Physlib.SpaceAndTime.Time.TimeTransMan

For any two points $t_1$ and $t_2$ in the time manifold $\text{TimeTransMan}$, if their real-valued representations are equal, i.e., $\text{val}(t_1) = \text{val}(t_2)$, then the points themselves are equal ($t_1 = t_2$). Here, $\text{val} : \text{TimeTransMan} \to \mathbb{R}$ is the map used to identify points in the manifold with the real numbers and induce its topology.

The topological space structure on the time manifold $\text{TimeTransMan}$ is defined as the topology induced from the real numbers $\mathbb{R}$ by the map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$.

The map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$, which identifies points in the time manifold with real numbers and induces its topology, is surjective. That is, for every $r \in \mathbb{R}$, there exists a point $t \in \text{TimeTransMan}$ such that $\text{val}(t) = r$.

The range of the map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$, which identifies points in the time manifold with real numbers, is the set of all real numbers $\mathbb{R}$.

The map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$ is an inducing map. This means that the topology on the time manifold $\text{TimeTransMan}$ is the induced topology from the real numbers $\mathbb{R}$ via the map $\text{val}$.

The map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$ is injective.

The map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$, which identifies points in the time manifold $\text{TimeTransMan}$ with real numbers, is an open embedding. This means that $\text{val}$ is a continuous injection and is an open map onto its image in $\mathbb{R}$.

For any subset $s \subseteq \text{TimeTransMan}$, $s$ is an open set in the time manifold if and only if its image under the map $\text{val}$, denoted by $\text{val}(s)$, is an open subset of the real numbers $\mathbb{R}$.

The map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$ defines a homeomorphism between the time manifold $\text{TimeTransMan}$ and the real numbers $\mathbb{R}$. This identification is based on the topology of $\text{TimeTransMan}$ being induced by the real numbers through $\text{val}$.

The manifold $\text{TimeTransMan}$ is equipped with the structure of a charted space modeled on the real numbers $\mathbb{R}$. This structure is defined by an atlas containing a single global chart, which is the homeomorphism $\text{val} : \text{TimeTransMan} \to \mathbb{R}$.

The space $\text{TimeTransMan}$ has the structure of a real analytic ($C^\omega$) manifold modeled on the real numbers $\mathbb{R}$ with the standard model with corners $\mathcal{I}(\mathbb{R}, \mathbb{R})$. This manifold structure is induced by the coordinate map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$, which serves as a global chart.

The map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$, which assigns a real number to each point in the time manifold, is a $C^\omega$ (real analytic) map from the manifold $\text{TimeTransMan}$ to the real line $\mathbb{R}$. Both spaces are considered as manifolds modeled on $\mathbb{R}$ with the standard model with corners $\mathcal{I}(\mathbb{R}, \mathbb{R})$.

The map $\text{val} : \text{TimeTransMan} \to \mathbb{R}$ is a real analytic ($C^\omega$) diffeomorphism between the time manifold $\text{TimeTransMan}$ and the real numbers $\mathbb{R}$. Both the manifold $\text{TimeTransMan}$ and the real line $\mathbb{R}$ are considered with respect to the standard model with corners $\mathcal{I}(\mathbb{R}, \mathbb{R})$.

This definition provides an additive action of the real numbers $\mathbb{R}$ on the time manifold $\text{TimeTransMan}$. For a real number $p \in \mathbb{R}$ and a time point $t \in \text{TimeTransMan}$, the resulting point $p + t$ is defined such that its underlying real-valued representation is the sum of $p$ and the underlying value of $t$ (i.e., $(p + t).\text{val} = p + t.\text{val}$).

For any real number $p \in \mathbb{R}$ and any point in time $t \in \text{TimeTransMan}$, the underlying real-valued representation of the point obtained by the additive action of $p$ on $t$, denoted $(p +_{\text{v}} t).\text{val}$, is equal to the sum of $p$ and the real-valued representation of $t$, denoted $t.\text{val}$.

