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This chapter gives the basic formulas of Restricted Relativity and General Relativity with a presentation of the mathematical tools accessible to uninitiated people.
Through the paragraph Definitions, we will also show that many "mysteries" of space-time are unfounded and result from some clumsiness which do not facilitate the understanding of Relativity. We can list :
"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 (t = t '= 0). At this initial 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 would be 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.
"The instant universe is unobservable. It appears as a Space-time where each observed object is seen at a space point and at a time point that is not the same for all space points." [AND Théorie - Partie 1]
Until the end of the 19th century, classical mechanics founded by Galileo and Newton constituted an undisputed basis of physics.
In 1887 an American physicist Albert Michelson and his colleague Edward Morley showed that the light speed did not verify the Galilean law of addition of velocities. On the contrary, the light speed in the vacuum was independent of the motion of the emitting source.
At the end of the 19th century, a second enigma disrupted the certainties of the scientists. The famous equations of British James 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 a uniform 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 [POIN L'Etat].
Also in 1905 Albert Einstein published his theory of Restricted Relativity [EIN Zur_Elektrodynamik].
We consider a reference frame R' in uniform translation at the speed V with respect to a reference frame R (see Figure above).
The two reference frames have their origin O and O' which coincide at time t = 0.
Let an arbitrary point M of abscissae x' in R' and x in R.
The transformation of Galileo from R to R' can be written as follows :
(G1) x' = x - V t
(G2) t' = t
The transformation of Lorentz-Poincaré introduces a new entity to describe the physical phenomena : Space-time. This can be written as follows :
(L1) x' = gamma (x - V t)
(L2) t' = gamma (t - B x)
(L3) gamma = 1 / (1 - V^{2}/c^{2})^{1/2}, called "Lorentz factor"
(L4) B = V / c^{2}
where c is a constant (space-time structure constant) which is similar to a limiting speed and which appears during the 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.
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 (law of speeds composition, 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.
- The Restricted Relativity does not prohibit studying the view point of accelerated observers.
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]. 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 the Restricted Relativity.
In 2001 Jean Hladik published, with one of his colleagues Michel Chrysos, the first book on Restricted Relativity presented in this modern form [HLA Pour_comprendre].
Inspired by the works listed below in the Bibliography we present here an elegant and rigorous presentation of the Lorentz-Poincaré transformation only based on the four Poincaré's postulates.
Postulat 1 : Space is homogeneous and isotropic
Space has the same properties at every point and in every direction. In other words space is invariant by translation and rotation.
Postulate 2 : Time is homogeneous
The time is identical in every point of the same reference frame. All fixed clocks in a given reference frame must be strictly set at the same time. In other words time is invariant by translation.
Postulate 3 (Principle of Restricted Relativity) : The laws of physical phenomena must be the same either for a fixed observer or for an observer entrained in a uniform translation movement.
The form of the equations which describe the mechanical phenomena is invariant by changing the reference frame by uniform translation.
Postulate 4 : Causality must be respected
When a phenomenon A is the cause of a phenomenon B, then A must occur before B in any reference frame.
The postulates of space and time homogeneity induce that the desired transformation is linear of the following form :
(Ha) x' = C(V) x + D(V) t
(Hb) t' = E(V) t + F(V) x
where the four functions C, D, E and F are to be determined.
The particular point M = O' correspond to : x' = 0 and x = V t
Equations (H) can be rewritten as follows :
(C1a) x' = gamma (x - V t)
(C1b) t' = gamma (A t - B x)
The unknowns become gamma, A and B which are three functions dependent only of V. Namely : gamma = gamma(V) ; A = A(V) ; B = B(V).
When V = 0 we must have : x' = x and t' = t corresponding to the identity transformation and it can be deduced that :
(C2) gamma(0) = 1
The postulate of space isotropy induces that the form of the equations is invariant by reflection (x -> -x ; x' -> -x' ; V -> -V) corresponding to the passage of the " -R " reference frame to the " -R' " reference frame. From this it can be deduced that :
(C3a) gamma(V) = gamma(-V)
(C3b) A(V) = A(-V)
(C3c) B(V) = - B(-V)
The postulate of form invariance induces that the form of the equations is invariant by inverse transformation (x' <-> x ; t' <-> t ; V <-> -V) corresponding to the exchange of the reference frames R and R'. From this it can be deduced that :
(C4a) x = gamma(-V) (x' + V t')
(C4b) t = gamma(-V) (A(-V) t' - B(-V) x')
From relations (C1)(C3) it can be deduced that :
(C5a) A = 1
(C5b) gamma^{2} (1 - V B) = 1
It remains to determine the unknown B.
The postulate of form invariance induces that the form of the equations is invariant by composition of the transformations (R -> R') and (R' -> R"). From relation (C5a) it can be deduced that :
(C6a) x" = gamma(U) (x' - U t')
(C6b) t" = gamma(U) (t' - B(U) x')
where U is the uniform translation speed of R" with respect to R'.
Let W the uniform translation speed of R" with respect to R.
From relation (C1) it can be deduced that :
(C7a) W = (V + U) / (1 + U B)
(C7b) B(U) / U = B / V
The relation (C7a) is the law of speeds composition.
The relation (C7b) shows that B is of the form :
(C8) B(V) = b V
where b is any constant (negative, zero or positive).
From particular relation (C2) the relation (C5b) can be written :
(C9) gamma^{2} = 1 / ( 1 - b V^{2}) avec gamma > 0
From relations (C8)(C9) the equations (C1) can be written :
(C10a) x' = (x - V t) / (1 - b V^{2})^{1/2}
(C10b) t' = (t - b V x) / (1 - b V^{2})^{1/2}
(C10c) b V^{2} < 1
It remains to determine the unknown b.
Let M1 and M2 two any points of the reference frame R.
From relation (C10b) it can be deduced that :
(t2' - t1')/(t2 - t1) = ( 1 - b V ((x2 - x1)/(t2 - t1)) ) / (1 - b V^{2})^{1/2}
The postulate of causality induces that the sign of the time interval (t2 - t1) in R must not change during the passage in (t2'- t1') in R'. This can be written :
(C11) b V (x2 - x1)/(t2 - t1) < 1
If b is negative this relation is not satisfied for any values of V, (x2 - x1) and (t2 - t1). The causality assumption is not respected for the case b < 0.
If b is positive or zero it can be written in the following form :
(C12) b = 1 / u^{2} > 0 where u is a positive constant similar to a speed.
From relation (C12) the relation (C10c) can be written :
(C13) V / u < 1
The constant u is similar to a limiting speed. Whatever the values of (x2 - x1) and (t2 - t1) it can be deduced that :
(C14) ((x2 - x1) / (t2 - t1)) / u < 1
From relations (C12) (C13) (C14) the relation (C11) is verified. The causality assumption is respected for the case b ≥ 0.
Note that some authors such J. HLADIK arrive at the same conclusion (b ≥ 0) without using the postulate of causality.
In practice the mathematical limit u is taken appropriately equal to the light speed c in the vacuum.
The Restricted Relativity applies only to reference frames in uniform translation and in a Space-time where the gravitational effects are completely neglected.
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).
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 gravitational interaction on a microscopic scale involving the quantum character of the elementary particles.
The fundamental equations of General Relativity, called Einstein equations or equations of the gravitational field, connect a local deformation of the geometry of Space-time with the presence of local tensions (see Figure above).
These equations can be seen as a generalization of the law of elasticity of Hooke in a weakly deformed continuous medium for which the deformation of an elastic structure is proportional to the tension exerted on this structure.
Einstein equations are written :
(E1) Eab = KHI Tab
with : Eab = Rab - (1/2) gab R + Λ gab
Note that some authors present these equations with the minus sign in front of Λ instead of the plus sign.
Eab 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.
a and b are the indices of the different tensors with a and b ranging from 0 to 3
KHI is the gravitational coupling coefficient : KHI = 8 π G / c^{4} (in kg^{-1}.m^{-1}.s^{2}). This coefficient was chosen so as to verify the Poisson equation of the Newtonian gravitation as a particular case of Einstein equations (see Newtonian Limit). KHI represents an extraordinarily small elasticity of space-time (equal to about 2,1 10^{-43} kg^{-1}.m^{-1}.s^{2}).
G is the universal gravitational constant : G = 6,6726 10^{-11} kg^{-1}.m^{3}.s^{-2}
c is the light speed in the vacuum : c = 2,99792458 10^{8} m.s^{-1}
gab is the Metric tensor which is solution of Einstein equations. The 16 gab components of this tensor are called gravitational potentials.
Rab est le Ricci tensor producted by Contraction of the Curvature tensor.
R is the Scalar curvature producted by Contraction of the Ricci tensor.
Λ is the cosmological constant of dimension m^{-2} and may be negative, zero or positive. Λ was introduced by Einstein only later in applications to cosmology. The problem of the planets motion, considered as particles in an empty space around the sun (Schwarzschild metric), is solved by taking Λ = 0 and Tab = 0. In cosmology, the universe model (Friedmann-Lemaitre-Robertson-Walker metric) 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).
Equivalent equations :
By contracting the Einstein equations by the inverse Metric tensor g^{ab}, the Scalar curvature R is related to the Energy-impulse tensor Tab by the relation :
(E2) R = -KHI T + 4 Λ
where T is the trace of the Energy-impulse tensor : T = g^{ab} Tab = T^{a}a
By replacing this relation in the Einstein equations (E1), we find the following equivalent equations :
(E3) Rab = KHI (Tab - (1/2) gab T) + Λ gab
In the particular case where Tab = 0 (vaccum space) and Λ = 0, the Ricci tensor Rab is zero.
Note that the Ricci tensor can be zero without the Curvature tensor being it (case of a gravitation field in the vacuum with Λ = 0).
Properties of Einstein equations :
Einstein equations satisfy the following constraints :
- Postulate : the equations are not demonstrated on the basis of more fundamental principles. This is the whole genius of Einstein to have postulated them.
- Principle of equivalence (local equivalence between gravitational field and acceleration field) : the equations respect the Principle of equivalence.
- Principle of general relativity (invariance of the physical laws in any change of reference frame): the equations are Covariant and keep thus the same form in any change of coordinates. This is the extraordinary power of tensorial formalism : once written in tensorial form (according to Tensoriality criteria), a physical law necessarily has a form independent of the coordinates system.
- Conservative tensors : the members of the equations are both conservative (zero Divergence) to respect the principle of local conservation of impulse and energy.
- Zero curvature to infinity : the equations induce zero gravitation, and therefore zero curvature, when the coordinates tend towards the infinite (far from any attractive mass). Space-time becomes the flat space-time of Restricted Relativity with its Minkowski metric.
- Newtonian Gravitation : the equations have as their particular case the Poisson equation of the Newtonian limit.
- Simplicity : Although General Relativity is not the only relativistic theory, it is the simplest that is consistent with the experimental data. However several questions remain open : the most fundamental one is to succeed in formulating a complete and coherent theory of quantum gravitation.
