Physical Model Archives - Intuitively Creative Consulting

The Metric Tensor

May 13, 2019August 12, 2019
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From a projective geometrical perspective, the points within a curvilinear $n-$ dimensional physical spacetime may be viewed as a subset of points, denoted as ${{x}^{I}}$ , referred to rectilinear coordinates axes in $m>n$ dimensions. Using upper case Roman letters to label the rectilinear coordinate indices, the components of the metric tensor ${{g}_{IJ}}$ of the rectilinear system are constants.

Let $d{{x}^{I}}$ be the coordinate differential that describes the same vector as $d{{x}^{\mu }}$ . The invariant square of the differential interval is given as

$d{{s}^{2}}={{g}_{IJ}}d{{x}^{I}}d{{x}^{J}}$

Given the earlier discussion (see the post called “Coordinate Transformations”), the differential $d{{x}^{I}}$ can be expressed in terms of the fields in the $n-$ dimensional spacetime,

$d{{x}^{I}}=\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }d{{x}^{\mu }}$

which uses the transformation law,

$x_{,\mu }^{I}=\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }$

The squared differential interval is then expressed as

$d{{s}^{2}}={{g}_{IJ}}\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }d{{x}^{\mu }}\sum\nolimits_{f'}{\left( \partial {{C}^{J}}/\partial {{f}^{\gamma '}}\ \right)}{f}'_{,\nu }^{\gamma }d{{x}^{\nu }}$

As the squared differential interval is form-invariant the metric tensor in the $n-$ dimensional spacetime may be written as follows:

${{g}_{\mu \nu }}={{g}_{IJ}}\sum\nolimits_{f}{\left( \partial {{C}^{I}}/\partial {{f}^{\lambda }}\ \right)}f_{,\mu }^{\lambda }\sum\nolimits_{f'}{\left( \partial {{C}^{J}}/\partial {{f}^{\gamma '}}\ \right)}{f}'_{,\nu }^{\gamma }$

In this way, transformation laws that are functionals of the matter fields associated with emergent spacetime points yield a metric tensor that is a functional of the matter fields – the metric of matter.

Physical Model

Emergence of Points

May 6, 2019August 12, 2019
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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

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

Physical Model

Vectors

April 21, 2019August 12, 2019
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The coordinate of an event in a given system of coordinates is not the same thing as a vector between the origin of the coordinate system and the coordinate of the event; the coordinate itself is not a vector. The difference between the coordinates of two infinitesimally separated events is a vector. If the coordinates of both events are both transformed linearly to coordinates in a second coordinate system, each of the coordinates transforms with the same transformation, which may be a linear non-homogeneous transformation. However, the additive constant that accounts for the non-homogeneity of the transformation law is the same for both of the coordinates. So, the additive constants cancel each other when the difference between them is formed. Thus, the linear non-homogeneous transformation of the coordinates becomes a linear homogeneous transformation of the coordinate differential, the vector. This is a fundamental aspect of the theory of relativity. The coordinate differential is representative of all true vectors. Vectors transform under linear homogeneous transformations even when the two coordinate systems are related by a linear non-homogeneous transformation.

Vectors are tensors of the first rank. A second rank tensor can be thought of as a vector whose components are vectors. Higher rank tensors can be formed similarly. Tensors of any rank can be expressed in different coordinate systems by applying the appropriate linear homogeneous transformation. By describing natural events using field theories expressed with tensors, the field theories are invariant to linear transformations between coordinate systems.

Physical Model

Coordinates

April 20, 2019August 12, 2019
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Physical theories extant describe natural events using the concept of fields. Fields are described mathematically by so-called laws that prescribe the characteristics of the fields such that the fields yield meaningful representations of experimental results. To obtain experimental results, experiments establish local coincidence of events in spacetime. Thus, the spacetime coordinates are implicit in physical theories.

An essential characteristic of the mathematical prescription is the requirement that theories and their predictions be invariant to changes in the coordinate systems. Transformation laws ensure this characteristic. Transformations include gauge transformations and coordinate transformations. Proper transformations within acceptable theories leave predictions invariant.

Experimental properties are expressed in magnitudes and units. Proper predictions of theory must, then, incorporate units consistently. Different systems of units exist. These systems are interconvertible. The units are related to the physical properties by which experiments and theories work together to represent nature.

The coordinate of an event is not unique. Rather, the coordinate depends upon the system of coordinates. Coordinate systems are characterized by an origin and a metric. The transformation that expresses the coordinate of an event in one coordinate system in terms of the coordinates of another coordinate system is a coordinate transformation.

Two coordinate systems that differ only in origin are related through a non-homogenous coordinate transformation. That is, the difference between the coordinate of an event in one system and the coordinate of that event in the other coordinate system is a non-zero constant (non-homogeneous).

Physical Model

February 26, 2019May 6, 2019
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This ongoing blog concerns the physical model of phenomena that is consistent with the underpinnings of the philosophical model. The physical model is approached from a qualitative picture and from a mathematical picture. The goal is to arrive at new experimentally verifiable conclusions, yet preserve well-established physical models.