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Relativity

Preamble ( Paragraph Start / Next )

This page gives the basic formulas of Restricted Relativity and General Relativity with a presentation of the mathematical tools accessible to uninitiated people.
This page also includes a detailed Lexicon of terms used in Relativity.
Notations on this page :
- The keywords have their first letter indicated in uppercase and are defined in the Lexicon.
- The authors quoted are referenced in square brackets under the reference [AUTHOR Title Page]. See Bibliography.
- The Mathematical notations are consistent with rare exceptions to those of Eric Gourgoulhon, Research Director at the CNRS [GOU, Relativité Restreinte][GOU, Relativité Générale].
- The Sign conventions are classically those of Misner, Thorne and Wheeler (MTW), with a Metric tensor of Signature (-, +, +, +), a Curvature tensor defined by Rⁱjkl = Γⁱjl,k + .. ., and a Ricci tensor defined by Rij = R^kikj

Contents of this page ( Paragraph Previous / Next )

1. What is Relativity ?
   1. The three stages of relativity
   2. Time relativity
   3. The "mysteries" of space-time
2. Restricted Relativity
   1. Historical background
   2. Lorentz-Poincaré transformation
   3. Proof
3. General Relativity
   1. Historical background
   2. Einstein Field Equation
   3. Solution of Einstein Field Equation
   4. Solution of Einstein Field Equation with Schwarzschild metric
   5. Solution of Einstein Field Equation with Friedmann-Lemaitre-Robertson-Walker metric
   6. Spectral shifts
4. Lexicon
5. Bibliography

1. What is Relativity ? ( Paragraph Previous / Next )

1.1. The three stages of relativity ( Previous / Next Subparagraph )

The relativity idea does not date from Einstein but finds its origin in the Galileo works.

We consider two Observers in relative motion whose reference frames are in rectilinear translation with uniform speed with respect to each other. These reference frames are called inertial.

• In Galilean relativity, there is invariance of the classic mechanics laws by inertial reference frame change.
  Time is absolute and the Observers will each find the same results in its reference frame by doing the same experiments of classic Mechanics.
  
  
• In Restricted Relativity (abbreviated "RR"), the invariance in Galilean relativity extends to the laws of Electromagnetism.
  Time is no longer absolute and becomes inseparable from space (notion of Space-time). The Galileo transformation becomes the Lorentz-Poincaré transformation. The classic mechanics laws become the Relativistic mechanics laws (but excluding Newtonian gravitation).
  Note that Restricted Relativity does not prohibit studying the view point of accelerated Observers. Eric Gourgoulhon's book [GOU, Relativité Restreinte] is thus one of the only ones to treat first with the measurements made in Space-time by any Observer (which can be accelerated or in rotation) and then to treat the particular case of the Inertial Observers.
  
  
• In General Relativity (abbreviated "GR"), the invariance in Restricted Relativity extends to gravitation and to any type of reference frame change (uniform or accelerated).
  Newtonian gravitation becomes relativistic gravitation (Einstein Field Equation) and the physics laws are put in tensorial form in order to be independent of the coordinate system.
  

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

1.2. Time relativity ( Previous / Next Subparagraph )

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

Imagine two Observers O and O' in relative movement with respect to each other, who wish to set their watches by exchange of optical signals. Suppose that the two watches are synchronized by any means so that they indicate the same time at the same initial instant. At this instant each Observer 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.
And Poincaré explains : "The transmission duration is not the same in both directions since the Observer O, for example, goes ahead of the optical propagation emanating from O', while the Observer O' flees the propagation emanating from O. The watches will indicate what can be called local time of each Observer, so that one of them will delay on the other. It does not matter since we have no way of seeing it..." [POI L'Etat, p.311]
The indicated time is the same for both Observers only in the case of Observers fixed with respect to each other or in the thought hypothesis of a light having an infinite speed.

We can therefore conclude that : "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]

1.3. The "mysteries" of space-time ( Previous / Start Subparagraph )

Many "mysteries" of Space-time are unfounded and result from clumsiness which do not facilitate the reading and understanding of Relativity. We can list :

• 1. Different viewpoints :
  For example, the Time concept is differently used (without being contradictory) by the authors. For mathematicians, time is reduced to a purely operative variable. For physicists, time represents, on the contrary, a physical time which is the real time measured by spatially located clocks able to synchronize by light signals exchange.
  Another example : Position and Image of a moving object are often confused.
  
  
• 2. Rigor lack in the use of words :
  For example, Time and Duration are sometimes confused, the duration being the time interval between two events.
  Same for Proper-Time and apparent-Time, which are two distinct times measured under different conditions.
  Same for physical time and biological time, which are also two distinct times.
  Another example : Position and Image of a moving object are often confused.
  
  
• 3. Sign conventions not explained :
  Some authors implicitly use other sign conventions than the classic Conventions of sign (MTW), which changes the sign of some Tensors (including that of the Metric tensor (g)) and/or that of the Cosmological constant (Λ) and/or that of the gravitational coupling coefficient (K).
  
  
• 4. Excessive writing simplification :
  Some authors simplify the formulas writing by eliminating some constants. The verification of the formulas homogeneity is no longer possible and can give place to errors in the numerical application of these formulas.
  For example, the light speed (c), the universal gravitational constant (G) and/or the dielectric permittivity in vacuum (ε₀) are sometimes arbitrarily set to 1.
  
  
• 5. Abusive generalization :
  Many authors mention only part of the applicability conditions of formulas. The published results then wrongly inherit a general status.
  For example, many formulas in General Relativity implicitly concern an empty space (of any non-gravitational field) with a zero Cosmological constant.
  
  
• 6. Insufficiently pedagogical presentation :
  Few authors produce an index of notations used.
  Few authors rigorously respect the vectorial notation that distinguishes scalar quantities (non-bold letters), vectorial quantities (bold letters) and tensorial quantities (bold capital letters).
  Few authors place the (often long) proof of formulas after their statement, ideally in a deported insert.
  Few authors produce examples and illustrative diagrams (geometric interpretations, for example) in addition of the text.
  
  
• 7. Borrowed text without source reference :
  Few authors indicate the reference of borrowed text extracts (author, title, page).
  
  
• 8. Non-free access to Internet documents :
  Few authors (teachers or researchers) give free access to their course on the Internet.
  

2. Restricted Relativity ( Paragraph Previous / Next )

2.1. Historical background ( Previous / Next Subparagraph )

Until the end of the 19th century, classic 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 Clerk 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 an uniform rectilinear translation.
Should not the Galilean principle be, if not abandoned, at least rehabilitated ?
In 1905 Jules Henri Poincaré laid the fundamental foundations of Restricted Relativity which erased at once all the anxieties of physicists about these two enigmas [POI L'Etat].
Also in 1905 Albert Einstein published his theory of Restricted Relativity [EIN Zur_Elektrodynamik].

Note some surprising conclusions among others :
- If two luminous particles move away from each other, their relative speed is equal to c and not 2c (speeds composition law, see below).
- Since the light speed is slowed down in various media according to their refractive index n, it is possible to accelerate particles that go faster than light in the same medium.

2.2. Lorentz-Poincaré transformation ( Previous / Next Subparagraph )

Hendrik Antoon Lorentz gave an imperfect version of this tranformation in 1899 and then 1904. Jules Henri Poincaré published the correct equations in 1905, baptizing them with the name of Lorentz.

We consider two reference frames R and R' in uniform rectilinear translation with respect to each other at the velocity v parallel to the x and x' axis (see Figure above).
The two reference frames have their origin O and O' which coincide at time t = 0.
Let M be an arbitrary point or event having spatio-temporal coordinates (x, y, z, t) in R and (x', y' = y, z' = z, t') in R'.

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

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

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.

The inverse transformation consists of replacing v by -v in relations (L1) to (L4) :

When the velocity v has any direction with respect to the x axis, the special transformation of Lorentz-Poincaré is written as follows (by noting r = (x, y, z) and r' = (x', y', z')) :

2.3. Proof ( Previous / Start Subparagraph )

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 (Poincaré's postulates), without need for reference to electromagnetism [LEV One_more]. The Einstein's postulate on the invariance of the light speed in all reference frames then appears as a simple consequence of the Lorentz-Poincaré transformation describing 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-Poincaré transformation only based on the four Poincaré's postulates.

3. General Relativity ( Paragraph Previous / Next )

3.1. Historical background ( Start / Next Subparagraph )

Restricted Relativity applies only to reference frames in uniform rectilinear translation and in a Space-time where the gravitational effects are completely neglected.
In 1915 Albert Einstein elaborated the General Relativity with the help of various mathematicians [EIN Die_Grundlage]. He completely rethinks the notion of Newtonian gravitation which being propagated instantaneously is no longer compatible with the existence of a limiting speed. He also postulate that all laws of Nature must have the same form in all reference frames whatever their state of motion (uniform or accelerated).

3.2. Einstein Field Equation ( Previous / Next Subparagraph )

Definition

The fundamental equation of General Relativity, called Einstein Field Equation (EFE) or Einstein equation of gravitational field, connects a local deformation of the geometry of Space-time with the presence of local tensions (see Figure above).
John Archibald Wheeler, American specialist of General Relativity, summarizes this state as follows : "Matter tells space-time to bend and space-time tells matter how to move".

This equation can be seen as a generalization of the elasticity law 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. This equation is written in tensorial form as follows :

Note that some authors present this equation with the minus sign in front of Λ instead of the plus sign.

gab is the Metric tensor which is solution of Einstein Field Equation. The 16 gab components of this Tensor are called gravitational potentials.

Sab is the Einstein tensor which measures the local deformation of the Space-time geometry. There is no gravitational force in General Relativity since this deformation of Space-time takes its place. This Tensor has the 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 ρ 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.

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

R is the Scalar curvature producted by Contraction of the Ricci tensor.

a and b are the indexes of the different Tensors with a and b ranging from 0 to 3

K is the gravitational coupling coefficient : K = 8 π G c^-4 (in kg^-1.m^-1.s²). This coefficient was chosen so as to verify the Poisson equation of the Newtonian gravitation as a particular case of Einstein Field Equation (see Newtonian Limit). K represents an extraordinarily small elasticity of Space-time (equal to about 2.1 10^-43 kg^-1.m^-1.s²).

G is the universal gravitational constant : G = 6.67408 10^-11 kg^-1.m³.s^-2

c is the light speed in the vacuum : c = 2.99792458 10⁸ m.s^-1

Λ is the Cosmological constant of dimension m^-2 and may be negative, zero or positive. The problem of the planets motion, considered as material particles in an empty space around the sun (Schwarzschild space-time), is solved by taking Λ = 0 and Tab = 0. In cosmology, the universe model (Friedmann-Lemaitre-Robertson-Walker space-time) is determined by a priori non-zero Λ value and the universal space is considered as filled with a real gas of galaxies with density ρ and pressure p = 0 (Standard cosmological model).

Equation presented differently

This equation is sometimes presented with the "minus" sign in front of Λ and/or in front of K.
It depends on the author conventions taken for the Signature of the Metric tensor, the definition of Curvature tensor and the definition of Ricci tensor.

Equivalent equation

By contracting the Einstein Field Equation by the inverse Metric tensor g^ab, the Scalar curvature R is related to the Energy-impulse tensor Tab by the relation :
    (E2) R = -K T + 4 Λ
where T is the trace of the Energy-impulse tensor : T = g^ab Tab = T^aa
By replacing this relation in the Einstein Field Equation (E1), we find the following equivalent equation :

This equivalent equation may be more practical in certain cases, for example when we are interested in the weak gravitational field limit and we can replace gab by the Minkowski metric without significant loss of precision.
In the particular case where Tab = 0 (vaccum space) and Λ = 0, the Ricci tensor Rab is zero.
A solution of Einstein Field Equation of vaccum space with Λ = 0, like the Schwarzschild solution, is therefore a Metric whose Ricci tensor is identically zero. On the other hand, the Curvature tensor is not zero, except in the case of the trivial solution constituted by the Minkowski metric (flat Space-time of Restricted Relativity). cf [GOU, Relativité générale, p.119].

Properties of Einstein Field Equation

3.3. Solution of Einstein Field Equation ( Previous / Next Subparagraph )

The 16 components of the Sab Einstein tensor are function only of the gravitational potentials gab and their first and second derivatives. These components are linear with respect 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 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 Schwarzschild metric (S1, S2...) describes the geometry around the masses (M1, M2...), these masses can be a star, a planet or a Black hole.
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 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 Restricted Relativity.

3.4. Solution of Einstein Field Equation with Schwarzschild metric ( Previous / Next Subparagraph )

Definition

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, θ, φ) with respect to two parameters μ and α only functions of r.
These gab allow to calculate the components of the Ricci tensor (Rab) and then, by Contraction, the Scalar curvature (R). See calculations detailed below.