The components of the Eab Einstein tensor are function only of the gravitational potentials gab and their first and second derivatives. These components are linear 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 Friedmann-Lemaitre-Robertson-Walker metric (F) is used in cosmology to describe the universe evolution at large scales. It is the main tool leading to the construction of the standard cosmological model : the Big Bang theory.
The Schwarzschild metric (S1, S2...) describes the geometry around the masses (M1, M2...).
The Minkowski Metric (K) describes the geometry away from the large masses, on the asymptotically flat part of the previous metrics, according to a tangent Euclidean space-time of the Restricted Relativity.
Under the hypothesis that the gravitational field is static and centrally symmetrical (Schwarzschild metric) as the case of Sun and many stars, the gravitational potentials gab are expressed in spherical coordinates (r, 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 equations then are 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 ds^{2} is finally completely determined as follows :
g00 = -(1 - r^{*}/r)
g11 = 1 / (1 - r^{*}/r)
g22 = r^{2}
g33 = r^{2} sin^{2}[θ]
gij = 0 for i and j taken different between 0 and 3
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 approximation. See Newtonian Limit.
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 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. The space-time described by the Schwarzschild metric 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. When their mass m is very small compared to the mass M of the central body of the Schwarzschild metric, it can be proved that their trajectories (orbits) are plane and become ellipses when r tends to infinity.
Detailed calculation of components g^{ab}, Rab, R, Eab, α and μ [GOU Relativité_Générale, p.117] :
In the case of a gravitational field with static and centrally symmetry (Schwarzschild metric), the gravitational potentials gij of the Metric tensor are the following :
g00 = -e^{2 μ}
g11 = e^{2 α}
g22 = r^{2}
g33 = (r^{2}) sin^{2}[θ]
gij = 0 for i and j taken different between 0 and 3
where μ and α are only functions of r.
The gravitational potentials g^{ij} of the inverse Metric tensor are then the following such that : g^{ij} gjk = δ^{i}k
where δ is the Kronecker symbol.
g^{00} = -e^{-2 μ}
g^{11} = e^{-2 α}
g^{22} = 1/r^{2}
g^{33} = (1/r^{2}) sin^{-2}[θ]
g^{ij} = 0 for i and j taken different between 0 and 3
The Christoffel symbols Γ^{i}jk are then written by the relations : Γ^{i}jk = (1/2) g^{il} (glk_{,j} + glj_{,k} - gjk_{,l})
Γ^{0}01 = Γ^{0}10 = μ'
Γ^{1}00 = e^{2 (μ - α)} μ' ; Γ^{1}11 = α' ; Γ^{1}22 = -r e^{-2 α} ; Γ^{1}33 = -r sin^{2}[θ] e^{-2 α}
Γ^{2}12 = Γ^{2}21 = 1/r ; Γ^{2}33 = -cos[θ] sin[θ]
Γ^{3}13 = Γ^{3}31 = 1/r ; Γ^{3}23 = Γ^{3}32 = 1/ tan[θ]
where μ' = dμ/dr and α' = dα/dr
The other Christoffel symbols are all zero.
The Rij components of Ricci tensor are then written by the relations : Rij = R^{k}ikj = Γ^{k}ij_{,k} - Γ^{k}ik_{,j} + Γ^{k}kl Γ^{l}ij - Γ^{k}jl Γ^{l}ik
R00 = e^{2 (μ - α)} ( μ" + (μ')^{2} - μ' α' + 2 μ'/r )
R11 = -μ" - (μ')^{2} + μ' α' + 2 α'/r
R22 = e^{-2 α} ( r (α' - μ') - 1 ) + 1
R33 = sin^{2}[θ] R22
The other components Rij are all zero.
The Scalar curvature is then written by the relation : R = g^{ij} Rij
R = 2 e^{-2 α} ( -μ" - (μ')^{2} + μ' α' + 2 (α' - μ')/r + (e^{2 α} - 1)/r^{2} )
In the case of Λ = 0, the Einstein tensor is then producted by the relation : Eab = Rab - (1/2) gab R
E00 = (1/r^{2}) e^{2 (μ - α)} (2 r α' + e^{2 α} - 1 )
E11 = (1/r^{2}) (2 r μ' - e^{2 α} + 1 )
E22 = r^{2} e^{-2 α} ( μ" + (μ')^{2} - μ' α' + (μ'- α')/r )
E33 = sin^{2}[θ] E22
The other components Eij are all zero.
The Einstein equations are then written by the relation : Eab = KHI Tab
E00 = KHI T00
E11 = KHI T11
E22 = KHI T22
E33 = KHI T33
0 = KHI Tij for i and j taken different between 0 and 3
In the case Tab = 0, the Einstein equations then reduce to the 3 following equations :
2 r α' + e^{2 α} - 1 = 0
2 r μ' - e^{2 α} + 1 = 0
μ" + (μ')^{2} - μ' α' + (μ'- α')/r = 0
The first equation is integrated into :
α = -(1/2) ln[ 1 - r^{*}/r]
where r^{*} is a constant.
By replacing this α value into the second equation, this one is integrated into :
μ = (1/2) ln[ 1 - r^{*}/r] + b_{0}
where b_{0} is a constant.
The zero of the gravitational field at infinity (so as to ensure an asymptotically flat metric with μ = 0 when r tends to infinity) requires that : b_{0} = 0.
By replacing these α and μ values in the third equation, this one is always satisfied.
We finally find :
g00 = -(1 - r^{*}/r)
g11 = 1/(1 - r^{*}/r)
Under the hypothesis that Space-time is spatially homogeneous and isotropic (Friedmann-Lemaitre-Robertson-Walker metric), the gravitational potentials gab are expressed in spherical coordinates (r, 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-Pulse 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 equations are 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 KHI c^{4} + (1/3) Λ c^{2}
(F2) a"/a = -(1/6) (ρ + 3 p/c^{2}) KHI c^{4} + (1/3) Λ c^{2}
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) c^{2} where w is a constant that is equal to -1 (quantum vacuum), 0 (zero pressure) or 1/3 (electromagnetic radiation).
This equation of state, associated with the two equations (F1) and (F2), gives a remarkable relation linking ρ(t) and a(t) :
(Q0) ρ(t) a(t)^{3(1 + w)} = ρ_{0} a_{0}^{3(1 + w)} = constant
where ρ_{0} and a_{0} 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) :
(Q1) (a')^{2} + k c^{2} = A a^{-(1 + 3 w)} + B a^{2}
(Q1a) A = (1/3) ρ_{0} (a_{0})^{3(1 + w)} KHI c^{4} = constant
(Q1b) B = (1/3) Λ c^{2}
This differential equation is analytically integrated for w = 0 or 1/3 (with any Λ and k), which completely determines a(t) and the metric ds^{2} as follows :
g00 = -1
g11 = a(t)^{2} (1 - k r^{2})^{-1}
g22 = a(t)^{2} r^{2}
g33 = a(t)^{2} r^{2} sin^{2}[θ]
gij = 0 for i and j taken different between 0 and 3
The first Friedmann equation (F1) is often presented in the condensed form :
k (c/a)^{2} / H(t)^{2} = Ω + Ω_{v} - 1
where :
H(t) = Hubble parameter (of dimension s^{-1}) = a'/a that accounts for the universe expansion. See Hubble law
Ω(t) = density parameter (dimensionless) = (8/3) π G ρ(t) / H(t)^{2}
Ω_{v}(t) = reduced cosmological constant (dimensionless) = (1/3) Λ c^{2} / 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 black energy. See Standard cosmological model.
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).
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 and Proof below).
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 followed by an accelerated contraction occurring after the 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 & gt; 0) and (k = 0) corresponds to the Standard cosmological model when the pressure is zero (w = 0).
- The curves C5, and again C1, are related to case (0 < Λ < Λ_{F}) and (k > 0). They correspond to two possible behaviors : an open model of non-Big Bang type (decelerated contraction followed by accelerated expansion after the minimum point M2), and a closed model with a maximum point M1.
- The curves C3, and again C4, are related to singular case (Λ = Λ_{F}) and (k > 0). They correspond to two possible behaviors : a static model (Einstein static universe) and an open model with a particular inflexion point which is also a point with horizontal tangent.
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 [KHA Some_exact_solutions] :
Λ_{F} = 3 (k/m)^{m} ( (1/n) A c^{-2} )^{-n}
n = 2/(1 + 3 w) > 0
m = n + 1
Λ_{F} is linked to the singular scale factor a_{F} as follows :
Λ_{F} = 3 (k/m) a_{F}^{-2}
a_{F}^{2/n} = (A c^{-2})(m/n)(1/k)
By expressing the constant A at this particular inflexion point such as ρ_{0} = ρ_{F} and a_{0} = a_{F}, we find the expressions of Λ_{E} and aE of the Einstein static universe (with k = 1) :
Λ_{E} = (1/n) ρ_{E} KHI c^{2} = (1/2)(1 + 3 w) ρ_{E} KHI c^{2}
aE^{-2} = (1/3)(m/n)(1/k) ρ_{E} KHI c^{2} = (1/2)(1/k)(1 + w) ρ_{E} KHI c^{2}
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) = aE ; ρ(t) = ρ_{E} ; p(t) = pE
where aE, ρ_{E} and pE are constants.
The second Friedmann equation (F2) then becomes : Λ = Λ_{E}
where : Λ_{E} = (1/2)(ρ_{E} + 3 pE / c^{2}) KHI c^{2}
Λ_{E} is the singular cosmological constant of Einstein which characterizes a static universe.
Note that outside a vacuum (ρ_{E} = pE = 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 / aE^{2} = (1/2)(ρ_{E} + pE / c^{2}) KHI c^{2}
If the cosmic fluid satisfies the strict low energy condition then : ρ_{E} + pE / c^{2} > 0 and therefore necessarily : k > 0, so : k = 1
The curve a(t) is thus a constant (see curve C3 in Figure 1 above) :
a(t) = aE = ( (1/2)(ρ_{E} + pE/c^{2}) KHI c^{2} )^{-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) = a_{0} e^{H0 (t - t0)}
where a_{0} 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} + c^{2}
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 a_{0}, ρ_{0} > 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 2 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 a_{0}, ρ_{0} > 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 2 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} - c^{2}
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 a_{0}, ρ_{0} > 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 2 above for k = 1).
Note that the curve goes from the "Big Bang" point (t - ti = 0) to the "Big Crunch" point (t - ti = 2 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 a^{2}
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 a_{0}, ρ_{0} > 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 2 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 2 above for k = -1) with deceleration (a" < 0) but leading to an inflection point I (a" = 0). The second phase is again an open curve but with acceleration (a" > 0) (see curve C4 in Figure 1 above).
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 c^{2} = 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 a_{0}, ρ_{0} > 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 2 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 a_{0}, ρ_{0} > 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 2 above for k = 0).
