Jon's Big Theory

The universe is an anti de-sitter space within an all but equal and opposite de sitter space. The zero point energy density of the vacuum is masked by the zero point curvature of space-time.
The universe has three uniform dimensions of space and a minimally non-uniform dimension of time.
Taking ${\displaystyle u}$ as the co-ordinate of the fourth dimension and ${\displaystyle t}$ as a measure of pure time then, in natural units, ${\displaystyle u}$ is defined by
${\displaystyle u^{2}=t^{2}-1}$
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Derivatives of u and t

from ${\displaystyle \ \ \ u^{2}=t^{2}-1\ \ \ }$ it follows that
${\displaystyle u=\pm \left(t^{2}-1\right)^{1/2}}$
${\displaystyle {\frac {du}{dt}}=\pm {\tfrac {1}{2}}\left(t^{2}-1\right)^{-1/2}\left(2t\right)=\pm {\frac {t}{u}}}$
${\displaystyle \left({\frac {du}{dt}}\right)^{2}={\frac {t^{2}}{u^{2}}}}$
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Change Of Co-ordinates

The general formula for a change of co-ordinate of a tensor F_t is

${\displaystyle F_{t}=F_{x}{\dfrac {\partial x}{\partial t}}+F_{y}{\dfrac {\partial y}{\partial t}}+F_{z}{\dfrac {\partial z}{\partial t}}+F_{u}{\dfrac {\partial u}{\partial t}}}$
since t is only dependant on u, and is fully dependant on u, this simplifies to
${\displaystyle F_{t}=F_{u}{\dfrac {du}{dt}}}$
likewise for ${\displaystyle F_{tt}}$
${\displaystyle F_{tt}=F_{uu}\left({\dfrac {du}{dt}}\right)^{2}}$
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Einstein's field equations

Einstein's field equations may be written
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }}$
alternatively, using zero point values we get
${\displaystyle G_{\mu \nu }+G_{\mu \nu }^{^{_{ZP}}}=\kappa T_{\mu \nu }+\kappa T_{\mu \nu }^{^{_{ZP}}}}$
Subtracting the second equation from the first we get
${\displaystyle \Lambda g_{\mu \nu }-G_{\mu \nu }^{^{_{ZP}}}=-\kappa T_{\mu \nu }^{^{_{ZP}}}}$
Thus ahowing the cosmological constant to be the difference of the two zero point values.

If we assume that the curvature components are in the fourth dimension and that the energy components are in proper time then

${\displaystyle G_{uu}+G_{uu}^{^{_{ZP}}}=\kappa T_{tt}+\kappa T_{tt}^{^{_{ZP}}}}$

incorporating the cosmological constant

${\displaystyle \Lambda g_{uu}-G_{uu}^{^{_{ZP}}}=-\kappa T_{uu}^{^{_{ZP}}}}$

and adding together

${\displaystyle \Lambda g_{uu}-G_{uu}^{^{_{ZP}}}+G_{uu}+G_{uu}^{^{_{ZP}}}=\kappa T_{tt}+\kappa T_{tt}^{^{_{ZP}}}-\kappa T_{uu}^{^{_{ZP}}}}$

${\displaystyle G_{uu}+\Lambda g_{uu}=\kappa T_{tt}+\kappa T_{tt}^{^{_{ZP}}}-\kappa T_{uu}^{^{_{ZP}}}}$

${\displaystyle G_{uu}\left({\frac {du}{dt}}\right)^{2}+\Lambda g_{uu}\left({\frac {du}{dt}}\right)^{2}=\kappa T_{tt}\left({\frac {du}{dt}}\right)^{2}+\kappa T_{tt}^{^{_{ZP}}}\left({\frac {du}{dt}}\right)^{2}-\kappa T_{uu}^{^{_{ZP}}}\left({\frac {du}{dt}}\right)^{2}}$

${\displaystyle G_{tt}+\Lambda g_{tt}=\kappa T_{tt}{\frac {t^{2}}{u^{2}}}+\kappa T_{tt}^{^{_{ZP}}}{\frac {t^{2}}{u^{2}}}-\kappa T_{tt}^{^{_{ZP}}}}$

${\displaystyle G_{tt}+\Lambda g_{tt}=\kappa T_{tt}{\frac {t^{2}}{u^{2}}}+\kappa T_{tt}^{^{_{ZP}}}\left({\frac {t^{2}}{u^{2}}}-1\right)}$

${\displaystyle G_{tt}+\Lambda g_{tt}=\kappa T_{tt}{\frac {t^{2}}{u^{2}}}+{\frac {\kappa }{u^{2}}}T_{tt}^{^{_{ZP}}}}$


So the vacuum solution is
${\displaystyle G_{tt}+\Lambda g_{tt}={\frac {\kappa }{u^{2}}}T_{tt}^{^{_{ZP}}}}$

Assuming the cosmological constant to be negligable, we derive the Friedman equation for the vacuum.

