View of the world (physically rather than philosophically)
Introduction
Have you ever thought of how can we describe the basic principles of our world (or universe), especially in a physical or mathematical way?
The thought itself seems like a philosophical problem (and is actually thought over by philosophers for thousands of years). However, maybe it can be interesting to think it over in another perspective.
Note that most of the definitions used below are different from the popular definitions!
Galilean world
Here is the basic principle of the Galilean world:
Principle 1. The world is a Galilean structure with $3$-dimensional space and $1$-dimensional time.
Here is the definition of a Galilean structure. A Galilean structure with $\chi$-dimensional space and $\iota$-dimensional time is a $3$-tuple $\left(\mathscr A,\tau,\rho\right)$ with the following principles:
- $\mathscr A$ is a $\nu$-dimensional affine space associated with the vector space $\mathbb R^\nu$, where $\nu\coloneqq\chi+\iota$;
- $\tau:\mathbb R^\nu\rightarrow\mathbb R^\iota$ is a linear mapping;
- For $a,b\in\mathscr A$ such that $\tau\!\left(a-b\right)=0$, the mapping $\rho$ satisfies $\rho\!\left(a,b\right)=\sqrt{\left(a-b\right)^2}$.
To make the physical meanings of the above mathematical stuff clear, we
- call $\mathscr A$ the universe or a Galilean space,
- call the points in the universe the events,
- call $\tau$ the time,
- say two events $a,b$ are simultaneous iff $\tau\!\left(a-b\right)=0$,
- call $\rho\!\left(a,b\right)$ the distance between simultaneous events $a,b$.
Here is the second principle of the Galilean world:
Principle 2 (Galileo’s principle of relativity). Laws of nature remain the same under Galilean transformation.
Here is the definition of a Galilean transformation. An affine transformation $g$ over the Galilean space $\mathscr A$ is called a Galilean transformation iff both of the following are satisfied:
- $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=\tau\!\left(ga-gb\right)$ (preservation of intervals of time),
- $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=0\Rightarrow \rho\!\left(a,b\right)=\rho\!\left(ga,gb\right)$ (preservation of distance between simultaneous events).
Galilean transformations form a group (why?) called the Galilean group, which is an $\left(\iota+\frac{\chi\left(\chi+3\right)}2\right)$-dimensional Lie group (why?).
$\begin{align*} \mathscr A&\coloneqq\mathbb R^\nu,\\ \tau&\coloneqq\left(t,x\right)\mapsto t,\\ \rho&\coloneqq\left(a,b\right)\mapsto\sqrt{\left(a-b\right)^2} \end{align*}$ is a Galilean structure (why?). Here $\mathscr A$ is called the Galilean coordinate space.
The following transformations on the Galilean coordinate space are Galilean transformations (why?):
- $\left(t,x\right)\mapsto\left(t,x+vt\right)$, where $v\in\mathbb R^\chi$ (uniform motion),
- $\left(t,x\right)\mapsto\left(t+s,x+d\right)$, where $s\in\mathbb R^\iota$ and $d\in\mathbb R^\chi$ (translation),
- $\left(t,x\right)\mapsto\left(t,Gx\right)$, where $G\in\mathrm O\left(\chi\right)$ (rotation).
Every Galilean transformation of the Galilean coordinate space can be represented uniquely as the composition of a rotation, a translation, and a uniform motion (why?).
All Galilean spaces with the same dimensions are isomorphic to each other (why?).
In fact, the two principles above are not enough to build up the whole classical mechanics. We need to define motion, velocity, and acceleration in our $3+1$ universe. Then, to describe how the motion is determined, we need the following principle:
Principle 3 (Newton’s principle of determinacy). The motion is uniquely determined by initial positions and initial velocities.
With this principle, we can conclude that the motion can be depicted by Hamilton’s principle $\delta\int\mathcal L\!\left(q,\dot q,t\right)=0$ (why?), which leads to Euler–Lagrange equation $\frac{\mathrm d}{\mathrm dt} \frac{\partial\mathcal L}{\partial\dot q}= \frac{\partial\mathcal L}{\partial q}$ (why?).