Proof of the general shape of the curves a(t) according to Friedmann equations :
Friedmann equations (F1) and (F2) are written in the form :
(Q1) (a')^{2} + k c^{2} = A a^{-(1 + 3 w)} + B a^{2}
(Q2) a"/a = -F a^{-3(1 + w)} + B
(Q1a) A = (1/3) ρ_{0} (a_{0})^{3(1 + w)} KHI c^{4}
(Q1b) B = (1/3) Λ c^{2}
(Q2a) F = (1/2) (1 + 3 w) A
In the standard case where ρ > 0 and w > (-1/3), A is positive and we deduce that :
1. When a tends to 0, the relation (Q1) induces that the quantity (a') tends to the infinity corresponding to the primordial universe explosion (Big Bang theory).
2. When a tends to infinity, the relation (Q1) induces that :
(Q3) If Λ is non-zero, B is non-zero and the quantity (a')^{2} behaves as the quantity (B a^{2}) when B is positive.
(Q4) If Λ is zero, the quantity (a')^{2} behaves like the quantity (-k c^{2}) when k is negative and like the quantity (A a^{-(1 + 3 w)}) when k is zero.
3. When a' is zero :
(Q5) the relation (Q1) is satisfied only for the following values combinaisons (Λ, k, w, A) :
Λ < 0
(Λ = 0) and (k > 0)
(0 < Λ < Λ_{F}) and (k > 0)
(Λ = Λ_{F}) and (k > 0)
with : Λ_{F} = 3 (k/m)^{m} ( (1/n) A c^{-2} )^{-n}
n = 2/(1 + 3 w) > 0
m = n + 1
We deduce the following results illustrated by the curves C1 to C5 in Figure 1 above :
4. If Λ is negative :
4.1. The relation (Q2) induces that the quantity (a") is always negative. The evolution of a(t) is decelerated, with no inflection point (a" = 0).
4.2. B is negative. The relation (Q5) induces that a(t) reaches a maximum (a'= 0 ; point M1 on curve C1) for which : (-B) a^{3(1 + w)} + k c^{2} a^{(1 + 3 w)} - A = 0
5. If Λ is zero :
5.1. The relation (Q2) induces that the quantity (a") is always negative. The evolution of a(t) is decelerated, with no inflection point (a" = 0).
5.2. If k is negative, the relation (Q4) induces that a(t) tends to the straight line a(t) = c (-k)^{1/2} t when a tends to infinity (curve C2).
5.3. If k is zero, the relation (Q4) induces that a(t) tends to the curve a(t) = ((1/j) A^{1/2} t)^{j} with j = ( (2/3) (1 + w)^{-1} ) when a tends to infinity (curve C2).
5.4. If k is positive, the relation (Q5) induces that and a(t) reaches a maximum (a' = 0 ; point M1 on curve C1) for which : a^{(1 + 3 w)} = (1/k) A c^{-2}
6. If Λ is positive, B is positive :
6.1. The relation (Q2) induces that the quantity (a") is first negative (decelerated evolution) then becomes positive (accelerated evolution) after passing through an inflection point (a" = 0 ; point I on curve C4) for which : a_{I} ^{3(1 + w)} = (F/B).
6.2. The relation (Q3) induces that a(t) tends to the exponential curve a(t) = exp [B^{1/2} t] when a tends to infinity.
6.3. Singular case : when Λ equals Λ_{F}, with positive k, the relation (Q5) induces that the curve a(t) has a point with horizontal tangent (a' = 0) that coincides with the inflection point I. This model has two types of possible behavior : a static model (Einstein static universe) for which a(t) = constant (curve C3), and an open model with an inflection point for which a' = a" = 0 at the point a_{I} = a_{F} (curve C4).
6.4. When Λ is less than Λ_{F}, with positive k, the relation (Q5) induces that the curve a(t) has two extremums (a' = 0 ; points M1 and M2). This model has two types of possible behavior : a closed model (a" < 0) with a maximum point in M1 (curve C1), and an open model (a" > 0) with a minimum point in M2 (curve C5), the respective inflection points I1 and I2 being fictitious and rejected in the forbidden band (a1 < a < a2). Note that the open model is not a Big Bang model.
Detailed calculation of components g^{ab}, Rab, R, Eab, Tab and a(t) [GOU Relativité_Générale, p.195] :
In the case of spatially homogeneous and isotropic Space-time (Friedmann-Lemaitre-Robertson-Walker metric), the gravitational potentials gij of the Metric tensor are the following :
g00 = -1
g11 = a^{2} (1 - k r^{2})^{-1}
g22 = a^{2} r^{2}
g33 = a^{2} r^{2} sin^{2}[θ]
gij = 0 for i and j taken different between 0 and 3
where k is a constant (0, 1 or -1) and a is a function of t only.
The gravitational potentials g^{ij} of the inverse Metric tensor are then the following such that : g^{ij} gjk = δ^{i}k
where δ is the Kronecker symbol.
g^{00} = -1
g^{11} = a^{-2} (1 - k r^{2})
g^{22} = a^{-2} (1/r^{2})
g^{33} = a^{-2} (1/r^{2}) sin^{-2}[θ]
g^{ij} = 0 for i and j taken different between 0 and 3
The Christoffel symbols Γ^{i}jk are then written by the relations : Γ^{i}jk = (1/2) g^{il} (glk_{,j} + glj_{,k} - gjk_{,l})
Γ^{0}11 = a a' (1/c)/(1 - k r^{2}) ; Γ^{0}22 = a a' r^{2} (1/c) ; Γ^{0}33 = a a' r^{2} (1/c) sin^{2}[θ]
Γ^{1}01 = Γ^{1}10 = a' (1/c)(1/a) ; Γ^{1}11 = k r / (1 - k r^{2}) ; Γ^{1}22 = -r (1 - k r^{2}) ; Γ^{1}33 = -r (1 - k r^{2}) sin^{2}[θ]
Γ^{2}02 = Γ^{2}20 = a' (1/c)(1/a) ; Γ^{2}12 = Γ^{2}21 = 1/r ; Γ^{2}33 = -cos[θ] sin[θ]
Γ^{3}03 = Γ^{3}30 = a' (1/c)(1/a) ; Γ^{3}13 = Γ^{3}31 = 1/r ; Γ^{3}23 = Γ^{3}32 = 1/ tan[θ]
where a' = d(a)/dt
The other Christoffel symbols are all zero.
The Rij components of Ricci tensor are then written by the relations : Rij = R^{k}ikj = Γ^{k}ij_{,k} - Γ^{k}ik_{,j} + Γ^{k}kl Γ^{l}ij - Γ^{k}jl Γ^{l}ik
R00 = -3 a" (1/a)(1/c^{2})
R11 = (a a" + 2 a'^{2} + 2 c^{2} k)(1/c^{2})/(1 - k r^{2})
R22 = (a a" + 2 a'^{2} + 2 c^{2} k) (r/c)^{2}
R33 = sin^{2}[θ] R22
The other components Rij are all zero.
The Scalar curvature is then written by the relation : R = g^{ij} Rij
R = 6 (1/c^{2})( (a"/a) + (a'/a)^{2} + (c/a)^{2} k )
The Einstein tensor is then producted by the relation : Eab = Rab - (1/2) gab R + Λ gab
E00 = R00 + (R/2) - Λ
E11 = ( (2b + a"/a)/c^{2} - 3 (b + a"/a)/c^{2} + Λ ) a^{2} /(1 - k r^{2})
E22 = E11 r^{2} (1 - k r^{2})
E33 = E22 sin^{2}[θ]
The other components Eij are all zero.
For a Perfect Fluid of density ρ and pressure p, the Energy-Pulse Tensor is then producted by the relation : Tij = (c^{2} ρ + p) ui uj + p gij
The hypothesis of spatial isotropy induces that the observer is co-mobile with the fluid.
The hypothesis of spatial homogeneity also induces that ρ and p are quantities function of t only.
The expression of Tij are written :
T00 = ρ c^{2}
T11 = p a^{2} /(1 - k r^{2})
T22 = T11 r^{2} (1 - k r^{2})
T33 = T22 sin^{2}[θ]
The other components Tij are all zero.
The Einstein equations are then written by the relation : Eab = KHI Tab
E00 = KHI T00
E11 = KHI T11
E22 = KHI T22
E33 = KHI T33
0 = Eij = KHI Tij = 0 for i and j taken different between 0 and 3
The Einstein equations then reduce to the 2 following equations :
b = (1/3) ρ KHI c^{4} + (1/3) Λ c^{2}
(1/2) b + a"/a = (1/2) Λ c^{2} - (1/2) p KHI c^{2}
By replacing the first equation in the second one, we find Friedmann equations :
(F1) (a'/a)^{2} + k (c/a)^{2} = (1/3) ρ KHI c^{4} + (1/3) Λ c^{2}
(F2) a"/a = -(1/6) (ρ + 3 p/c^{2}) KHI c^{4} + (1/3) Λ c^{2}
Deriving the first equation with respect to t and replacing a" in the second one, we find the following simple relation :
d(ρ)/dt = -3 (a'/a)(ρ + p/c^{2})
In the case where the cosmic fluid has a equation of state such as : p(t) = w ρ(t) c^{2}, this relation becomes :
d(ρ)/(ρ) = -3 (1 + w)(da/a)
which integrates into :
ρ(t) = ρ_{0} (a_{0} / a(t))^{3(1 + w)}
where ρ_{0} and a_{0} are two constants (index 0 generally corresponding to current data).
By replacing this expression of ρ(t) into the first Friedmann equation (F1), we find a differential equation that is a function of a(t) only :
(Q1) (a')^{2} + k c^{2} = A a^{-(1 + 3 w)} + B a^{2}
(Q1a) A = (1/3) ρ_{0} (a_{0})^{3(1 + w)} KHI c^{4}
(Q1b) B = (1/3) Λ c^{2}
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 law which induces a cosmological spectral shift due to an effect of distance from the source.
This shift is always directed towards red.
To explain these very profound phenomena of physics, General Relativity has had to go through the successive generalizations of Space-time notion :
- Euclidean space-time to interpret the Doppler-Fizeau effect.
- Curved space-time to interpret the Einstein effect.
- Space-time with variable curvature to interpret the Hubble law.
Notions used in this chapter, listed alphabetically :
The aberration of light is the difference between the incidence directions of the same light ray perceived by two observers in relative motion.
In the case of a light source S1 seen by an observer S' in movement with respect to S1 (velocity V), the light emanating from S1 appears to come from S2 and not from S1 (see Figure above).
In the case of rain falling vertically on the ground, the pedestrian who walks in the rain (velocity V) must tilt his umbrella forward if he does not wish to be wet.
Let S be an observer of a reference frame R and S' an observer of a reference frame R' in uniform translation of velocity V with respect to R.
u is the unit vector of the propagation SS'.
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 : tan^{2}[θ/2] = (1 - cos[θ])/(1 + cos[θ]), we have the equivalent relation :
tan[θ'/2] = ( (1 + V/c)/( 1 - V/c) )^{1/2} tan[θ/2]
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).
When V is small compared to c, there is no aberration effect (θ' = θ).
