Simple Considerations on the Cosmological Redshift

Transcription

1 Apeiron, Vol. 5, No. 3, July 8 35 Simple Considerations on the Cosmologial Redshift José Franiso Garía Juliá C/ Dr. Maro Mereniano, 65, Valenia (Spain) jose.garia@dival.es Generally, the osmologial redshift is onsidered as an indiation of the expansion of the universe. However, we are going to onsider, using very elemental arguments, that this physial phenomenon ould be due to the gravitational field, and that the universe ould be stati and flat. Key words: Light redshift, gravity.. Introdution In the theory of the Big Bang, the redshift of the light emitted from distant galaxies is interpreted as a Doppler effet and then onsidered as an indiation of the expansion of the universe, following the law of Hubble. However, we are going to onsider, using very simple arguments, that the nature of that redshift ould be related with the gravity and not with that expansion.. The light redshift as a Doppler effet 8 C. Roy Keys In. 35

2 Apeiron, Vol. 5, No. 3, July 8 36 We suppose a light soure that emits a photon and that it is moving with relative onstant veloity v with respet to an observer, then from the speial relativity theory (SRT) of Einstein we have that t t v Δ = Δ (.) being Δ t and Δ t the intervals of time in the moving and rest frames, respetively, and the veloity of the light in the vauum. From (.), whih is the equation of the dilatation of the time, onsidering frequenies, we have v = ν ν ν = ν v (.) being ν and ν the light frequenies emitted and observed of the photon, respetively. Developing (.) ν ν v ( ) (( )( )) + v v v v ν = ν ν v = = + = + v For v << v ν = ν v (.3) + If the light soure is moving away from the observer, v is positive, ν < ν, and the light is red shifted. If the light soure is moving toward the observer, v is negative, ν > ν, and the light is blue 36 8 C. Roy Keys In.

3 shifted. Apeiron, Vol. 5, No. 3, July Gravitational redshift For a partile of mass m in a gravitational field with potential ϕ we have that E = m (3.) V = mϕ (3.) being E and V the relativisti and potential energies of the partile, respetively. Therefore, for a photon (assuming an effetive mass E m = ) E = hν (3.3) hν V = ϕ (3.4) being h the onstant of Plank. As the gravitational field is onservative, then hν T + V = hν + ϕ = onst. (3.5) hν hν hν + ϕ = hν + ϕ (3.6) ϕ ϕ ν + = ν + ν + ϕ = ν + ϕ (3.7) 8 C. Roy Keys In. 37

4 Apeiron, Vol. 5, No. 3, July 8 38 where T = hν is the kineti energy of the photon, ν and ν the light frequenies emitted and observed, respetively, ϕ and ϕ the potentials in the points of emission and observation, respetively. The gravitational potential ϕ varies with the inverse of the distane and always is ϕ <, only ϕ ( ) =. If ϕ < ϕ, then ν < ν, and the < hν. As V inreases (it is less negative) light is red shifted, and hν then T dereases. But, if ϕ > ϕ, then ν shifted. Now, introduing the light redshift parameter z, we have where z > when ν < ν ; For ϕ << z ν ν ν > ν, and the light is blue = = (3.8) ν ν z + ν = ν, ν ν = z +, ν ν ν z = = = ν ν z + z + ν ν ν + ϕ ϕ ϕ = = = ν ν + ϕ + ϕ ϕ ϕ z = ν ν ν = z + ν ν z ϕ ϕ = = ν z + 4. The osmologial redshift (3.9) (3.) (3.) 8 C. Roy Keys In. 38

5 Apeiron, Vol. 5, No. 3, July 8 39 Within the framework of the general relativity theory (GRT) of Einstein, we assume that in a gravitational field the interval ds has the form ds = a τ b dx + dy + dz = a τ b dr (4.) ( ) ( ) ( ) being τ the proper time, a ( τ ) the sale fator, b a onstant to infer, x, y and z the spae oordinates and r the distane. In [] (p. 3), b is an integration onstant. Note that for b =, (4.) is the square of the spae-time interval of Robertson and Walker with k =, being k the urvature onstant []. In [] (p. 3), k = is the only possible value. From, ds =, we obtain that the veloity of the light is dr = (4.) a( τ ) b And then b dr = a( τ ) Now, from [] (pp. 5-6), we put the observation point at the origin of the referene system, r =. We onsider two photons emitted from the point r in the times τ and τ + and that arrive at the point r = in the times τ and τ +, respetively. Then, for the photon emitted at the moment τ and that arrives at the moment τ we have that τ bdr b b b b r = = dr = r = r = ( τ ) [ ] ( ) r a τ r r Also, for the photon emitted at the moment τ + and that arrives 8 C. Roy Keys In. 39

