^{1/4} (t - ti)^{1/2}

For k = 1 : a(t) = E c ( 1 - (1 - (1/E)(t - ti))^{2} )^{1/2}

with E = (A*)^{1/2} c^{-2}

where a0, rho0 and ti are constants.

The curves a(t) are similar to the curves of the Friedmann model (see Figure 2 above for k = -1, 0 and 1).

The constant ti is generally set to 0 by an original choice of the coordinate t.

*
Proof of the general shape of the curves a(t) according to Friedmann equations :
Friedmann equations (F1) and (F2) are written in the form :
(Q1) (a')^{2} + k c^{2} = A a^{-(1 + 3 w)} + B a^{2}
(Q2) a"/a = -F a^{-3(1 + w)} + B
with A = (1/3) rho0 (a0)^{3(1 + w)} KHI c^{4}
B = (1/3) LAMBDA c^{2}
F = (1/2) (1 + 3 w) A
In the standard case where rho > 0 and w > (-1/3), we deduce from relations (Q1) and (Q2) that :
1. When a tends to 0, the relation (Q1) induces that the quantity (a') tends to the infinity corresponding to the primordial universe explosion (Big Bang theory).
2. When a tends to infinity, the relation (Q1) induces that :
(Q3) If LAMBDA is non-zero, B is non-zero and the quantity (a')^{2} behaves as the quantity (B a^{2}) only when B is positive.
(Q4) If LAMBDA is zero, the quantity (a')^{2} behaves like the quantity (-k c^{2}) only when k is negative or zero.
3. When a' is zero :
(Q5) the relation (Q1) is satisfied only for the following values combinaisons (LAMBDA, k, w, A) :
LAMBDA < 0
(LAMBDA = 0) and (k > 0)
(0 < LAMBDA < LAMBDA_F) and (k > 0)
(LAMBDA = LAMBDA_F) and (k > 0)
with : LAMBDA_F = 3 (k/m)^{m} ( (1/n) A c^{-2} )^{-n}
n = 2/(1 + 3 w) > 0
m = n + 1
We deduce the following results illustrated by the curves C1 to C5 in Figure 1 above :
4. If LAMBDA is negative :
4.1. The relation (Q2) induces that the quantity (a") is always negative. The evolution of a(t) is decelerated, with no inflection point (a" = 0).
4.2. B is negative. The relation (Q5) induces that a(t) reaches a maximum (a'= 0 ; point M1 on curve C1) for which : (-B) a^{3(1 + w)} + k c^{2} a^{(1 + 3 w)} - A = 0
5. If LAMBDA is zero :
5.1. The relation (Q2) induces that the quantity (a") is always negative. The evolution of a(t) is decelerated, with no inflection point (a" = 0).
5.2. The relation (Q4) induces that :
5.2.1. If k is negative or zero, a(t) tends to the straight line a(t) = c (-k)^{1/2} t when a tends to infinity (curve C2).
5.2.2. If k is positive, the relation (Q5) induces that and a(t) reaches a maximum (a' = 0 ; point M1 on curve C1) for which : a^{(1 + 3 w)} = (1/k) A c^{-2}
6. If LAMBDA is positive, B is positive :
6.1. The relation (Q2) induces that the quantity (a") is first negative (decelerated evolution) then becomes positive (accelerated evolution) after passing through an inflection point (a" = 0 ; point I on curve C4) for which : aI ^{3(1 + w)} = (F/B).
6.2. The relation (Q3) induces that a(t) tends to the exponential curve a(t) = exp [B^{1/2} t] when a tends to infinity.
6.3. Singular case : when LAMBDA equals LAMBDA_F, with positive k, the relation (Q5) induces that the curve a(t) has a point with horizontal tangent (a' = 0) that coincides with the inflection point I. This model has two types of possible behavior : a static model (Einstein static universe) for which a(t) = constant (curve C3), and an open model with an inflection point for which a' = a" = 0 at the point aI = aF (curve C4).
6.4. When LAMBDA is less than LAMBDA_F, with positive k, the relation (Q5) induces that the curve a(t) has two extremums (a' = 0 ; points M1 and M2). This model has two types of possible behavior : a closed model (a" < 0) with a maximum point in M1 (curve C1), and an open model (a" > 0) with a minimum point in M2 (curve C5), the respective inflection points I1 and I2 being fictitious and rejected in the forbidden band (a1 < a < a2). Note the open model is not a Big Bang model.
*

In the case of

g00 = -1

g11 = a

g22 = a

g33 = a

gij = 0 for i and j taken different between 0 and 3

where k is a constant (0, 1 or -1) and a is a function of t only.

The gravitational potentials g

where DELTA is the Kronecker symbol.

g

g

g

g

g

The Christoffel symbols GAMMA

GAMMA

GAMMA

GAMMA

GAMMA

where a' = d(a)/dt

The other Christoffel symbols are all zero.

The Rij components of Ricci tenseur are then written by the relations : Rij = R

R00 = -3 a" (1/a)(1/c

R11 = (a a" + 2 a'

R22 = (a a" + 2 a'

R33 = sin

The other components Rij are all zero.

The Ricci curvature is then written by the relation : R = g

R = 6 (1/c

The Einstein tensor is then producted by the relation :

E00 = R00 + (R/2) - LAMBDA

E11 = ( (2b + a"/a)/c

E22 = E11 r

E33 = E22 sin

The other components Eij are all zero.

For a Perfect Fluid of density rho and pressure p, the Energy-Pulse Tensor is then producted by the relation : Tij = (c

The hypothesis of spatial isotropy induces that the observer is co-mobile with the fluid.

The hypothesis of spatial homogeneity also induces that rho and p are quantities function of t only.

The expression of Tij are written :

T00 = rho c

T11 = p a

T22 = T11 r

T33 = T22 sin

The other components Tij are all zero.

The Einstein equations are then written by the relation :

E00 = KHI T00

E11 = KHI T11

E22 = KHI T22

E33 = KHI T33

Eij = KHI Tij for i and j taken different between 0 and 3

The Einstein equations then reduce to the 2 following equations :

b = (1/3) rho KHI c

(1/2) b + a"/a = (1/2) LAMBDA c

By replacing the first equation in the second one, we find

(a'/a)

a"/a = -(1/6) (rho + 3 p/c

Deriving the first equation relative to t and replacing a" in the second one, we find the following simple relation :

d(rho)/dt = -3 (a'/a)(rho + p/c

In the case where the

d(rho)/(rho) = -3 (1 + w)(da/a)

which integrates into :

rho(t) = rho0 (a0 / a(t))

where rho0 and a0 are two constants (index 0 generally corresponding to current data).

By replacing this expression of rho(t) into the first Friedmann equation (F1), we find a differential equation that is a function of a(t) only :

(a')

with A = (1/3) rho0 (a0)

and B = (1/3) LAMBDA c

General Relativity successfully explains three types of fundamental spectral shifts [AND Theory - Part 2] :

The **Doppler-Fizeau effect** which induces a spectral shift due to a **speed effect** of the light source relative to the observer.

This shift is directed indifferently towards blue or red depending on whether speed is an approach speed or distance speed but whose transverse effect is always directed towards red.

The **Einstein effect** which induces a spectral shift of gravitational origin due to the **effect of a mass** close to the source.

Radiation emitted in an intense gravitational field is observed with a shift that is always directed towards red.

The **Hubble law** which induces a cosmological spectral shift due to an **effect of distance** from the source.

This shift is always directed towards red.

To explain these very profound phenomena of physics, General Relativity has had to go through the successive generalizations of Space-time notion :

- **Euclidean space-time** to interpret the Doppler-Fizeau effect.

- **Curved space-time** to interpret the Einstein effect.

- **Space-time with variable curvature** to interpret the Hubble law.

Notions used in this chapter, **listed alphabetically** :

The aberration of light is the difference between the incidence directions of the same light ray perceived by two observers in relative motion.

In the case of a light source S1 seen by an observer S' in movement relative to S1 (velocity **V**), the light emanating from S1 appears to come from S2 and not from S1 (see Figure above).

In the case of rain falling vertically on the ground, the pedestrian who walks in the rain (velocity **V**) must tilt his umbrella forward if he does not wish to be wet.

Let S be an observer of a reference frame R and S' an observer of a reference frame R' in uniform translation of velocity **V** relative to R.

**u** is the unit vector of the propagation **SS'**.

If the propagation **u** makes with the velocity **V** an angle theta in R and theta' in R', then we have the relation :

cos[theta'] = (cos[theta] - V/c) / (1 - cos[theta] V/c)

Using the relation : tan^{2}[theta/2] = (1 - cos[theta])/(1 + cos[theta]), we have the equivalent relation :

tan[theta'/2] = ( (1 + V/c)/( 1 - V/c) )^{1/2} tan[theta/2]

**So we always have : theta' > theta, as if the light received by the mobile observer concentrated on its movement direction**.

When the propagation **u** is parallel to the velocity **V** in the reference frame R (theta = 0 or Pi), then the formula reduces to : cos[theta'] = 1 or -1, which induces : theta' = 0 or Pi, and there is no aberration effect.

When **u** is perpendicular to **V** in the reference frame R (theta = Pi/2), then the formula reduces to : cos[theta'] = -V/c, which induces : theta' > Pi/2 (and the pedestrian must tilt his umbrella forward).

When V is small relative to c, there is no aberration effect (theta' = theta).

For a vector space of dimension n having for base vectors the set (**e1**, **e2**... **en**), the Christoffel symbols GAMMA^{i}jk represent the basic vectors evolution as a function of their partial derivative.

Using the Convention of partial derivative and the Convention of summation, this is written : **ej**,k = GAMMA^{i}jk **ei**

GAMMA is symmetric relative to the lower index : GAMMA^{i}jk = GAMMA^{i}kj

GAMMA can be written as a function of the components gij of the Metric tensor :

GAMMA^{i}jk = (1/2) g^{il} (glk,j + glj,k - gjk,l)

*Proof :
By deriving gij = ei.ej relative to x^{k}, we find :
gij,k = (ei,k).ej + ei.(ej,k) = (GAMMA^{l}ik el).ej + ei.(GAMMA^{l}jk el)
This is written :
gij,k = GAMMA^{l}ik glj + GAMMA^{l}jk gil
A circular permutation of the three indices i, j, k then gives the following two equalities :
gki,j = GAMMA^{l}kj gli + GAMMA^{l}ij gkl
gjk,i = GAMMA^{l}ji glk + GAMMA^{l}ki gjl
We then find by linear combination :
gij,k + gki,j - gjk,i = 2 GAMMA^{l}kj gil
By multiplying the two members by g^{mi} and using the relation g^{mi} gil = DELTA^{m}l, we find :
GAMMA^{m}kj = (1/2) g^{mi} (gij,k + gki,j - gjk,i)
By renaming the indices (i in l and m in i), we finally find :
GAMMA^{i}jk = (1/2) g^{il} (glk,j + glj,k - gjk,l)
*

The contraction operation of the index of a mixed component of a tensor consists in choosing two indices, one covariant and the other contra-variant, then in equalizing and summing them relative to this twice repeated index.