In the particular case of a gravitational field in vacuum (when the Energy-impulse tensor (Tab) is zero) and a zero Cosmological constant (Λ = 0), Einstein Field Equation then is reduced to a system of two differential equations of the functions μ and α. Their integration gives the expressions μ and α. See calculation detailed below.
The Schwarzschild metric ds² is finally completely determined as follows :

where r^* 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 : r^* = 2 G M c^-2, producted by comparing the Schwarzschild g00 with the g00 of the Newtonian limit. In the case of Soleil (with M_Soleil = 1.9891 10³⁰ kg), r* is very small and is egal to : 3.0 km
The particular values r = 0 and r = r^*, 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 which we always have : r >> r*.
For Black holes the singularity r = r^* 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. Schwarzschild space-time is thus asymptotically flat.

We finally write and solve the equations of Geodesics which describe the movement of free particles in the space considered, that is when these particles (material systems or photons) are not subjected to an external force other than gravitation in the context of General Relativity. See Geodesic of a material body and Geodesic of a photon.

Proof

3.5. Solution of Einstein Field Equation with Friedmann-Lemaitre-Robertson-Walker metric ( Previous / Next Subparagraph )

Friedmann equations

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, θ, φ) with respect 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 Scalar curvature (R).
By choosing a Perfect Fluid model for the Energy-impulse Tensor (Tab), its components then can be calculated as a function of the pressure p and the density ρ of the physical environment that fills the space.
The Einstein Field Equation is then reduced to a system of two differential equations of the functions a(t), ρ(t) and p(t), called Friedmann equations :
    (F1) (a'/a)² + k (c/a)² = (1/3) ρ K c⁴ + (1/3) Λ c²
    (F2) a"/a = -(1/6) (ρ + 3 p c^-2) K c⁴ + (1/3) Λ c²
The system is completed by giving to cosmic fluid an equation of state as p = p(ρ). An example of a frequently used equation of state is : p(t) = w ρ(t) c² 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 ρ(t) and a(t) :
    (Q0) ρ(t) a(t)^{3(1 + w)} = ρ₀ a₀^{3(1 + w)} = constant
    where ρ₀ 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) :

This differential equation is analytically integrated for w = -1, 0 or 1/3 (with any Λ and k), which completely determines a(t) and the metric ds² as follows :

The first Friedmann equation (F1) is often presented in the condensed form :
    1 + Ω_k = Ω + Ω_v
    where :
H(t) = Hubble parameter (of dimension s^-1) = a'/a that accounts for the universe expansion. See Hubble-Lemaître law
Ω_k(t) = reduced curvature (dimensionless) = k (c/a)² / H(t)²
Ω(t) = density parameter (dimensionless) = (8/3) π G ρ(t) / H(t)²
Ω_v(t) = reduced Cosmological constant (dimensionless) = (1/3) Λ c² / H(t)²
q(t) = deceleration parameter (dimensionless) = -a a"/ (a')² = -1 - H'(t)/H(t)²
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 Dark energy.

The Friedmann second equation (F2) is also written in the form :
    (Q2) a"/a = -F a^{-3(1 + w)} + B
    (Q2a) F = (1/2) (1 + 3 w) A
Note that the relation (Q2) is also found immediately by derivation of the relation (Q1).

General shape of the curves a(t)

In the standard case where ρ > 0 and w > (-1/3), we then deduce from relations (Q1) and (Q2) the general shape of the curves a(t) for any Λ and k (see Figure 1 above with Proof below, or Figure 2 above cf [HAR Cosmology, table p.367] or [LUM, table p.153]).
All these curves, except two, represent Big Bang models for which a(t) tends to 0 when t tends to 0 :
- The curve C1 corresponding to case (Λ < 0), or case (Λ = 0) and (k > 0), corresponds to a closed model (decelerated expansion with a maximum point M1).
- The curve C2 corresponding to case (Λ = 0) and (k ≤ 0) correspond to an open model (decelerated expansion).
- The curve C4 corresponding to case (Λ > 0) and (k ≤ 0), or to the case (Λ > Λ_F) and (k > 0) correspond to an open model with inflection point I (decelerated expansion followed by accelerated expansion). The sub-case (Λ > 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 case (0 < Λ < Λ_F) and (k > 0). They correspond to two possible behaviors : an open model of non-Big Bang type (accelerated expansion with a minimum point M2), and a closed model with a maximum point M1.
- The curve C3 is related to singular case (Λ = Λ_F) and (k > 0). It corresponds to a static model (Einstein static universe) for which a(t) = constant. This model is unstable to density disturbances ρ with a local tendency to expansion (curve C5 after point M2) or to contraction (curve C1 before point M1).
Note that these curves represent a subset of curves listed by Harrison [HAR Classification].

Λ_F is the singular Cosmological constant of Friedmann which is written (see Proof below) :
    (Q6) Λ_F = (1/2)(1 + 3 w) ρ_F K c²
    (Q6a) a_F = ( (1/2)(1/k)(1 + w) ρ_F K c² )^-1/2
For k = 1 and ρ_F = ρ_E, we get the expressions Λ_E and a_E of the Einstein static universe :
    Λ_E = (1/2)(ρ_E + 3 p_E c^-2) K c²
    a_E = ( (1/2)(ρ_E + p_E c^-2) K c² )^-1/2

Simple solutions for a(t)

Some particularly simple solutions for a(t) are presented below (index 0 generally corresponding to data to date).
Apart from the first two solutions, the others are almost all Big Bang models presented according to the values of parameters w, then Λ then k.

1. Einstein static universe
It is the static cosmological model with : a(t) = a_E ; ρ(t) = ρ_E ; p(t) = p_E
where a_E, ρ_E and p_E are constants.
The second Friedmann equation (F2) then becomes : Λ = Λ_E
where : Λ_E = (1/2)(ρ_E + 3 p_E c^-2) K c²
Λ_E is the singular Cosmological constant of Einstein which characterizes a static universe.
Note that outside a vacuum (ρ_E = p_E = 0), a static solution can exist only with a non-zero Cosmological constant.
By replacing this value of Λ in the first Friedmann equation (F1), we find :
    k / a_E² = (1/2)(ρ_E + p_E c^-2) K c²
If the cosmic fluid satisfies the strict low energy condition then : ρ_E + p_E 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) = a_E = ( (1/2)(ρ_E + p_E c^-2) K c² )^-1/2

2. De Sitter Space-time
It is the cosmological model of the vacuum (ρ = p = 0) with Λ > 0 and k = 0 (flat curvature).
The first Friedmann equation (F1) then becomes : (a'/ a) ² = (H0)²
    with H0 = B^1/2 = c (Λ / 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 and is not a Big Bang model.

3. Friedmann model with open curvature
It is the cosmological model without pressure (w = 0) with Λ = 0 and k = -1 (open curvature)
By replacing these values in differential equation (Q1), we find :
    a'² = A a^-1 + c²
    where A = A(w = 0) according to the relation (Q1a)
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, ρ₀ > 0 and ti are constants, ti being generally set to 0 by an original choice of the coordinate t.
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 3 above for k = -1).

4. Friedmann model with flat curvature (or Einstein-De Sitter Space-time)
It is the cosmological model without pressure (w = 0) with Λ = 0 and k = 0 (flat curvature)
By replacing these values in differential equation (Q1), we find :
    a'² = A a^-1
    where A = A(w = 0) according to the relation (Q1a)
This equation is integrated into :
    a(t) = ( (1/j) A^1/2 (t - ti) )^j
    with j = 2/3
    where a₀, ρ₀ > 0 and ti are constants, ti being generally set to 0 by an original choice of the coordinate t.
The curve a(t) is a power function (see Figure 3 above for k = 0).

5. Friedmann model with closed curvature
It is the cosmological model without pressure (w = 0) with Λ = 0 and k = 1 (closed curvature)
By replacing these values in differential equation (Q1), we find :
    a'² = A a^-1 - c²
    where A = A(w = 0) according to the relation (Q1a)
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 π
    where a₀, ρ₀ > 0 and ti are constants, ti being generally set to 0 by an original choice of the coordinate t.
The term (t - ti) expressed more simply as a function of (a) in the form :
    For t - ti < π (D/c) : t - ti = (D/c) ( Arccos[1 - (a/D)] - ((a/D)(2 - (a/D)))^1/2 )
    For t - ti > π (D/c) : t - ti = 2 π (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 with respect to the value t - ti = π (D/c) (see Figure 3 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 π (D/c)) through an expansion phase (a' > 0) and then a contraction phase (a' < 0).

6. Model without pressure (w = 0) with non-zero Λ
The exact solution of this model is given by [KHA Some_exact_solutions].

7. Model without pressure (w = 0) with non-zero Λ and k = 0 (flat curvature)
By replacing these values in differential equation (Q1), we find :
    a'² = A a^-1 + B a²
    where A = A(w = 0) and B given by relations (Q1a) and (Q1b)
This equation is integrated into :
    if Λ < 0 : a(t) = (-A/B)^1/3 sin^2/3[ (3/2) (-B)^1/2 (t - ti) ]
    if Λ > 0 : a(t) = (A/B)^1/3 sinh^2/3[ (3/2) B^1/2 (t - ti) ]
    where a₀, ρ₀ > 0 and ti are constants, ti being generally set to 0 by an original choice of the coordinate t.
If Λ < 0, the curve a(t) is similar to the closed curve of the Friedmann model (see Figure 3 above for k = 1).
If Λ > 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 3 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).

8. Model for electromagnetic radiation (w = 1/3) with non-zero Λ
The exact solution of this model is given by [KHA Some_exact_solutions].

9. Model for electromagnetic radiation (w = 1/3) with Λ = 0
By replacing these values in differential equation (Q1), we find :
    a'² + k c² = A a^-2
    where A = A(w = 1/3) according to the relation (Q1a)
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, ρ₀ > 0 and ti are constants, ti being generally set to 0 by an original choice of the coordinate t.
The curves a(t) are similar to the curves of the Friedmann model (see Figure 3 above for k = -1, 0 and 1).

10. Model with w > (-1/3), Λ = 0 and k = 0 (flat curvature)
By replacing these values in differential equation (Q1), we find :
    a'² = A a^{-(1 + 3 w)}
    where A = A(w) according to the relation (Q1a)
This equation is integrated into :
    a(t) = ( (1/j) A^1/2 (t - ti) )^j
    with j = (2/3) (1 + w)^-1 < 1
    where a₀, ρ₀ > 0 and ti are constants, ti being generally set to 0 by an original choice of the coordinate t.
The curve a(t) is a power function having a parabolic branch along the time axis when t tends to infinity (see Figure 3 above for k = 0).

Proofs

3.6. Spectral shifts ( Previous / Start Subparagraph )

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 with respect 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-Lemaître 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-Lemaître law.

4. Lexicon ( Paragraph Previous / Next )

Notions used in this page, listed alphabetically :

1. (light) Aberration
2. Angular momentum of a particle mesured by an observer
3. Antisymmetry and symmetry
4. Base change
5. Big Bang
6. Black hole
7. Celestial reference frame and Terrestrial reference frame
8. Christoffel symbols
9. Convention of partial derivative
10. Convention of summation (called "Einstein convention")
11. Cosmological constant
12. Covariance and contravariance
13. Covariance of a physical law (or principle of General Relativity)
14. Covariant derivative (or gradient)
15. Cross product
16. Curvature
17. Curvature tensor (or Riemann-Christoffel tensor)
18. Dark energy
19. Dark matter
20. Dilation of durations (or Slowing down of moving clocks)
21. Direct and indirect base
22. Divergence
23. Doppler effect (or Doppler-Fizeau effect)
24. Dual (space and base)
25. Einstein effect (or gravitational spectral shift or gravitational Doppler effect)
26. Einstein Field Equation (EFE)
27. Einstein tensor
28. Electromagnetic tensor (or Maxwell tensor or Faraday tensor)
29. Elementary tensor operators
30. Energy and impulse of a particle mesured by an observer
31. Energy-impulse tensor
32. Energy-impulse tensor of ElectroMagnetic field
33. Energy-impulse tensor of Perfect Fluid
34. Friedmann equations
35. Friedmann-Lemaitre-Robertson-Walker metric (or FLRW metric)
36. Galileo transformation
37. Geodesic
38. Geodesic of a material body
39. Geodesic of a photon
40. GPS
41. Gravitational mirage (or gravitational lens)
42. Hubble-Lemaître law
43. Image of an moving object
44. Index contraction (or tensor trace)
45. Index increasing
46. Index lowering
47. Inertial (or galilean) reference frame
48. Kronecker symbol
49. Lengths contraction (or FitzGerald-Lorentz contraction)
50. Levi-Civita symbol
51. Levi-Civita tensor
52. Light-cone
53. Linear form
54. Local reference frame
55. Local rest space of the observer
56. Lorentz factor
57. Lorentz-Poincaré transformation
58. Mathematical notations
59. Maxwell equations
60. Metric tensor (or fundamental tensor)
61. Minkowski metric
62. MTW sign conventions
63. Multiplicity of proper times (or Desynchronization of perfect clocks)
64. Natural base
65. Newtonian limit
66. Observer
67. Orthonormal base
68. Partial derivative (convention)
69. Poisson equation
70. Principle of equivalence
71. Quadri-acceleration
72. Quadri- electric power
73. Quadri- electromagnetic potential
74. Quadri-force
75. Quadri-impulse
76. Quadri-position
77. Quadri-rotation
78. Quadri-vector
79. Quadri-velocity
80. Relativistic mechanics
81. Relativistic metric
82. Ricci tensor
83. Scalar curvature (or invariant scalar)
84. Scalar product and norm
85. Schwarzschild metric
86. Signature
87. Simultaneity
88. Space-time
89. Standard cosmological model
90. Standard time
91. Summation (convention)
92. Symmetry and antisymmetry
93. Tensor
94. Tensor product
95. Tensoriality criteria
96. Time and Duration
97. Trajectory
98. Twin paradox (or clocks paradox or Langevin traveler)
99. Universe age
100. Universe history
101. Universe line
102. Vector type
103. Wormhole

(light) Aberration

Figure 1 : Phenomenon of light aberration.
Figure 2 : Distortion of the celestial sphere. The four figures correspond to different values of the speed v of the moving observer O' (v = 0, 0.3c, 0.6c, 0.9c) [GOU, Relativité Restreinte, p.162].

