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This chapter gives the basic formulas of Restricted 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's Equations of Restricted Relativity. For many physicists, time is a **physical, real time**, measured by means of spatially located clocks capable of synchronizing by exchanging light signals.

- Unlawful application of the Lorentz-Poincare's 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 (light flash, laser ...) 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 is not in fact 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 JamesMaxwell** 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} and B = V / c^{2}**

where c is a constant (space-time structure constant) which is similar to a limiting speed and which appears during the demonstration 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's 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 demonstration of the Lorentz-Poincare's 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**

(Hc) **with V > 0 or V = 0**

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's 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's equations are written :

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

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

**Eab** is the Einstein's 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's equation of the Newtonian gravitation as a particular case of Einstein's 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's equations. The 16 gab components of this tensor are called **gravitational potentials**.

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

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

LAMBDA is the **cosmological constant** of dimension m^{-2}. It 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, is solved by taking LAMBDA = 0 and **Tab** = **0**. In cosmology the universe model 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.

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's 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's 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's 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's 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's 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's metric (S1, S2...) describes the geometry around the masses (M1, M2...).

The Minkowski's 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's 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's tensor (**Rab**) and then, by Contraction, the Ricci's curvature (R). See calculations detailed below.

In the particular case of a **zero cosmologic constant** (LAMBDA = 0) and a gravitational field **in vacuum** (when the Energie-impulse tensor (**Tab**) is zero), Einstein's 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's 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's 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's metric expressed in spherical coordinates. The space-time described by the Schwarzschild's 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's 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's 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's 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's 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's symbols are all zero.

The Rij components of Ricci's 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's 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's 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's 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's 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's 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's tensor (**Rab**) and then, by Contraction, the Ricci's 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's equations are then reduced to a system of two differential equations of the functions a(t), rho(t) and p(t), called **Friedmann's 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 **state equation of cosmic fluid** as p = p(rho). An example of a frequently used state equation is : p = w rho c^{2} where w is a constant that is equal to -1 (**quantum vacuum**), 0 (**zero pressure**) or 1/3 (**electromagnetic radiation**).

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

(a')^{2} + k c^{2} = A a^{-(1 + 3 w)} + B a^{2}

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

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

This differential equation is analytically integrated under certain conditions for w, k and LAMBDA, which completely determines a(t) and the metric ds2. See calculation detailed below.

For rho0 > 0 and w > -(1/3), note that the quantity a' tends to infinity when a tends to 0, corresponding to the primordial explosion of the universe (**Big Bang theory**).

Note some particularly simple solutions for a(t) :

**Einstein's Universe**

It is the **static cosmological model** with : a(t) = a0 ; rho(t) = rho0 ; p(t) = p0

where a0, rho0 and p0 are three constants.

The second Friedmann's equation (F2) then becomes : LAMBDA = (1/2) (rho0 + 3 p0 / c^{2}) KHI c^{2}

Note that outside a vacuum (rho0 = p0 = 0), a static solution can exist only with a non-zero cosmological constant (LAMBDA).

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

k / a0^{2} = (1/2) KHI c^{2} (rho0 + p0 / c^{2})

If the cosmic fluid satisfies the **strict low energy condition** then : rho0 + p0 / c^{2} > 0 and therefore necessarily : k > 0, so : k = 1

a(t) is thus determined as follows :

a(t) = a0 = ( (1/2) KHI c^{2} (rho0 + p0/c^{2}) )^{-1/2}

**De Sitter's Space-time**

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

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

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

This equation is integrated into :

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

where a0 is a constant.

** Einstein-De Sitter's Space-time**

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

If the cosmic fluid has a state equation of type : p = w rho c^{2}, then : w = 0 and rho(t) = rho0 (a0 / a(t))^{3} according to the calculation detailed below.

By replacing this value of rho(t) into the first Friedmann's equation (F1), we find :

a a'^{2} = (4/9) (a0)^{3} / (T0)^{2}

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

This equation is integrated into :

a(t) = a0 (t/T0 + b0)^{2/3}

where a0, rho0 and b0 are constants.

