User:Pao.christ/sandbox1

to be inserted in Friedman's equations before density parameters

CORRECTIONS TO FRIEDMAN' S EQUATIONS

An alternative set of equations to Friedman's has been proposed. They are based on  isotropy and homogeneity. The energy conservation equation follows by considering a spherical ball of given matter density and a mass m at its surface. Its motion is determined only by the matter inside. The dimensions of the ball disappear since the expansion rate H enters as  $$v^2\simeq H^2 r^2\simeq G\rho r^2 $$. Since $$H^2$$ has dimensions $$t^{-2}$$, $$\rho $$ can only be time dependent. Hence no constant is allowed in such a homogeneous equation.

$$ (\frac{\dot\chi}{\chi})^2 - 2 G \tilde\rho=  0  \qquad \quad (1) $$

where $$\tilde \rho$$ stands for the usual $$ (4\pi /3) \rho$$ and $$\chi$$ for the scale factor. So this one is essentially equivalent to the corresponding Friedman's.

But in an expanding Universe the potential is not a state function so that in its derivative an extra term due to the mass variation appears in addition to the standard newtonian acceleration. Indeed in the determining the acceleration the evaluation of the time dependence of the density $$\rho$$, appearing in $$\rho\chi^2 $$ from the energy equation, enters. If matters is conserved i.e.

$$ dM/dt = 0 = d/dt (\chi^3 \rho(t) ) = 3 \rho \chi^2 d\chi /dt + \chi ^3 d\rho/dt $$

which yields the familiar density dependence from homogeneity

$$ d\rho/\rho = - 3 d\chi/\chi $$

proper to General Relativity, one obtains the standard Newtonian like relation

$$ \frac{\ddot \chi}{\chi}   =  -  G \rho $$

If on the other hand for the equally homogeneous

$$ d\rho/\rho = - 2 d\chi/\chi $$

embodying the linear relation between radius and mass $$ dM/ M = d\chi/ \chi $$, which accounts for the time dependence of the age of the Universe, the previous result gets compensated by a counterterm  resulting in

$$ \ddot \chi  = 0      \qquad    (2) $$

Of course in this case

$$ \dot \chi  = c    $$

The second formulation of the acceleration equation is also backed up by the constancy in time (zero time derivative) of the black hole condition

$$\frac{GM}{c^2 r} =1 $$ which confirms Feynmamn's observation and his warning that ' we are here not dealing with an ordinary problem but with a cosmological one.' This term, additionally justified because the cosmological potential is not a state function, which predicts  a steady expansion  represents the "misterious force"  (dark energy) which balances gravitational attraction. So self energy is seen to provide the repulsive  force since it increases  the total energy when particles move away.  This is the missing dark energy at present represented by the cosmological constant.'''

This predicts the linear expansion of the Universe

Fig. 1) The Universe evolution. The lines R =  ct represent the borders of the causally connected expanding Universe. A different history than presently proposed is  evident. First, inflation which was invoked to account for homogeneity (with problems of causality) is not needed. Also the recent time development which seems to present peculiar features : a deceleration (due to gravitational attraction) followed  by a later acceleration phase, does not show up in the present approach.

The local invariant interval of a homogeneous isotropic expanding Universe in the Lemaitre- Hubble-Painleve-Gullstrand reads

$$ ds^2 = c^2 dt^2 - \chi^2(t) dx^2 $$ $$y = \chi x $$ and the Hubble parameter $$  H  = H(t) = \frac {\dot \chi}{ \chi}$$.

Thus $$ ds^2 = c^2 dt^2 - \chi ^2(t) [ (\frac {dy}{\chi} - \frac{\dot \chi }{\chi^2 } y dt )^2 ]  $$

or $$ ds^2 = dt^2 ( c^2 - (\dot y - H y )^2 ) $$

So the original space part of the invariant interval has been transformed in a velocity dependent one.

Here $$ H y = v(t,y)$$represents the velocity of expansion of the point y at the time t and light propagation results as shown in Fig. 2).



Fig. 2) Light propagation in the (t, y) plane. Because of the vector composition of the local relativistic invariant light cone with the frame velocity determined by the varying Hubble parameter $$H= \frac{1}{t}$$ (thick arrow), light deviates more and more when emitted at former times (with an analogous effect to light deviation in a static gravitational field). At $$ y= y^{m}$$ the Hubble velocity becomes smaller than the transverse light component thus allowing all the "light" emitted at the Big Bang to reach the earth at different times. Since $$  y= y_{0}^{m}$$  is bigger than $$ y= y'^{m}$$  the maximal world "dimensions" identified with the Hubble radius increase with time. Not in scale.

This coordinate system based on a  cosmological extension of the Painleve Gullstrand coordinates shows in addition that the proper time of relativistically receding galaxies is of course  longer than ours. Thus they are shown to live longer than naively expected.