The aberration of light is the difference between the incidence directions of the same light ray perceived by two Observers in relative motion.
It is not a purely relativistic effect because the aberration results from the finite time of propagation of a signal, as shown by the example of the pedestrian who walks in the rain : the rain falling vertically on the ground, the pedestrian must tilt his umbrella forward if he does not wish to be wet.
Let O be an Observer of a reference frame R and O' an Observer of a reference frame R' in uniform rectilinear translation of velocity v with respect to R.
In the case of a light source M seen by O', the light emanating from M appears to come from M' and not from M (see Figure 1 above).

u is the unit vector of the light propagation MO.
If the propagation u makes with the velocity v an angle θ in R and θ' in R', then we have the relation :

    cos[θ'] = (cos[θ] - v/c) / (1 - cos[θ] v/c)

Using the relation : tan²[θ/2] = (1 - cos[θ])/(1 + cos[θ]), we have the equivalent relation :

So we always have : θ' > θ, 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 (θ = 0 or π), then the formula reduces to : cos[θ'] = 1 or -1, which induces : θ' = 0 or π, and there is no aberration effect.
- When u is perpendicular to v in the reference frame R (θ = π/2), then the formula reduces to : cos[θ'] = -v/c, which induces : θ' > π/2 (and the pedestrian must tilt his umbrella forward).
    For a terrestrial Observer O' (v = 30 km.s^-1), the Image of a star located at the north pole of the ecliptic (θ = π/2) thus describes in a year a circle of radius 20'' around this pole. Not to be confused with the parallax which is the view angle of a star when Earth travels its orbit and which is at most 0.77'' for the nearest star (Proxima Centauri) [GOU, Relativité Restreinte, p. 162].
- When v is small compared to c, there is no aberration effect (θ' = θ).

The aberration phenomenon is visualized very well by considering a uniform grid on the celestial sphere of the Observer O (see Figure 2 above). The Observer O' sees directions appearing out of the field for the Observer O, until seeing the two celestial poles at the same time.

Angular momentum of a particle mesured by an observer

The angular momentum vector σC of a particle M relative to a given point C and mesured by an Observer O in his Local reference frame at time τ is given by the following relation (see Figure above) [GOU, Relativité Restreinte, p.322] :

P is the Impulse vector of the particle.
u0 and E_u0 are the Quadri-velocity of the Observer and his Local rest space E_u0.
"x_u0" is the Cross product operator between two any vectors of E_u0, what is written : CM x_u0 P = Ε(u0, CM, P, .)
where :
Ε is the Levi-Civita tensor
Ε(u0, CM, P, .) is the vector representing the Linear form Ε(u0, CM, P, z) for the Scalar product g.

σC belongs to E_u0 and its components are of dimension kg.m².s^-1

Antisymmetry

See Symmetry and antisymmetry.

Base change

Let [A] be the transition matrix from base {ei} to base {e'k} such that : e'k = Aⁱk ei, and [B] = [A^-1] the inverse transition matrix such that : ei = Bⁱk e'k
For any tensor T of order 2, using the multilinearity of T and the properties of Covariance and contravariance, the components of the tensor T' are given by the following laws :
T'ij = A^ki A^lj Tkl
T' ^ij = Bⁱk B^jl T^kl
T' ⁱj = Bⁱk A^lj T^kl
The base change transforms the tensor T into a tensor T' whose components are linear combinations of the components of the origin tensor.

For any tensor T p times contravariant and q times covariant, the general transformation law is as follows [GOU, Relativité Restreinte, p. 476] :

Big Bang

Definition :
The Big Bang is a word with a double meaning :
- Event marking the beginning of the expansion of the universe, without this prejudging the existence of an "initial moment" or a beginning to its history.
- Standard cosmological model describing the evolution of the universe from an infinitely dense and incredibly hot state (10¹³ °C) to the present.
The Big Bang, wrongly called a "primordial explosion", is in fact a "primordial expansion" such that the scale factor a(t) tends towards 0 when t tends towards 0. There is no ejection of matter from of a place in the universe but expansion of space reduced to a point, which takes with it all the objects of the universe.

Simplified chronology (see [The secrets of space] and http://scphysiques.free.fr/2nde/documents/Histoire_atome/bigbang.html ) :
- Between 0 and 10^-43 seconds after the Big Bang, corresponding to "Planck time", a sort of incompressible temporal quantum according to quantum physics, our physics is silent but scientists believe that at the end of this period gravity separated itself from the other forces of nature (strong nuclear force, weak nuclear force and electromagnetic force).
- Between 10^-43 and 10^-6 seconds after the Big Bang, scientists describe the evolution of the universe based on hypotheses. The Big Bang can only be described according to known physics equations from 10^-6 seconds.
- Around 10^-32 seconds after the Big Bang, one hypothesis is that the universe was a "soup" of elementary particles and antiparticles. Some still exist today in the form of quarks, antiquarks and bosons like gluons. Others no longer exist, such as gravitons (hypothetical particles that transmit gravity) and Higgs bosons which transmit mass to other particles.
- 300 000 years after the Big Bang, when the temperature dropped to around 2700°C, the first hydrogen and helium atoms were formed. The electrons then being linked to the atoms, matter and radiation become decoupled. Photons can then travel through the universe in the form of radiation. They reach Earth not as visible light but as low-energy photons from the cosmic microwave background.
- 200 million years after the Big Bang, the first stars appeared, made almost exclusively of hydrogen and helium. During their lives and deaths, these early stars create new chemical elements from nuclear fusion in the hot cores of these stars, such as carbon, oxygen, silicon, and iron. These chemical elements are scattered throughout space and other galaxies.
- The second and third generations of stars form later in this enriched interstellar medium. They create even more chemical elements sent back into the interstellar medium via solar winds and supernova explosions.
- The Universe history then continues with the creation of our Sun, the Earth and life on Earth. Note that the Earth will never receive visible light before the combustion of the first stars.
- Today the universe is extremely sparse (a few atoms per cubic meter) and cold (-271°C).

Black hole

A black hole is a celestial object so compact that the intensity of its gravitational field prevents any form of matter or radiation from escaping from it.
The static black hole (without rotation) is described by the Schwarzschild Metric.
The rotating black hole is described by the Kerr Metric which generalizes the Schwarzschild Metric.

Celestial reference frame and Terrestrial reference frame

In geophysics the celestial reference frame is a reference system whose orthogonal axes (x, y, z) are defined using reference points sufficiently distant to appear fixed on the celestial sphere. These distant objects are extragalactic radio-sources.
Several celestial reference frames exist (cf [ZIEGLER, Modélisation, p.24]) :
The International Celestial Reference System (ICRS), adopted in 1997 by the International Astronomical Union (IAU), has its origin in the barycenter of the Solar System in accordance with General Relativity. Its pole (z-axis) is currently approximately in the direction of the Polar Star (Polaris). The origin of straight ascents (x-axis) is in the direction defined from quasar 3C 273, in the direction of the vernal point which is the position of the Sun seen from Earth at the spring equinox. The third axis (y-axis) is orthogonal to the first two and oriented so that the three axes form a direct direction trihedron.
The Barycentric Celestial Reference System (BCRS), defined in 2000 and 2006 by the IAU, merges with the ICRS in terms of orientation and origin but also defines a Relativistic metric for the coordinate system. The proper time of BCRS is called Barycentric Coordinate-Time (BCT).
The Geocentric Celestial Reference System (GCRS) differs from the BCRS in its origin, which is the mass center of Earth. The proper time of GCRS is called Geocentric Coordinate-Time (GCT).
These three reference frames have in common a conventional and theoretically invariable orientation (non-rotating with respect to the stars). They do not follow the rotation of the Earth over time as the terrestrial reference frames would do.

Among the terrestrial reference frames, the current standard is the International Terrestrial Reference System (ITRS) which is in daytime co-rotation with the Earth and originates from its center of mass. The proper time of ITRS is called Terrestrial Time (TT).

Christoffel symbols

For a vector space of dimension n having for basis vectors the set (e1, e2... en), the Christoffel symbols Γⁱjk (called "of second kind") 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 :

Warning : Although possessing three indexes, the Christoffel symbols of second kind are not mixed Tensors of order 3 because they do not satisfy the Tensoriality criteria. For example, the index lowering : Γjik = glj Γ^lik is not possible, which is false considering that : glj Γ^lik = Γijk
In contrast, they appear abundantly in expressions which represent Tensors (for example : Covariant derivative, Divergence, Ricci Tensor).

Γⁱjk can be written as a function of basis vectors of Dual space :

Note that there is another Christoffel symbols Γijk (called "of first kind") defined by the relation :

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

Properties :
Γijk is Symmetric with respect to the two extreme indexes : Γijk = Γkji
Γⁱjk is Symmetric with respect to the lower index : Γⁱjk = Γⁱkj
Derivative of metric : gij_,k = Γijk + Γjik
Index contraction (cf [GOU, Relativité restreinte, p.506]) : Γⁱji = (-g)^-1/2 ((-g)^1/2),j
    with : g = Determinant of the matrix gij associated with Metric Tensor g
The Christoffel symbols are all zero only in the particular case of Restricted Relativity (Minkowski metric) with Cartesian coordinates, for which the components gij are all constant.

Convention of partial derivative

In order to lighten the expressions of the derivatives of functions dependent on n variables f(x¹, x²... xⁿ), we denote the partial derivatives in the following forms :

Convention of summation (called "Einstein convention")

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 = x¹ e1 + x² e2 + ... + xⁿ en = ∑_{k=1, 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 :

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 indexes. For example, in the sum a^km 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^km b^m for n = 3 represents the system of equations :
c1 = a¹1 b¹ + a¹2 b² + a¹3 b³
c2 = a²1 b¹ + a²2 b² + a²3 b³
c3 = a³1 b¹ + a³2 b² + a³3 b³
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 indexes. For example, a^km b^m ck for n = 4 represents a sum of 16 terms :
a^km b^m ck = a¹1 b¹ c1 + a¹2 b² c1 + a¹3 b³ c1 + a¹4 b⁴ c1 + ... + a²1 b¹ c2 + ... + a⁴4 b⁴ c4

The convention of summation also applies to Partial derivative (example : g^il,l) and to Covariant derivative (example : F^kl_;l), although the derivative symbol is not a covariant index [DENIS, Cours, p.25].

