Essay:Hubble redshift in Einstein's universe
- REDIRECT Hubble redshift in Einstein's universe
This essay is an original work of W. Jim Jastrzebski, file 160-12.htm JimJast 13:00, 27 August 2011 (EDT) Unofficial PhD program supervised by prof. Józef Namysłowski, The Institute of Theoretical Physics, University of Warsaw, Room 139, Hoża 69, 00-681 Warsaw, Poland
It is shown that the observed features of Hubble redshift can be explained within the framework of Einstein's general relativity. The observed Hubble redshift could be attributed to thus far unnoticed mechanism of time dilation coupled to curvature of space. Einstein's universe regains its status as a viable model predicting cosmological observations such as the apparent expansion of space and the observed acceleration of this expansion.
Derivation of Hubble "constant"
Consider Einstein's homogeneous universe filled with dust (with a particle of dust corresponding to a galaxy). 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 "constant" depends on the distance between the place in deep space and the observer.
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 = 4πGρ / 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 / d2r, 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 / dr2 = ΛE ...... (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 Einstein's 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 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)
For Einstein's universe of density Hubble constant Ho = 70km / s / Mpc. After expanding the Hubble "constant", H(t) into Taylor series around t = 0 the acceleration of this apparent expansion is approximately equal to
- ...... (7)
which agrees within one standard deviation with 1998 observations by the Supernova Cosmology Project team.
The analysis of Hubble time dilation (HTD) can be carried out using Einstein's general relativity and the law of conservation of energy.
The observed HTD can be attributed to the geometry of spacetime of the stationary homogeneous dust universe. Such a model may bo oversimplified, however, it does reproduce the essential properties of HTD, and allows one to estimate the measured properties of the 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) that expresses the essential transition from Newtonian approach in which space and time are distinct, to a general relativistic spacetime. Excelent agreement of the calculated acceleration of apparent expansion of this model universe with the measured value, lends additional support for the model.
Over the years the author has benefited from discussions with many people too numerous to to list them here. The members of the faculty of the Institute of Theoretical Physics, Warsaw, Poland were kind enough to spare their time and offer critique which helped in clarifying the ideas expressed in the above manuscript.
- ↑ See Essay:Demystified gravitation for step-by-step derivation of eq. (1)
- ↑ 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
- ↑ 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, [href="http://supernova.lbl.gov/]"