For example, for a tensor **U** of order three whose mixed components are u^{ij}k, we find : w^{i} = u^{ik}k = u^{i1}1 + u^{i2}2 + ... u^{in}n

The quantities w^{i} (**contracted components** of the tensor **U**) form the components of a tensor **W** of order one.

Note that the "matrix product" operator is a particular case of the tensor product **U ^{i}j * V^{k}l** contracted in the form : w

In order to lighten the expressions of the derivatives of functions dependent on n variables f(x^{1}, x^{2}... x^{n}), we denote the partial derivatives in the following forms :

dk(f) = d(f)/d(x^{k}) = f,k

djk(f) = d^{2}(f)/(dx^{j} dx^{k})

For a vector space of dimension n having as its basis vectors the set (**e1**, **e2**... **en**), any vector **x** of this space can be written : **x** = x^{1} **e1** + x^{2} **e2** + ... + x^{n} **en** = Sum_for_k_ranging_from_1_to_n [x^{k} **ek**]

In order to simplify this writing we use a notation convention consisting in deleting the symbol "Sum" which is written in condensed form : **x** = x^{k} **ek** where the index k (called **mute index**) always varies from 1 to n.

The summation is done on the index provided that they are **repeated respectively up and down in the same monomial term**.

When the **prime symbol** is used to distinguish two distinct bases of the same vector space, we can further simplify the notation by placing the prime symbol on the index rather than on the vector: **x** = x'^{k} **e'k** = x^{k'} **ek'**

Some terms in a sum may have several indices. For example, in the sum a^{k}m b^{m}, the summation is done on the index m. The index k (called **free index**) characterizes a particular term.

For example the equation ck = a^{k}m b^{m} for n = 3 represents the system of equations :

c1 = a^{1}1 b^{1} + a^{1}2 b^{2} + a^{1}3 b^{3}

c2 = a^{2}1 b^{1} + a^{2}2 b^{2} + a^{2}3 b^{3}

c3 = a^{3}1 b^{1} + a^{3}2 b^{2} + a^{3}3 b^{3}

There is no summation here on the index k which is found alone in the same monomial term.

When the monomial term has **several mute index** the summation takes place simultaneously on all these indices. For example, a^{k}m b^{m} ck for n = 4 represents a sum of 16 terms :

a^{k}m b^{m} ck = a^{1}1 b^{1} c1 + a^{1}2 b^{2} c1 + a^{1}3 b^{3} c1 + a^{1}4 b^{4} c1 + ... + a^{2}1 b^{1} c2 + ... + a^{4}4 b^{4} c4

For a vector space of dimension n having for base vectors the set (**e1**, **e2**... **en**), we call **contra-variant components** of a vector **x** the numbers x^{i} such that : **x** = x^{i} **ei**, and **covariant components** the numbers xj such that : xj = **x.ej** (see Figure above).

The contra-variant (respectively covariant) name derives from the fact that these components are transformed by base changing in a contrary (respectively identical) manner to that of the basic vectors.

The **contra-variant components** are noted with **higher indices**.

The **covariant components** are noted with **lower indices**.

We have the following relations :

xj = x^{i} gij

x^{i} = xj g^{ij}

**x.y** = g^{ij} xi yi

When index vary from 0 to 3, Greek letters (such as alpha or beta) are often used rather than Latin letters (such as i or j).

Note that for orthonormal base there is no difference between the covariant and contra-variant components of a Tensor.

For each tensor **U** of order 2 of components u^{ij}, its covariant derivative **Grad(U)** is the tensor of order 3 of the following components :

u^{ij};k = u^{ij},k + u^{lj} GAMMA^{i}lk + u^{il} GAMMA^{j}lk

where GAMMA are the Christoffel symbols.

The curvature tensor is a symmetric tensor of order four. Using the Convention of partial derivative, its componants have as expression :

R^{i}jkl = GAMMA^{i}jl,k - GAMMA^{i}jk,l + GAMMA^{i}mk GAMMA^{m}jl - GAMMA^{i}ml GAMMA^{m}jk

where GAMMA are the Christoffel symbols.

In Cartesian coordinates, all the components are of dimension m^{-2}

This tensor has the following properties :

Antisymmetry : R^{i}jkl = -R^{i}jlk

Swapping indices alone : Rijkl = -Rjikl = -Rijlk

Swapping of indices two by two : Rijkl = Rklij

For each tensor **U** of order 2 of components u^{ij}, its divergence **Div(U)** is the tensor of order 1 producted by contracting one of the index of the Covariante derivative with the derivation index. Its components are as follows :

u^{ij};j = u^{ij},j + u^{lj} GAMMA^{i}lj + u^{il} GAMMA^{j}lj

where GAMMA are the Christoffel symbols.

The Doppler effect is the frequency change of a periodic phenomenon induced by the movement of the emitter relative to the receiver. In the case of sound waves, for example, the sound emitted by an approaching car is sharper than the sound emitted when it moves away.

Let us take the general case in Restricted Relativity of a light wave propagating at the wave speed c.

If f is the frequency of the wave perceived by an observer S of a reference frame R, then any observer S' of the reference frame R' in uniform translation of velocity **V** relative to R will perceive this same wave at the following frequency f'.