Pictures above from left to right : Riemann, Christoffel and Ricci-Curbastro
For a vector space of dimension n having for basis vectors the set (e1, e2... en), the Christoffel symbols Γ^{i}jk (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 : ej_{,k} = Γ^{i}jk ei
Γ^{i}jk is symmetric with respect to the lower index : Γ^{i}jk = Γ^{i}kj
Although possessing three indices, the Christoffel symbols of second kind are not mixed tensors of order 3 because they do not satisfy the Tensoriality criteria. Nevertheless, they appear abundantly in expressions which represent tensors (for example : Covariant derivative, Divergence, Ricci Tensor).
Note that there is another Christoffel symbols Γijk (called "of first kind") defined by the relation : Γijk = glj Γ^{l}ik, where the coefficients gjl are the components of the Metric tensor.
Γ^{i}jk can be written as a function of basis vectors of Dual space : Γ^{i}jk = e^{i}.ej_{,k}
Proof : e^{i}.ej_{,k} = e^{i}.(Γ^{i}jk ei) = Γ^{i}jk δ^{i}i = Γ^{i}jk
Γ^{i}jk can be written as a function of the components gij of the Metric tensor : Γ^{i}jk = (1/2) g^{il} (glk_{,j} + glj_{,k} - gjk_{,l})
Proof :
By deriving gij = ei.ej with respect to x^{k}, we find :
gij_{,k} = (ei_{,k}).ej + ei.(ej_{,k}) = (Γ^{l}ik el).ej + ei.(Γ^{l}jk el)
This is written :
gij_{,k} = Γ^{l}ik glj + Γ^{l}jk gil
A circular permutation of the three indices i, j, k then gives the following two equalities :
gki_{,j} = Γ^{l}kj gli + Γ^{l}ij gkl
gjk_{,i} = Γ^{l}ji glk + Γ^{l}ki gjl
We then find by linear combination :
gij_{,k} + gki_{,j} - gjk_{,i} = 2 Γ^{l}kj gil
By multiplying the two members by g^{mi} and using the relation g^{mi} gil = δ^{m}l, we find :
Γ^{m}kj = (1/2) g^{mi} (gij_{,k} + gki_{,j} - gjk_{,i})
By renaming the indices (i in l and m in i), we finally find :
Γ^{i}jk = (1/2) g^{il} (glk_{,j} + glj_{,k} - gjk_{,l})
The contraction operation of the index of a mixed component of a tensor consists in choosing two indices, one covariant and the other contra-variant, then in equalizing and summing them with respect to this twice repeated index.
For example, for a tensor U of order 3 whose mixed components are U^{ij}k, we find : W^{i} = U^{ik}k = U^{i1}1 + U^{i2}2 + ... U^{in}n
The quantities W^{i} (contracted components of the tensor U) form the components of a tensor W of order 1.
Note that the "matrix product" operator is a particular case of the tensor product U^{i}j * V^{k}l contracted in the form : W^{i}l = U^{i}k V^{k}l
In order to lighten the expressions of the derivatives of functions dependent on n variables f(x^{1}, x^{2}... x^{n}), we denote the partial derivatives in the following forms :
f_{,i} = d_{i}(f) = d(f)/d(x^{i})
f_{,i,j} = d_{ij}(f) = d^{2}(f)/(dx^{i} dx^{j})
Δf = Laplacian of f = div(grad(f)) = f_{,1,1} + f_{,2,2} + ... + f_{,n,n}
For a vector space of dimension n having as its basis vectors the set (e1, e2... en), any vector x of this space can be written : x = x^{1} e1 + x^{2} e2 + ... + x^{n} en = Sum_for_k_ranging_from_1_to_n [x^{k} ek]
In order to simplify this writing we use a notation convention consisting in deleting the symbol "Sum" which is written in condensed form : x = x^{k} ek where the index k (called mute index) always varies from 1 to n.
The summation is done on the index provided that they are repeated respectively up and down in the same monomial term.
When the prime symbol is used to distinguish two distinct bases of the same vector space, we can further simplify the notation by placing the prime symbol on the index rather than on the vector: x = x'^{k} e'k = x^{k'} ek'
Some terms in a sum may have several indices. For example, in the sum a^{k}m b^{m}, the summation is done on the index m. The index k (called free index) characterizes a particular term.
For example the equation ck = a^{k}m b^{m} for n = 3 represents the system of equations :
c1 = a^{1}1 b^{1} + a^{1}2 b^{2} + a^{1}3 b^{3}
c2 = a^{2}1 b^{1} + a^{2}2 b^{2} + a^{2}3 b^{3}
c3 = a^{3}1 b^{1} + a^{3}2 b^{2} + a^{3}3 b^{3}
There is no summation here on the index k which is found alone in the same monomial term.
When the monomial term has several mute index the summation takes place simultaneously on all these indices. For example, a^{k}m b^{m} ck for n = 4 represents a sum of 16 terms :
a^{k}m b^{m} ck = a^{1}1 b^{1} c1 + a^{1}2 b^{2} c1 + a^{1}3 b^{3} c1 + a^{1}4 b^{4} c1 + ... + a^{2}1 b^{1} c2 + ... + a^{4}4 b^{4} c4
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 tensor form : W = 0, or what amounts to the same : B = C with W = B - C (W, B and C being tensors of the same type), any change of reference frame transforms this equation into the tensor form : W' = 0
which does not change the form of the physical law.
If we take for example Newton's law : F = m γ, the covariant expression of this law is then in contra-variant components [AMI Initiation_aux_Tenseurs, p.18] :
F^{i} = m v^{i}_{;t}
with : v^{i} = dx^{i}/dt
The term v^{i}_{;t} is the Covariant Derivative of the velocity dx^{i}/dt with respect to time t, which is :
v^{i}_{;t} = v^{i}_{;k} (dx^{k}/dt) = v^{i}_{,k} (dx^{k}/dt) + Γ^{i}jk v^{j} v^{k} = dv^{i}/dt + Γ^{i}jk v^{j} v^{k}
where Γ^{i}jk are the Christoffel symbols.
Note that the Christoffel symbols are all zero only in the particular case of Restricted Relativity (Minkowski metric) with Cartesian coordinates.
If moreover the force F derives from a potential in the form : F = -grad(Φ), then F can be written in contra-variant components :
F^{i} = -g^{ij} dΦ/dx^{j} = m ( d^{2}x^{i}/dt^{2} + Γ^{i}jk (dx^{j}/dt) (dx^{k}/dt) )
where the g^{ij} are the contra-variant components of the Metric tensor.
In this tensor form, Newton's law will remain form-invariant under any coordinates transformation (spherical coordinates for example).
Proof of the covariance :
Let us take the example of a physical law written in the form of the tensor equation : W = 0, W being a mixed tensor of order 2.
Let [A] be the transition matrix of base {ei} to base {e'k} and [B] = [A^{-1}] the inverse transition matrix.
The base change transforms (see Tensor) the components W^{i}j = 0 of this equation into : W' ^{i}j = B^{i}k A^{l}j W^{k}l = 0
which does not change the form of the physical law.
For a vector space E of dimension n having for basis vectors the set (e1, e2... en), we call (see Figure above) :
contra-variant components of a vector x the numbers x^{i} such that : x = x^{i} ei, and
covariant components the numbers xj such that : xj = x.ej.
The contra-variant (respectively covariant) name derives from the fact that these components are transformed by base change in a inverse (respectively identical) manner to that of the basic vectors.
The contra-variant components are noted with higher indices.
The covariant components are noted with lower indices.
We have the following relations :
xj = x^{i} gij
x^{i} = xj g^{ij}
x.y = g^{ij} xi yi
When index vary from 0 to 3, Greek letters (such as α or μ) are often used rather than Latin letters (such as i or j).
Note that the covariant components xj can also be defined in a Duale base in the form : x = xj e^{j}, where the e^{j} are the basis vectors of the Dual vector space E* (see Figure above).
For orthonormal base there is no difference between the covariant and contra-variant components.
Proof of the contra-variant (respectively covariant) name :
Consider the base change from base {ei} = (e1, e2... en) to base {e'k} = (e'1, e'2... e'n).
Let [A] be the transition matrix from base {ei} to base {e'k}.
The elements of [A] are Aik such that : e'k = A^{i}k ei
The higher index is the line index of the matrix. The lower index is the column index of the matrix.
Let [B] = [A^{-1}] be the inverse transition matrix from base {e'k} to base {ei} such that : ei = B^{i}k e'k
For the contra-variant components, we have :
x = x^{i} ei
x = x' ^{k} e'k = x' ^{k} (A^{i}k ei) = (A^{i}k x' ^{k}) ei
Hence : x^{i} = A^{i}k x' ^{k}
And so : x' ^{i} = B^{i}k x^{k}
The contra-variant components are transformed by base change in an inverse manner to that of the basic vectors (with inverse transition matrix [B]).
For the covariant components, we have :
xj = x.ej
x'k = x.e'k = x.(A^{j}k ej) = A^{j}k (x.ej) = A^{j}k xj
The contra-variant components are transformed by base change in an identical manner to that of the basic vectors (with transition matrix [A]).
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.
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) :
For scalar : u_{;l} = u_{,l}
For contra-variant vector : u^{m}_{;l} = u^{m}_{,l} + u^{r} Γ^{m}rl
For covariant vector : ui_{;l} = ui_{,l} - ur Γ^{r}il
For contra-variant tensor of order 2 : U^{mn}_{;l} = U^{mn}_{,l} + (U^{rn} Γ^{m}rl + U^{rm} Γ^{n}rl)
For covariant tensor of order 2 : Uij_{;l} = Uij_{,l} - (Urj Γ^{r}il + Uri Γ^{r}jl)
For mixed tensor of order 2 : U^{m}i_{;l} = U^{m}i_{,l} + U^{r}i Γ^{m}rl - U^{m}r Γ^{r}il
...
For mixed tensor of order 5 : U^{mn}ijk_{;l} = U^{mn}ijk_{,l} + (U^{rn}ijk Γ^{m}rl + U^{rm}ijk Γ^{n}rl) - (U^{mn}rjk Γ^{r}il + U^{mn}rik Γ^{r}jl + U^{mn}rij Γ^{r}kl)
where Γ^{i}jk are the Christoffel symbols.
The word "curvature" has several meanings in Relativity :
- Curvature tensor (or Riemann-Christoffel tensor) of Space-time, which is a tensor of order 4.
- Ricci Tensor of Space-time, which is a tensor of order 2.
- Scalar curvature R of Space-time, which is a tensor of order 0 (scalar).
- Curvature k* of the spatial hypersurfaces (related to the spatial curvature parameter k), which is a scalar used in the Friedmann-Lemaitre-Robertson-Walker Metric. Not to be confused with the Scalar curvature R of Space-time.
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 the Restricted Relativity.