According to Principle 2, for a closed system, its Euler–Lagrange equation should remain unchanged after a Galilean transformation (in a specific coordinate system, which in most cases is the Cartesian coordinate system utilized by Galilean coordinate space) acts on it, from which we can see that the universe is time-homogeneous (invariance under time translation), space-homogeneous (invariance under space translation), and space-isotropic (invariance under space rotation).
The rest (deriving the Lagrangian for some typical mechanical systems, and solving them) is just the normal classical mechanics, and is not related to the topic today.
Einsteinian world
We can build up the Einsteinian world similarly as we have done for the Galilean world.
Principle 1. The world is an Einsteinian structure with $3$-dimensional space and $1$-dimensional time.
Here is the definition of an Einsteinian structure. An Einsteinian structure with $\chi$-dimensional space and $\iota$-dimensional time is a $4$-tuple $\left(\mathscr A,\tau,\sigma,\rho\right)$ with the following principles:
- $\mathscr A$ is a $\nu$-dimensional affine space associated with the vector space $\mathbb R^\nu$, where $\nu\coloneqq\chi+\iota$;
- $\tau:\mathbb R^\nu\rightarrow\mathbb R^\iota$ and $\sigma:\mathbb R^\nu\rightarrow\mathbb R^\chi$ are linear mappings;
- The linear mapping $a\mapsto\left(\tau\!\left(a\right), \sigma\!\left(a\right)\right): \mathbb R^\nu\rightarrow\mathbb R^\nu$ has full rank;
- $\forall a,b\in\mathscr A:\rho\!\left(a,b\right)= \sqrt{\tau\left(a-b\right)^2-\sigma\left(a-b\right)^2}$.
To make the physical meanings of the above mathematical stuff clear, we
- call $\mathscr A$ the universe or an Einsteinian space,
- call the points in the universe the events,
- call $\tau$ the time,
- call $\sigma$ the space,
- call $\rho\!\left(a,b\right)$ the spacetime interval between events $a,b$.
Here is the second principle of the Einsteinian world:
Principle 2 (Einstein’s principle of relativity). Laws of nature remain the same under extended Poincaré transformation.
Here is the definition of a Poincaré transformation. An affine transformation $g$ over the Einsteinian space $\mathscr A$ is called a Poincaré transformation iff $\forall a,b\in\mathscr A: \rho\!\left(a,b\right)=\rho\!\left(ga,gb\right)$. Well, the definition is much simpler than that of Galilean transformation.
Poincaré transformations form a group (why?) called the Poincaré group, which is a $\frac{\nu\left(\nu+1\right)}2$-dimensional Lie group (why?).
$\begin{align*} \mathscr A&\coloneqq\mathbb R^\nu,\\ \tau&\coloneqq\left(ct,x\right)\mapsto t,\\ \sigma&\coloneqq\left(ct,x\right)\mapsto x,\\ \rho&\coloneqq\left(a,b\right)\mapsto \sqrt{\tau\!\left(a-b\right)^2-\sigma\!\left(a-b\right)^2} \end{align*}$ is an Einsteinian structure (why?), where the constant $c\in\mathbb R$ is called the speed of light. Here $\mathscr A$ is called the Minkowski space.
The following transformations on the Minkowski space are Poincaré transformations (why?):
- $a\mapsto a+d$, where $d\in\mathbb R^\nu$ (translation),
- $a\mapsto Ga$, where $G\in\mathrm O\!\left(\iota,\chi\right)$ is an indefinite orthogonal matrix (rotation).
Every Poincaré transformation of the Minkowski space can be represented uniquely as the composition of a translation and a rotation (why?).
All Einsteinian spaces with the same dimensions are isomorphic to each other (why?).
The rest is just the same as what we have done with Galilean world. You can find that the Einsteinian world is also space-homogeneous, time-homogeneous, and space-isotropic. Further more, it is time-isotropic.
Aristotelian world (imagination)
Although the Aristotelian world is not real, we can think of what it may look like.
Principle 1. The world is an Aristotelian structure with $3$-dimensional space and $1$-dimensional time.
