A New Interpretation of the Hubble Law

We propose a new interpretation of Hubble law. Waves are observed in the observer space-time. It defines the observer proper time T. Space-time is composed of three spatial dimensions and three temporal parameters: proper-time s of the observed object, proper time T of the observer and integration time t (currently considered as relative time). Time origin is the birth of the universe. So, universe is stable; it can be seen as the comobile space of expansion theory. When changing space-time from the source to the observer, waves are seen cooling; this explains the redshift effect. The distance is defined as the product of the delay time with the local speed of light of the observer. The mistake between t and T can explain why universe is viewed as not only in expansion but also in acceleration whereas we think it is stable.


Introduction
Up to now, physics laws are based on Einstein relativity, established in 1905.Hubble has discovered the correlation between redshift and distance [1].Cosmology is based on an interpretation of the redshift as expansion [2] through the ΛCDM model.
This article proposes a new approach to explain Hubble law.Other alternatives to expansion have been proposed such as Zwicky in 1929 [3].Here, we propose a model without expansion and based on a new space-time notion.It defines the local proper time of the observer.Space-time relationship is changing from the observed source to the observer.
First we present the model and then we discuss the interpretation of redshift applied to black body radiation.We compare the proposed model to measured data in the distance modulus-redshift diagram.At the end we explain why, in present interpretation, universe is viewed expanding and moreover accelerating [4] whereas we think it is stable.

Presentation
Using Euclidean space, the observer position is x o .His proper time is T. Time begins at universe origin.The observer detects light from an object in position x e , with a proper time s.Their separation distance is d e o x x x = − .t is the integration time.We define the time law, derived from the Einstein theory, of the cooling theory presented in one dimension form such as: , knowing that ( ) c T doesn't vary much with time T (T ~ 13.7 × 10 9 years from [5]).This model gives the idea that light speed decreases with time.That is a corpuscular point of view not well suited.It is better to view it as a space-time relation that generates a cooling phenomenon of waves as will be explained later.
In this model, product ( ) c T T ⋅ is constant.This suggests that universe radius r u is constant.It defines the Euclidean space.All present physics remains locally valid, only long scales interpretations are modified and explained without matter effects.Physical quantities and laws applied to matter in each space-time are universals.This model is compatible with the restricted theory, using u c r T = .As we will see in Section 2.2, it is compatible with black body theory because it generates a cooling phenomenon.
More than speed light, ( ) c T represents the relationship between space and time.The proposed model sug- gests that the space-time is bounded to the observer.He analyzes waves in his own space-time.The delay time between light emission and reception is

Application to the Black Body Radiation
The black body theory is local.It can be applied to a volume as little as possible.So it is not affected by the proposed theory.The Planck black body radiation law is: where c is the speed of light in vacuum, h is the Planck constant, k B is the Boltzmann constant, K 1 is the temperature in Kelvin (˚K), λ is the wavelength in meters (m).( ) We write again the Planck law assuming radiation has been emitted at the proper time s.In Equation (2.2A), we replace c by the local space-time relation at the time s of light emission, so u c r s = (from Equation (2.1B) applied to the source).In the source space-time: We make appear the observer proper time T in the above expression.
In the observer space-time u c r T = and we define the redshift z parameter so that We deduce: ) is the source radiation interpreted as a cooling black body in the observer space-time.The temperature decreases from K 1 to K 2 with ( ) Equation (2.2D) suggests that product K.T is constant.Today, on the cosmic background radiation temperature from [7] with K = 2.725 K and T = 13.7 × 10 9 years from [5], we get K.T = 1.18 × 10 18 K.s.Now, in Equation (2.2C), the factor z + 1 is gathered with wavelength, so we get In Equation (2.2E), cooling generates two effects:  The first one is the redshift of the source spectrum, wavelengths are divided by the (z + 1) factor.
 The second one is the total reduction of the radiance which is divided by the ( ) factor.This can explain the Tolman effect [8].
In the expansion theory, the ( ) is explained by the adiabatic dilution when volume increases.Energy conservation law must be applied in the source space-time.The z + 1 factor is the change of proper time from s to T. Wavelengths increase and frequencies decrease, this can explain the time dilatation effect [9].

The Hubble Law
In the present expansion theory distance x is related to redshift z using FLWR metric [10] and for z ~ 0, [11] an empirical law (from 280 measurements) illustrated by Let us now analyze the Hubble law under the proposed cooling theory.We first deal with the distance.We deduce from Equation (2.2B) ( ) The distance x from object to observer is defined by Equation (2.1D) and using Equation (2.1B)  ( ) Using Equation (2.3E) and Equation (2.3F) ( ) For z ~ 0, Equation (2.3G) and Equation (2.1B) gives x T z c = ⋅ ⋅ .We find again for z ~ 0 the same law has the one from the expansion theory Equation (2.3A).
Let's now deal with the radiation.When measuring a radiation, we assume that it follows a law such as where L is the intrinsic radiation, x is the distance from object to observer.That is due to the assumption of the energy conservation so the product 2 obs F x ⋅ remains constant.But, we have shown that propagation, in the cooling theory reduces the flux by a factor ( ) where L 0 is the apparent intrinsic radiation.We deduce from Equation (2.3G), Equation (2.3H) and Equation (2.3I): ( ) ( ) The magnitude is defined by , so, ( ) The distance modulus μ is defined by μ = m − m 0 .So, ( ) Figure 2 presents the empirical model from [3] Equation (2.3B) and the proposed cooling theory Equation (2.3J).Distance unit is in tenth of parsecs with 6 420 10 u r = × tenth of parsec [5].We note the very good agreement between the two curves.The remaining deviation is less than 0.2 distance modulus.This can be seen as a validation of the proposed theory which uses only time and space-time relationship.

What We Think Is Wrong in Present Expansion Model
The Hubble parameter H and the acceleration parameter 0 q are defined by x is defined as the product of delay time τ by the local space-time relation c(T), with ( )

Figure 1 .
The fitted curve is: