From Unexpected Minkowskian Solution of General Relativity with Cosmological Constant to the Accelerating Universe
aa r X i v : . [ phy s i c s . g e n - ph ] M a r From Minkowskian Solution of General Relativity with CosmologicalConstant to the Accelerating Universe
Dr. Yves Pierseaux ([email protected])November 20, 2018
Abstract
A Minkowskian solution of the equation of General Relativity (as written by Einstein in 1915) is trivialbecause it simply means that both members of the equation are equal to zero. However, if alternatively, oneconsiders the complete equation with a non-zero constant Λ (Einstein 1917), a Minkowskian solution is nolonger trivial because it amounts to impose a constraint on the right hand side of the equation (i.e. a non-nullstress-energy tensor). If furthermore one identifies (as usual) this tensor to the one of a perfect fluid, one findsthat this fluid has a positive energy density and a negative pressure that depend on the three constants of theequation (i.e. gravitational constant G , cosmological constant Λ and velocity of light c). When doing that( § GR (an enigmatic non-baryonicMinkowskian fluid).Can one build a model of this object on the basis of a dynamical equilibrium between the effectivegravitational attraction due to the positive energy density versus the negative pressure repulsion? We proposeto study such a model, where the (enigmatic) fluid is assumed to exist only in a limited sphere whose surfaceacts like a ”test body” sensitive to the gravitational field created by the fluid. No static equilibrium exists,but a pseudoNewtonian ”dynamical equilibrium” §
2) can be reached if the pseudoEuclidean fluid is in stateof expansion. Up to there, we have simply constructed a model of an ”abstract Universe” (i.e. the limitedsphere: There is no fluid outside this sphere !) that gives to a (purely mathematical) constant Λ a concretephysical meaning.We discover finally that our expanding fluid has not only dynamical properties ( §
3) but also opticalproperties that are connected with Doppler Redshift ( § Let us consider Einstein’s basic equation ([A. Einstein 1915]) of General Relativity ( GR ) completed by a positivemathematical constant Λ > g µν Riemanian metric, G µν Einstein’s curvature tensor, T µν stress-energy tensor, G gravitational constant and c lightvelocity): G µν + Λ g µν = 8 πGc T µν (1)In order to discover the physical meaning of this constantΛ >
0, let us simplify with G µν = 0 the equation (1)by introducing Minkowskian metric g µν = η µν = −
100 00 −
10 000 − . We obtain a tensor of MinkowskianVacuum (2): T V ACUUMµν = Λ c πG η µν (2)Let us now associate to this tensor (2) the one of a perfect relativistic fluid:1 V ACUUMµν ≡ ( p + ρ ) u µ u ν c − pη µν (3)Minkowskian Vacuum is then simulated by an enigmatic fluid with a positive density of energy ρ = Λ c πG and anegative pressure : p = − Λ c πG T V ACUUMµν = T GRµν = ρ
000 0 p
00 00 p p = Λ c πG −
100 00 −
10 000 − p + ρ = 0 (4)Our enigmatic Minkowskian fluid becomes a physical object in the framework of (complete) GR . Before theexamination of the physical properties of our fluid determined by three basic constant (Λ , G and c ) ( § REMARK 1 Our non-usual fluid (4) cannot be confused with usual perfect fluid (4-SR) in theframework of standard Special Relativity ( SR ) . In this case, we have u µ u ν = 0 in proper system for allcomponents except for purely temporal components u u = c = 1) T SRµν = p + ρ
000 0000 0000 0000 − p
000 0 − p
00 00 − p − p = ρ
000 0 p
00 00 p p p + ρ = 0 (4 SR) Any relativistic usual perfect fluid has a positive pressure p ( ≤ p ≤ w em ) . In this way our enigmatic fluid isno longer a usual fluid in ”immutable Minkowskian Vacuum” (standard SR ) but it is the (Classical) MinkowskianContinuum itself . REMARK 2 Our non-usual (classical) fluid (2) cannot be confused with usual (quantum) blackenergy (2bis) in the framework of Cosmology.