The constant b0 is generally set to 0 by an original choice of the coordinate t subject to adjusting the constants a0 and rho0 to a0 = a(t = T0) and rho0 = rho(t = T0).

** Einstein-De Sitter's Space-time with a non-zero cosmological constant**

It is the **cosmological model without pressure (p = 0) with k = 0 (flat curvature) and LAMBDA different from zero**

If the cosmic fluid has a state equation of type : p = w rho c^{2}, then : w = 0 and rho(t) = rho0 (a0 / a(t))^{3} according to the calculation detailed below.

By replacing this value of rho(t) into the first Friedmann's equation (F1), we find :

a a'^{2} = A + B a^{3}

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

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

This equation is integrated into :

a(t) = (A/B)^{1/3} sinh^{2/3}[(3/2) B^{1/2} t + b0]

where a0, rho0 and b0 are constants.

The constant b0 is generally set to 0 by an original choice of the coordinate t subject to adjusting the constants a0 and rho0.

Since a(t) tends to 0 when t tends to 0, note that Einstein-De Sitter's Space-time (with any LAMBDA) is a **model with Big-Bang** contrary to Einstein's Universe and De Sitter's Space-time.

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

k (c/a)^{2} / H(t)^{2} = OMEGA + OMEGAv - 1

where :

H(t) = **Hubble's parameter** (of dimension s^{-1}) = a'/a. H(t) increases very slowly with time because the Universe is expanding (a'(t) > 0). Its current value H0 called **Hubble's constant**) is about 70 (km/s)/Mpc, with 1 pc = 1 parsec = 3,2616 light-years = 3,085677581 10^{16} m

OMEGA(t) = **density parameter** (dimensionless) = (8/3) Pi G rho(t) / H(t)^{2}

OMAGAv(t) = **reduced cosmological constant** (dimensionless) = (1/3) LAMBDA c^{2} / H(t)^{2}

q(t) = **deceleration parameter** (dimensionless) = -a a"/ (a')^{2} = -1 - H'(t)/H(t)^{2}

*
Detailed calculation of components g ^{ab}, Rab, R, Eab, Tab and a(t) [GOUR Relativité Générale] :*

In the case of spatially homogeneous and isotropic Space-time (Friedmann-Lemaitre-Robertson-Walker's metric), the gravitational potentials gij of the Metric tensor are the following :

g00 = -1

g11 = a^{2} (1 - k r^{2})^{-1}

g22 = a^{2} r^{2}

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

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^{ij} of the inverse Metric tensor are then the following such that : g^{ij} gjk = DELTA^{i}k

where DELTA is the Kronecker's symbol.

g^{00} = -1

g^{11} = a^{-2} (1 - k r^{2})

g^{22} = a^{-2} (1/r^{2})

g^{33} = a^{-2} (1/r^{2}) sin^{-2}[theta]

g^{ij} = 0 for i and j taken different between 0 and 3

The Christoffel's 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}11 = a a' (1/c)/(1 - k r^{2}) ; GAMMA^{0}22 = a a' r^{2} (1/c) ; GAMMA^{0}33 = a a' r^{2} (1/c) sin^{2}[theta]

GAMMA^{1}01 = GAMMA^{1}10 = a' (1/c)(1/a) ; GAMMA^{1}11 = k r / (1 - k r^{2}) ; GAMMA^{1}22 = -r (1 - k r^{2}) ; GAMMA^{1}33 = -r (1 - k r^{2}) sin^{2}[theta]

GAMMA^{2}02 = GAMMA^{2}20 = a' (1/c)(1/a) ; GAMMA^{2}12 = GAMMA^{2}21 = 1/r ; GAMMA^{2}33 = -cos[theta] sin[theta]

GAMMA^{3}03 = GAMMA^{3}30 = a' (1/c)(1/a) ; GAMMA^{3}13 = GAMMA^{3}31 = 1/r ; GAMMA^{3}23 = GAMMA^{3}32 = 1/ tan[theta]

where a' = d(a)/dt

The other Christoffel's symbols are all zero.