Cosmological constant

The history of Cosmological constant Λ is particularly eventful :
- In 1915 Einstein publishes his Equation of General Relativity, without a cosmological constant Λ.
- In 1917 Einstein adds the parameter Λ to his Equation when he realizes that his theory implies a dynamic Universe for which space is function of time. He then gives this constant a very particular value to force his Universe model to remain static and eternal (Einstein static universe), which he will later call "the greatest stupidity of his life".
- In 1922 the Russian physicist Alexander Friedmann mathematically shows that Einstein Field Equation (whatever Λ) remains valid in a dynamic Universe.
- In 1927 the Belgian astrophysicist Georges Lemaître shows that the Universe is in expansion by combining General Relativity with some astronomical observations, those of Hubble in particular.
- In 1931 Einstein finally accepts the theory of an expanding Universe and proposed, in 1932 with the Dutch physicist and astronomer Willem de Sitter, a model of a continuously expanding Universe with zero cosmological constant (Einstein-De Sitter Space-time).
- In 1998 two teams of astrophysicists led, one by Saul Perlmutter, the other by Brian Schmidt and Adam Riess, carrie out measurements on distant supernovae and show that the speed of galaxies recession in relation to the Milky Way increases over time. The Universe is in accelerated expansion, which requires having a strictly positive Λ. The Universe would contain a mysterious Dark energy producing a repulsive force that counterbalances the gravitational braking produced by the matter contained in the Universe (see Standard cosmological model).
    For this work, Perlmutter (American), Schmidt (American-Australian) and Riess (American) jointly received the Nobel Prize in physics in 2011.
- In 2015 the value of the cosmological constant is estimated at Λ = 1,11 10^-52 m^-2

Covariance and contravariance

The notions of covariant and contravariant vectors apply to non-orthonormal referentials, which explains why they are not addressed in secondary or even higher mathematics courses. For Orthonormal base there is no difference between covariant and contravariant components.
For a vector space E of dimension n having for basis vectors the set (e1, e2... en), we call (see Figure above) :

The contravariant components are noted with higher indexes.
The covariant components are noted with lower indexes.
The contravariant (respectively covariant) name derives from the fact that these components are transformed by base change in an inverse (respectively identical) manner to that of the basic vectors.
When the indexes vary from 0 to 3, Greek letters (such as α or μ) are often used rather than Latin letters (such as i or j).
We have the following relations :

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

Covariance of a physical law (or principle of General Relativity)

The general covariance of a physical law has nothing to do with the Covariance of the tensor components. The word "covariance" does not refer to the covariant index of a Tensor but only indicates a writing of the physical law which remains form-invariant under any coordinates transformation (invariance by diffeomorphism) [AMI Initiation_aux_Tenseurs, p.18].
When the physical law can be written in the tensorial form : T = 0, or what amounts to the same : P = Q with T = P - Q (T, P and Q being Tensors of the same type), any change of reference frame transforms this equation into the tensorial form : T' = 0, which does not change the form of the physical law.

Example of Newton's law : F = m γ
The covariant expression of this law is then in contravariant components [AMI Initiation_aux_Tenseurs, p.18] :
    Fⁱ = m vⁱ_;t
with : vⁱ = dxⁱ/dt
The term vⁱ_;t is the Covariant Derivative of the velocity dxⁱ/dt with respect to time t, which is :
    vⁱ_;t = vⁱ_;k (dx^k/dt) = vⁱ_,k (dx^k/dt) + Γⁱjk v^j v^k = dvⁱ/dt + Γⁱjk v^j v^k
where Γⁱjk are the Christoffel symbols.
If moreover the force F derives from a potential in the form : F = -grad(Φ), then F can be written in contravariant components : Fⁱ = g^ij Fj = -g^ij dΦ/dx^j
where g^ij are the inverse components of the Metric tensor.
Newton's law is then written in the following tensorial form :

In this tensorial form, Newton's law will remain form-invariant under any coordinates transformation (spherical coordinates for example).

Another example : Maxwell equations.

Covariant derivative (or gradient)

The covariant derivative is the general expression of the derivative which remains form-invariant under any coordinates transformation. It helps to make the physical laws Covariant.
According to the general procedure stated by Einstein [BARRAU, Relativité générale, p.64], "we find the correct physical laws by replacing the usual derivatives by covariant derivatives".

Using the Convention of partial derivative and the Convention of summation, for each Tensor U of order (n), its covariant derivative is the tensor of order (n + 1) of the following components (denoted u _;l or D_l u) :

where Γⁱjk are the Christoffel symbols.

Cross product

The cross product of any two vectors v and w in an n-dimensional vector space is the Tensor T = v x w of order n-2 which components are as follows :

where εi...jk is the Levi-Civita symbol

Vector space E of dimension 3 :
The cross product is a vector T of components :
In covariant components : Ti = εijk v^j w^k
In contravariant components : Tⁱ = g^il Tl = g^il εljk v^j w^k
g^ij are the inverse components of the Metric tensor.

Vector space E of dimension 4 [GOU, Relativité Restreinte, p.87] :
In the Local rest space of the observateur E_u (see Figure in Local reference frame), the cross product (noted x_u) between any two vectors v and w of E_u is a vector which is written : v x_u w = Ε(u, v, w, .)
where :
Ε is the Levi-Civita tensor
Ε(u, v, w, .) is the vector representing the Linear form Ε(u, v, w, z) for the Scalar product g.

Curvature

The word "curvature" has several meanings in Relativity :

The first three curvatures depend only on the gravitational potentials gab and their first and second derivatives with respect to the coordinates. In Minkowski Metric, they are all zero and correspond to the flat Space-time of Restricted Relativity.

Curvature tensor (or Riemann-Christoffel tensor)

Figures above from left to right : Georg Friedrich Bernhard Riemann and Elwin Bruno Christoffel

The curvature tensor is a Tensor of order 4. It is the most complete measure of the local deformation of a curved Space-time. It has 4⁴ = 256 components.
According to the Classic Sign Conventions (MTW) and using the Convention of partial derivative, its componants have as expression :

where Γⁱjk are the Christoffel symbols.

The number of independent components of the curvature tensor is equal to 20 (instead of 256).
In Cartesian coordinates, all the components are of dimension m^-2
The Space-time is called "flat" when the curvature tensor is zero.

Warning : Unlike classic MTW sign conventions, some authors define the Tensor of curvature by the opposite of the Tensor of curvature above [GOU, Relativité générale, p.110].

Properties :
Rⁱjkl is Antisymmetric with respect to the two last indexes : Rⁱjkl = -Rⁱjlk
Rijkl is Antisymmetric with respect to the two first and last indexes : Rijkl = -Rjikl = -Rijlk
Rijkl is Symmetric with respect to the two pairs of extreme indexes : Rijkl = Rklij = Rjilk
    where : Rijkl = gim R^mjkl
First identity of Bianchi : Rⁱjkl + Rⁱklj + Rⁱljk = 0, also written : Rⁱ[jkl]
Second identity of Bianchi : Rⁱjkl_;m + Rⁱjlm_;k + Rⁱjmk_;l = 0, also written : Rⁱj[kl_;m]
Identity of Ricci :
    For any Contravariant vector v we have : Rⁱjkl v^j = vⁱ_;l;k - vⁱ_;k;l
    Curvature tensor thus measures the non-commutativity of double Covariant derivatives [GOU, Relativité générale, p.109].
Another identity [SEN, Relativité générale, p.62] :
    For any doubly Contravariant Tensor T we have : Rⁱjkl T^jm + R^mjkl T^ij = T^im_;l;k - T^im_;k;l

Dark energy

Dark energy is an hypothetical global repulsive force uniformly filling the entire universe, which tends to accelerate the universe expansion (therefore requiring : Λ > 0).
See Standard cosmological model.

Dark matter

Dark matter is a hypothetical material providing extra gravity that galaxies need to avoid breaking apart during their rotation.
See Standard cosmological model.

Dilation of durations (or Slowing down of moving clocks)

See simple explanation in Time relativity, simple definition in Time, rigorous definition in Lorentz factor and practical application in GPS.

Direct and indirect base

If ei are the basic vectors of Space-time, the base {ei} is called [GOU, Relativité Générale, p.21] :

where Ε is the Levi-Civita tensor.

Divergence

The divergence of a vectors field accounts for the infinitesimal variation of the volume (or electric charge) around a point. The divergence of a Tensor generalizes this notion.
The divergence of a Tensor U of order (n) is the tensor Div(U) of order (n - 1) producted by contracting one of the index of the Covariant derivative with the derivation index.
For a twice contravariant Tensor, there are two possible divergences : right divergence (component U^ij _;j) and left divergence (component U^ij _;i). The divergences are equal only if the tensor is Symmetric or Antisymmetric.

Example : Einstein tensor Sab has zero divergence (S^ab_;a = 0).

Doppler effect (or Doppler-Fizeau effect)

The Doppler effect is the frequency change of a periodic phenomenon induced by the movement of the emitter with respect 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 O of a reference frame R, then any Observer O' of the reference frame R' in uniform rectilinear translation of velocity v with respect to R will perceive this same wave at a frequency f' different from f. The expression of f' is the following in different cases.
u is the unit vector of the light propagation MO (see Figure in Aberration).
γ is the Lorentz facteur

(D1) Longitudinal Doppler effect (u parallel to v) :
    f' = f γ (1 - (v.u)/c)
When v is small compared to c, we find the approximate non-relativistic formulas :
    f' = f (1 - (vr.u)/c) for mobile receiver (velocity vr) and immobile emitter with respect to the propagation medium
    f' = f / (1 - (ve.u)/c) for immobile receiver and mobile emitter (velocity ve = -v) with respect to the propagation medium
    f' = f (1 - (vr.u)/c) / (1 - (ve.u)/c) for mobile receiver (velocity vr) and mobile emitter (velocity ve) with respect to the propagation medium (relative velocity vr - ve = v).

(D2) Transverse Doppler effect at the emission (u perpendicular to v dans R) :
    f' = f γ

(D3) Transverse Doppler effect at the reception (u perpendicular to v dans R') :
    f' = f γ^-1

(D4) Doppler effect (general formula) :
If the light propagation u makes with the velocity v an angle θ in R and θ' in R' (see Figure in Aberration), then we have the relation :

the relation between the angles θ and θ' being given by the Aberration formula.
For θ = θ' = 0° or 180°, we find the formula (D1) with shift directed towards red or blue according to whether the Observer of R' moves away or approaches the light source of R.
For θ = 90°, we find the formula (D2) with shift directed towards blue.
For θ' = 90°, we find the formula (D3) with shift directed towards red.
When v is small compared 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) with respect to the propagation medium (relative velocity vr - ve = v).

Dual (space and base)

The dual space E* of a vector space E is the space of coordinated Linear forms on E.
E* is also a vector space and of same dimension (n) as E.
If ej are the basic vectors of E and eⁱ the basic vectors of E*, then we have the following relations :

where δ is the Kronecker symbol

Any vector x is therefore expressed in each base as follows :
    x = xi eⁱ
    x = xⁱ ei
where xi and xⁱ are respectively the Covariant and contravariant components of the vector x.

Geometric interpretation (see Figure above) :
The vector eⁱ is orthogonal (with respect to the Scalar product) to all vectors ej having different index j (eⁱ.ej = 0) and has a scalar product equal to 1 with the vector ej having the same index (eⁱ.ei = 1).

Properties :
ei = gij e^j
eⁱ = g^ij ej
where gij and g^ij are respectively the direct and inverse components of Metric tenseur

Einstein effect (or gravitational spectral shift or gravitational Doppler effect)

A frequency produced by a light 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 :

where r^* is the gravitational radius (r^* = 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 effect has an impact in everyday life : if it were not taken into account in the gravitational Earth field, the GPS positioning system would be completely inoperative !

Einstein Field Equation (EFE)

See Einstein Field Equation

Einstein tensor

The Einstein tensor (Sab = Rab - (1/2) gab R + Λ gab) measures the local deformation of the chronogeometry of Space-time and represents its curvature at a given point. It is a Tensor of order 2, Symmetric and with zero Divergence (S^ab_;a = 0).
Its components are given by Einstein Field Equation.
In Cartesian coordinates, all the components are of dimension m^-2

Electromagnetic tensor (or Maxwell tensor or Faraday tensor)

The notations are those in Maxwell Equations.
In Restricted Relativity, the Lorentz force (F_LORENTZ = q (E + v x B)) is written in a tensorial form whose components are the following [GOU, Relativité Restreinte, p.538] :

where f_LORENTZ is the Quadri-force of Lorentz and u is the Quadri-velocity of the particle.

Fij is the electromagnetic tensor. It is a Antisymmetric Tensor of order 2.
In Cartesian coordinates, all the components are of dimension m^-1.V or C^-1.N or kg.m.s^-3.A^-1.
In the Local reference frame of the Observer, and according to the Signature convention (-, +, +, +), these components are written [GOU, Relativité Restreinte, p.541] as follows :

where :
Ei = spatial components of the electric field E
Bⁱ = spatial components of the magnetic field B
δⁱj = Kronecker symbol
εijkl = Levi-Civita symbol with 4 indexes

By index increasing (see Elementary tensor operators), we find the components of the Tensors Fⁱj and F^ij in the following form :

Fⁱj = g^ik_MINK Fjk
Fⁱi for i ≥ 0 = 0
Fⁱ0 for i > 0 = F⁰i = Ei
F²1 = -F¹2 = -c B³
F³1 = -F¹3 = c B²
F³2 = -F²3 = -c B¹

F^ij = g^il_MINK F^jl
Fⁱⁱ for i ≥ 0 = 0
Fⁱ⁰ for i > 0 = -F⁰ⁱ = -Ei
F²¹ = -F¹² = -c B³
F³¹ = -F¹³ = c B²
F³² = -F²³ = -c B¹

g^ij_MINK are the inverse components of the Minkowski metric.