Standard method in Cosmology consists in associating asupplementary stress-tensor T Λ µν to a cosmological constant (CC Λ ) in the second member of (1) in order to havea second contribution to G µν : (with c=1): G µν = 8 πGT µν − Λ g µν = 8 πG ( T µν − T Λ µν ) with T Λ µν = Λ8 πG g µν = ρ Λ g µν (2 Riemann) By associating T µν stress-energy tensor to the one of a perfect relativistic fluid T Λ µν = ( p + ρ ) u µ u ν − pg µν weobtain usually a fluid (black energy of ”quantum vacuum” ) characterized by an unknown Riemanian metric g µν (2-Riemann, see 18-RW) whilst in (2) the metric is determined a priori Minkowskian. In standard Cosmology,Minkowskian limit can only be a trivial result of a very improbable compensation T µν − T Λ µν = 0 . This negative pressure of vacuum seems to be a stranger in SR . However, on the only basis of LT in 1905, Poincar´e (1853-1912)introduced such a negative pressure of ether in order to define a punctual electron by its electromagnetic field ([H. Poincar’e 1905]). Hesuspected the gravitational origin of his negative pressure but he did not succeed to find the exact relation between his ”gravitational”scalar field p and the gravitational constant G . For example we have free electromagnetic (em) isotropic field( w em , p em , p em , p em ) with null trace p em = + w em .According to Poincar´e, we have to add a special tensor to the tensor of em field in order to define Poincar´e’s electron:( c = 1): T electronµν = T emµν + p em η µν = ( p em + w em ) u µ u ν = w em u µ u ν , In proper system, we have: T electronµν =( w em , p em , p em , p em ) + (cid:0) w em , − p em , − p em , − p em (cid:1) = (cid:0) w em , , , (cid:1) . Poincar´e’s special tensor (non-null trace) is then w em
000 0 p P
00 00 p P p P corresponding to negative Poincar´e’s non-em pressure p Poincar ´ e = p P = − w em . This is the rea-son why, one admits usually (von Laue) that 3 / w em = m e ) comes from electromagnetic origin. Weshowed in “La Structure Fine de la Relativit´e Restreinte” that there is an enigmatic connection between Poincar´e’s gravitationalelectron (1905) and Einstein’s (gravitational) photon (light complex or photon, 1905, see note 10). Let us note that the ”vacuum” T Λ µν in (2-Riemann) depends on g µν ( G µν = 0) that indicates a presence of usual matter whilstthe vacuum in (2) is radically without any usual (baryonic) matter. Thermodynamical Properties of Minkowskian Fluid and UnstableStatic Model
Basic condition ρ + p = 0 (4) gives a new physical interpretation of Minkowskian metric as a fluid. g µν = η µν ⇔ ρ + p = 0 = h (4-bis)The geodesic of a material point is usually determined in Minkowskian space-time as a straight line. But herewe have a point of space-time continuum itself. In order to discover physical properties of our enigmatic fluidthe only possible point of departure is local thermodynamical properties given by (4bis):where h is null densityof enthalpy. Given that ρ is a density of energy of fluid, we have by integration a finite volume V with a finiteenergy U : U + pV = 0 = H (5)By differentiation we obtain: dU + pdV = 0 = dH = h (6)that seems trivially return to (4bis) with reduction of element of volume dV = 0 ( dH = hdV = 0). Usually itis claimed that Minkowskian vacuum would be static ( dV = 0). Let us consider, at flat Minkowskian limit, anEuclidean sphere of fluid: V = 43 πR U = 43 πρR (7)At Minkowskian limit we have also to take into account Einstein’s relation of ”materialization” of energy: U = M c M = 43 π ρc R (8)How can we test the behavior (static or not static) of such an Euclidean Sphere of fluid? Let us consider a testpoint (infinitesimal pseudomass µ = dM ) on the surface of sphere. We have to introduce gravitational constantbecause ρ VACUUM = Λ c πG . We suggest then to study a Newtonian model where the Minkowskian fluid is assumedto exist only in a limited sphere whose surface acts like a ”test body” sensitive to the gravitational field createdby the fluid. The surface is submitted to gravitational attractive potentialΦ = − GMR = − πG ρc R U P = − GM µR = − πG ρc R µ (9)Then the surface of fluid will collapse towards the center of sphere given that we have only attractive potentialenergy. So a static finite sphere of our fluid is unstable (∆ V = V
0) and Minkowskian solution (4bis) seemsimpossible. We rediscover in this way that standard immutable Minkowskian vacuum must be defined withoutgravitation.