6 at the moment τ + we have that Consequently Apeiron, Vol. 5, No. 3, July 8 33 τ + τ+ b dr b r = = a ( τ ) τ τ τ + d = a r ( τ ) a( τ ) τ τ+ ( τ ) ( τ) = ( τ + τ ) ( τ + τ) F F F d F d ( τ + τ) ( τ) = ( τ + τ ) ( τ ) F d F F d F ( ) = ( ) F τ F τ a = ( τ ) a( τ ) dθ dθ And using the light angular frequenies, ω = and ω =, being θ the angle, we have dθ dθ = a τ ω a τ ω a ( ) ( ) ( ) = a( ) τ ω τ ω ( τ ) ( τ ) a ω = ω (4.3) a Hene, the light angular frequeny emitted ω = πν is not equal 8 C. Roy Keys In. 33

7 Apeiron, Vol. 5, No. 3, July 8 33 to the light angular frequeny observed ω = πν. Now, for the light redshift parameter, we have ω ω ω a ( τ ) z = = = (4.4) ω ω a τ And >. Also, z > if ω > ω, that is, if hν > hν. But also we have that ds = a τ b dx + dy + dz = a τ b dr z > if a( τ ) a( τ ) ( ) ( ) ( ) ( ) = d (4.5) being d the length of an ar measured in the surfae of radius a ( τ ) and dr is the length of an angle multiplied by b, that is, ar = radius angle or d = a( τ ) bdr. Note that now the veloity of the light would be bdr = (4.6) a τ τ In the Big Bang model, a ( τ ) inreases with the time, and as > τ, then a( τ ) > a( τ ), therefore ω > ω and also ( τ ) ( ) z > and d = a bdr inreases, hene the universe expands, and the light is red shifted. On the other hand, for the stationarity of (4.5), it would be required that a ( τ ) b = onst., whih implies that b =. Also, as the sale fator a ( τ ) is onsidered like a urvature radius it is infinite for a flat surfae [3] (p. 479). 8 C. Roy Keys In. 33

8 a a Apeiron, Vol. 5, No. 3, July 8 33 Then, in our flat ( a ( τ ) = ) and stationary ( b = ( τ ) = = undetermined and also ( τ ) ( τ ) 8 C. Roy Keys In. ) model, d = a b dr = dr = undetermined, therefore the universe does not expand. Then, z > only beause the gravity an make that ω > ω (when ϕ > ϕ ), equation (3.), whih is derived for a weak gravitational field in the framework of the GRT in [3] (p. 349). ω a ( τ ) Now, from [3] (p. 486), we put = in the form ω a ( τ ) ω a ( τ ) = ω a τ Δ τ, therefore where ( ) ω a( Δ ) ( ) ( ) ( ) τ τ da τ Δ = = Δ τ = H ω a τ a τ da ( ) ( τ ) τ H = τ is the onstant of Hubble and Δ = Δ τ. a d Then ω ω ω H Δ = = (4.7) ω ω and making ω ω v = (4.8) ω we have the law of Hubble v = H Δ (4.9) being v the veloity of reession. 33

9 Apeiron, Vol. 5, No. 3, July Note that for v <<, the Lorentz transformation: x ( x vt) v v v =, t = t x, is onverted into the Galileo transformation: x = x vt, t = t, hene dx dx v =, dt dt and for the ase of the light, = v. Then, for v << (whih is in aordane with a weak gravitational field) v ω ω πν πν ν ν λ λ v = = = = ω πν ν λ being λ the wavelength. And from (3.9) and (4.8) z v = (4.) z + v For z <<, we have, from (4.), z = and v <<, and, from (3.), ϕ ϕ = z. In the Big Bang model, the valid equation is (4.), with (4.4) and (4.9), but in our flat ( a ( τ ) = ) and stationary ( b = ) model, the valid equation would be (3.), with (3.8). Note that for b = ( ) ( ) ( ) da τ da τ da τ v = H Δ = a( τ ) b Δ r = b Δ r = Δ r = a( τ) (4.) C. Roy Keys In.