The type $\text{TimeTransMan}$ is equipped with an additive group action of the real numbers $(\mathbb{R}, +)$. This action, denoted by $+_{\text{v}}$, satisfies the identity property $0 +_{\text{v}} t = t$ and the compatibility property $(p_1 + p_2) +_{\text{v}} t = p_1 +_{\text{v}} (p_2 +_{\text{v}} t)$ for all $p_1, p_2 \in \mathbb{R}$ and $t \in \text{TimeTransMan}$.

The less-than-or-equal-to relation $\le$ on the time manifold $\text{TimeTransMan}$ is defined such that for any two points $t_1, t_2 \in \text{TimeTransMan}$, $t_1 \le t_2$ if and only if their corresponding real values satisfy $t_1.\text{val} \le t_2.\text{val}$, where $.\text{val}$ is the map identifying the manifold with the real numbers $\mathbb{R}$.

For any two points $t_1, t_2$ in the time manifold $\text{TimeTransMan}$, the ordering relation $t_1 \le t_2$ holds if and only if $t_1.\text{val} \le t_2.\text{val}$, where $.\text{val}$ is the map that identifies the manifold with the real numbers $\mathbb{R}$.

The manifold of time, denoted by $TimeTransMan$, is non-empty.

Given a choice of units $x \in \text{TimeUnit}$ and two points $t_1, t_2 \in \text{TimeTransMan}$, the distance between these points in the units of $x$ is defined as: $$\text{dist}(x, t_1, t_2) = \frac{1}{x_{\text{val}}} |t_2.{\text{val}} - t_1.{\text{val}}|$$ where $t.{\text{val}} \in \mathbb{R}$ represents the underlying real-valued coordinate of a point in the time manifold, and $x_{\text{val}} \in \mathbb{R}_{>0}$ is the numerical value associated with the choice of units.

Given a choice of units $x \in \text{TimeUnit}$ and two points $t_2, t_1 \in \text{TimeTransMan}$, the signed difference between $t_2$ and $t_1$ in the units of $x$ is defined as: $$\text{diff}(x, t_2, t_1) = \begin{cases} \text{dist}(x, t_1, t_2) & \text{if } t_1 \le t_2 \\ -\text{dist}(x, t_1, t_2) & \text{if } t_1 > t_2 \end{cases}$$ where $\text{dist}(x, t_1, t_2)$ is the distance between $t_1$ and $t_2$ in units of $x$, and $\le$ is the natural ordering on the time manifold. This operation calculates the relative displacement $\frac{t_2.\text{val} - t_1.\text{val}}{x_{\text{val}}}$, where $t.\text{val} \in \mathbb{R}$ represents the coordinate of a time point and $x_{\text{val}} \in \mathbb{R}_{>0}$ represents the magnitude of the unit.

For any choice of units $x \in \text{TimeUnit}$ and any two points in time $t_1, t_2 \in \text{TimeTransMan}$, the signed difference between $t_1$ and $t_2$ in the units of $x$ is given by the difference of their underlying real coordinates scaled by the inverse of the unit's value: $$\text{diff}(x, t_1, t_2) = \frac{1}{x_{\text{val}}} (t_1.\text{val} - t_2.\text{val})$$ where $t.\text{val} \in \mathbb{R}$ represents the coordinate of a time point and $x_{\text{val}} \in \mathbb{R}_{>0}$ represents the magnitude of the unit.

For any choice of units $x \in \text{TimeUnit}$ and any time point $t \in \text{TimeTransMan}$, the signed difference between $t$ and itself in the units of $x$ is zero: $$\text{diff}(x, t, t) = 0$$

For a given choice of units $x \in \text{TimeUnit}$ and a fixed reference time point $t \in \text{TimeTransMan}$, the function mapping a time point $t' \in \text{TimeTransMan}$ to the signed difference $\text{diff}(x, t', t) \in \mathbb{R}$ is injective. That is, for any $t_1, t_2 \in \text{TimeTransMan}$, if $\text{diff}(x, t_1, t) = \text{diff}(x, t_2, t)$, then $t_1 = t_2$.

For a given choice of units $x \in \text{TimeUnit}$ and a fixed reference time point $t \in \text{TimeTransMan}$, the function that maps any time point $t' \in \text{TimeTransMan}$ to the signed difference $\text{diff}(x, t', t) \in \mathbb{R}$ is surjective. That is, for every real number $r \in \mathbb{R}$, there exists a time point $t' \in \text{TimeTransMan}$ such that $\text{diff}(x, t', t) = r$.