The existence of our fluid is directly connected with Minkowskian (Pseudo-Euclidean) space-time, where basicallythe time is not separated from space (2). Let us thus consider that thermodynamical differential dV variation ofvolume of fluid is a temporal variation dV ( t ): dU ( t ) + pdV ( t ) = 0 (10)In this way, equation (6) is no longer trivial. We have a variable volume V ( t ) coupled with a constant density U ( t ) = ρV ( t ) = 43 πρR ( t ) = M ( t ) c (11)Let us now consider that (prerelativistic) Newtonian law of gravitation is also variable with a temporal grav-itational potential Φ( t ) = − GM ( t ) R ( t ) . Our test point ( µ = dM ) at radial distance R ( t ) has therefore a positive Like in electrostatic we consider a test charge (mass) as small as possible in such a way that we have no modification of theelectrostatic field. dR ( t ) dt = ˙ R ( t ). Potential energy U P = − GµM ( t ) R ( t ) can be then now compensated by kinetics energy U C = µ ˙ R ( t ) : 12 ˙ R ( t ) − GM ( t ) R ( t ) = 0 = ⇒
12 ˙ R ( t ) − πG ρc R ( t ) = 0 (12)( µ disappears). This Pseudo-Newtonian model of Pseudo-Euclidean fluid is based on a dynamical equilibrium“sphere-test body” between attraction and repulsion. We obtain in this way a stability of expanding sphere witha radial enigmatic (Remark 3) ”escape velocity” ˙ R ( t ) > R ( t ) = R ( t ) c r πGρ = cR ( t ) r Λ3 (13)If we suppose a finite spherical volume of fluid in dynamical equilibrium then it is in exponential expanding(11). Escape velocity (13) disappears if and only if Λ = 0 . Physical meaning of mathematical constant Λ is nowclarified by Minkowskian solution that implies the introduction (from 13) of a GLOBAL SCALE FACTOR R ( t )(with Minkowskian metric, 2 or see 19): R ( t ) = R H e √ Λ3 t (14)with a constant of integration R (0) = R H that seems, at first sight, not depend on Λ.We can also define a constant of expansion of Fluid (Vacuum) that we suggest to note H Λ (15 left): H Λ = ˙ R ( t ) R ( t ) ρ VACUUM = 3 H c πG (15)together with a density inside the sphere (15 right). Our model suppose that there is no fluid ( ρ = 0) outside thesphere of fluid ( ρ VACUUM ). Given that the fluid simulates space-time continuum itself, there is nothing ( ρ = 0)outside the sphere. Everything happens as if our sphere was a “Universe”. By introducing mathematical constantΛ in (1) we are thus naturally led to a theory of Universe, i.e. a cosmological interpretation: R ( t ) = R H e H Λ t ⇔ R ( t ) = cH Λ e H Λ t (16)Our model explains then why Hubble’s expansion is necessarily a global expansion (no local observed effect ofexpansion. If the constant of integration R H (15-16), i.e. a global constant, is not equal to c/H (if ˙ R (0) = c, see Remark 3), there would exist two global constants of Hubble. This would be a nonsense. In order to havePseudo-Newtonian model of Pseudo-Euclidean fluid. without contradiction, we must have H Λ R H = c (16) .From ddt ( ˙ R ( t ) R ( t ) ) = R ( t ) ¨ R ( t ) − ˙ R ( t ) R ( t ) = ˙ H Λ ( t ) = 0 we can define from radial acceleration ¨ R ( t ) also a (standard)parameter of deceleration that we suggest to note q Λ : q Λ = − R ( t ) ¨ R ( t )˙ R ( t ) = − q Λ is here negative and implies thus an acceleration of expansion. Initial condition of (17) t = 0 are: R (0) ¨ R (0)˙ R (0) = R H α Λ c = 1 (17-bis)Initial conditions mean that R (0) = R H and ¨ R (0) = α Λ define “horizon values” exactly on the same way that˙ R (0) = c