Elementary tensor operators

Let U, V and T be Tensors of arbitrary order and valence bearing on the indexes i, j, k, l...
Using the Convention of summation, the following elementary operations are defined on these Tensors :

Energy and impulse of a particle mesured by an observer

Energy E and impulse vector P of a particle M, mesured by an observateur O in his Local reference frame at time τ, are given by the following relations (see Figure above) [GOU, Relativité Restreinte, p.275 to 278] :

L, p and m are the Universe line, Quadri-impulse and mass at rest (or proper mass) of the particle.
L0, u0 and a0 are the Universe line, Quadri-velocity and quadri-accelaration of the Observer.
γ is the Lorentz facteur
ω is the Quadri-rotation of the Observer O.
v is the velocity of point M with respect to the Observer O in his Local rest space E_u0.
"x_u0" is the Cross product operator between two any vectors of E_u0, what is written : ω x_u0 OM = Ε(u0, ω, OM, .)
where :
Ε is the Levi-Civita tensor
Ε(u0, ω, OM, .) is the vector representing the Linear form Ε(u0, ω, OM, z) for the Scalar product g.

The impulse vector P (of dimension kg.m.s^-1) is the orthogonal projection of vector p on E_u0 (with P.u0 = 0).
The energy E (of dimension J or kg.m².s^-2) satisfies the Einstein relation : E² = m² c⁴ + P.P c² which is simplified in : E = ||P|| c for a particle of zero mass (case of photon for example).

When L crosses L0 at the proper time τ0 (OM = 0) or when O is an inertial observer (a0 = ω = 0), these relations are simplified by :

The first relation expresses the equivalence between mass and energy. It was established in 1905 by Einstein [EIN Ist die Trägheit], a few months after his founding article on Restricted Relativity.
This relation can be written : E = E_mass + E_cin, where E_mass is the mass energy (E_mass = m c²) and E_cin the kinetic energy of the particle (E_cin = (γ - 1) m c²).
In non-relativistic limit (||v|| << c) with limited development of γ, it is found that : E_cin = (1/2) m v.v and : P = m v

Energy-impulse tensor

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 [GOU, Relativité Générale, p.112] :

It is a tensor of order 2, Symmetric and constructed so that its zero Divergence (T^ab_;a = 0) expresses in Continuum mechanics the two laws of conservation of impulse and energy (3 equations for the impulse vector and an equation for the energy).
In Cartesian coordinates, all the components are of dimension kg.m^-1.s^-2

Energy-impulse tensor of ElectroMagnetic field

The notations are those in Maxwell Equations.
The components of the Energy-impulse tensor (Tij_EM) of ElectroMagnetic field are the following [GOU, Relativité Restreinte, p.635] :

where Fij is the Electromagnetic tensor.

In Restricted Relativity (Minkowski metric), the calculations give in Cartesian coordinates [GOU, Relativité Restreinte, p.636] :
    T00_EM = energy density = (1/2) ε₀ (E.E + c² B.B)
    Ti0_EM = T0i_EM for i > 0 corresponding to (-1/c) times φ with φ = Poynting vector = (1/ μ₀) E x B
    Tij_EM for i and j > 0 corresponding to Sij = ε₀ ( (1/2) (E.E + c² B.B) δij - (Ei Ej + c² Bi Bj) )
where δ is the Kronecker symbol.

Properties :
The trace T_EM of the tensor Tij_EM is zero as follows : T_EM = g^ij Tij_EM = ε₀ (Fim g^ij F^mj - (1/4) g^ij gij Fkl F^kl) = ε₀ (Fim F^im - (1/4) 4 Fkl F^kl) = 0

Energy-impulse tensor of Perfect Fluid

A fluid is called "perfect" when the viscosity and thermal conduction effects can be neglected, which is the case in cosmology where the Universe expansion is assumed to be adiabatic (without heat exchange with the outside).
The components of the Energy-impulse tensor (Tij_FP) of Perfect Fluid are the following [GOU, Relativité Générale, p.114] :

where :
ρ c² and p represent respectively the energy density and the pressure of the fluid, both measured in the reference frame of the fluid.
u is the unit field which represents at each point the Quadri-velocity of a fluid particle (with ui = gik u^k and uj = gjk u^k).

When the Observer is co-mobile with the fluid, the calculations give in Cartesian coordinates [GOU, Relativité Générale, p.114] :
    T00_PF = ρ c²
    Ti0_PF for i > 0 = T0i_PF = 0
    Tij_PF for i and j > 0 corresponding to Sij = p δij
where δ is the Kronecker symbol.

The Perfect Fluid satisfies the low energy condition when : (ρ ≥ 0) and (ρ c² + p ≥ 0), and the dominant energy condition when : (ρ c² ≥ |p|).

Friedmann equations

See Solution of Einstein Field Equation with Friedmann-Lemaitre-Robertson-Walker metric

Friedmann-Lemaitre-Robertson-Walker metric (or FLRW metric)

Figures above from left to right : Alexander Alexandrowitsch Friedmann, Georges Lemaître, Howard Percy Robertson and Arthur Geoffrey Walker

The Friedmann-Lemaitre-Robertson-Walker metric is a Relativistic metric corresponding to a spatially homogeneous and isotropic Space-time.
In spherical coordinates (r > 0, colatitude θ = [0, π], longitude φ = [0, 2 π]) (see Figure in Space-time), the metric is written as follows for any coordinate system (ct, r, θ, φ) and by taking the Signature convention (-, +, +, +) :

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 curvature factor or universe radius (a(t) > 0).
The coordinate r is dimensionless and the radius (a) has the dimension of a length.
The case k = 0 corresponds to a flatness of spatial sections and not of Space-time.

The gravitational potentials gij then are the following :

The sign of d(a)/dt informs about the universe evolution : positive if expansion, negative if contraction and zero if static.
The spatial coordinates (xⁱ) then describe spatial hypersurfaces of Euclidean type (for k = 0), hyperspherical type (for k = 1) and hyperbolical type (for k = -1), whose the spatial curvature k* is constant and is equal to : k* = 6 k a(t)^-2

Galileo transformation

See Lorentz-Poincaré transformation

Geodesic

For a given Relativistic metric a geodesic is the curve (or trajectory) of the shortest distance between two given points.
Geodesics thus describe the movement of free particles (material systems or photons), that is when they are not subjected to an external force other than gravitation in the context of General Relativity.
John Archibald Wheeler, American specialist of General Relativity, says : "Matter tells space-time to bend and space-time tells matter how to move".

If we choose to parameterize the trajectory by the proper Time τ of the particle satisfying to : ds² = -c² dτ², the geodesic tensor equations are written as follows [GOU, Relativité Générale, p.48] :

If the Metric tensor g is known (and therefore Γ), this equation constitutes a system of 4 differential equations of the second order for the 4 functions xⁱ. According to Cauchy theorem, this system admits a unique solution if the following initial conditions are fixed :
    xⁱ(0) = four arbitrary constants
    (dxⁱ/dτ)(0) = uⁱ₀
    uⁱ₀ being four constants satisfying : gij uⁱ₀ u^j₀ = -c²

In Restricted Relativity (Minkowski metric) with Cartesian coordinates, the coefficients gij are all constant, which cancels all the Christoffel symbols. The equations of the geodesics are reduced to : d²xⁱ / dτ² = 0 whose solutions are the ordinary straight lines : xⁱ(τ) = aⁱ(τ) τ + bⁱ

Geodesic of a material body

Trajectory :
The trajectory of a material body in a gravitational field with spherical symmetry (Schwarzschild metric) is a like-time Geodesic [GOU, Relativité Générale, p.71].
We suppose that the mass m of the material body is very small compared to the mass M of the central body of the Schwarzschild metric and that the Observer is placed in the equatorial plane z = 0 of the central body (θ = π/2, see Figure in Space-time).
The trajectory of the material body is flat and given by the following differential equation [GOU, Relativité Générale, pp80, 74, 72, 63, 64] :

where :
r* = Schwarzschild radius (r* = 2 G M c^-2)
M = mass of the central body
ε (in m².s^-2) = c² (1 - r*/r) dt/dτ = -c² ut.u = -c² g00 u⁰
l (in m².s^-1) = r² sin²[θ] dφ/dτ = c uφ.u = c g33 u³
l_crit = 3^1/2 r* c
u = Quadri-velocity of the material body
ut and uφ = vectors of the Natural base associated respectively with stationarity and azimuthal symmetry of the metric.
τ = proper Time of the material body.

Note that the two quantities ε and l are constant along the Geodesic.
In Newtonian limit (when the material body is infinitely far from the central body), ε and l can be interpreted as follows :
    ε = c² + E0/m = energy per unit mass of the material body, measured by a static Observer.
    l = angular momentum (relative to the z axis) per unit mass of the material body, measured by a static Observer.
    E0 = mechanical energy of the material body (sum of kinetic energy and gravitational potential energy).

According to the coupled values of l and ε, the trajectory of the material body is then as follows [DEV, Un peu de physique, Trajectoire d'une particule] :
In the case where : |l| < l_crit, the material body is irreparably attracted by the central body.
In the case where : |l| > l_crit, the material body avoids the central body by continuing its course or, on the contrary, goes into orbit around.
In the case of an orbit, the trajectory can be a perfect circle or a quasi-ellipse with advance of the periastron (see Figure above). At the Newtonian limit, the trajectory is a stable ellipse that verifies Kepler's laws.

Periastron :
The periastron is the point of the trajectory closest to the central body (see Figure above).
The advance δφ_per of the periastron (at first order of r*/r) is as follows [GOU, Relativité Générale, p.82] :

where :
a = half-major axis of the ellipse
e = eccentricity of the ellipse

In the case of the periastron advance around the Sun (perihelion) of the planet Mercury (with M_Sun = 1.9891 10³⁰ kg, a_Mercury = 5.79 10¹⁰ m, e_Mercure = 0.206), we find : δφ_per = 5.0 10^-7 rad. Since the orbital period of Mercury is 88 days, the cumulative effect after a century is : Δφ_per = 43"
This result was given by Einstein from the publication of his General Relativity theory in 1915 [EIN Die_Grundlage].

Geodesic of a photon

Figures above from left to right : Figure 1 = Deflection of light rays for large b ; Figures 2 to 4 from [DEV A little physics - Trajectory of a ray of light] = Deflection of light rays according to the value of b

Trajectory :
The trajectory of a photon in a gravitational field with spherical symmetry (Schwarzschild metric) is a like-light Geodesic [GOU, Relativité Générale, p.82].
We suppose that the Observer is placed in the equatorial plane z = 0 of the central body of the metric (colatitude θ = π/2, see Figure in Space-time).
The photon trajectory is flat and given by the following differential equation [GOU, Relativité Générale, p.85, 86] :

where :
r* = Schwarzschild radius (r* = 2 G M c^-2)
M = mass of the central body
b = impact parameter for photons coming from infinity (see Figure 1 above)
b_crit = (3/2) 3^1/2 r*

In the case where : b > b_crit, the photons coming from infinity will approach the central body and return to infinity. The trajectory looks like a hyperbola for the large values of b (see Figure 1 above) and can make several rounds of the central body before starting again for the values of b near b_crit (see Figure 2 above).
In the case where : b ≤ b_crit, the photons are trapped by the central body and a distant Observer can not receive them (see Figures 3 and 4 above).

This differential equation is integrated in the following form which is solved as an elliptic integral [DEV Un peu de physique - Trajectoire d'un rayon lumineux].

Periastron :
The periastron is the point of the trajectory closest to the central body (see Figure 1 above).
Its radius r_per is given by the following equation [GOU, Relativité Générale, p.86] :

This cubic equation is solved as follows :

Deflection :
In the case where : b >> b_crit and at first order of r*/b, the deflection δφ of the light ray from a straight line path (see Figure 1 above) is then as follows [GOU, Relativité Générale, p.86, 87] :

In the case of a trajectory skimming the Sun surface (b = R_Sun = 6.96342 10⁸ m, with M_Sun = 1.9891 10³⁰ kg), we find : r* = 3,0 10³ m and δφ = 1.75"
This deflection was highlighted by Arthur Eddington and his team, measuring the stars position near the solar disk during the eclipse of 1919. It is this event that made Einstein famous among the general public.
The deflection of light rays is at the origin of the phenomenon of Gravitational mirage.