The Rij components of Ricci's 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 = -3 a" (1/a)(1/c^{2})

R11 = (a a" + 2 a'^{2} + 2 c^{2} k)(1/c^{2})/(1 - k r^{2})

R22 = (a a" + 2 a'^{2} + 2 c^{2} k) (r/c)^{2}

R33 = sin^{2}[theta] R22

The other components Rij are all zero.

The Ricci's curvature is then written by the relation : R = g^{ij} Rij

R = 6 (1/c^{2})( (a"/a) + (a'/a)^{2} + (c/a)^{2} k )

The Einstein's tensor is then producted by the relation : Eab = Rab - (1/2) gab R + LAMBDA gab

E00 = R00 + (R/2) - LAMBDA

E11 = ( (2b + a"/a)/c^{2} - 3 (b + a"/a)/c^{2} + LAMBDA ) a^{2} /(1 - k r^{2})

E22 = E11 r^{2} (1 - k r^{2})

E33 = E22 sin^{2}[theta]

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

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^{2}

T11 = p a^{2} /(1 - k r^{2})

T22 = T11 r^{2} (1 - k r^{2})

T33 = T22 sin^{2}[theta]

The other components Tij are all zero.

The Einstein's equations are then written by the relation : Eab = KHI Tab

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's equations then reduce to the 2 following equations :

b = (1/3) rho KHI c^{4} + (1/3) LAMBDA c^{2}

(1/2) b + a"/a = (1/2) LAMBDA c^{2} - (1/2) p KHI c^{2}

By replacing the first equation in the second one, we find Friedmann's equations :

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

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

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^{2})

In the case where the cosmic fluid has a state equation of type : p = w rho c^{2}, this relation becomes :

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

which integrates into :

rho(t) = rho0 (a0 / a(t))^{3(1 + w)}

where rho0 and a0 are two constants.

By replacing this expression of rho(t) into the first Friedmann's equation (F1), we find a differential equation that is a function of a(t) only :

(a')^{2} + k c^{2} = A a^{-(1 + 3 w)} + B a^{2}

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

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

For a vector space of dimension n having for base vectors the set (**e1**, **e2**... **en**), the Christoffel's 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)

*Demonstration :
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's 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's 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's 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 of a light wave propagating at the wave speed c.

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

Longitudinal Doppler effect (parallel propagation to **V**) : f' = f gamma (1 - V/c)

Transverse Doppler effect (perpendicular propagation to **V**) : f' = f gamma

*Demonstration in Restricted Relativity [ANN Electricité 2] :
*

The Einstein's 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's equations.

In Cartesian coordinates, all the components are of dimension m

The notations are those in Maxwell's Equations.

In Restricted Relativity, the Lorentz's 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's 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's 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's 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's 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's 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's 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|).

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

In spherical coordinates (r, theta, phi) 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**

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's metric : ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dr^{2} + r^{2} (d(theta)^{2} + sin^{2}[theta] d(phi)^{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's symbols.

In Restricted Relativity (Minkowski's metric) with Cartesian coordinates, the coefficients gij are constant, which cancels all the Christoffel's 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}

The expression of the Kronecker's symbol DELTA is as follows :

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

The expression of the Levi-Civita's 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's transformation

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

Maxwell's 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 current 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's 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's symbol.

The following results can be proved :

g

gij;k = g

gij g

g

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

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

The Ricci's 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's 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's tensor is also zero.

*Demonstration :
By contracting the Einstein's equations by the inverse Metric tensor g^{ab}, the Ricci's 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's 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's 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's theorem).

In spherical coordinates (r, theta, phi) 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

*Demonstration :
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.

*Demonstration :
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's equations (L2) :
t'2 - t'1 = gamma ( (t2 - t1) - B (x2 - x1) ) = -gamma B (x2 - x1)
gamma and B being given by equation (L3).
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 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 local or relative or "measured" 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**.

*Demonstration 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's 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*

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.
- HLADIK J., Pour comprendre simplement la théorie de la Relativité, Ellipses, 2005.
- HLADIK J., Initiation à la Relativité restreinte et générale, Ellipses, 2013.
- LEVY-LEBLONG J.M., One more derivation of the Lorentz transformation, 30 April 1975.
- 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 : March 24, 2017.