For a given choice of units $x \in \text{TimeUnit}$ and a fixed reference time point $t \in \text{TimeTransMan}$, the function mapping a time point $t' \in \text{TimeTransMan}$ to the signed difference $\text{diff}(x, t', t) \in \mathbb{R}$ is a bijection.

Given a choice of time unit $x \in \text{TimeUnit}$, a duration $r \in \mathbb{R}$, and a point in time $t \in \text{TimeTransMan}$, the function returns the point in the time manifold that is separated from $t$ by a signed difference of $r$ in the units of $x$. Formally, $\text{addTime}(x, r, t)$ is defined as the inverse of the signed difference function with respect to its first argument, i.e., it is the unique point $t' \in \text{TimeTransMan}$ such that $\text{diff}(x, t', t) = r$.

For any choice of time unit $x \in \text{TimeUnit}$, any duration $r \in \mathbb{R}$, and any time point $t \in \text{TimeTransMan}$, the point in the time manifold obtained by adding the duration $r$ to $t$ in the unit $x$ is the point whose coordinate is $x_{\text{val}} \cdot r + t.\text{val}$. That is, $\text{addTime}(x, r, t) = \langle x_{\text{val}} \cdot r + t.\text{val} \rangle$, where $x_{\text{val}}$ is the magnitude of the unit $x$ and $t.\text{val}$ is the coordinate of the time point $t$.

For any choice of time unit $x \in \text{TimeUnit}$, any duration $r \in \mathbb{R}$, and any time point $t \in \text{TimeTransMan}$, the coordinate of the point in the time manifold obtained by adding the duration $r$ to $t$ in the unit $x$ is equal to $x_{\text{val}} \cdot r + t.\text{val}$, where $x_{\text{val}}$ is the magnitude of the unit $x$ and $t.\text{val}$ is the coordinate of the time point $t$.

Given an origin point $t_0 \in \text{TimeTransMan}$, a choice of time unit $x \in \text{TimeUnit}$, and a point in time $t \in \text{TimeTransMan}$, the function returns the point in time that is the same distance away from $t_0$ as $t$ is, but in the opposite direction. This is calculated by finding the signed difference $\Delta \tau = \text{diff}(x, t_0, t)$ and adding this duration to the origin, resulting in $\text{addTime}(x, \Delta \tau, t_0)$. Physically, this represents a reflection of the point $t$ across the origin $t_0$ relative to the metric defined by $x$.

Given an origin $t_0 \in \text{TimeTransMan}$ and a point in time $t \in \text{TimeTransMan}$, the function returns the point in the time manifold that is the same distance away from $t_0$ as $t$ is, but in the opposite direction. This operation corresponds to a reflection of the point $t$ across the reference point $t_0$. Although the definition is implemented using a default unit of time, the resulting point is independent of the choice of metric on the manifold.

For any origin point $t_0 \in \text{TimeTransMan}$, any choice of time unit $x \in \text{TimeUnit}$, and any time point $t \in \text{TimeTransMan}$, the negation of $t$ around $t_0$ is equal to the negation of $t$ around $t_0$ calculated using the unit $x$: $$\text{neg}(t_0, t) = \text{negMetric}(t_0, x, t)$$ Here, $\text{neg}(t_0, t)$ represents the point in the time manifold that is the same distance from $t_0$ as $t$ is but in the opposite direction, and $\text{negMetric}(t_0, x, t)$ is the specific implementation of this reflection using the metric defined by unit $x$.

Given a chosen origin $t_0 \in \text{TimeTransMan}$ and a choice of time unit $x \in \text{TimeUnit}$, the function `toTime` is a homeomorphism between the time manifold $\text{TimeTransMan}$ and the space $\text{Time}$. The forward map sends a point $t \in \text{TimeTransMan}$ to the signed difference $\text{diff}(x, t, t_0)$, effectively representing the time point as a duration relative to the origin. Its inverse map takes a duration $r \in \text{Time}$ and returns the point in $\text{TimeTransMan}$ obtained by adding $r$ to the origin $t_0$ in units of $x$, denoted as $\text{addTime}(x, r, t_0)$.