Pictures above from left to right : Riemann, Christoffel and Ricci-Curbastro
The curvature tensor is a symmetric tensor of order 4. It is the most complete measure of the local deformation of a curved space-time. It has 4^{4} = 256 components. Using the Convention of partial derivative, its componants have as expression :
R^{i}jkl = Γ^{i}jl_{,k} - Γ^{i}jk_{,l} + Γ^{i}mk Γ^{m}jl - Γ^{i}ml Γ^{m}jk
where Γ^{i}jk are the Christoffel symbols.
In Cartesian coordinates, all the components are of dimension m^{-2}
The Space-time is called "flat" when the curvature tensor is zero.
This tensor has the following properties :
Antisymmetry : R^{i}jkl = -R^{i}jlk
Swapping indices alone : Rijkl = -Rjikl = -Rijlk
Swapping of indices two by two : Rijkl = Rklij
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}]
See Time.
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 contra-variant 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.
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 S of a reference frame R, then any observer S' of the reference frame R' in uniform translation of velocity V with respect to R will perceive this same wave at the following frequency f'.
u is the unit vector of the propagation SS' (see Figure above called "Aberration").
(D1) Longitudinal Doppler effect (u parallel to V) :
f' = f gamma (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 gamma
(D3) Transverse Doppler effect at the reception (u perpendicular to V dans R') :
f' = f gamma^{-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', then we have the relation :
f' = f gamma (1 - cos[θ] V/c) = f gamma^{-1} (1 + cos[θ'] V/c)^{-1}
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).
Partial proof [ANN Electricité_2] :
Longitudinal Doppler effect (see Figure in Lorentz-Poincaré Transformation) :
The equation of the light wave propagating in the direction Ox is as follows for the observer bound to R :
s(x, t)= s0 cos[ 2 π f (t - x/c) ]
For the observer bound to R', it becomes s(x', t') using the inverse Lorentz-Poincaré transformation :
(L1') x = gamma (x' + V t')
(L2') t = gamma (t' + B x')
(L3) gamma = 1 / (1 - V^{2}/c^{2})^{1/2}
(L4) B = V / c^{2}
So :
s(x', t')= s0 cos[ 2 π f gamma (t'(1 - V/c) + x'(B - 1/c)) ]
The frequency f' perceived is thus :
f' = f gamma (1 - V/c)
The longitudinal Doppler effect is said to be first order because it depends on (1 - V/c). It causes a decrease in frequency for V > 0 (leakage of the observer with respect to the wave) and an increase in the opposite case.
Transverse Doppler effect at the emission (see Figure in Lorentz-Poincaré Transformation) :
The equation of the light wave propagating in the direction Oy is the following for the observer bound to R :
s(y, t)= s0 cos[ 2 π f (t - (y/c)) ]
For the observer bound to R', it becomes s (x', y', t') using the inverse Lorentz-Poincaré transformation :
(L0') y = y'
(L2') t = gamma (t' + B x')
So :
s(x', y', t')= s0 cos[ 2 π f gamma (t' + B x' - gamma^{-1} (y'/c)) ]
The frequency f' perceived is thus :
f' = f gamma
The transverse Doppler effect is said to be second order.
The dual space E* of a vector space E is the space of linear applications (or forms) on E.
For E of dimension n having for basis vectors the set {ei} = (e1, e2... en), then the set of basis vectors of E* is denoted {e^{i}} = (e^{1}, e^{2}... e^{n}) and verifies the following relations :
e^{i}.ej = δ^{i}j
where δ is the Kronecker symbol
Geometric interpretation (see Figure above) : The vector dual e^{i} is orthogonal (with respect to the Scalar product) to all vectors ej having different index j (e^{i}.ej = 0) and has a scalar product equal to 1 with the vector ej having the same index (e^{i}.ei = 1).
Any vector x is therefore expressed in each base in the form :
x = xi e^{i}
x = x^{i} ei
where xi and x^{i} are respectively the Covariant and contra-variant components of the vector x.
Proof of covariant components :
The first form induces that : x.ej = (xi e^{i}).ej = xi δ^{i}j = xj
which is indeed the definition of the covariant components xj of the vector x.
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 :
f' = f ( (1 - r^{*}/r)/(1 - r^{*}/r') ) ^{1/2}
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 tensor (Eab) measures the local deformation of the chrono-geometry of Space-time and represents its curvature at a given point. It is a tensor of order 2, symmetric and with zero Divergence (E^{ab}_{;a} = 0).
Its components are given by Einstein equations.
In Cartesian coordinates, all the components are of dimension m^{-2}
The notations are those in Maxwell Equations.
In Restricted Relativity, the Lorentz force (F_LORENTZ = q (E + v x B)) is written in a tensor form whose components are the following [GOU Relativité_Restreinte, p.538] : fi_LORENTZ = q Fij u^{j}
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 tensor of order 2.
In Cartesian coordinates, all the components are of dimension m^{-1}.V ou C^{-1}.N ou kg.m.s^{-3}.A^{-1} and are written [GOU Relativité_Restreinte, p.541] :
Fii for i ≥ 0 = 0
Fi0 for i > 0 = -F0i = Ei
F21 = -F12 = -c B^{3}
F31 = -F13 = c B^{2}
F32 = -F23 = -c B^{1}
Ei and B^{i} are respectively the spatial components of the electric field E and magnetic field B.
By increasing of index (see Tensor operators), we find the components of the tensors F^{i}j and F^{ij} in the following form :
F^{i}j = g^{ik}_MINK Fjk
F^{i}i for i ≥ 0 = 0
F^{i}0 for i > 0 = F^{0}i = Ei
F^{2}1 = -F^{1}2 = -c B^{3}
F^{3}1 = -F^{1}3 = c B^{2}
F^{3}2 = -F^{2}3 = -c B^{1}
F^{ij} = g^{il}_MINK F^{j}l
F^{ii} for i ≥ 0 = 0
F^{i0} for i > 0 = -F^{0i} = -Ei
F^{21} = -F^{12} = -c B^{3}
F^{31} = -F^{13} = c B^{2}
F^{32} = -F^{23} = -c B^{1}
Let U, V and W be Tensors of arbitrary order and valence bearing on the indices i, j, k, l...
Using the Convention of summation, the following elementary operations are defined on these tensors :
- Sum (W^{i}jk = U^{i}jk + V^{i}jk) of components : W^{i}jk = U^{i}jk + V^{i}jk
- Product by a scalar s (W^{i}jk = s U^{i}jk) of components : W^{i}jk = s U^{i}jk
- Scalar product (W = U^{ij}.Vij) of component : W = U^{ij} Vij
- Tensor product (W^{i}jkl = U^{i}j * Vkl) of components : W^{i}jkl = U^{i}j Vkl
- Covariant derivative
- Divergence
- index increasing :
A lower index can be changed to a higher index by multiplication with the inverse Metric tensor g^{ij}. Examples :
U^{i}k = g^{ij} Ujk
U^{ik} = g^{il} g^{km} Ulm
- index lowering :
A higher index can be changed to a lower index by multiplication with the Metric tensor gij. Examples :
Uik = gij U^{j}k
Ulm = gjl gkm U^{jk}
U^{k}lm = glp U^{kp}m
- Contraction of index
- Base change. See 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 :
T00 : energy density or pressure or c^{2} times the density
T0j for j > 0 : (-c) times the component i of the relativistic impulse density (momentum impulse density) or (-1/c) times the component i of the energy flow (Poynting vector φ for electromagnetic field)
Tij for i and j > 0 : spatial components of the stress tensor (Sij)
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}
The notations are those in Maxwell Equations.
The components of the Energy-impulse tensor (Tab_EM) of ElectroMagnetic field are the following [GOU Relativité_Restreinte, p.635] :
Tij_EM = ε_{0} (Fim F^{m}j - (1/4) gij Fkl F^{kl})
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) ε_{0} (E.E + c^{2} B.B)
Ti0_EM = T0i_EM for i > 0 corresponding to (-1/c) times φ with φ = Poynting vector = (1/ μ_{0}) E x B
Tij_EM for i and j > 0 corresponding to Sij = ε_{0} ( (1/2) (E.E + c^{2} B.B) δij - (Ei Ej + c^{2} Bi Bj) )
where δ is the Kronecker symbol.
A fluid is called "perfect" when the viscosity and thermal conduction effects can be neglected, which is the case in cosmology.
The components of the Energy-impulse tensor (Tab_FP) of Perfect Fluid are the following [GOU Relativité_Générale, p.114] :
Tij_PF = (ρ c^{2} + p) ui uj + p gij
where :
c^{2} ρ 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 c^{2}
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 c^{2} ≥ -p), and the dominant energy condition when : (ρ c^{2} ≥ |p|).
See Solution of Einsteins equations with Friedmann-Lemaitre-Robertson-Walker metric
The metric of Alexander Alexandrowitsch Friedmann, Georges Lemaitre, Howard Percy Robertson and Arthur Geoffrey Walker is a Relativistic metric corresponding to a spatially homogeneous and isotropic Space-time.
In spherical coordinates (r > 0, θ = [0, π], φ = [0, 2 π]) this metric is written by taking the sign convention (- + + +) :
ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dr^{2} (1 - k r^{2})^{-1} + r^{2} (dθ^{2} + sin^{2}[θ] dφ^{2}) )
where k is a constant called space curvature parameter that can be flat (k = 0), closed (k = 1) or open (k = -1) ;
and a(t) is a function of t only, called scale factor or curvature factor or universe radius (a(t) > 0).
The coordinate r is dimensionless and the radius (a) has the dimension of a length.
The gravitational potentials gij then are the following :
g00 = -1 ; g11 = a(t)^{2} (1 - k r^{2})^{-1} ; g22 = a(t)^{2} r^{2} ; g33 = a(t)^{2} r^{2} sin^{2}[θ] ; gij = 0 for i and j taken different between 0 and 3
The sign of d(a)/dt informs about the universe evolution : positive if expansion, negative if contraction and zero if static.
The spatial coordinates (x^{i}) then describe spatial hypersurfaces of Euclidean type (for k = 0), hyperspherical type (for k = 1) and hyperbolical type (for k = -1), whose spatial curvature k* is constant and is equal to : k* = 6 k a(t)^{-2}
For k = 0 we find the Minkowski metric : ds^{2} = -c^{2}dt^{2} + a(t)^{2} ( dr^{2} + r^{2} (dθ^{2} + sin^{2}[θ] dφ^{2}) )
Proof [GOU Relativité_Générale, p.194] :
A spatially homogeneous and isotropic space-time is equivalent to a maximally symmetric space of dimension 3 (or spatially constant curvature k*) with three possible types of maximally symmetric spaces according to the value of k* (not proofed here):
If k* = 0, space is the space R^{3} provided with the standard Euclidean metric.
Its metric is the following :
gij dx^{i} dx^{j} = dr^{2} + r^{2} (dθ^{2} + sin^{2}[θ] dφ^{2})
If k* > 0, space is the hypersphere S^{3} which is the part of R^{4} defined by : x^{2} + y^{2} + z^{2} + w^{2} = 1
where (x, y, z, w) denotes a generic element of R^{4}.
This definition is the three-dimensional transposition of the definition of the two-dimensional sphere S^{2} in R^{3}.