Apparent radius :
For an Observer located at infinity, the apparent radius Ra of a star of radius R and that of a Black hole of radius r* (considered to delimit its surface) then is [GOU, Relativité Générale, p.299, 300] :

The star always appears bigger than it is (Ra > R).
For the Black hole, the photons with low impact parameter (b < b_crit) are trapped by the Black hole and the Observer receives no photon. So he observes on the sky background a black disk of apparent radius almost three bigger than it is (Ra = 2.60 r*).

GPS

GPS (Global Positioning System), originally known as NAVSTAR System (NAVigation System by Timing And Ranging), is a very precise three-dimensional positioning system (latitude, longitude, altitude) with worldwide coverage, developed by the United States Department of Defense (DoD).
GPS is relative to the worldwide WGS84 geodetic system (see Meteo - Appendix 2) and provides an accuracy of 10 m horizontally and 30 m vertically for the basic public service.
The European GPS counterpart is the GALILEO system, operational since 2016.
GALILEO is relative to the European GTRF geodetic system which is compatible with WGS84 to less than 1 m. GALILEO provides an accuracy of 4 m horizontally and 8 m vertically for the basic public service.

Detail about GPS (cf [GOU, Relativité Générale, p. 203-205]) :
The GPS is based on a set of at least 24 satellites distributed 4 by 4 in 6 circular orbits of orbital period T = 12 h and radius r_sat = 26561 km = 4.16 R_Earth (with R_Earth = 6378 km and M_Earth = 5.97 10²⁴ kg). See Figure above.
Each satellite (i) carries a cesium atomic clock and emits a radio signal including its emission date (ti) and its three spatial coordinates ri (deduced ephemeris related to its orbit).
Any Observer on Earth who receives at the same time (t) the signals of at least four satellites can then calculate his position r. If Space-time was flat (Minkowski metric), the observer should simply solve the system of 4 equations : ||r - ri|| = c (t - ti). But in reality, it is essential to consider three types of temporal shifts :
1. Imprecision of cesium atomic clocks :
To have a precision of one meter on the position r, it is necessary a precision on the dates ti of the order of 3 ns (dt = 1 m / c). The stability of cesium atomic clocks is such that : dt/t = 10^-13. dt = 3 ns is thus reached in t = 10 h approximately. To achieve the required accuracy, the clocks are then adjusted several times a day using signals sent from the ground.
2. Durations dilation :
This durations dilation constitutes the transverse (second order) Doppler effect. The first-order Doppler effect in v/c affects only the signal frequency and not the content of the delivered message (date and position of the emitter).
As the satellites are moving relative to the observer, their orbital speed is : v = (G M_Earth / r_sat)^1/2 = 3.87 km.s^-1. Hence the result : dt/t = γ - 1 = 8.3 10^-11 (where γ is the Lorentz factor). If no correction is applied, dt = 3 ns is reached in 30 seconds !
3. Einstein effect :
Since the satellites are four times higher than the observer on the ground, Einstein effect therefore applies with an offset : dt/t = G M_Earth c^-2 (1/R_Earth - 1/r_sat) = 5.3 10^-10
If no correction is applied, dt = 3 ns is reached in 6 seconds, which would correspond to a positioning error of 14 km in a day !
To take into account these two relativistic shifts, the proper frequency of the atomic clocks is corrected before the radio signal emission towards the Earth (see [GOU, Relativité Générale, p. 205, 211]).

Gravitational mirage (or gravitational lens)

The deflection of light rays is at the origin of the phenomenon of gravitational mirage.
This phenomenon can take different aspects depending on the alignment of the Observer, the mass that deforms (galaxy for example) and the observed source (quasar for example). We can see perfect rings (called Einstein rings), circles arcs or simply images multiplied from the source (see Figures above).
Not only do these gravitational lenses deflect but amplify the deflected light, which fascinates astronomers.

Hubble law (or Hubble-Lemaître law)

The Hubble-Lemaître law states that galaxies move away from each other at an expansion speed v approximately proportional to their distance 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 10¹⁶ 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.

Image of an moving object

[GOU, Relativité Restreinte, p. 163-171]

- For the same Observer O at a fixed instant τ of his proper Time, we must not confuse the position (x, y, z) of an object M and its perceived image. This image or photograph is generated by all the photons arriving from M to O. The Lengths contraction concerns in particular the position of the object and not its image.
- For a cube that moves perpendicularly to the sight line of an Inertial observer, he perceives the image of a cube that has been rotated by a given angle θ (see Figure 1 above). He perceives the back face of the cube in addition to the face facing him. This apparent rotation is not relativistic and results from the finite time of light propagation. Relativity intervenes only through the contraction of the face oriented towards the observer according to the Lorentz-Poincaré Transformation.
- The image of a moving sphere is a sphere that seems to have undergone a rotation and presents its opposite face to the movement (see Figure 2 above). If we take into account only the finite time of light propagation in a (Galilean) non-relativistic theory, a sphere would appear elongated in the direction of the movement. It is thanks to the Lengths contraction that it appears exactly spherical. Only the angular size of the circle can change.
- Some astrophysical movements may be superluminous, with apparent speeds greater than c. These superluminous apparent speeds are generated by subluminic movements in a direction close to the sight line. Thus the micro-quasar GRS 1915+105 was observed in 1994 in our galaxy with an apparent speed 1.25 c corresponding to an material ejection at the speed 0.92 c at an angle θ of 70°.

Index contraction (or tensor trace)

The contraction operation of the index of a mixed component of a Tensor consists in choosing two indexes, one covariant and the other contravariant, then in equalizing and summing them with respect to this twice repeated index.
For example, for a Tensor U of order 3 whose mixed components are U^ijk, the contraction on the indexes j and k gives the Tensor Tⁱ = U^ikk = Uⁱ¹1 + Uⁱ²2 + ... Uⁱⁿn
The quantities Tⁱ (contracted components of the tensor U) form the components of a tensor T of order 1.
Note that the "matrix product" operator is a particular case of the Tensor product Uⁱj ⊗ V^kl contracted in the form : Tⁱl = Uⁱk V^kl

Another examples :

Case of the Curvature tensor Rⁱjkl : the contraction on the indexes i and k gives the Ricci tensor : Rjl = R^kjkl
Warning : If the same letter is kept for the resulting tensor (after deletion of the identical indexes), then the notation may become ambiguous. For the Tensor Rⁱjlk (different from Rⁱjkl), the contraction on the indexes i and k gives the Tensor R^kjlk = Rjl, i.e. the opposite of the Ricci Tensor despite the identical notation Rjl.

Case of the Curvature tensor Rijkl : the Index increasing of the index i gives, after contraction on the indexes m and i, the Tensor g^mi Rijkl = R^miijkl = R^mjkl
Warning : If the same letter is kept for the resulting tensor (after deletion of the identical indexes), then the notation may become ambiguous. For the Tensor Rjikl (different from Rijkl), its contraction with g^mi gives the Tensor g^mi Rjikl = R^mijikl = R^mjkl, i.e. the opposite of the Tensor R^miijkl despite the identical notation R^mjkl.

Index increasing

A lower index can be changed to a higher index by multiplication with the inverse Metric tensor g^ij then Index contraction. Examples :
Uⁱk = g^ij Ujk
U^ik = g^il g^km Ulm
Warning : If the same letter is kept for the resulting tensor (after deletion of the identical indexes), then the notation may become ambiguous (see Index contraction).

Index lowering

A higher index can be changed to a lower index by multiplication with the Metric tensor gij then Index contraction. Examples :
Uik = gij U^jk
Ulm = gjl gkm U^jk
U^klm = glp U^kpm
Warning : If the same letter is kept for the resulting tensor (after deletion of the identical indexes), then the notation may become ambiguous (see Index contraction).

Inertial (or galilean) reference frame

In classic physics, an inertial reference frame (called "stationary system" by Einstein [EIN Zur_Elektrodynamik]) is a reference system in which the inertia principle is verified, that is to say that any free material body (on which the resultant forces is zero) is in rectilinear translational movement (without rotation) with uniform speed, the immobility being a special case.
A material body in rotational movement or non-rectilinear translational movement or non-uniform speed constitutes a non-inertial reference frame.

In Restricted Relativity, a body (or Observer) is called "inertial" when its Local reference frame is constant along its Universe line, i.e. it verifies the following relation [GOU, Relativité Restreinte, p.91, 92] :

where τ is the proper Time of the body
eα are the four basic vectors of the Local reference frame linked to the body.

A body is called "inertial" also when both its trajectory is a straight line of Minkowski's Space-time (zero quadri-acceleration a) and its Quadri-rotation ω is zero.
a and ω are quantities measurable by the observer contrary to his Quadri-velocity u [GOU, Relativité restreinte, p.90].

Kronecker symbol

The expression of the Kronecker symbol δ is as follows :

Warning : the Kronecker symbol is not a Tensor. For example, the index lowering : δij = gik δ^kj is not possible, which is false considering that : gik δ^kj = gij

Property :
Index replacement in any Tensor T : Tjk = δⁱj Tik

Lengths contraction (or FitzGerald-Lorentz contraction)

In Restricted Relativity, if a bipoint of given proper length (l0) is fixed in a reference frame, then its length in any other reference frame is less than the proper length and is called apparent length (l).
Some authors talk about Lengths contraction. This contraction is not physical but simply results from measurements that only make sense with respect to a given reference frame.
The lengths contraction intervenes only in the direction of the relative movement between reference frames. There is no contraction in the orthogonal directions.

Levi-Civita symbol

The expression of the Levi-Civita symbol ε is as follows (not to be confused with the Levi-Civita tensor) :

When any two indexes are interchanged, equal or not, the symbol is negated : ε...i...l... = -ε...l...i...

For 3 indexes (i,j,k) we have :
    εijk = +1 for 012 or 120 or 201
    εijk = -1 for 021 or 102 or 210
For 4 indexes (i,j,k,l) we have :
    εijkl = +1 for 0123 or 0231 or 0312 or 1032... or 3210
    εijkl = -1 for 0132 or 0213 or 0321 or 1023... or 3201

ε allows to express many vectorial operations in a compact form. In a vector space of dimension 3 and with the Convention of summation, it can be written for example :
- Cross product (w = u x v) of components : wⁱ = ε^ijk uj vk
- Curl (w = curl(u)) of components : wⁱ = ε^ijk uk_,j
- Determinant (d = det(u,v,w)) of component : d = ε^ijk ui vj wk = εijk uⁱ v^j w^k

Levi-Civita tensor

The Levi-Civita tensor Ε is a Tensor of order 4 which is written as follows, for any quadruplet of vectors (u, v, w, z) [GOU, Relativité Restreinte, p.20, 21, 482 to 487] :

Its 256 components in any base {ei} are as follows :

Properties :
- Ε is Antisymmetric (or alternating) : the quantity Ε(u, v, w, z) is 0 when two of its arguments are equal. For example : Ε(u, v, u, w) = 0
- In any Orthonormal and Direct base : Ε(u, v, w, z) = det(u, v, w, z)
- Ε only has one independent component Ε0123 = μ (-g)^1/2
- The quantity (Εijkl Ε^ijkl) is egal to -24
- If ei are the basis vectors : Ε(e0, e1, e2, e3) = μ (-g)^1/2
- In any Orthonormal base : Ε(e0, e1, e2, e3) = μ

Light-cone

The light-cone is a fundamental notion of Restricted Relativity allowing the distinction between a past event, a future event and an inaccessible event (in the past or in the future).
If a light signal starts from origin point O to point M of coordinates (x, y, z), then the set of trajectories of the light rays from O in the Space-time will be a hypercone of equation : c² t² = x² + y² + z², called Light-cone (see Figure above where x represents in a simplified way the three spatial coordinates x, y and z).

Linear form

Figures above from left to right : René Maurice Fréchet and Frigyes Riesz

Any linear form ω is an application that matches a number ω(u) to any vector u, and this in a linear way [GOU, Relativité Restreinte, p.21] :

ω(u) is sometimes noted <ω, u> with the bra-ket notation as in quantum mechanics.

Examples : if a and b are vectors of a vector space E, the linear applications that match the vector u respectively to the scalars u.a and det(u, a, b) are both linear forms.

Properties :
- There is only one vector w that represents the Linear form ω of the Dual space (E*) for the Scalar product.
- The vector w satisfies the relation : w.z = ω(z) for any vector z
- In any base {ei} of E, its Covariant components are written : wi = w.ei = ω(ei) and its contravariant components are written : wⁱ = g^ij wj = g^ij ω(ej) = ω(g^ij ej) = ω(eⁱ)
- Notation [GOU, Relativité Restreinte, p.87, 538] : The vector w is sometimes noted : w = ω(.)