**u** is the unit vector of the propagation **SS'** (see Figure above named "Aberration").

(E1) Longitudinal Doppler effect (**u** parallel to **V**) :

f' = f gamma (1 - (**V.u**)/c)

When V is small relative to c, we find the approximate non-relativistic formulas :

f' = f (1 - (**Vr.u**)/c) for mobile receiver (velocity **Vr**) and immobile emitter relative to the propagation medium

f' = f / (1 - (**Ve.u**)/c) for immobile receiver and mobile emitter (velocity **Ve** = **-V**) relative to the propagation medium

f' = f (1 - (**Vr.u**)/c) / (1 - (**Ve.u**)/c) for mobile receiver (velocity **Vr**) and mobile emitter (velocity **Ve**) relative to the propagation medium (relative velocity **Vr - Ve** = **V**).

(E2) Transverse Doppler effect at the emission (**u** perpendicular to **V** dans R) :

f' = f gamma

(E3) Transverse Doppler effect at the reception (**u** perpendicular to **V** dans R') :

f' = f gamma^{-1}

(E4) Doppler effect (general formula) :

If the light propagation **u** makes with the velocity **V** an angle theta in R and theta' in R', then we have the relation :

f' = f gamma (1 - cos[theta] V/c) = f gamma^{-1} (1 + cos[theta'] V/c)^{-1}

the relation between the angles theta and theta' being given by the Aberration formula.

For theta = theta' = 0° or 180°, we find the formula (E1) with shift directed towards red or blue according to whether the observer of R' moves away or approaches the light source of R.

For theta = 90°, we find the formula (E2) with shift directed towards blue.

For theta' = 90°, we find the formula (E3) with shift directed towards red.

When V is small relative to c, we find the approximate non-relativistic formula :

f' = f (1 - (**Vr.u**)/c) / (1 - (**Ve.u**)/c) for mobile receiver (velocity **Vr**) and mobile emitter (velocity **Ve**) relative to the propagation medium (relative velocity **Vr - Ve** = **V**).

*Partial proof [ANN Electricité 2] :
*

A frequency produced by a source in a gravitational field is decreased (red shifted) when it is observed from a place where gravity is less. This is a pure General Relativity effect and not a shift by Doppler effect.

By using the Schwarzschild metric centered on a massive mass (mass M) with spherical symmetry, and in the particular case of a zero cosmological constant and a gravitational field in vacuum, the observed frequency f' at the radial distance r' is a function of the produced frequency f at the radial distance r according to the law :

f' = f ( (1 - psi/r)/(1 - psi/r') ) ^{1/2}

where psi is the **gravitational radius** (psi = 2 G M/c^{2}).

When the observer is situated in a place of gravitation less than the source place (r' > r), we find (f' < f) corresponding to the observation of a shift directed towards red.

The Einstein tensor (**Eab**) measures the local deformation of the chrono-geometry of Space-time and represents its curvature at a given point. It is a tensor of order two, symmetric and with zero Divergence (**E ^{ab};a** =

Its components are given by Einstein equations.

In Cartesian coordinates, all the components are of dimension m

The notations are those in Maxwell Equations.

In Restricted Relativity, the Lorentz force (**F_LORENTZ** = q **E** + **v** x **B**) is written in a tensor form whose components are the following : Fi_LORENTZ = q Fij (v^{j} /c)

**Fij** is the electromagnetic tensor. It is a tensor of order 2.

In Cartesian coordinates, all the components are of dimension m^{-1}.V ou C^{-1}.N ou m.kg.s^{-3}.A^{-1} and are written :

Fii = 0 for i = 0 or i > 0

Fi0 = -F0i = E^{i} for i > 0

F21 = -F12 = -c B^{3}

F31 = -F13 = c B^{2}

F32 = -F23 = -c B^{1}

E^{i} and B^{i} are respectively the spatial components of the electric field **E** and magnetic field **B**.

By increasing of index (see Tensor operators), we find the components of the tensors **F ^{i}j** and

F

F

F

F

F

F

F

F

F

F

F

F

The Energy-impulse tensor (**Tab**) can take very varied forms depending on the distribution of matter or energy. For example : the tensor of the perfect fluid or that of electromagnetism.

Its components have the following meaning :

T00 : energy density or pressure or c^{2} times the density

T0j for j > 0 : (-c) times the component i of the relativistic impulse density (momentum impulse density) or (-1/c) times the component i of the energy flow also called **Poynting vector** (**PHI**)

Tij for i and j > 0 : spatial components of the **stress tensor** (**Sij**)

It is a tensor of order two, symmetric and constructed so that its zero Divergence (**T ^{ab};a** =

In Cartesian coordinates, all the components are of dimension m

The notations are those in Maxwell Equations.

The components of the Energy-impulse tensor (**Tab_EM**) of ElectroMagnetic field are the following :

Tij_EM = eps0 (Fim F^{m}j - (1/4) gij Fkl F^{kl})

where :

**Fij** is the Electromagnetic tensor.

With a Minkowski metric, the calculations give in Cartesian coordinates :

T00_EM = energy density = (1/2) eps0 (**E.E** + c^{2} **B.B**)

Ti0_EM = T0i for i > 0 corresponding to (-1/c) times **PHI** with **PHI** = (1/ mu0) **E x B**

Tij_EM for i and j > 0 corresponding to Sij = eps0 ( (1/2) (**E.E** + c^{2} **B.B**) DELTAij - (Ei Ej + c^{2} Bi Bj) )

where DELTA is the Kronecker symbol.