defines an “horizon value” (Remark 3). We obtain a basic minimal relativistic acceleration . “Pseudo” because Newton’s law depending on time is not the usual point of view. We adopt in this way, in accordance with aGR point of view, that we can extend Newton’s law of gravitation from material baryonic particles m to points of space M. Our pseudo-Newtonian model, entirely based on (12), cannot be confused with historical Friedman’s Newtonian model. Friedmanconsidered a sphere of material fluid with variable radius R ( t ) and material density ρ ( t ) but, unlike (12), with constant mass M ofUniverse M = V ( t ) ρ ( t ) = πR ( t ) ρ ( t ) ⇒ dMdt = ddR ( ρR ) = 0. He considered a test body at radial distance with radial velocity ˙ R ( t )and therefore he obtained an equation with E = 0 (in standard notation K = - Em =0): E = E c + E p = m ˙ R ( t ) − πGmρ ( t ) R ( t ) =0 ⇒ ( ˙ R ( t ) R ( t ) ) = πGρ c . With critical constant density he obtained H ( t ) = H and therefore (14). Nevertheless in Friedman’sprerelativistic Newtonian model the total mass M of Universe is constant whilst, in our Pseudo-Newtonian model, there is a typicallyrelativistic variable mass of vacuum (c=1) M ( t )= M H e H Λ t with M (0) = M H = c G R H = c G q . We rediscover here in the framework of GR and SR, Milgrom’s idea of minimal acceleration.
4e deduce a pseudoNewtonian scalar field of gravitational force with a global principle of equivalence”acceleration-gravitation” ¨ R ( t )2 = G M ( t ) R ( t ) . Our dynamical Universe supposes then, with ˙ R (0) − G M (0) R (0) = 0, aninitial linear density M (0) R (0) = G c (together with an initial force G c and power of expansion G c ). REMARK 3 An important objection could be formulated at this stage: Our Pseudo-Newtonianmodel would not be a Pseudo-Euclidean model because our basic equation (12) uses a non-relativisticform of energy .Everybody knows how to write kinetics energy for a material particle mc ( γ ( β ) − . Here we have not amaterial point but a point of fluid in the framework of GR. Escape velocity ˙ R ( t ) for such a point can be as largeas we wish (not limited by c ). Pseudo-Newtonian equation (12) is a relativistic equation because velocity of lightplays a basic role. In fact, the escape velocity ˙ R ( t ) , of a point of space itself, is limited by c but not in usualmeaning β < c with domain of variation [0 , c [ . Indeed, in order to avoid a supplementary constant R H (14) , ifwe admit for the initial velocity (13) ˙ R (0) = cR H q Λ3 = c , the domain ] c, ∞ [ of variation of ˙ R ( t ) is limited by c . We rediscover in this way a basic tachyonic Pseudo-Euclidean “light-space-time” structure. We have to expectthen optical properties of fluid. In Cosmology our model is very near the model of de Sitter’s empty ( ρ = p = 0) Universe. The latter is alsoin exponential expansion a ( t ) = Ae √ Λ3 t with H Λ = ˙ a ( t ) a ( t ) and acceleration q = −
1. Lemaitre’ scale factor a ( t ) isintroduced in de Sitter’s metric (23)([W. de Sitter 1917]). ds = c dt − a ( t )( dr + r dθ + r sin θdφ ) (18)whilst our scale factor R ( t ) is globally induced from (12). In de Sitter’s model, the constant A (” A = 1”) in a ( t ) = Ae √ Λ3 t is then not a global constant determined by initial condition of the problem (like R H in 16). With condition of radiality dθ = dφ = 0 we have respectively de Sitter’s metric and Minkowski’s metric( a ( t ) = 1 with R ( t ) = 0 ds = c dt − a ( t ) dr