# Robertson-Walker Solution and Friedmann's equations

Discussion in 'Alien Hub' started by waitedavid137, Apr 25, 2020.

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1. ### waitedavid137Honorable

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So let's start by writing down the Robertson-Walker solution for a universe of uniform matter distribution.
$ds^2 = dct^2 -a^2 \left ( \frac{dr^2}{1-Kr^2}+r^2 d\Omega^2 \right )$
where $a$ is a function of $t$.
And
$K=\frac{k}{R{_{0}}^{2}}$
and
$k=-1,0,1$
They constructed this as a model in such a way that the matter distribution around an observer would look the same from his vantage point, no matter where he was located in that universe. Despite clumping of matter as we observe as galaxies etc... it is a small variation from the statistical behavior of the overall matter, including especially cosmic microwave background radiation, so they expected this to be a good model for the average behavior of the universe.
Calculation from that spacetime yields the following Einstein tensor elements,
$R{^{0}}_{0}-\frac{1}{2}g{^{0}}_{0}R=\frac{3}{c^2}\frac{\dot{a}^2 +Kc^2}{a^2}$
$\begin{matrix} R{^{1}}_{1}-\frac{1}{2}g{^{1}}_{1}R=\frac{2}{c^2}\frac{\ddot{a}}{a}+\frac{1}{3}G{^{0}}_{0}\\ \\R{^{2}}_{2}-\frac{1}{2}g{^{2}}_{2}R=\frac{2}{c^2}\frac{\ddot{a}}{a}+\frac{1}{3}G{^{0}}_{0} \\ \\R{^{3}}_{3}-\frac{1}{2}g{^{3}}_{3}R=\frac{2}{c^2}\frac{\ddot{a}}{a}+\frac{1}{3}G{^{0}}_{0}\end{matrix}$
all other elements are zero.
Now Einstein's field equations are

$G{^{\mu}}_{\nu}=R{^{\mu}}_{\nu}-\frac{1}{2}g{^{\mu}}_{\nu}R=g{^{\mu}}_{\nu}\lambda +\frac{8\pi G}{c^4}T{^{\mu}}_{\nu}$
where
$\left [ g{^{\mu}}_{\nu} \right ]=\left [ \delta {^{\mu}}_{\nu} \right ]=\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$
So the first equation I wrote for the $G{^{0}}_{0}$ element calculated for the spacetime's Einstein tensor given that element for his field equations results in
$\frac{3}{c^2}\frac{\dot{a}^2 +Kc^2}{a^2}=\lambda +\frac{8\pi G}{c^4}T{^{0}}_{0}$
which can be rewritten as
$\frac{\dot{a}^2 +Kc^2}{a^2}=\frac{8\pi G\left ( \frac{T{^{0}}_{0}}{c^2} \right )+\lambda c^2}{3}$
This is one of the two Friedmann equations, where a lot of folk define $\rho$
$\rho =\frac{T{^{0}}_{0}}{c^2}$
Next take the second equation I wrote for the $G{^{1}}_{1}$ element calculated for the spacetime's Einstein tensor given that element for his field equations results in
$\frac{2}{c^2}\frac{\ddot{a}}{a}+\frac{1}{3}G{^{0}}_{0}=\lambda +\frac{8\pi G}{c^4}T{^{1}}_{1}$
Define $p$ for pressure as
$p=-T{^{1}}_{1}=-T{^{2}}_{2}=-T{^{3}}_{3}$
Now insert this and the result for $G{^{0}}_{0}$ to arrive at
$\frac{2}{c^2}\frac{\ddot{a}}{a}+\frac{1}{3}\left ( \lambda+\frac{8\pi G}{c^4}T{^{0}}_{0} \right )=\lambda -\frac{8\pi G}{c^4}p$
This can be rewritten
$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\left ( \frac{T{^{0}}_{0}}{c^2}+\frac{3p}{c^2} \right )+\frac{\lambda c^2}{3}$
which is the other of Friedmann's equations.