10 Apeiron, Vol. 5, No. 3, July whih implies that the model is also stati. The equation (4.) gives the value of the osmologial redshift due to the expansion of the universe, in whih the spae itself is expanding following the law of Hubble. But, the equation (3.) gives the value of the light redshift aused by the fore of attration of the gravity, in a stati universe. Now, from [3] (pp ), in the ontext of the limit of the flat model of Friedmann, as for our model, a ( τ ) b = onst. and a ( τ ) = and b =, we substitute a( τ ) b by a time funtion e ( τ ) and put the interval ds like ds = e τ dx + dy + dz (4.) ( ) ( ) and for eah τ, e ( τ ) has a onstant value that an be redued to the unity by means of a simple transformation of oordinates (proportional to ). For low values of the pressure P it is found onst. 3 onst. that ρ e ( τ ) = onst., being ρ = the mass density, then τ 3 e ( τ ) = onst. τ. And, for low values of τ it is P = ρ, and it is 3 4 found that ρ e ( τ ) = onst., then ( τ ) e ( ) = and ρ ( ) 8 C. Roy Keys In. e = onst. τ. For τ =, =, whih would be the mass density of a volume of matter of radius zero. Therefore, with these equations our flat ( a ( τ ) = ), stationary ( b = ) and stati ( v = ) model of the universe would be in the limit R = and of the flat model of Friedmann (τ =, ( ) 334

11 Apeiron, Vol. 5, No. 3, July GM GM v = ve ( ) = = =, being G the gravitational R ( ) GM onstant of Newton, and ve = R ( τ ) veloity, the radius and the mass of the universe, respetively). 8 C. Roy Keys In. R ( τ ) and M the esape 5. Disussion For v <<, we an onsider that the light is red shifted beause its veloity would be v, then the kineti energy of the photon, hν, dereases. But v an be a relative veloity away from the observer (Doppler effet), or the reession veloity (Hubble law), or a veloity related with the fore of attration of the gravity and obtained by omparison between (3.) and (4.) ϕ ϕ v = (5.) If the gravitational field is muh weaker in the point of observation ϕ GM than in the point of emission, then ϕ << ϕ and v = =, R being M and R the mass and the radius of the light soure, respetively. Now, for two supposed idential galaxies, at rest, at distanes d and d, with apparent radii R and R, and for z <<, we have, for an expanding universe, that v H d d = = (5.) v H d d 335

12 and, for a stati universe, when ϕ but, as Apeiron, Vol. 5, No. 3, July << ϕ, that GM v R R = = (5.3) v GM R R R R d = (5.4) d GM GM then, both veloities, v = H d and v = d R = Rd, would give v the same value of the redshift, z =. Therefore, both red shifts would be interhangeable apparently, but the one due to the reession veloity exists only if the universe is expanding, while the one aused by the veloity related with the gravity exists always. This ould be favorable to the stati universe model. Although, for high values of the z parameter, it would be required to use a veloity for high gravitational fields. ϕ GM Note that the veloity v = = would be the esape R veloity of a photon, beause from (3.5), and adapting the esape E hν veloity onept for a photon (with m = = ), we have hveph λ + h λ ϕ = 8 C. Roy Keys In

13 Apeiron, Vol. 5, No. 3, July v eph ϕ GM = = (5.5) R v eph being the esape veloity of a photon of wavelength λ emitted by a light soure of mass M and radius R. 6. Conlusion We have seen first the light redshift as a relativisti (SRT) Doppler effet, where the light is red shifted by a relative onstant veloity away from the observer. After, we have seen the gravitational redshift, where the light is red shifted by the gravity and the law of onservation of the energy. And for last, in a GRT ontext, we have seen the osmologial redshift, in two models: the Big Bang and in our flat, stationary and stati as a limit of the flat model of Friedmann. In the Big Bang model the light is red shifted by the expansion of the universe, in whih the spae itself is expanding following the law of Hubble. In our proposed flat, stationary and stati model the light is red shifted by the fore of attration of the gravity, and the expansion of the universe would not be real but only apparent. In both models the values of the osmologial redshift are the same. Hene, the expansion of the universe would not be the only explanation of this physial phenomenon. Referenes [] A. A. Logunov, The Theory Of Gravity, arxiv: gr-q/5v (). [] Marelo Samuel Berman, Is the universe a white-hole?, arxiv: physis/67v3 (7). [3] L. D. Landau and E. M. Lifshitz, Teoría lásia de los ampos, seond edition, Editorial Reverté, S. A., Barelona (973). 8 C. Roy Keys In