Its metric is the following :
gij dx^{i} dx^{j} = dΧ^{2} + sin^{2}[Χ] (dθ^{2} + sin^{2}[θ] dφ^{2})
with Χ = [0, π]
If k* < 0, space is the hyperbolic space H^{3} which is the upper sheet of the two-sheeted hyperboloid of dimension 3 defined in R^{4} by : x^{2} + y^{2} + z^{2} - w^{2} = -1
Its metric is the following :
gij dx^{i} dx^{j} = dρ^{2} + sinh^{2}[ρ] (dθ^{2} + sin^{2}[θ] dφ^{2})
with ρ > 0
By setting r = sin[Χ] = sinh[ρ], these three metrics are written in a common form :
gij dx^{i} dx^{j} = dr^{2} (1 - k r^{2})^{-1} + r^{2} (dθ^{2} + sin^{2}[θ] dφ^{2})
with k = 0 for Euclidean space, k = 1 for the hypersphere and k = -1 for hyperbolic space.
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".
The geodesic tensor equations are written as follows :
(d^{2}x^{i} / dp^{2}) + Γ^{i}lk (dx^{k}/dp) (dx^{l}/dp) = 0
where p is the curvilinear abscissa (or affine parameter) along the trajectory
and Γ^{i}jk are the Christoffel symbols.
We can choose for p the proper Time τ of the particle satisfying to : ds^{2} = -c^{2} dτ^{2}
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 x^{i}. According to Cauchy theorem, this system admits a unique solution if the following initial conditions are fixed :
x^{i}(0) = four arbitrary constants
(dx^{i}/dp)(0) = u^{i}_{0}
u^{i}_{0} being four constants satisfying : gij u^{i}_{0} u^{j}_{0} = -c^{2}
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 : d^{2}x^{i} / dp^{2} = 0 whose solutions are the ordinary straight lines : x^{i}(p) = a^{i}(p) p + b^{i}
The law of Edwin Powell Hubble states that galaxies move away from each other at an expansion speed v approximately proportional to their distance d :
v = H(t) d
where H(t) is the Hubble parameter used in particular in the Friedmann equations.
The value to date of H(t) (called Hubble constant H0) is about 70 (km/s)/Mpc, with 1 pc = 1 parsec = 3,2616 light-years = 3,085677581 10^{16} m
The speed v is not a physical speed. It only reflects the Space-time expansion which causes an global movement of the universe galaxies. The Earth thus "retreats before the light" because the space-time expands.
Any distant galaxy having the same proper Time as the observer (called cosmic time), there is no relative time effect (Doppler-Fizeau effect) on its radiation period but a simple differential delay effect on the radiation period received.
The own movements acquired by the galaxies superimpose to this global movement because of their gravitational interactions with their neighbors.
The expression of the Kronecker symbol δ is as follows :
δ^{i}j = δij = δ^{ij} = 1 for i = j and 0 otherwise.
Warning : the Kronecker symbol is not a tensor. We can not write for example that : δij = gik δ^{k}j = gij, which is wrong.
The expression of the Levi-Civita symbol ε is as follows :
εijkl... = ε^{ijkl...} =
0 if two ou more indices (i,j,k,l...) are equal
+1 if (i,j,k,l...) is an even permutation of (1,2,3,4...)
-1 if (i,j,k,l...) is an odd permutation of (1,2,3,4...)
When any two indices are interchanged, equal or not, the symbol is negated :
ε...i...l... = -ε...l...i...
For 3 indices (i,j,k) we have :
εijk = +1 for 123 or 231 or 312
εijk = -1 for 132 or 213 or 321
For 4 indices (i,j,k,l) we have :
εijkl = +1 for 1234 or 1342 or 1423 or 2143 ... or 4321
εijkl = -1 for 1243 or 1324 or 1432 or 2134 ... or 4312
ε allows in particular to express many vectorial operations in a compact form :
- Vector 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
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 : c^{2} t^{2} = x^{2} + y^{2} + z^{2}, called Light cone (see Figure above where x represents in a simplified way the three spatial coordinates x, y and z).
Any vector v of the Space-time is called :
- time-like when the Scalar product v.v < 0. This is the case of the vector tangent to the trajectory of a non-zero mass particle. Two events of Space-time can be connected by information going at a speed lower than light speed.
- light-like (or light vector or isotropic vector) when the Scalar product v.v = 0 with v ≠ 0. This is the case of the vector tangent to the trajectory of a zero mass particle (photon for example). Two events of Space-time can be connected by information going at the light speed.
- space-like when the Scalar product v.v > 0. This is the case of the vector neither time-like nor light-like. Two events of Space-time can not be connected by information going at the light speed.
By definition the time-like, light-like and space-like vectors are situated respectively inside, on the surface and outside the light cone.
Hendrik Antoon Lorentz gave an imperfect version of these equations in 1899 and then 1904. Jules Henri Poincaré published the correct equations in 1905, baptizing them with the name of Lorentz. See Lorentz-Poincaré transformation
Any particle of charge q and velocity v, subjected to an electric field E and to a magnetic field B, undergoes the Lorentz force F_LORENTZ = q (E + v x B)
The equations of James Clerk Maxwell specify the evolution of electromagnetic fields E and B. In vacuum they writte as follows :
div(E) = ρ / ε_{0}
curl(E) = - dB/dt
div(B) = 0
curl(B / μ_{0}) = j + ε_{0} dE/dt
the two densities ρ and j being connected by the relation of the conservation of the charges : 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)
μ_{0} is the permeability of vacuum : μ_{0} = 4 π 10^{-7} kg.m.A^{-2}.s^{-2}
ε_{0} is the dielectric permittivity in vacuum : ε_{0} = 1 / (μ_{0} c^{2})
c is the light speed in vacuum (c = 2,99792458 10^{8} m.s^{-1}
The vector operators used are the following :
v1.v2 and v1 x v2 : scalar product and vector product of any two vectors v1 and v2.
div(v) and curl(v) : divergence and curl of any vector v.
It can be proved (arduously) that these equations are invariant with respect to the Lorentz-Poincaré equations.
For a vector space of dimension n having for basis vectors the set (e1, e2... en), we denote the Scalar product of two basic vectors in the form : gij = ei.ej
The Metric tensor is the tensor gij whose components are gij. It is a tensor of order 2, symmetric and 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 : g^{ij} gjk = g^{i}k = δ^{i}k
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
The following results can be proved :
g^{ij} gij = n
gij;k = g^{ij};k = 0
gij g^{ij}_{,k} = -g^{ij} gij_{,k}
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 equations under the conditions Λ = 0 and Tab = 0 (since the Curvature tensor is zero, so also the Ricci tensor Rab).
The coordinates are the following taking the convention of sign (- + + +) :
In Cartesian coordinates : ds^{2} = -c^{2}dt^{2} + dx^{2} + dy^{2} + dz^{2}
corresponding to the gravitational potentials gij such that : g00 = -1 ; g11 = 1 ; g22 = 1 ; g33 = 1 ; gij = 0 for i and j taken different between 0 and 3.
This metric gij_MINK has the following properties : g^{ij} = g^{ji} = gij = gji
In spherical coordinates (r > 0, θ = [0, π], φ = [0, 2 π]) : ds^{2} = -c^{2}dt^{2} + dr^{2} + r^{2} dθ^{2} + r^{2} sin^{2}[θ] dφ^{2}
corresponding to the gravitational potentials gij such that : g00 = -1 ; g11 = 1 ; g22 = r^{2} ; g33 = r^{2} sin^{2}[θ] ; gij = 0 for i and j taken different between 0 and 3.
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 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 [EIN Sur_electrodynamique, para.4]".
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, engineer and 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 classical cinematic framework." [SPA Einstein_et_la_revolution, p.23]
- Henri Bergson, renowned 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 apparent durations given by Poincaré [POI L'Etat, p.311]).
The Poisson equation of Newtonian gravitation (Δψ = 4 π G ρ) is a particular case of the Einstein equations which correspond to a spatially isotropic space-time, containing a non-relativistic perfect fluid (p << ρ c^{2}), in a weak gravitational field (|ψ| << c^{2}) and without cosmological constant (Λ = 0).
This Newtonian limit provides the coupling coefficient (KHI = 8 π G/c^{4}) used in the Einstein equations as well as the Schwarzschild gravitational radius (r^{*} = 2 G M c^{-2}) used in the Solution of Einsteins equations with Schwarzschild Metric.
Complete proof (about the ten Einstein equations) :
In a weak gravitational field, we can always find a system of coordinates (x^{i} for i = 0 to 3) = (ct, x, y, z) where the metric components are written according to sign convention (- + + +) :
(N1a) ds^{2} = -A^{2} c^{2}dt^{2} + A^{-2} (dx^{2} + dy^{2} + dz^{2})
(N1b) A = 1 + (ψ/c^{2})
where ψ is the Newtonian gravitational potential (ψ = -G M/r) satisfying : |ψ| << c^{2}
We note by o(B) the function "Small o of the quantity B in the neighborhood of 1" which is the negligible function of Landau.
This metric is written at the first order of ψ/c^{2} in the following equivalent form :
(N2a) ds^{2} = -(1 + B + o(B^{2})) c^{2}dt^{2} + (1 - B + o(B^{2})) (dx^{2} + dy^{2} + dz^{2})
(N2b) B = 2 (ψ/c^{2}) = -2 G M c^{-2} / r = B(r)
Note the following useful relations :
(N3a) r B_{,i} = r dB/dx^{i} = r (dB/dr)(dr/dx^{i}) = -B (x^{i}/r) = o(B)
(N3b) r^{2} B_{,i,j} = r^{2} d(B_{,i})/dx^{j} = -x^{i} r^{2} d(B r^{-2})/dx^{j} = 3 B (x^{i}/r) (x^{j}/r) = o(B)
(N3c) Schwartz Theorem : B_{,i,j} = B_{,j,i}
(N3d) B_{,i,0} = B_{,0} = 0
The gravitational potentials gij of the Metric tensor are then the following :
(P1a) g00 = -(1 + B + o(B^{2})) = -1 + o(B)
(P1b) g11 = g22 = g33 = (1 - B + o(B^{2})) = 1 + o(B)
(P1c) gij = 0 for i and j taken different between 0 to 3
The gravitational potentials gij of the inverse Metric tensor are then as follows : g^{ij} gjk = δ^{i}k
where δ is the Kronecker symbol.
(P2a) g^{00} = 1/g00 = -1 + o(B)
(P2b) g^{11} = g^{22} = g^{33} = 1/g11 = 1 + o(B)
(P2c) g^{ij} = 0 for i and j taken different between 0 and 3
Note the following useful relation :
(P3) r gii_{,k} = -r B_{,k} + o(B^{2})
The Christoffel symbols Γ^{i}jk are then written by the relations : Γ^{i}jk = Γ^{i}kj = (1/2) g^{il} (glk_{,j} + glj_{,k} - gjk_{,l})
Given the relation (P2c), these relations are simplified by :
(S1a) Γ^{i}jk = Ki Gijk
(S1b) Ki = (1/2) g^{ii}
(S1c) Gijk = gik_{,j} + gij_{,k} - gjk_{,i}
Four distinct cases are to be considered according to the values of i, j and k.