Last edited: Apr 25, 2020
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Heat Death? To clarify, Is Heat death the expected conclusion to this model? With uniform distribution, What prevents heat death in this model? Eventually, Entropy wins. dissipation exists between transformations of energy. This model has a life span. They all do I suppose.

My question is this to you David, In a universe where Time is something that can be manipulated. How does time Effectively function? Let me assume the universe dissipates because of heat death, At this point in time, Does time stop existing? could someone from a time after the universe ended, Travel backward to when the universe did exist? Or since time is relative, Would time cease to exist without the presence of energy/mass or a universe?

Could something travel backward in time from a point when this universe was completely annihilated and make it to this universe? Or would Time travel from such a point be impossible because there was no mass or energy or gravity that was left of this place?

If this question is too contrived I'm sorry. I ask weird questions.

Last edited: Apr 26, 2020
3. ### waitedavid137Honorable

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Heat death and the model, I'll brush over, then to say, its common understanding is blown out of the water by Penrose who took me many years to start to understand.
So consider the first order Friedmann equation
$\frac{\dot{a}^2 +Kc^2}{a^2}=\frac{8\pi G\left ( \frac{T{^{0}}_{0}}{c^2} \right )+\lambda c^2}{3}$
The scale factor $a$ at this point can have the behavior that it expands according to this time forever, or that it at some finite time according to this time coordinate comes to stop expanding and starts to recollapse. Early the dominate term determining the behavior of $\dot{a}$ for our universe is the $T{^{0}}_{0}$ part, which is the energy density of the matter within the universe. In order for $\dot{a}$ to come to zero, ie for the universe to stop expanding and start to collapse at some finite time according to that time coordinate, there must therefor come to be a time when the energy density dissipates enough that it comes to be the case that
$\frac{8\pi G}{3}\frac{T{^{0}}_{0}}{c^2}+\frac{1}{3}\lambda c^2 -\frac{Kc^2}{a^2}=0$
In order for that to happen, either the cosmological constant $\lambda$ must be negative, or the universe must be sufficiently positively closed curved. Observations indicate that the cosmological constant is some small but positive value and as best we have been able to measure the universe is open flat. So the stop to a return collapse doesn't occur at a finite time according to that time coordinate which is then interpreted as heat death.
Then came Penrose.
Penrose figured out that the time and space coordinates can be scaled in such a way that events all the way out to infinity can be graphed onto a finite region in picturing them, while scaling the both in such the same way that lightlike paths remain 45 degree angle lines in the picture. This is called a Penrose diagram. What we see from the diagram is that those paths extend right through the boundaries of the picture where this time coordinate goes infinite.
Once crossing the boundary of the picture, one must define some scale for an entirely new region. Aside from time and space being scaled in such the same way as to keep those angles for the paths 45 degrees, there is nothing that requires that the scaling be the same as our first region. He proposed that what over all scales the time and space obey for the next region then are probabilistic. As such every new region extended to has a chance that an observer in the middle of it would recon that the photons came from some early extremely hot dense state that he would interpret according to his standards of spacetime coordinates as a big bang. That means that every now and again the photons cross into a region where the state happens to support life and the intelligent life in that region is left wondering why conditions are appropriate for life. The truth then is that you don't always yield regions that you get life, you just do every so often, and those are the regions that you then have life that can wonder about its condition.
So the universe eventually winds up cold and dead. Expanding forever. But then after forever, sometimes the radiation from it is found in a dense state that starts up new life which interprets the state as the result of a big bang and finds that their universe will come to the same doom.
This model is based on a uniform matter distribution. In order to consider time travel and such, one must look at nonuniform cases of the matter distribution. In the real world the matter isn't perfectly uniform, so one can certainly do that. This model then just becomes a large scale approximation to the overall behavior of the universe.

Last edited: Apr 26, 2020
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