${\displaystyle G_{uu}=3{\frac {\left(da/du\right)^{2}+kc^{2}}{a^{2}}}}$
${\displaystyle G_{uu}\left({\frac {du}{dt}}\right)^{2}=3{\frac {\left(da/du\right)^{2}+kc^{2}}{a^{2}}}\left(du/dt\right)^{2}}$
${\displaystyle G_{tt}=3{\frac {\left(da/dt\right)^{2}}{a^{2}}}+3{\frac {kc^{2}}{a^{2}}}\left({\frac {du}{dt}}\right)^{2}=3{\frac {{\dot {a}}^{2}}{a^{2}}}+3{\frac {kc^{2}}{a^{2}}}\left({\frac {du}{dt}}\right)^{2}}$
${\displaystyle G_{tt}={\frac {\kappa }{u^{2}}}T_{tt}^{^{_{ZP}}}}$
${\displaystyle 3{\frac {{\dot {a}}^{2}}{a^{2}}}+3{\frac {kc^{2}}{a^{2}}}\left({\frac {du}{dt}}\right)^{2}={\frac {\kappa }{u^{2}}}T_{tt}^{^{_{ZP}}}}$
for an intrinsic curvature of zero
${\displaystyle 3{\frac {{\dot {a}}^{2}}{a^{2}}}={\frac {\kappa }{u^{2}}}T_{tt}^{^{_{ZP}}}}$

Assuming that in fundamental units ${\displaystyle \ T_{tt}^{^{_{ZP}}}=1}$

${\displaystyle 3{\frac {{\dot {a}}^{2}}{a^{2}}}={\frac {\kappa }{u^{2}}}}$

${\displaystyle {\frac {{\dot {a}}^{2}}{a^{2}}}={\frac {\kappa }{3}}{\frac {1}{u^{2}}}}$

${\displaystyle {\frac {\dot {a}}{a}}={\sqrt {\frac {\kappa }{3}}}{\frac {1}{u}}}$

${\displaystyle \int {\frac {\dot {a}}{a}}dt={\sqrt {\frac {\kappa }{3}}}\int {\frac {1}{u}}dt}$

${\displaystyle ln\left(a\right)={\sqrt {\frac {\kappa }{3}}}\ ln\left(t+u\right)\ +\ constant\ of\ integration}$

${\displaystyle a\propto \left(t+u\right)^{\sqrt {\kappa /3}}}$

${\displaystyle a\propto \left(t+{\sqrt {t^{2}-1}}\right)^{\sqrt {\kappa /3}}}$

So there we have it. Inflation, slow roll, with a graceful exit, straight from the vacuum; providing that ${\displaystyle \kappa }$ in natural units is more than three.
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What is the value of ${\displaystyle \kappa }$ in natural units?

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Time Has A Begining, The Fourth Dimension Does Not.

The universe has three uniform dimensions of space and a minimally non-uniform dimension of time.
Taking ${\displaystyle u}$ as the co-ordinate of the fourth dimension and ${\displaystyle t}$ as a measure of cosmological time i.e. proper time for all things cosmological, then in natural units, ${\displaystyle u}$ is defined by
${\displaystyle u^{2}=t^{2}-1}$
If we look at the universe with all the dimensions of space "edge-on" then we will simply see a line representing the fourth dimension.
Here we see a line representing the fourth dimension with all the dimensions of space seen "edge on".
The line is continuous, it does not have a beggining or an end.
The graph shows the relationship between the fourth dimension and pure time.
Time does have a beggining.
At the south pole every direction is north.
At u=0 every direction is forwards in time.
When space and time are added together the square of the result is equal to the difference of the squares of the parts. If there is more time added than space then the result is time. If there is more space added than time then the result is space. If the amount of time is equal to the amount of space then the result is Null, neither time nor space. Note 1 second is equivalent to 299 792 458 metres. The fourth dimension is a dimension of time at all points except u=0, where it is Null, and proper time changes direction.
A universe in two halves. Not the big bounce.