Local reference frame

In Restricted Relativity, the local reference linked to a body (or Observer O) along its Universe line is a quadruplet of vectors (e0, e1, e2, e3) defined in any point (or event) of this Universe line and satisfying the following properties [GOU, Relativité Restreinte, p.60, 79, 80] :
    - (e0, e1, e2, e3) is an Orthonormal and Direct base, called Serret-Frenet tetrad.
    - e0 is tangent to the Universe line of the body (condition : e0 = u), where u is the Quadri-velocity of the body.
    - the field (e0, e1, e2, e3) is class C¹, i.e. the base (e0, e1, e2, e3) varies infinitesimally when we passe from a given point to a neighboring point on the Universe line.

The three vectors e1, e2 and e3 are therefore orthogonal to e0 = u. They belong to the Local rest space E_u of the Observer O (see Figure above).
Note that e1 can be choosen tangent to the curvature a of the Universe line of the body (condition : e1 = a/||a||), where a is the quadri-acceleration of the body. The length 1/||a|| is called curvature radius of the Universe line at point O.

Properties :
    d(e0)/dτ = c a
    d(e1)/dτ = c ||a|| e0 + c T1 e2
    d(e2)/dτ = -c T1 e1 + c T2 e3
    d(e3)/dτ = -c T2 e2
τ is the proper Time of the body
T1 and T2 are respectively the first and the second torsion of the Universe line of the body.

Local rest space of the observer

Let an Observer O of Universe Line L and Quadri-velocity u (see Figure above).
His local rest space is the vector hyperplane E_u orthogonal to u.

Lorentz factor

Let an Observer O of Universe Line L0 and Quadri-velocity u0 (see Figure above).
Let a particle M of Universe Line L and Quadri-velocity u.
It is assumed that L is situated in the neighborhood of L0, in the sense that L can be described in the Local reference frame of O, which means that the spatial distance between L0 and L is always much smaller than ||a0||^-1, where a0 is the Quadri-acceleration of O.
Let an infinitesimal increment dτ0 of the Proper time τ0 of O and dτ the Proper time of the particle when it moves from M(τ0) to M(τ0 + dτ0) along its universe line.
The Lorentz factor γ of the particle M with respect to the Observer O is defined by the following relation [GOU, Relativité Restreinte, p.99 à 109] :

The following relations are then demonstrated :

ω is the Quadri-rotation of the Observer O.
v is the velocity of point M with respect to the Observer O in his Local rest space E_u0.
"x_u0" is the Cross product operator between two any vectors of E_u0, what is written : ω x_u0 OM = Ε(u0, ω, OM, .)
where :
Ε is the Levi-Civita tensor
Ε(u0, ω, OM, .) is the vector representing the Linear form Ε(u0, ω, OM, z) for the Scalar product g.

When L crosses L0 at the proper time τ0 (OM = 0) or when O is an inertial observer (a0 = ω = 0), these relations are simplified by :

When L crosses L0 at the proper time τ0 (OM = 0) or when the Quadri-acceleration a0 of O is zero (in particular if O is an Inertial observer (a0 = ω = 0), then it is proved that : γ ≥ 1. Given the definition of γ as ratio of two proper durations, this property is called Durations dilation.

Lorentz-Poincaré transformation

See Lorentz-Poincaré transformation

Mathematical notations

The main mathematical notations used in this page are as follows :

Elementary tensor operators :
Operators synthesis
, d grad div rot Δ Δ ◊ : Convention of partial derivative
a^km b^m ck : Convention of summation (called "Einstein convention")
; : Covariant Derivative
Div : Divergence
* : Dual (space and base)
. ||x|| : Scalar product and norme
⊗ : Tensor product
x : Cross product

Greace alphabetic symbols :
Λ : Cosmological constant
γ : Lorentz factor
ω : Linear form
η : Minkowski matrix
Γ : Christoffel symbols
δ : Kronecker symbol
ε : Levi-Civita symbol
ε₀ : Dielectric permittivity in the vacuum

Latin alphabetic symbols :
a : Universe Radius
c : Light speed in the vacuum
Ε : Levi-Civita tensor
E_u : Local rest space of the observer
F : Electromagnetic tensor
g : Metric tensor
G : Universal gravitational constant
K : Gravitational coupling coefficient
k : Spacially constant curvature
O : Observer
R : Ricci tensor
R : Scalar curvature
r* : Gravitational radius
S : Einstein tensor
T : Energy-impulse tensor

Maxwell equations

Non-relativistic electromagnetism :

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)

The equations of James Clerk Maxwell specify the evolution of electromagnetic fields E and B. In vacuum with sources (point electric charges and/or their electric powers), these fields writte as follows :

The two densities ρ and J are connected by the relation of the charges conservation (see proof below) :
    (R5) div(J) + dρ/dt = 0

E is the electric field (in m^-1.V or C^-1.N or kg.m.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)
ρ is the Electric charge density (in m^-3.s.A)
μ₀ is the permeability of vacuum : μ₀ = 4 π 10^-7 kg.m.A^-2.s^-2
ε₀ is the dielectric permittivity in vacuum such that :
(R0) ε₀ = 1 / (μ₀ c²)
c is the light speed in vacuum (c = 2.99792458 10⁸ m.s^-1
The operators used are the following :
    v1.v2 and v1 x v2 : Scalar product and Cross product of any two vectors v1 and v2.
    div(v) and curl(v) : divergence and curl of any vector v (see Convention of partial derivative).
    Δ : Laplacian operator (see Convention of partial derivative).
    ◊ : Alembertian operator (see Convention of partial derivative).

Relativistic electromagnetism :

Electrical and magnetic potentials :
The Electric potential V and the Magnetic potential A are defined from relations (R2) and (R3) by the following relations (see proof below) :
    (R6) rot(A) = B
    (R7) -grad(V) = E + d/dt(A)
If A satisfies the relation (R6), then any potential A' = A + grad(Ψ) also satisfies (R6). The magnetic potential A is therefore determined at within the gradient of a scalar field Ψ.
Likewise if two electric potentials V and V' satisfie the relation (R7), then we can write :
-grad(V' - V) = d/dt(A' - A)
We can deduce :
grad(V' - V + d/dt(Ψ)) = 0
V' = V - d/dt(Ψ) + f(t)
The electric potential V is therefore determined at within a time function f(t).

Lorenz gauge condition :
Replacing of the relations (R6) and (R7) in relations (R1) and (R4) gives :
(Δ - c^-2 d²/dt²)(V) = -ρ/ε₀ - d/dt(C)
(Δ - c^-2 d²/dt²)(A) = -μ₀ J + grad(C)
C = div(A) + c^-2 d/dt(V)
Since the potential A is determined at within the gradient of a scalar field ψ, we can arbitrarily write :
    (R8) C = 0
This relation is called Lorenz gauge condition (of the Danish physicist Ludvig Valentin Lorenz, not to be confused with the Dutch physicist Hendrik A. Lorentz), which ultimately gives :
    (R9) ◊(V) = -ρ/ε₀
    (R10) ◊(A) = -μ₀ J
The three relations (R8), (R9) and (R10) constitute a group of equations that contain the same information as the Maxwell equations.
These relations are invariant by Lorentz-Poincaré transformation (see proof below). it is the same for the Maxwell equations.

Resolution of the equations :
In vacuum and without any source (ρ = 0 and J = 0), the relations (R9) and (R10) show that the electromagnetic field is described by two Alembert's equations called wave equations.
In the general case of a distribution of sources P located in a domain D carrying electric charges and currents, or whose density decreases sufficiently rapidly with distance, the solutions of the equations (R9) and (R10) are the following, with the additional conditions V(∞) = 0 and A(∞) = 0 :
    (R11) V(M, t) = ∑_D[ (1/ε₀) (4 π/||PM||) ρ(P, t - ||PM||/c) dτ ]
    (R12) A(M, t) = ∑_D[ μ₀ (4 π/||PM||) J(P, t - ||PM||/c) dτ ]
These solutions are called delayed potentials because the potentials values at a point M at a time t depend on those of the sources at all the points P at the previous instant (t - ||PM||/c).

Proofs :

Metric tensor (or fundamental tensor)

For a vector space E of dimension n having for basis vectors the set (e1, e2... en), we denote the Scalar product of two basic vectors in the form :

The Metric tensor is the Tensor gij of order 2, whose components are gij. It is a bilinear application of given Signature, that matches the vector pair (u, v) to the scalar : u.v.
gij is a Symmetric Tensor with zero Divergence (g^ab_;a = 0). Its 16 components gij (for i and j taken between 0 and 3) are called gravitational potentials. These are functions of x, y, z and t
In Cartesian coordinates, all the components are dimensionless.

The Inverse Metric tensor is the Tensor g^ij such that :

where δ is the Kronecker symbol.
The component g^ij can be calculated as follows (Cramer rule) :
g^ij = Cofactor_ji / g
with g = Determinant of the matrix gij
Cofactor_ij = (-1)^i+j Minor_ij
Minor_ij = Determinant of the sub-matrix given by deleting the row i and the column j in the matrix gij

Properties :
Duality : g^ij = eⁱ.e^j
g^ij gij = n
gij;k = g^ij;k = 0
gij g^ij_,k = -g^ij gij_,k
g < 0 in any vector base [GOU, Relativité Restreinte, p.484].

Minkowski metric

The metric of Hermann Minkowski is a Relativistic metric corresponding to the flat Space-time of Restricted Relativity. This metric is solution of the Einstein Field Equation under the conditions Λ = 0 and Tab = 0 (since the Curvature tensor is zero, so also the Ricci tensor Rab).

In Cartesian coordinates, the metric is written as follows for any coordinate system (ct, x, y, z) and by taking the Signature convention (-, +, +, +) :

This metric gij_MINK has the following properties : g^ij = g^ji = gij = gji


In spherical coordinates (r > 0, colatitude θ = [0, π], longitude φ = [0, 2 π]) (see Figure in Space-time), the metric is written as follows for any coordinate system (ct, r, θ, φ) and by taking the Signature convention (-, +, +, +) :

MTW sign conventions

In Relativity, the classic sign conventions are those of Misner, Thorne and Wheeler [MTW, Gravitation, 1973], also called "Landau-Lipschitz Spacelike Convention" (LLSC),
with a Metric tensor of Signature (-, +, +, +), a Curvature tensor defined by Rⁱjkl = Γⁱjl,k + .. ., and a Ricci tensor defined by Rij = R^kikj
If we define by (C1)(C2)(C3)(C4) the following four signs, these authors then classify most of existing conventions according to the three independent signs (C1)(C2)(C4).
    (C1) Signature (-, +, +, +)
    (C2) Curvature tensor defined by Rⁱjkl = Γⁱjl,k + ...
    (C3) Ricci tensor defined by Rij = R^kikj
    (C4) "Einstein sign" which is the sign in front of the second member K Tab of Einstein Field Equation and corresponding to (C4) = (C2)(C3)

These classic sign conventions are also used by Caroll (Spacetime and Geometry, 2004), Hartle (Gravity, 2003), Hawking & Ellis (The large scale structure of space-time, 1973), Poisson (A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics, 2004) and Gourgoulhon ([GOU, Relativité restreinte, pp 8][GOU, Relativité générale, pp 24, 109, 111]).

Multiplicity of proper times (or Desynchronization of perfect clocks)

The desynchronization of perfect clocks is the most extraordinary and counter-intuitive prediction of Restricted Relativity. The most astounding version of this prediction is the Twin paradox.
This effect was described for the first time by Einstein in 1905 in the following form : "If at point A there are two synchronized clocks and we move one of them at a constant speed v according to a closed curve which amounts to A, the displacement being completed in t seconds, then the mobile clock will delay by (1/2) t (v/c)² seconds on the immobile clock when it arrives at A (neglecting the fourth order of v/c and higher orders) [EIN Sur_electrodynamique, para.4]".
Einstein indicates however that this result is valid if "we make the hypothesis that the result obtained for a polygonal line is also true for a curved line".
Thus, two clocks associated with distinct Universe lines will generally be desynchronized when they intersect. More precisely, two perfect clocks (that is, never deregulating) and ideally synchronized clocks will shift if they are separated before reuniting them again. The number of seconds counted will differ between the two clocks, the cumulative time depending on the trajectory followed by each clock. If the reference frame is not Inertial, the offset will be an advance or a delay.

But what about the rational explanation of this effect ? Opinions are rare and divided, including :
- Pierre Spagnou, scientific author, teacher at ISEP, explains : "The effect is chrono-geometric : the clocks are not disrupted ; the cause is that the temporal "lengths" are not conserved during our movements, unlike the classic cinematic framework." [SPA Einstein_et_la_revolution, p.23]
- Henri Bergson, French philosopher, explains : "We are told that if two identical and synchronous clocks are in the same place in the reference system, if one is moved and brought back close to the other at the end of time t (time of the system), this one will delay on the other clock. In reality, it would be necessary to say that the moving clock exhibits this delay at the precise instant when it touches, still moving, the immobile system, and where it will go in. But as soon as it goes in, it will indicate the same time as the other..." [BER Durée, p.208].
It should be noted that we have not yet found an explanation of this effect which is intuitively satisfactory (as is the example of the Dilatation of durations given by Poincaré [POI L'Etat, p.311]).