A fluid is called "perfect" when the viscosity and thermal conduction effects can be neglected, which is the case in cosmology.

The components of the Energy-impulse tensor (**Tab_FP**) of Perfect Fluid are the following :

Tij_PF = (c^{2} rho + p)(vi /c)(vj /c) + p gij

where :

c^{2} rho and p represent respectively the energy density and the pressure of the fluid, both measured in the reference frame of the fluid.

**v** is the field which represents at each point the quadri-velocity of a fluid particle.

With a Minkowski metric and when the observer is **co-mobile** with the fluid, the calculations give in Cartesian coordinates :

T00_PF = rho c^{2}

Ti0_PF = T0i for i > 0 = 0

Tij_PF for i and j > 0 corresponding to Sij = p DELTAij

where DELTA is the Kronecker symbol.

The Perfect Fluid satisfies the **low energy condition** when : (rho = 0 or > 0) and (rho c^{2} = -p or > -p), and the **dominant energy condition** when : (rho c^{2} = |p| or > |p|).

See Solution of Einsteins equations with Friedmann-Lemaitre-Robertson-Walker metric

The Friedmann-Lemaitre-Robertson-Walker metric is a Relativistic metric corresponding to a spatially homogeneous and isotropic Space-time.

In spherical coordinates (r > 0, theta = [0, Pi], phi =[0, 2 Pi]) this metric is written by taking the sign convention (- + + +) :

ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dr^{2} (1 - k r^{2})^{-1} + r^{2} (d(theta)^{2} + sin^{2}[theta] d(phi)^{2}) )

where k is a constant called **space curvature parameter** that can be flat (k = 0), closed (k = 1) or open (k = -1) ;

and a(t) is a function of t only, called **scale factor or universe radius** (a(t) > 0).

The coordinate r is dimensionless and the radius (a) has the dimension of a length.

The gravitational potentials gij then are the following :

g00 = -1 ; g11 = a(t)^{2} (1 - k r^{2})^{-1} ; g22 = a(t)^{2} r^{2} ; g33 = a(t)^{2} r^{2} sin^{2}[theta] ; gij = 0 for i and j taken different between 0 and 3

The sign of d(a)/dt informs about the universe evolution : positive if expansion, negative if contraction and zero if static.

The coordinates (x^{i}) then describe spatial hypersurfaces of Euclidean type (for k = 0), spherical or elliptical type (for k = 1) and hyperbolical type (for k = -1).

For k = 0 we find the Minkowski metric : ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dr^{2} + r^{2} (d(theta)^{2} + sin^{2}[theta] d(phi)^{2}) )

*Proof [GOUR Relativité Générale] :
A spatially homogeneous and isotropic space-time is equivalent to a maximally symmetric space of dimension 3 (or spatially constant curvature k*) with three possible types of maximally symmetric spaces according to the value of k* (not proofed here):
If k* = 0, space is the Euclidian space R^{3} of metric :
ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dr^{2} + r^{2} (d(theta)^{2} + sin^{2}[theta] d(phi)^{2}) )
If k* > 0, space is the hypersphere S^{3} of metric :
ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dKsi^{2} + sin^{2}[Ksi] (d(theta)^{2} + sin^{2}[theta] d(phi)^{2}) )
If k* < 0, space is the hyperbolic space H^{3} of metric :
ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dRho^{2} + sinh^{2}[Rho] (d(theta)^{2} + sin^{2}[theta] d(phi)^{2}) )
with Ksi = [0, Pi] and Rho > 0
By setting r = sin[Ksi] = sinh[Rho], these three metrics are written in a common form :
ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dr^{2} (1 - k r^{2})^{-1} + r^{2} (d(theta)^{2} + sin^{2}[theta] d(phi)^{2}) )
with k = 0 for Euclidean space, k = 1 for the hypersphere and k = -1 for hyperbolic space.
Note that the curvature k* is then : k* = 6 k a(t)^{-2}
*

Geodesics describe the movement of free particles, that is when they are not subjected to external force (other than gravitation in the context of General Relativity).

For a given Metric a geodesic is the curve (or trajectory) of the shortest distance between two given points.

The motion of material systems and photons in space-time is described by the geodesic tensor equations. With the Relativistic metric, they are written :

(d^{2}x^{i} / ds^{2}) + GAMMA^{i}lk (dx^{k}/ds) (dx^{l}/ds) = 0

where GAMMA are the Christoffel symbols.

In Restricted Relativity (Minkowski metric) with Cartesian coordinates, the coefficients gij are constant, which cancels all the Christoffel symbols. The equations of the geodesics are reduced to : d^{2}x^{i} / ds^{2} = 0 whose solutions are the ordinary straight lines : x^{i}(s) = a^{i}(s) s + b^{i}

Hubble law states that galaxies move away from each other at an expansion speed v approximately proportional to their distance d :

v = H(t) d

where H(t) is the **Hubble parameter** used in particular in the Friedmann equations.

The value to date of H(t) (called **Hubble constant H0**) is about 70 (km/s)/Mpc, with 1 pc = 1 parsec = 3,2616 light-years = 3,085677581 10^{16} m

The speed v is not a physical speed. It only reflects the Space-time expansion which causes an global movement of the universe galaxies. The Earth thus retreats before the light because the space-time expands.

Any distant galaxy having the same proper Time as the observer (called **cosmic time**), there is no relative time effect (Doppler-Fizeau effect) on its radiation period but a simple differential delay effect on the radiation period received.

The own movements acquired by the galaxies superimpose to this global movement because of their gravitational interactions with their neighbors.

