Conservation of energy in quantum gravity

W. Jim Jastrzebski, file 162-1.htm
Unofficial PhD program supervised by prof. Józef Namysłowski
Institute of Theoretical Physics of University of Warsaw
Room 139, Hoża 69, 00-681 Warszawa, Poland
 
It has been shown in this paper that the principle of conservation of energy implies that
  1. universe is stationary and its apparent expansion is simulated by the "Hubble time dilation",
  2. Hubble "constant" of stationary dust universe of constant density, is Ho=c/R, where c is the speed of light and R is the radius of curvature of space,
  3. radius of curvature of space R is then ca. 4.3 Gpc,
  4. density of space is determind as ca. 6×10-27kg/m3 for Hubble "constant" measured as Ho=70km/s/Mpc,
  5. terms of Taylor series of Hubble "constant" H(t) around t=0, starting from second term observed in 1998 by "Supernova Cosmology Project" team as equal dH/dt|t=0 = -Ho2/2, simulate the accelerating expansion of space,
  6. acceleration of cosmic probes Pioneers 10 and 11 (Pioneer effect), are predicted as ca. 7×10-10m/s2,
  7. spacetime is flat (Minkowski),
  8. metric tensor of space time is non symmetric and degenerate, and assumed to be dτ2 = exp(-2r/R)dt2 + 2sinh(2r/R)dtdr/c - exp(2r/R)dr2/c2, which for r << R approximates to Minkowski's dτ2 = dt2 - dr2/c2. Furtheremore
  9. true distances from the Earth to quasars are orders of magnitude smaller than thought, and
  10. "repulsive gravitation" does not exist and
  11. consequently "dark energy" and "cosmological constant" don't exist as physical entities either, and last but not least:
  12. the relativistic gravitation is a quantum theory, with unknown yet basic carriers of gravitational energy. The gravitational energy of a particle is E=mc2 and can't change by smaller quantities than ΔE=hν, where h is Planck constant and ν is frequency of gravitational wave (frequency of its shmutrinos, as Feynman called them, or whatever caries gravitational energy from one object to another.
Introduction

It is shown that mechanism of Hubble redshift is consistent with the principle of conservation of energy. It is done through the calculation of the amount of dynamical friction of photons. It turns out that it explains and/or rejects total of 12 so far unexplaned phenomena that are not explained by the big bang hypothesis

Derivation of Hubble redshift in static universe

Consider Einstein's homogeneous universe filled with dust. Let photons move through this dust interacting with it only gravitationally. We will assume that energy conservation holds and that Newton's approximation can be applied. With these assumptions one can readily calculate energy transfer from photons to dust. To the observer at some distance from the light source this energy transfer will manifest itself as a change in wavelength, which is exactly what was observed by Hubble. The relativistic interpretation of this result allows the derivation of Hubble redshift (HR), including discovery that Hubble's constant depends on the distance between the place in deep space and the observer [2].

Let Ed = Eo - E be the gravitational energy acquired by the dust due to gravitational interaction between dust and photons of energy E and initial energy Eo and let Λ = 4π G ρ / c2, where G is Newtonian gravitational constant, ρ is density of dust, and c is speed of light (which makes accidentally Λ equal to the cosmological constant of Einstein's universe or 1/R2, where R is radius of Einstein's universe). The linear density of Newtonian gravitational force acting on dust (force per unit length), which is identically equal to d2E/dr2, where r is distance travelled by photons, can be written, using relativistic relation between mass and energy (mass = Energy / c2) as 4πGρ(Eo - Ed) / c2 leading to equation
  d2E/dr = Λ E (1)
Substituting 1 / R2 for Λ and solving the equation with initial conditions E(r = 0) = Eo and (dE/dr)(r = 0) = - Eo / R (meaning selecting a solution that makes physical sense) one gets
  E / Eo = exp( - r / R ) (2)
Since in general relativity (EGR) there is nothing else but time dilation and the curvature of space as the media controlling gravitation, EGR interpretation of the above result is that time is running slower at a distance from (any) observer according to relation
  dτ/dt = exp( - r / R ) (3)
where τ is proper time in deep space and t is coordinate time at observer. The effect might be called Hubble Time Dilation (HTD) in honor of its discoverer, and as distinguished from the gravitational time dilation predicted by Einstein. After differentiating the above equation at r = 0 we get a relation between the HTD in deep space (d2τ/dtdr)2 and the curvature of space Λ = 1 / R2 as
  (d2τ/dtdr)2 - Λ = 0 (4)
and it suggests the existence of antisymmetric part of Ricci tensor in time domain, named here tentatively Hμν or Hubble Tensor (HT), such that Hμν + Rμν = 0 indicating that the spacetime is intrinsically flat as proposed by Narlikar and Arp [2] and required by the law of conservation of energy.

It follows from equation (2) or (3) equivalently, that the redshift, produced by HTD, is equal to
  Z = (Eo - E) / E = exp( r / R ) - 1 (5)
simulating the expansion of space, with the Hubble "constant" of this apparent expansion at r = 0
  Ho = c / R (6)
After expanding the Hubble "constant", H(r), into Taylor series around r = 0 the acceleration of this apparent expansion is approximately equal to
  dH / dr = - Ho2 / 2 (7)
This value agrees within one standard deviation with 1998 observations by the Supernova Cosmology Project team [1].

