Emergence of Points

May 6, 2019August 12, 2019
by admin

The primordial vacuum is envisaged to be an infinite set of points that are indistinguishable one from the other. The points of any physical spacetime, the physical points, differ from those of the primordial vacuum in that they are distinguishable one from the other. The concept is presented first through a relatively simple equation. The equation considered expresses that the coordinate, $x$ , of a physical point is determined by constraints $C$ on a function of the coordinate itself $F\left( x \right)$ such that

$x = C\left[ {F\left( x \right)} \right]$

Such constraints are functionals of the coordinates. The function $F\left( x \right)$ may be considered to be expressible as the sum of functions $f\left( x \right)$ such that

$x = C\left[ {\sum\nolimits_f {f\left( x \right)} } \right]$

Let lower case Greek suffixes denote the coordinate directions in a particular spacetime, e.g. ${x^\mu }$ . The constraint and the functions also take on such a suffix. The chain rule is then used to denote the differential of the coordinate in this spacetime as

$d{x^\mu } = \sum\nolimits_f {\left( {{\raise0.7ex\hbox{${\partial {C^\mu }}$} \!\mathord{\left/ {\vphantom {{\partial {C^\mu }} {\partial {f^\nu }}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial {f^\nu }}$}}} \right)} \left( {{\raise0.7ex\hbox{${\partial {f^\nu }}$} \!\mathord{\left/ {\vphantom {{\partial {f^\nu }} {\partial {x^\lambda }}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial {x^\lambda }}$}}} \right)d{x^\lambda }$

Used here is the summation over repeated upper and lower suffixes, in this case $\nu$ and $\lambda$ . This relationship serves to define a Kronecker delta function (KDF) $\delta _\lambda ^\mu$ ,

$\delta _\lambda ^\mu = \sum\nolimits_f {\left( {{\raise0.7ex\hbox{${\partial {C^\mu }}$} \!\mathord{\left/ {\vphantom {{\partial {C^\mu }} {\partial {f^\nu }}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial {f^\nu }}$}}} \right)} \left( {{\raise0.7ex\hbox{${\partial {f^\nu }}$} \!\mathord{\left/ {\vphantom {{\partial {f^\nu }} {\partial {x^\lambda }}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\partial {x^\lambda }}$}}} \right)$

That coordinate differentials $d{x^\mu }$ are non-zero means that the spacetime has extension in the coordinate direction ${x^\mu }$ . When viewed as a matrix, the KDF has diagonal elements that are all equal to one. If the spacetime is characterized by $n$ directions in which non-zero extension can be measured, corresponding to $\mu = 0,1,...,n - 1$ , then the trace of the KDF equals $n$ , and

$\delta _\mu ^\mu = n$

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