The expression of the Kronecker symbol DELTA is as follows :

DELTA^{i}k = DELTAik = DELTA^{ik} = 1 for i = k and 0 otherwise.

The expression of the Levi-Civita symbol EPSILON is as follows :

EPSILONijkl... = EPSILON^{ijkl...} =

0 if two ou more indices (i,j,k,l...) are equal

+1 if (i,j,k,l...) is an even permutation of (1,2,3,4...)

-1 if (i,j,k,l...) is an odd permutation of (1,2,3,4...)

When any two indices are interchanged, equal or not, the symbol is negated :

EPSILON...i...l... = -EPSILON...l...i...

For 3 indices (i,j,k) we have :

EPSILONijk = +1 for 123 or 231 or 312

EPSILONijk = -1 for 132 or 213 or 321

For 4 indices (i,j,k,l) we have :

EPSILONijkl = +1 for 1234 or 1342 or 1423 or 2143 ... or 4321

EPSILONijkl = -1 for 1243 or 1324 or 1432 or 2134 ... or 4312

EPSILON allows in particular to express many vectorial operations in a compact form :

- Vector product (**w** = **u** x **v**) of components : wi = EPSILON^{ijk} uj vk

- Curl (**w** = curl(**u**)) of components : wi = EPSILON^{ijk} uk,j

- Determinant (d = det(**u**,**v**,**w**)) of component : d = EPSILON^{ijk} ui vj wk

See Lorentz-Poincare transformation

Any particle of charge q and velocity **v**, subjected to an electric field **E** and to a magnetic field **B**, undergoes the Lorentz force **F_LORENTZ** = q **E** + **v** x **B**

Maxwell equations specify the evolution of electromagnetic fields **E** and **B**. In vacuum they writte as follows :

div(**E**) = rho / eps0

curl(**E**) = - d**B**/dt

div(**B**) = 0

curl(**B** / mu0) = **j** + eps0 d**E**/dt

the two densities rho and **j** being connected by the relation of the conservation of the charges : div(**j**) + d(rho)/dt = 0

**E** is the electric field (in m^{-1}.V or C^{-1}.N or m.kg.s^{-3}.A^{-1})

**B** is the magnetic field (in T or kg.s^{-2}.A^{-1})

**j** is the electric power density (in m^{-2}.A)

**v** is the particle velocity (in m.s^{-1})

q is the electric charge (in C or s.A)

rho is the electric charge density (in m^{-3}.s.A)

mu0 is the permeability of vacuum : mu0 = 4 Pi 10^{-7} m.kg.A^{-2}.s^{-2}

eps0 is the dielectric permittivity in vacuum : eps0 = 1 / (mu0 c^{2})

c is the light speed in vacuum (c = 2,99792458 10^{8} m.s^{-1}

The vector operators used are the following :

**v1**.**v2** and **v1** x **v2** : scalar product and vector product of any two vectors **v1** and **v2**.

div(**v**) and curl(**v**) : divergence and curl of any vector **v**.

It can be proved (arduously) that these equations are invariant relative to the Lorentz-Poincare equations.

For a vector space of dimension n having for base vectors the set (**e1**, **e2**... **en**), we denote the Scalar product of two basic vectors in the form : gij = **ei.ej**

The Metric tensor is the tensor **gij** whose components are gij. It is a tensor of order two, symmetric and with zero Divergence (**g ^{ab};a** =

In Cartesian coordinates, all the components are dimensionless.

The

where DELTA is the Kronecker symbol.

The following results can be proved :

g

gij;k = g

gij g

g

The Minkowski metric is a Relativistic metric corresponding to the flat space-time of Restricted Relativity. The coordinates are the following taking the convention of sign (- + + +) :

In Cartesian coordinates : ds^{2} = -c^{2}dt^{2} + dx^{2} + dy^{2} + dz^{2}

corresponding to the gravitational potentials gij such that : g00 = -1 ; g11 = 1 ; g22 = 1 ; g33 = 1 ; gij = 0 for i and j taken different between 0 and 3.

This metric gij_MINK has the following properties :

g^{ij} = g^{ji} = gij = gji

g^{i}j = DELTA^{i}j

where DELTA is the Kronecker symbol.

In spherical coordinates : ds^{2} = -c^{2}dt^{2} + dr^{2} + r^{2} dtheta^{2} + r^{2} sin^{2}[theta] dphi^{2}

corresponding to the gravitational potentials gij such that : g00 = -1 ; g11 = 1 ; g22 = r^{2} ; g33 = r^{2} sin^{2}[theta] ; gij = 0 for i and j taken different between 0 and 3.

If (ds) is the distance (or interval) between two events infinitely close to the Space-time, then the Relativistic metric is the square of this distance and is written : ds^{2} = gij dx^{i} dx^{j}

In the curved space-time of General Relativity this metric is written in Cartesian coordinates :

ds^{2} = g00 (c dt)^{2} + g01 (c dt) dx + g02 (c dt) dy + g03 (c dt) dz +

g10 dx (c dt) + g11 dx^{2} + g12 dx dy + g13 dx dz +

g20 dy (c dt) + g21 dy dx + g22 dy^{2} + g23 dy dz +

g30 dz (c dt) + g31 dz dx + g32 dz dy + g33 dz^{2}

The coefficients gij are the components of the Metric tensor.

The Ricci curvature is a number (R) of dimension m^{-2} producted by Contraction of the Ricci tensor in the form :

R = g^{ij} Rij = R^{i}i

The Ricci tensor (**Rab**) is a symmetric tensor of order two producted by Contraction of the Curvature tensor on the first and third index.