de Sitter ds = c dt − dr M inkowski CC ⇒ R ( t ) (19)that are both particular cases of non-static ([Y. Pierseaux (2010)]) radial Robertson-Walker’s metric ( dy = dz =0) ds = c dt − a ( t )1 − Kr dr (20)with local (in metric) parameter of Gaussian curvature K = 0.Let us now introduce the limit of light velocity with ds = 0 first in flat metric of de Sitter ( K = 0): dt − a ( t ) dr = 0 de Sitter ( c = 1)Usually one deduces from de Sitter model the following formula of Redshift z = △ λλ by Doppler effect in GR z = a ( t reception ) a ( t emission ) (21)(with standard notations of the time of emission of radial photon from a remote galaxy towards time of receptionin our galaxy). Moreover with two usual cosmological measurable parameters H and q according to the followingstandard development into series:1 + z ⋍ Hr + (1 + 12 q ) H r + . = 1 + β s + (1 + 12 q ) β s + .. (22)where β s = Hr is standard law of Hubble (with radial comobile distance r ). Recall that β s = vc (s for space) isnot the velocity between two galaxies (two material β m points) but a velocity β s = Hr between the “points”(elements of volume) of space itself occupied by galaxies. In de Sitter’s case we have thus: In the model of Einstein-de Sitter (1932) there is also a Parabolic-Euclidean solution ( K = 0) with Critical density. Neverthelessit is a model (with Λ = 0) for usual matter. This model is today obsolete because usual matter seems occupy only 1% of criticaldensity. Hoyle’s Steady State is also based on metric (18-de Sitter) with a null CC. Minkowskian fluid involves a perfect cosmologicalprinciple (Hoyle-Bondi). z ⋍ β s + 12 β s .. (23)Let us now introduce (inferior) velocity of light in our model. Optical property of our fluid is given by Minkowskianlimit ( dt − dr = 0) of GR: 1 + z = R ( t reception ) R ( t emission ) = k s = s β s − β s (24)It is logical to admit a special relativistic development (Einstein’s standard Doppler radial factor for material point k m = q β m − β m ) where z can be as large that we wish( β s < . Equation (24) involves then a “GR interpretation”(velocity of space itself) of Einstein’s Doppler formula. With q = − k s = k Bondi = s β s − β s ⋍ β s + 12 β s + 12 β s + 38 β s ... (25)Our conjecture (24) here suggest then the conjecture that the parameters of accelerating universe is given byfamous “Bondi’s factor” at any order (25). For the coherence of our model of points of space without baryonic mass, we need for the light a null restmass in such a way that in perfect fluid (in 3), we have ” p + ρ = 0” in front of the term of four-velocity u µ u ν c of“particle” (see note 2 Poincar´e’s electron) . We showed the existence of a simple unexpected global Minkowskian solution of Einstein’s complete (with CC)equation of GR. The logical sequence from Pseudo-Euclidean solution (2) towards the Pseudo-Newtonian Fluid(12) is the following ( c = 1): T V IDEµν = Λ8 πG η µν = ⇒ p + ρ = 0 = ⇒ dU ( t ) + pdV ( t ) = 0 = ⇒
12 ˙ R ( t ) − πGρR ( t ) = 0Minkowskian metric (infinitesimal interval) involves (with CC) then a global scale factor R ( t ). We wonder ifwe can introduce such a scale factor in a finite interval in a next paper[Y. Pierseaux (2013)]. From relativisticpseudoNewtonian equation (12), we deduce here dynamical properties H Λ = ˙ R ( t ) R ( t ) , q Λ = − R ( t ) ¨ R ( t )˙ R ( t ) = − z = R ( t reception ) R ( t emission ) = k with Bondi’s factor reinterpreted as a Redshift in GR .Dynamical ( § § §