Given the relations (P3)(P2b)(P1c), these cases are written at the first order of B (without term o(B^{2})) :
Case 1 where (i = 0) and (j = 0)
Examples : (ijk) = (000), (001), (010)
Gijk = g0k,0 + g00_{,k} - g0k,0 = g00_{,k} = -B_{,k}
Γ^{0}0k = Γ^{0}k0 = (-1/2) g^{00} B_{,k}
Case 2 where (i ≠ 0) and (j = i)
Examples : (ijk) = (101), (110), (111)
Gijk = gik_{,i} + gii_{,k} - gik_{,i} = gii_{,k} = -B_{,k}
Γ^{i}ik = Γ^{i}ki = (-1/2) g^{11} B_{,k}
Case 3 where (i ≠ j) and (j = k)
Examples : (ijk) = (011), (100)
Gijk = gik_{,k} + gik_{,k} - gkk_{,i} = -gkk_{,i} = B_{,i}
Γ^{i}kk = (1/2) g^{ii} B_{,i} = (1/2) g^{11} B_{,i}
Case 4 where (i, j and k all different)
Examples : (ijk) = (012), (102)
Gijk = gik_{,j} + gij_{,k} - gjk_{,i} = 0
Γ^{i}jk = 0
Given the relations (P2a)(P2b)(N3a)(N3b), note the following useful relations :
(S2a) r Γ^{i}jk = o(B)
(S2b) r^{2} Γ^{i}jk_{,l} = o(B)
(S2c) Γ^{i}jk_{,0} = 0
The components Rij of Ricci Tensor are then written by the relations : Rij = R^{k}ikj = (Γ^{k}ij_{,k} - Γ^{k}ik_{,j}) + (Γ^{k}kl Γ^{l}ij - Γ^{k}jl Γ^{l}ik)
Given the relations (S2a)(S2b), these relations are simplified by :
(T1) r^{2} Rij = r^{2} Rji = r^{2} (Γ^{k}ij_{,k} - Γ^{k}ik_{,j}) + o(B^{2})
Hence the expression of each Rij at the first order of B :
Component R00 :
Given the relation (S2c) and Case 3 above, we find :
S1 = Γ^{k}00_{,k} = Γ^{0}00_{,0} + [Γ^{k}00_{,k}]for_k≠0 = 0 + Sum_for_k≠0[(1/2) g^{11} B_{,k,k}] = (1/2) g^{11} ΔB
S2 = Γ^{k}0k_{,0} = 0
R00 = S1 - S2 = (1/2) g^{11} ΔB = (1/2) ΔB
Component Rii for (i≠0) :
Given the relation (S2c) and Cases 2, 3 and 1 above, we find :
S1 = Γ^{k}ii_{,k} = Γ^{i}ii_{,i} + Γ^{0}ii_{,0} + [Γ^{k}ii_{,k}]for_k≠i_and_k≠0 = (-1/2) g^{11} B_{,i,i} + 0 + Sum_for_k≠i_and_k≠0[(1/2) g^{11} B_{,k,k}] = (-1/2) g^{11} (2 B_{,i,i} - ΔB)
S2 = Γ^{k}ik_{,i} = Γ^{0}i0_{,i} + [Γ^{k}ik_{,i}]for_k≠0 = (-1/2) g^{00} B_{,i,i} + Sum_for_k≠0[(-1/2) g^{11} B_{,i,i}] = (-1/2) (g^{00} + 3 g^{11}) B_{,i,i}
Rii for (i≠0) = S1 - S2 = (1/2) (g^{00} + g^{11}) B_{,i,i} + (1/2) g^{11} ΔB = (1/2) ΔB = R00
Component Rij for (i≠j) :
Given the relations (S2c)(N3d)(N3c) and Cases 4, 2 and 1 above, we find :
S1 = Γ^{k}ij_{,k} = S11 + S12
S11 = [Γ^{k}ij_{,k}]for_k≠i_and_k≠j = 0 + 0
S12 = Γ^{i}ij_{,i} + Γ^{j}ij_{,j}
Case A : If (i=0) and (j≠0) : S12 = Γ^{0}0j_{,0} + Γ^{j}0j_{,j} = 0 + (-1/2) g^{11} B_{,0,j} = 0
Case B : If (i≠0) and (j=0) : S12 = Γ^{i}i0_{,i} + Γ^{0}i0_{,0} = (-1/2) g^{11} B_{,0,i} + 0 = 0
Case C : If (i≠0) and (j≠0) : S12 = Γ^{i}ij_{,i} + Γ^{j}ij_{,j} = (-1/2) g^{11} B_{,j,i} + (-1/2) g^{11} B_{,i,j} = (-1/2) 2 g^{11} B_{,i,j}
S1 = S11 + S12 = S12
S2 = Γ^{k}ik_{,j} = S21 + S22
S21 = Γ^{0}i0_{,j} = (-1/2) g^{00} B_{,i,j}
S22 = [Γ^{k}ik_{,j}]for_k≠0 = 3 (-1/2) g^{11} B_{,i,j}
S2 = S21 + S22 = (-1/2) (g^{00} + 3 g^{11}) B_{,i,j}
Rij for (i≠j) = S1 - S2 =
Case A : 0 - (-1/2) (g^{00} + 3 g^{11}) B_{,0,j} = 0
Case B : 0 - (-1/2) (g^{00} + 3 g^{11}) B_{,i,0} = 0
Case C : (1/2) (g^{00} + g^{11}) B_{,i,j} = 0
Rij for (i≠j) = 0 whatever Case A, B or C.
The Scalar curvature is then written by the relation : R = g^{ij} Rij
Given the relation (P2c), R is simplified by :
(C1) R = g^{ii} Rii
Hence the expression of R at the first order of B :
R = g^{00} R00 + Sum_for_i≠0[g^{ii} Rii] = g^{00} R00 + g^{11} (3 R00) = 2 R00 = ΔB
The Einstein tensor is then written by the relation : Eab = Rab - (1/2) gab R + Λ gab
By replacing in this relation the expressions found for gij, g^{ij}, Rij and R, we find at the first order of B :
E00 = R00 - (1/2) g00 R + Λ g00 = ΔB - Λ
Eii for (i≠0) = Rii - (1/2) g11 R + Λ g11 = Λ
Eij for (i≠j) = Rij - (1/2) gij R + Λ gij = 0
The Energy-impulse tensor of Perfect Fluid of density ρ and pressure p is then written by the relation : Tij = (c^{2} ρ + p) ui uj + p gij
In the case of a spatially isotropic space-time containing a non-relativistic perfect fluid (p << ρ c^{2}), then Tij is written :
T00 = ρ c^{2}
The other components Tij are all zero.
The Einstein equations are then written by the relation : Eab = KHI Tab and give at the first order of B :
ΔB - Λ = E00 = KHI T00 = KHI ρ c^{2}
Λ = Eii for (i≠0) = KHI Tii = 0
0 = Eij for (i≠j) = KHI Tij = 0
Given the relation (N2b), the first equation is written :
Δψ = (1/2) KHI ρ c^{4} + (1/2) Λ c^{2}
In the case where the cosmological constant is zero (Λ = 0), the ten Einstein equations are reduced to a single equation (Δψ = (1/2) KHI ρ c^{4}). By choosing a coupling coefficient KHI equal to : KHI = 8 π G/c^{4}, we then find the Poisson equation of the Newtonian gravitation (Δψ = 4 π G ρ).
By comparing the g00 of the Schwarzschild metric (g00 = -(1 - r^{*}/r)) with the g00 of the Newtonian limit (g00 = -(1 + B)), the Schwarzschild gravitational radius is found by : r^{*} = 2 G M c^{-2}
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 :
Δψ = 4 π G ρ
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}.m^{3}.s^{-2}).
ψ has the dimension of m^{2}.s^{-2}
Proof
If g is the gravitational field produced at a point O of mass m by a spherical source S of mass M situated at the distance r from O, we have the following relations :
F = m g
g = -u G M / r^{2}
g = -grad(ψ)
ψ = -G M / r
F = gravitational force exerted in O
u = unit vector of the line SO
The field g is otherwise characterized by the two laws :
div(g) = -4 π G ρ
curl(g) = -curl(grad(ψ)) = 0
in an analogous manner to the electric field E with respect to the electrical potential V without magnetic field B :
div(E) = ρ_charge / ε_{0}
curl(E) = 0
The first law (div(g) = -4 π G ρ) results from the Gaussian theorem (for any closed surface S with volum V and outgoing normal vector n : Sum_for_S[ g.n dS ] = -4 π G M = Sum_for_V[ -4 π G ρ dV ]) associated with the divergence theorem (Sum_for_S[ g.n dS ] = Sum_for_V[ div(g) dV ]).
Hence the result :
-4 π G ρ = div(g) = div(-grad(ψ)) = -Δψ
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. By "inertial" we mean "whose trajectory is a line of Minkowski's space-time and whose Quadri-rotation is zero".
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] :
a = (1/c) du/dτ = a^{i} ei
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 (see Light cone)
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] :
f = dp/dτ
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 c^{2} a + c (dm/dτ) u
f.u = -c (dm/dτ)
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] :
p = m c u
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 = -m^{2} c^{2}
See Space-time.
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] :
u = (1/c) dx/dτ = u^{i} ei
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 a time-like vector type : u.u = -1 (see Light cone)
If (ds) is the distance (or interval) between two events infinitely close to the Space-time, then the Relativistic metric is the square of this distance and is written : ds^{2} = gij dx^{i} dx^{j}
In the curved space-time of General Relativity this metric is written in Cartesian coordinates :
ds^{2} = g00 (c dt)^{2} + g01 (c dt) dx + g02 (c dt) dy + g03 (c dt) dz +
g10 dx (c dt) + g11 dx^{2} + g12 dx dy + g13 dx dz +
g20 dy (c dt) + g21 dy dx + g22 dy^{2} + g23 dy dz +
g30 dz (c dt) + g31 dz dx + g32 dz dy + g33 dz^{2}
The coefficients gij are the components of the Metric tensor.
Pictures above from left to right : Riemann, Christoffel and Ricci-Curbastro
The Ricci tensor (Rab) is a symmetric tensor of order 2 producted by Contraction of the Curvature tensor on the first and third index. It is a tensor that also measures the local deformation of space-time but incompletely.
Using the Convention of partial derivative, its components are the following :
Rij = R^{k}ikj = Γ^{k}ij_{,k} - Γ^{k}ik_{,j} + Γ^{k}kl Γ^{l}ij - Γ^{k}jl Γ^{l}ik
where Γ^{i}jk are the Christoffel symbols.