Using the Convention of partial derivative, its components are the following :

Rij = R^{k}ikj = GAMMA^{k}ij,k - GAMMA^{k}ik,j + GAMMA^{k}kl GAMMA^{l}ij - GAMMA^{k}jl GAMMA^{l}ik

where GAMMA are the Christoffel symbols.

In Cartesian coordinates, all the components are of dimension m^{-2}

In the particular case where the cosmological constant (LAMBDA) and the Energy-impulse tensor are both zero, then the Ricci tensor is also zero.

*Proof :
By contracting the Einstein equations by the inverse Metric tensor g^{ab}, the Ricci curvature is related to the Energy-impulse tensor by the relation : R = - KHI T^{a}a + 4 LAMBDA
By replacing this relation in the Einstein equations, we find the following equivalent equations :
Rab = KHI (Tab - (1/2) gab T^{a}a) + LAMBDA gab
The tensor Rab is zero when LAMBDA = 0 and Tab = 0
*

The scalar product of two arbitrary vectors **x** and **y** is written:
**x.y** = x^{i} yi = xi y^{i} = g^{ij} xi yj = gij x^{i} y^{i}

where the coefficients gij are the components of the Metric tensor.

The norm ||**x**|| of any vector **x** is the square root of the absolute value of the scalar product of **x** by itself :

||**x**|| = (|**x.x**|)^{1/2}

The Schwarzschild metric is a Relativistic metric corresponding to the **static gravitational field with central symmetry**. This is the case of the Sun and many stars. The central body is spherically symmetrical and not necessarily static (for example, a pulsating star that oscillates radially or a star that collapses into a black hole while maintaining its spherical symmetry). The gravitational field must be static even if it is not static in the area where the matter is located. Note that the gravitational field is necessarily static in spherical symmetry and **in vacuum** (Birkhoff theorem).

In spherical coordinates (r > 0, theta = [0, Pi], phi =[0, 2 Pi]) this metric is written by taking the sign convention (- + + +) :

ds^{2} = -e^{2 nu} c^{2}dt^{2} + e^{2 alpha} dr^{2} + r^{2} d(theta)^{2} + r^{2} sin^{2}[theta] d(phi)^{2}

where nu and alpha are only functions of r.

The gravitational potentials gij then are the following :

g00 = -e^{2 nu} ; g11 = e^{2 alpha} ; g22 = r^{2} ; g33 = r^{2} sin^{2}[theta] ; gij = 0 for i and j taken different between 0 and 3

*Proof [GOUR Relativité Générale] :
The spherical center symmetry of the field allows to write the metric in the following form :
ds*

In Restricted Relativity, it is shown that two events located in different places can be **simultaneous** in one reference frame without being in another. The notion of simultaneity loses its universal character.

*Proof :
Let two simultaneous events (x1, y1, z1, t1) and (x2, y2, z2, t2 = t1) in the referential frame R. In the reference frame R' in uniform translation relative to R, the duration (t'2 - t'1) between these two same events is written taking into account the Lorentz-Poincare equations (L2) :
t'2 - t'1 = gamma ( (t2 - t1) - B (x2 - x1) ) = -gamma B (x2 - x1)
gamma and B being given by equations (L3) ans (L4).
When the two events are not located at the same points, the spatial difference (x2 - x1) in R is not zero. The temporal difference (t'2 - t'1) in R' is therefore not zero despite the simultaneity (t2 = t1) of the two events in R.
*

Space-time is a four-dimensional space where time is no longer a separate quantity independent of space but a variable playing the same role as spatial variables. The notion of Simultaneity is no longer universal.

In this space-time a **point** or **event** **x**(x, y, z, t) is identified by a four-dimensional vector **x** called **quadrivector** or **4-vector** whose components are denoted :

- In Cartesian coordinates : x^{0} = ct ; x^{1} = x ; x^{2} = y ; x^{3} = z

- In spherical coordinates : x^{0} = ct ; x^{1} = r ; x^{2} = theta ; x^{3} = phi

The cosmological model that best describes the history and behavior of the observable universe is the **standard model of cosmology** (or **Big Bang model** or **ACDM model** meaning "Lambda - Cold Dark Matter").

This model represents an universe (see curve C4 in Figure 1 above) :

- **spatially homogeneous and isotropic** on a large scale (thus also with **constant spatial curvature**). See Friedmann-Lemaitre-Robertson-Walker metric

- filled with a **perfect fluid** of **generally zero pressure** (galaxy gas corresponding to : w = 0) and density rho composed of **hot (relativistic or radiation) matter** and **cold (non-relativistic) matter**.

- whose **spatial curvature is zero** (k = 0).

- which would contain, in addition to the ordinary matter, **dark matter** (surplus gravity which the galaxies need not to be discarded during their rotation) and **black energy** (global repulsive force which tends to accelerate the universe expansion and requiring : LAMBDA > 0).

- coming from a primordial explosion such that the scale factor a(t) tends to 0 when t tends to 0 (**Big Bang model**).

The term "tensor" was introduced by the physicist W. Voigt to represent the tensions in a solid.

A tensor is a function of the coordinates of space, defined in an n-dimensional space by n^{m} components, where m is the **order** of the tensor. The tensor of order 0 is a scalar and has a single component. The tensor of order 1 is a vector with n components. The tensor of order 2 is a square matrix with n^{2} components.