According to Aristotle’s theory about the natural place, the world has something like a “center”, so the world cannot be space-homogeneous. However, he admits the invariance of natural laws over time, so the world is still time-homogeneous. It may be also reasonable to assume that the world is space-isotropic.
Thus, our definition of the Aristotelian structure should be non-affine, and the Aristotelian transformations should be composed of rotation and time translation.
However, although the space is non-affine, the time is affine. This makes it tricky to mix space and time together into an “$\mathscr A$”. However, there is a workaround. We can define the universe still an affine space, while give it an origin. Since this origin only add limitations to space transformation instead of time transformation, we can make it an $\iota$-dimensional affine subspace instead of a single point.
Here is the definition of an Aristotelian structure. An Aristotelian structure with $\chi$-dimensional space and $\iota$-dimensional time is a $4$-tuple $\left(\mathscr A,\tau,o,\rho\right)$ with the following principles:
- $\mathscr A$ is a $\nu$-dimensional affine space associated with the vector space $\mathbb R^\nu$, where $\nu\coloneqq\chi+\iota$;
- $\tau:\mathbb R^\nu\rightarrow\mathbb R^\iota$ is a linear mapping;
- $o$ is an $\iota$-dimensional affine subspace of $\mathscr A$;
- For $a,b\in\mathscr A$ such that $\tau\!\left(a-b\right)=0$, the mapping $\rho$ satisfies $\rho\!\left(a,b\right)= \sqrt{\left(a-b\right)^2}$.
To make the physical meanings of the above mathematical stuff clear, we
- call $\mathscr A$ the universe or an Aristotelian space,
- call the points in the universe the events,
- call $\tau$ the time,
- call $o$ the center of space,
- say two events $a,b$ are simultaneous iff $\tau\!\left(a-b\right)=0$,
- call $\rho\!\left(a,b\right)$ the distance between simultaneous events $a,b$.
Then the task is to define the Aristotelian transformations.
Principle 2. Laws of nature remain the same under Aristotelian transformation.
Here is the definition of an Aristotelian transformation. An affine transformation $g$ over the Aristotelian space $\mathscr A$ is called an Aristotelian transformation iff all of the following are satisfied:
- $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=\tau\!\left(ga-gb\right)$,
- $\forall a,b\in\mathscr A: \tau\!\left(a-b\right)=0\Rightarrow \rho\!\left(a,b\right)=\rho\!\left(ga,gb\right)$.
- $\forall a\in o:ga\in o$.
Notice the third condition, which makes it different from a Galilean transformation.
Aristotelian transformations from a group (why?) called the Aristotelian group, which is an $\left(\iota+\frac{\chi\left(\chi-1\right)}2\right)$-dimensional Lie group (why?).
$\begin{align*} \mathscr A&\coloneqq\mathbb R^\nu,\\ \tau&\coloneqq\left(t,x\right)\mapsto t,\\ o&\coloneqq\mathbb R^\iota\times\left\{0\right\},\\ \rho&\coloneqq\left(a,b\right)\mapsto\sqrt{\left(a-b\right)^2} \end{align*}$ is an Aristotelian structure (why?). Here $\mathscr A$ is called the Aristotelian coordinate space.
The following transformations on the Aristotelian coordinate space are Aristotelian transformations (why?):
- $\left(t,x\right)\mapsto\left(t+s,x\right)$, where $s\in\mathbb R^\iota$ (time translation),
- $\left(t,x\right)\mapsto\left(t,Gx\right)$, where $G\in\mathrm O\!\left(\chi\right)$ (rotation).
Every Aristotelian transformation of the Aristotelian coordinate space can be represented uniquely as the composition of a time translation and a rotation (why?).
All Aristotelian spaces with the same dimensions are isomorphic to each other (why?).
After building up the Aristotle world, how can we develop the mechanics here? Maybe it can be interesting.
Other imaginations
Here are some other imaginations of a world:
- What about a space-anistropic universe?
- What about defining the spacetime interval by multiplying space interval and time interval?
- What about a time-heterogeneous universe?
- What about making laws of nature unchanged under uniform acceleration?
- What about…
These can be materials for science fiction (novels or video games).