In Cartesian coordinates, all the components are of dimension m^{-2}
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 = g^{ij} Rij = R^{i}i
The scalar product of two arbitrary vectors x and y is written:
x.y = x^{i} yi = xi y^{i} = g^{ij} xi yj = gij x^{i} y^{i}
where the coefficients gij are the components of the Metric tensor.
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}
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, θ = [0, π], φ = [0, 2 π]) this metric is written by taking the sign convention (- + + +) :
ds^{2} = -e^{2 μ} c^{2}dt^{2} + e^{2 α} dr^{2} + r^{2} dθ^{2} + r^{2} sin^{2}[θ] dφ^{2}
where μ and α are only functions of r.
The gravitational potentials gij then are the following :
g00 = -e^{2 μ} ; g11 = e^{2 α} ; g22 = r^{2} ; g33 = r^{2} sin^{2}[θ] ; gij = 0 for i and j taken different between 0 and 3
Proof [GOU Relativité_Générale, p.116] :
The spherical center symmetry of the field allows to write the metric in the following form :
ds^{2} = -N^{2} c^{2}dt^{2} + A^{2}dr^{2} + B^{2} (dθ^{2} + sin^{2}[θ] dφ^{2})
where the components N, A and B are functions of r and t.
The staticity of the field then allows to delete the dependence of t in these components.
The coordinate r can be otherwise choosen as the areolar radius of the invariance spheres related to the spherical symmetry.
This is written :
N(r,t) = N(r) = e^{μ}
A(r,t) = A(r) = e^{α}
B(r,t) = B(r) = r
In Restricted Relativity, it is shown that two events located in different places can be simultaneous in one reference frame without being in another. The notion of simultaneity loses its universal character.
Proof :
Let two simultaneous events (x1, y1, z1, t1) and (x2, y2, z2, t2 = t1) in the referential frame R. In the reference frame R' in uniform translation with respect to R, the duration (t'2 - t'1) between these two same events is written taking into account the Lorentz-Poincaré equations (L2) :
t'2 - t'1 = gamma ( (t2 - t1) - B (x2 - x1) ) = -gamma B (x2 - x1)
gamma and B being given by equations (L3) ans (L4).
When the two events are not located at the same points, the spatial difference (x2 - x1) in R is not zero. The temporal difference (t'2 - t'1) in R' is therefore not zero despite the simultaneity (t2 = t1) of the two events in R.
Space-time is a four-dimensional space where time is no longer a separate quantity independent of space but a variable playing the same role as spatial variables. The notion of Simultaneity is no longer universal.
In this space-time a point or event x(x, y, z, t) is identified by a four-dimensional vector x called Quadri-vector or 4-vector whose components are denoted :
- In Cartesian coordinates : x^{0} = ct ; x^{1} = x ; x^{2} = y ; x^{3} = z
- In spherical coordinates (r > 0, colatitude θ = [0, π], longitude φ = [0, 2 π]) : x^{0} = ct ; x^{1} = r ; x^{2} = θ ; x^{3} = φ
Note that r has no physical meaning. From a geometrical view point, r gives the area (A = 4 π r^{2}) of the invariance spheres related to the spherical symmetry (spheres with t = constant and r = constant). The coordinate r is sometimes called areolar radius [GOU Relativité_Générale, p.59].
If ei are the basic vectors of this space-time, the base {ei} is called orthonormal (with respect to the Scalar product g) when [GOU Relativité_Générale, p.26] :
g00 = e0.e0 = -1
gii = ei.ei = 1 for i = 1 to 3
gij = ei.ej = 0 for i and j taken different between 0 and 3
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.
The cosmological model that best describes the history and behavior of the observable universe is the standard model of cosmology (or Big Bang model or Λ-CDM model meaning "Λ Cold Dark Matter").
This model represents an universe (see curve C4 in Figure 1 above) :
- spatially homogeneous and isotropic on a large scale (thus also with constant spatial curvature). See Friedmann-Lemaitre-Robertson-Walker metric
- filled with a perfect fluid of generally zero pressure (galaxy gas corresponding to : w = 0) and density ρ composed of hot (relativistic or radiation) matter and cold (non-relativistic) matter.
- whose spatial curvature is zero (k = 0).
- which would contain, in addition to the ordinary matter, dark matter (surplus gravity which the galaxies need not to be discarded during their rotation) and black energy (global repulsive force which tends to accelerate the universe expansion and requiring : Λ > 0).
- coming from a primordial explosion such that the scale factor a(t) tends to 0 when t tends to 0 (Big Bang model).
The term "tensor" was introduced by the physicist Woldemar Voigt to represent mathematically the tensions in a solid (see Figure above).
A tensor is a multilinear function of the coordinates of space, defined in an n-dimensional vector space by n^{m} components, where m is the order of the tensor. The tensor of order 0 is a scalar and has a single component. The tensor of order 1 is a vector with n components. The tensor of order 2 is a square matrix with n^{2} components.
Each tensor also has a valence or type denoted (h, q) where h is the number of contra-variant index (indicated in the upper position) and q the number of Covariant index (indicated in the lower position) according to : m = h + q.
For any tensor W, its components can be contra-variant (example : W^{ijk}), covariant (example : Wijk) or mixed (example : W^{i}jk is a mixed tensor of order 3 with one contra-variant index i and two covariant index j and k).
The tensorial calculation has the advantage of being freed from all systems of coordinates and the results of the mathematical developments are thus invariant by change of reference frame (see Covariance of physical law).
Base change :
Let [A] be the transition matrix from base {ei} to base {e'k} such that : e'k = A^{i}k ei, and [B] = [A^{-1}] the inverse transition matrix such that : ei = B^{i}k e'k
For any tensor W of order 2, using the multilinearity of W and the properties of Covariance and Contra-variance, the components of the tensor are given by the following laws :
W'ij = A^{k}i A^{l}j Wkl
W' ^{ij} = B^{i}k B^{j}l W^{kl}
W' ^{i}j = B^{i}k A^{l}j W^{k}l
The base change transforms the tensor W into a tensor W' whose components are linear combinations of the components of the origin tensor.
For any tensor W h times contra-variant and q times covariant, the general law is as follows [GOU Relativité_Restreinte, p. 476] :
W' ^{i1... ih} j_{1}... j_{q} = (B^{i1} k_{1}) ... (B^{ih} k_{h}) (A^{l1} j_{1}) ... (A^{lq} j_{q}) W ^{k1... kh} l_{1}... l_{q}
In old books, a tensor is defined not as a multilinear application, but as an array of numbers W ^{i1... ih} j_{1}... j_{q} which transforms according to this law in a base change. See Tensoriality criteria.
Any mathematical object that satisfies one of the following tensorial criteria is a tensor.
Criterion 1 : any object defined intrinsically as a multilinear form (see Tensor).
Criterion 2 : any numbers table that transforms by base change according to the transformation formulas required (see Tensor).
Criterion 3 : any result of an elementary operation or a combination of Elementary tensor operations (Sum, Product ... Covariant derivative ... Base change).
Proper time and apparent time
Each reference body has its proper time. This proper time (or true time) is the time (τ) measured in the reference frame where the body is immobile. The apparent time (or improper or relative or observed or "measured" time) conversely is the time (t) measured in a mobile reference frame with respect to this proper reference frame.
Proper time and apparent time are therefore two distinct times measured under different conditions.
All measurements are made by fixed clocks in their reference frame and whose internal mechanism is generally insensitive to the reference frame movement. An atomic clock is an ideal clock because the time it provides does not depend very much on accelerations undergone which are very low compared to the centripetal acceleration of an electron around its atomic nucleus (about 10^{23} m.s^{-2}).
The proper time τ of a material particle along its trajectory is defined by the relation : dτ = (1/c) (-ds^{2})^{1/2}
where ds^{2} is the Relativistic metric.
Proper duration and apparent duration
In Restricted Relativity, for a given reference frame, the proper duration (d0) is the time interval that separates two events occurring at the same place in this reference frame. In any other reference frame, the duration is greater than the proper duration and is called apparent duration (d). Some authors speak of "Dilation of apparent durations" or "slowing down of moving clocks".
Biological time
See Twin paradox
Proof of relation : d > d0 [ANN Electricité_2] :
Let two events occurring in the reference frame R at the same place of coordinates (x, y, z) but at different instants t1 and t2. The (proper) duration separating them is : d0 = t2 - t1.
For an observer of the referential frame R' in uniform translation at the speed V with respect to R, the events occur at instants t1' and t2' given by the Lorentz-Poincaré equation (L2) :
t1'= gamma (t1 - B x)
t2'= gamma (t2 - B x)
and separated by the (apparent) duration : d = t2' - t1' = gamma (t2 - t1) = d0 / (1 - V^{2}/c^{2})^{1/2}
So : d > d0
In the twin paradox, one of the twins stays on Earth while his brother makes a space journey at a speed close to the light speed and then goes back to Earth. We must then consider three inertial references frames : the twin "at rest", the traveling twin on outward journey and the traveling twin on return journey. To date it is no longer disputed that the clocks of the two successive reference frames of the traveling twin indicate a total duration shorter than the one of the twin at rest (Multiplicity of proper times) and there is no longer a paradox. However few authors produce a demonstration of this phenomenon that is physically acceptable, with non-infinite Quadri-accelerations at the rupture points of the spatio-temporal trajectory of the traveling twin [GOU Relativité_Restreinte, p.41].
Note that the Principle of equivalence combined with the Multiplicity of proper times predicted by Restricted Relativity leads us to predict a multiplicity of proper times where there is a multiplicity of gravitational potentials. Two perfect clocks placed in places of different gravitational potentials will not record identical cumulative times. We can speak of "Twin paradox bis": two twins living at different altitudes will not record the same duration.
But what about the aging of the two twins ? Can we say that the traveling twin goes back to Earth "younger" than the sedentary twin ? Opinions are divided :
- Poincaré never speaks of biological time and asserts that "the properties of time are only those of clocks".
- Einstein asserts that the physical time measured by clocks is the time actually experienced by the inhabitants linked to a given reference frame.
- Thibault Damour, Phycisien of Relativity, asserts that even if the traveling twin goes back to Earth younger than the sedentary twin, he will not live any longer. The best image is cryogenics : instead of sending one of the twins into space, he was put in an ice block and then delivered. The proper time of the traveling twin is actually shorter than the one of the sedentary twin but he does not acquire an additional "biological time", for example the number of heartbeats.
- Most astrophysicists assert that talking about biological time has no scientific meaning.
The universe age is the time elapsed since the Big Bang. The best approximation to date is given by : 1 / H0
where H0 is the Hubble constant (See Hubble law),
giving an age of about 13 billion years.
The universe line (or spatio-temporal trajectory) of a material particle is a Space-time curve (or sequential path of events) corresponding to the history of the particle.
By definition, the universe lines of physical particles are always located inside the Light Cone at a given point (see Figure above). These are any curves of time-like type (see Light Cone). They are Geodesics only when the material particle is not subjected to any other interaction than that induced by the gravitational field.
The authors quoted in this chapter are referenced in square brackets under the reference [AUTHOR Title].