Each tensor also has a **valence** or **type** denoted (h, q) where h is the number of contra-variant index (indicated in the upper position) and q the number of Covariant index (indicated in the lower position).

For any tensor **W**, its components can be **contra-variant** (example : **W ^{ijk}**),

The tensorial calculation has the advantage of being freed from all systems of coordinates and the results of the mathematical developments are thus invariant by change of reference frame.

Let **U**, **V** and **W** be Tensors of arbitrary order and valence bearing on the indices i, j, k, l...

Using the Convention of summation, the following operations are defined on these tensors :

- Sum (**W ^{i}jk = U^{i}jk + V^{i}jk**) of components : w

- Product by a scalar s (

- Scalar product (

- Tensor product (

- Covariant derivative

- Divergence

- index increasing :

A lower index can be changed to a higher index by multiplication with the inverse Metric tensor

u

u

- index lowering :

A higher index can be changed to a lower index by multiplication with the Metric tensor

uik = gij u

ulm = gjl gkm u

u

- Contraction of index

Each reference body has its proper time. This **proper time** (or true time) is the time measured in the reference frame where the body is immobile. The **apparent time** (or improper or observed or "measured" or local or relative time) conversely is the time measured in a mobile reference frame relative to this proper reference frame.

All measurements are made by fixed clocks in their reference frame and whose internal mechanism is generally insensitive to the reference frame movement.

In Restricted Relativity, for a given reference frame, the **proper duration** (d0) is the time interval that separates two events occurring at the same place in this reference frame. In any other reference frame, the duration is greater than the proper duration and is called **apparent duration** (d). Some authors speak of "**durations dilation**" or "**slowing down of moving clocks**".

It is more accurate to say that **the proper and apparent times are times measured under different conditions**.

*Proof of relation : d > d0 [ANN Electricité 2] :
Let two events occurring in the reference frame R at the same place of coordinates (x, y, z) but at different instants t1 and t2. The (proper) duration separating them is : d0 = t2 - t1.
For an observer of the referential frame R' in uniform translation at the speed V relative to R, the events occur at instants t1' and t2' given by the Lorentz-Poincare equation (L2) :
t1'= gamma (t1 - B x)
t2'= gamma (t2 - B x)
and separated by the (apparent) duration : d = t2' - t1' = gamma (t2 - t1) = d0 / (1 - V*

The universe age is the time elapsed since the Big Bang. The best approximation to date is given by : 1 / H0

where H0 is the Hubble constant (See Hubble law),

giving an age of about 13 billion years.

Any vector **v** of the Space-time is called :

- **time-like** when the Scalar product **v.v** < 0. This is the case of the vector tangent to the trajectory of a non-zero mass particle (trajectory called "**universe line**"). Two events of Space-time can be connected by information going at a speed lower than light speed.

- **light-like** (or **light vector** or **isotropic vector**) when the Scalar product **v.v** = 0 with **v** different from **0**. This is the case of the vector tangent to the trajectory of a zero mass particle (photon for example). Two events of Space-time can be connected by information going at the light speed.

- **space-like** when the Scalar product **v.v** > 0. This is the case of the vector neither time-like nor light-like. Two events of Space-time can not be connected by information going at the light speed.

The authors quoted in this chapter are referenced in square brackets under the reference [AUTHOR Title].

- ANDRILLAT H., La théorie de la relativité générale, Partie 1 : les fondements, Bulletin de l'Union des Physiciens, N°760 Janvier 1994.
- ANDRILLAT H., La théorie de la relativité générale, Partie 2 : la méthode, Bulletin de l'Union des Physiciens, N°762 Mars 1994.
- ANNEQUIN R. et BOUTIGNY J., Electricité 2, Cours de sciences physiques, Vuibert, 1974.
- GOURGOULHON E., Relativité restreinte - Des particules à l'astrophysique, EDP Sciences et CNRS Editions, 2010.
- GOURGOULHON E., Relativité générale, Observatoire de Paris, Universités Paris 6, 7 et 11, Ecole Normale Supérieure, cours UE FC5, 2013-2014.
- HARRISON E.R., Classification of uniform cosmological models, Monthly Notices of the Royal Astronomical Society, Vol. 137, p.69-79, 1967. Correspondence with notation of this page : nu = 1 + w ; R = a ; t = ct ; G = G c
^{-2}; Cv = A c^{-2}. - HLADIK J., Pour comprendre simplement la théorie de la Relativité, Ellipses, 2005.
- HLADIK J., Initiation à la Relativité restreinte et générale, Ellipses, 2013.
- KHARBEDIYA L.I., Some exact solutions of the Friedmann equations with the cosmological term, In Russian : Astron. Zh. Akad Nauk SSSR, Vol. 53, 1145-1152, 1976. English translation by R.B. Rodman in Soviet Astronomy of the American Institute of Physics, Vol. 20, N°6, p.647-650, Nov.-Dec. 1976. Correspondence with notations of this page : case (r or d) = w (1/3 or 0) ; R = a ; t = ct ; h = k ; A or B = 3 A c
^{-2}; mu = KHI ; rhor or rhod = rho0 c^{-2}; LAMBDAcr = LAMBDA_F. - LEVY-LEBLONG J.M., One more derivation of the Lorentz transformation, American Journal of Physics, Vol. 44, N°3, March 1976.
- POINCARE H., L'Etat actuel et l'Avenir de la physique mathématique, Bulletin des sciences mathématiques, Vol. 28, p. 302-324, 1904.

Copyright © 2005 Régis Petit. Last page update : April 23, 2017.