Natural base

For any point M of coordinates (x⁰, x¹, x², x³) of Space-time, the natural base associated to point M is the set of vectors ui satisfying the following relation [GOU, Relativité Restreinte, p.492] [GOU, Relativité Générale, p.19] :

ui thus describes the increase of the coordinate xⁱ in the neighborhood of M (see Figure above) and is denoted vectorially : ui = d/d(xⁱ). Not to be confused with the partial derivative operator.

Properties :
- The vectors ui constitute a vector base of Space-time.
- Each vector ui is actually a vectors field which can be written as follows :

The vectors ur, uθ and uφ do not constitute an Orthonormal base of R³ for the usual Euclidean scalar product.
The associated orthonormal base (called "normalized natural base") is written : (ur, r^-1 uθ, (r sin[θ])^-1 uφ).

Newtonian limit

The Poisson equation of Newtonian gravitation (Δψ = 4 π G ρ) is a particular case of the Einstein Field Equation which corresponds to a spatially isotropic space-time, containing a non-relativistic perfect fluid (p << ρ c²), in a weak gravitational field (|ψ| << c²) and without Cosmological constant (Λ = 0).
This Newtonian limit provides the gravitational coupling coefficient (K = 8 π G c^-4) used in the Einstein Field Equation as well as the Schwarzschild gravitational radius (r^* = 2 G M c^-2) used in the Solution of Einstein Field Equation with Schwarzschild Metric.

Observer

An observer O is defined by the data of a Universe line L and a Local reference frame (e0, e1, e2, e3) along L (see Figure above) [GOU, Relativité restreinte, p.80].

The notion of observer is the cause of many misunderstandings especially on the part of philosophers. The relativistic observer does not see phenomena in the usual sense of the word 'see'. He collects position and time measurements at different points of his reference frame. This presupposes that a clock has been placed at each point of the reference frame and that this clocks set is synchronized.
An observation made by an observer O therefore consists in collecting, for an event M occurring in this reference frame, a values set for the position (x, y and z coordinates) and one for the time (t) [MAR Mécanique p.14].

Orthonormal base

If ei are the basic vectors of Space-time, the base {ei} is called orthonormal (relative to Scalar product g) when [GOU, Relativité Générale, p.26] :

The matrix [gij] of this particular base, noted [ηij], is called Minkowski matrix and corresponds to the Space-time of Restricted Relativity in Cartesian coordinates.

Partial derivative (convention)

See Convention of partial derivative.

Poisson equation

The Poisson equation was established by the French mathematician and physicist Simeon Denis Poisson.
In Newtonian universal gravitation, the (non relativistic) gravitational potential ψ is related to the density ρ by the Poisson equation :

where Δ is the Laplacian operator (see Convention of partial derivative).
and G is the universal gravitational constant (G = 6.6726 10^-11 kg^-1.m³.s^-2).
ψ has the dimension of m².s^-2

Principle of equivalence

The principle of equivalence is one of the fundamental principles of General Relativity. It generalizes the Newtonian principle of equality between gravitational mass and inertial mass by asserting that a gravitational field is locally equivalent to an acceleration field.
It is stated as follows [GOU, Relativité Restreinte, p.709] : Physical measurements made by an "Inertial" Observer in a uniform gravitational field are exactly the same as those performed by a uniformly accelerated Observer.

Quadri-acceleration

The quadri-acceleration or 4-acceleration of any point x of Space-time is the Quadri-vector a which measures the variation of the Quadri-velocities field u along the point trajectory. This vector of dimension m^-1 is defined by the following relation [GOU, Relativité Restreinte, p.38] :

where τ is the proper Time of the point
and ei are the basic vectors of the vector space of dimension 4.

This Quadri-vector has the following properties :
a is orthogonal to u : a.u = 0
a is either a zero vector or a space-like Vector type : a.a ≥ 0

Quadri- electric power

The quadri- electric power or 4- electric power is the Quadri-vector j defined by the following relation [GOU, Relativité Restreinte, p.580] :

Q is the total electric charge of a three-dimensional region R in the Local rest space E_u0 of an Inertial Observer O of Quadri-velocity u0. So Q represents the flow of the vector field j through R (see Figure above).
j has the dimension of a density of electric charge (in m^-3.A.s).

The orthogonal decomposition of j relative to u0 is then written :

ρ is the electric charge density measured by O (in m^-3.A.s).
J is the electric power density measured by O (in m^-2.A).

Quadri- electromagnetic potential

The quadri- electromagnetic potential ou 4- electromagnetic potential is the Quadri-vector A* defined by the following relation [GOU, Relativité Restreinte, p.594] :

Fij is the Electromagnetic tensor.
E_u0 is the Local rest space of an Inertial Observer O of Quadri-velocity u0 (see Figure above).
In Cartesian coordinates, all components of A* are of dimension V or kg.m².s^-3.A^-1

The orthogonal decomposition of A* relative to u0 is then written :

V is the electric potential relative to O (in V or kg.m².s^-3.A^-1).
A is the magnetic potential relative to O (in m^-1.s.V or kg.m.s^-2.A^-1).

Quadri-force

The quadri-force or 4-force of a material particle is the Quadri-vector f defined by the following relation [GOU, Relativité Restreinte, p.313] :

where p is the Quadri-impulse of the particle and τ its proper Time
This Quadri-vector is of dimension N or kg.m.s^-2.

This Quadri-vector has the following properties :
f = d(m c u)/dτ = m c² a + c (dm/dτ) u
f.u = -c (dm/dτ)

Quadri-impulse

The quadri-impulse or 4-impulse of a simple material particle (whithout spin or internal structure) is the Quadri-vector p defined by the following relation [GOU, Relativité Générale, p.37] :

where m is the mass at rest (or proper mass) of the particle and u is its Quadri-velocity
This Quadri-vector is of dimension kg.m.s^-1 (analogous to a linear momentum).

This Quadri-vector has the following properties :
p.p = -m² c²

Quadri-position

See Space-time.

Quadri-rotation

Figures above from left to right : Enrico Fermi and Arthur Geoffray Walker

The quadri-rotation or 4-rotation of any body of Space-time is the only Quadri-vector ω defined by the following relation [GOU, Relativité Restreinte, p.86 to 91] :

where τ is the proper Time of the body
eα are the four basic vectors of the Local reference frame linked to the body, choosing e1 tangent to the curvature a of the Universe line (condition : e1 = a/||a||)
u and a are respectively the Quadri-velocity and Quadri-acceleration of the body
"x_u" is the Cross product operator between two any vectors of the Local rest space of the observer E_u, what is written : ω x_u eα = Ε(u, ω, eα, .)
where :
Ε is the Levi-Civita tensor
Ε(u, ω, eα, .) is the vector representing the Linear form Ε(u, ω, eα, z) for the Scalar product g.

ω has the dimension of angular velocity (in s^-1).
Ω_FW is the Fermi-Walker Tensor. It affects only the components of the vectors in the plane (u, a) which is the osculating plane of the Universe line of the body.
Ω_rot is the Spatial rotation Tensor. Since Ω_rot(ω) = ω x_u ω = 0, this tensor acts only in the subspace (of dimension 2) of the vector hyperplane E_u orthogonal to ω. In other words, Ω_rot represents the spatial rotation of the trihedron (ei) in the (dimension 2) plane orthogonal to both u and ω.

This Quadri-vector has the following properties :
- ω is orthogonal to u since belonging to the vector hyperplane E_u : ω.u = 0
- ω is a space-like Vector type : ω.ω ≥ 0.
- For α = 0, we have : e0 = u, a.u = 0, u.u = -1, ω x_u u = Ε(u, ω, u, .) = 0, and we find that : d(e0)/dτ = d(u)/dτ = c a
- It is proved that : ω = c T2 e1 + c T1 e3, T1 and T2 being respectively the first and the second torsion of the Universe line of the body (see Local reference frame).

Quadri-vector

See Space-time.

Quadri-velocity

The quadri-velocity or 4-velocity of any point x of Space-time is the only unit Quadri-vector u which is tangent to the point trajectory and directed towards the future. This dimensionless vector is defined by the following relation [GOU, Relativité Restreinte, p.36] :

where τ is the proper Time of the point
and ei are the basic vectors of the vector space of dimension 4.

This Quadri-vector has the following property :
u is a unit time-like Vector : u.u = -1

Relativistic mechanics

Relativistic mechanics refers to mechanics compatible with the Restricted Relativity and General Relativity.
As in classic mechanics, the subject can be divided into two : the kinematics that describes the movement in terms of Quadri-position, Quadri-velocity, Quadri-rotation and Quadri-acceleration, and the dynamics that describes the movement cause in terms of Quadri-impulse, Angular momentum, Quadri-force and Energy.

In addition, when there is a change of Observer, some quantities are transformed : lengths (Lorentz-Poincaré transformation), velocities [GOU, Relativité restreinte, p.141], accelerations [GOU, Relativité restreinte, p.151], frequencies (Doppler effect), angles of observation (Aberration) and images (Image).

Relativistic metric

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 :

In the curved Space-time of General Relativity, this metric can be negative, zero or positive, and is written in Cartesian coordinates :
ds² = g00 (c dt)² + g01 (c dt) dx + g02 (c dt) dy + g03 (c dt) dz +
g10 dx (c dt) + g11 dx² + g12 dx dy + g13 dx dz +
g20 dy (c dt) + g21 dy dx + g22 dy² + g23 dy dz +
g30 dz (c dt) + g31 dz dx + g32 dz dy + g33 dz²
The coefficients gij are the components of the Metric tensor.
Note that the quantity ds² is always negative for a non-zero mass particle, the vector tangent to its Universe Line being a time-like Vector.

Ricci tensor

The Ricci tensor (Rab) is a Symmetric Tensor of order 2 producted by Contraction of the Curvature tensor. It is therefore a Tensor that measures the local deformation of Space-time but incompletely.

According to the Classic Sign Conventions (MTW) it is producted by Contraction of the Curvature tensor on the first and third index. Using the Convention of partial derivative, its components are the following :

where Γⁱjk are the Christoffel symbols.

In Cartesian coordinates, all the components are of dimension m^-2
Warning : the Ricci tensor can be zero without the Curvature tensor being it.

Warning : Unlike classic MTW sign conventions, some authors contract the Tensor of curvature on the first and fourth indexes, which amounts to defining the opposite of the Tensor of Ricci.

Scalar curvature (or invariant scalar)

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

R is actually the trace of the Ricci tensor.

Scalar product and norm

Scalar product of two vectors :

Using the Convention of summation, the scalar product of two arbitrary vectors x and y is written :

where the coefficients gij are the components of the Metric tensor.
Some authors also use the following two notations :
x.y = <x, y> = g(x, y)

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


Scalar product of two contravariant Tensors U and V of order 2 :

T = U^ij.V^kl of component : T = U^ij Vij

Schwarzschild metric

The metric of Karl Schwarzschild 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, colatitude θ = [0, π], longitude φ = [0, 2 π]) (see Figure in Space-time), the metric is written as follows for any coordinate system (ct, r, θ, φ) and by taking the Signature convention (-, +, +, +) :

where μ and α are only functions of r.
The gravitational potentials gij then are the following :

Signature

The (-, +, +, +) signature of the Metric tensor, called Lorentzian signature, means that there exists a basis of the vector space E such that the Scalar product g(u,v) is expressed as a function of the components uⁱ and v^j of the vectors u and v in the following form [GOU, Relativité Restreinte, p.8] :
    g(u,v) = -u⁰v⁰ + u¹v¹ + u²v² + u³v³

Warning : Unlike classic MTW sign conventions, some authors use the (+, -, -, -) signature instead of (-, +, +, +), which amounts to using -g instead of g.
The classic convention (-, +, +, +) has the advantage of not distinguishing between Quadri-vectors and tri-vectors because the Scalar product induced by g on the space-like hyperplanes (three-dimensional cuts of Space-time at t = constant) coincides with the Euclidean scalar product of three-dimensional space.

Simultaneity

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.

Space-time

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.

Definition [GOU, Relativité Restreinte, p.24] :
Space-time (called Minkowski space-time) is defined by the set of four mathematical objects :
- Real affine space of dimension 4, associated with a oriented vector space E (depending on whether the base is Direct or indirect)
- Scalar product of (-, +, +, +) Signature, associated with the Metric tensor g
- Light-cone with its two time arrows (past sheet and future sheet)
- Levi-Civita tensor E associated with the metric g

Quadri-vector and coordinates :
In this space-time of origin O fixed, a point or event x(x, y, z, t) is identified by a four-dimensional vector x called Quadri-vector or 4-vector whose components are denoted :