Quantum nature of gravitational energy

To show that gravitational energy of any object is its internal energy E = mc2 is enough to note that gravitational force in direction xi, Fi ::= - dE/dxi = mgi. For this purpose we show the full expression for internal energy of a particle, E = mc2(dτ/dt), where (dτ/dt)2 = g00, and so (dτ/dt) is the square root of time-time component of metric tensor of spacetime [4], and differentiate it with respect to displacement dxi getting exactly the gravitational force Fi = mgi. Now it becomes obvious that any exchange of gravitational energy between objects can be done only by exchanging hν quanta, demonstrating that relativistic gravitation is a quantum theory the same as electromagnetism or chromodynamics (that uses the same quanta, namely photons, and on occasions bosons while they are available in processes generating bosons. It might be noticed that transfer of gravitational energy in smaller units than hν quanta is forbidden.

Conclusions

The analysis of Hubble redshift (HR) can be carried out using Einstein's general relativity and the law of conservation of energy. The observed HR can be attributed to the time dilation in the universe which remains stationary as predicted by Einstein in 1917.

In addition to reproducing the measured properties of HR, the approach outlined in this paper allows direct calculation of the important parameters of our universe such as the average density of space and the acceleration of its apparent expansion. While the formula for Hubble redshift, equation (5), can be derived directly from equation (2) obtained using Newtonian approximation, it is equation (3), which expresses the essential transition from Newtonian approach in which space and time are distinct, to a general relativistic spacetime.

Due to particular value of variable Λ in equation (1) (as a half of contraction of Ricci tensor), Λ may be used to eliminate the "cosmological constant" from Einstein's field equation [5]. And as an additional feature of relativistic gravitation we may notice its quantum nature mediated by any observable particle of spin 0, 1/2, 1, 2 ... (even by a szmutrino if it axists).

Bobliography

  1. Kuznetsova N, Barbary K, et al, A New Determination of the High-Redshift Type Ia Supernova Rates with the Hubble Space Telescope Advanced Camera for Surveys, 2008, The Astrophysical Journal, Volume 673, Issue 2, The American Astronomical Society, pp. 981-998, Bibliographic Code: 2008 ApJ...673..981K,
    http://supernova.lbl.gov/
  2. Narlikar, J. & Arp, H. Flat spacetime cosmology - a unified framework for extragalactic redshifts, 1993, Astrophysical Journal, Part 1 (ISSN 0004-637X) vol. 405, no. 1, The American Astronomical Society, p. 51-56, Bibliographic code: 1993 ApJ...405...51N
  3. STScI-2009-08, Refined Hubble constant narrows possible explanations for dark energy,
    http://hubblesite.org/newscenter/archive/releases/2009/08/full/
  4. Landau, L. D., Lifszyc. E. M., 1973, "Teoria pola", Państwowe Wydawnictwo Naukowe, Wydanie II zmiennione, p. 285, eq. (88.9).
  5. Meissner, Krzysztof A., 2002, "Klasyczna Teoria Pola" ("The Classical Theory of Fields"), Wydawnictwo Naukowe PWN, ISBN-83-01-13717-7, p. 120.
  6. Misner, Charles W., Thorne, Kip S., Wheeler, John Archibald, 1973, "Gravitation", W. H. Freeman and Co, New York, p. 410-411.
Appendix A: Detailed derivation of equation (1)

Let c be speed of light, G the Newtonian gravitational constant, ρ the density of Einstein's dust universe (while one galaxy corresponds to one dust particle), the gravitational energy of dust gained through the gravitational interaction of light with dust ("dynamical friction") contained in the volume of a ball of dust of radius r be Ed(r). After photons of energy Eo are radiated out from arbitrarily placed center of coordinate system called "point zero" (point with radial coordinate r = 0) in homogeneous space, making the space inhomogeneous and therefore the Newtonian "gravitational force" is pushing the dust away from point zero and at distance r from point zero this force is equal, from Newtonian equation for gravitational force and energy
  Fparticle = - (d/dr)Eparticle(r) = ( G / c2 )[ Eo - Ed(r) ] mparticle / r2 (A.1)
where Eparticle(r) is gravitational energy of dust particle at distance r from point zero and mparticle is the mass of a particle of Einstein's dust universe.

Integrating over all dust particles in a spherical layer of dust of radius r and thickness dr we get mass of this layer of dust at distance r from point zero as
  mlayer = 4 π ρ r2 dr (A.2).

Substituting (A.2) for mparticle in (A.1) the total force that is the source of gravitational energy of dust of this leyer becomes
  Flayer(r) = - (d/dr)Elayer(r) = ( 4πGρ / c2 ) [ Eo - Ed(r) ] dr (A.3).

Integrating both sides of (A.3) over all spherical layers between point zero and r to get total energy of dust, and differentiating both sides with respect to r to get rid of the integral on the right side of this equation, we get
  - d2Ed(r) / dr2 = ( 4πGρ / c2 ) [ Eo - Ed(r) ] (A.4).

Substitutig Ed(r) = Eo - E(r) where E(r) is energy of photons at distance r from their source and Λ = 4πGρ / c2 we get
  d2E(r) / dr2 = Λ E(r) (A.5)
which is identical with eq. (1).