For any choice of origin $t_0 \in \text{TimeTransMan}$ and any time unit $x \in \text{TimeUnit}$, the homeomorphism $\text{toTime}(t_0, x)$ maps the origin point $t_0$ to $0$.

For any choice of origin $t_0 \in \text{TimeTransMan}$ and any time unit $x \in \text{TimeUnit}$, the inverse of the homeomorphism $\text{toTime}(t_0, x)$ maps the zero duration $0 \in \text{Time}$ to the origin point $t_0$. That is, $$(\text{toTime}(t_0, x))^{-1}(0) = t_0$$

Let $t_0 \in \text{TimeTransMan}$ be a chosen origin and $x \in \text{TimeUnit}$ be a choice of time unit. For any time point $t \in \text{TimeTransMan}$, the underlying real value of its image under the homeomorphism $\text{toTime}(t_0, x)$, denoted as $(\text{toTime}(t_0, x)(t)).\text{val}$, is equal to the signed difference $\text{diff}(x, t, t_0)$ between $t$ and $t_0$ in units of $x$.

For any origin point $t_0 \in \text{TimeTransMan}$, time unit $x \in \text{TimeUnit}$, and duration $r \in \text{Time}$, the inverse of the homeomorphism $\text{toTime}(t_0, x)$ maps $r$ to the point in the time manifold produced by adding the duration $r$ to the origin $t_0$ in the unit $x$. That is, $$(\text{toTime}(t_0, x))^{-1}(r) = \text{addTime}(x, r, t_0)$$ where $\text{toTime}(t_0, x) : \text{TimeTransMan} \cong_t \text{Time}$ is the homeomorphism identifying the manifold with the space of durations relative to an origin.

Let $t_0 \in \text{TimeTransMan}$ be a chosen origin and $x \in \text{TimeUnit}$ be a choice of time unit. Let $\text{toTime}(t_0, x) : \text{TimeTransMan} \to \text{Time}$ be the homeomorphism that maps a point in the time manifold to its coordinate representation. For any duration $r \in \mathbb{R}$ and any time point $\tau \in \text{TimeTransMan}$, the following identity holds: $$\text{toTime}(t_0, x)(\text{addTime}(x, r, \tau)) = \langle r \rangle + \text{toTime}(t_0, x)(\tau)$$ where $\text{addTime}(x, r, \tau)$ is the point in the time manifold obtained by adding the duration $r$ to $\tau$ in units of $x$, and $\langle r \rangle$ denotes the duration $r$ viewed as an element of the space $\text{Time}$.

Let $t_0 \in \text{TimeTransMan}$ be a choice of origin point in the time manifold and $x \in \text{TimeUnit}$ be a choice of time unit. For any two durations $t_1, t_2 \in \text{Time}$, the point in the manifold obtained by applying the inverse homeomorphism $(\text{toTime}(t_0, x))^{-1}$ to the sum of durations $t_1 + t_2$ is equal to the point reached by adding the signed difference between $(\text{toTime}(t_0, x))^{-1}(t_1)$ and $t_0$ to the point $(\text{toTime}(t_0, x))^{-1}(t_2)$ in the units of $x$: $$(\text{toTime}(t_0, x))^{-1}(t_1 + t_2) = \text{addTime}(x, \text{diff}(x, (\text{toTime}(t_0, x))^{-1}(t_1), t_0), (\text{toTime}(t_0, x))^{-1}(t_2))$$ where the homeomorphism $\text{toTime}(t_0, x) : \text{TimeTransMan} \cong_t \text{Time}$ identifies points in the manifold with durations relative to an origin, $\text{diff}$ calculates the signed duration between two points, and $\text{addTime}$ calculates the point reached by adding a duration to a reference point.

Let $t_0 \in \text{TimeTransMan}$ be a choice of origin and $x \in \text{TimeUnit}$ be a choice of time unit. For any two durations $t_1, t_2 \in \text{Time}$, the following identity holds: $$(\text{toTime}(t_0, x))^{-1}(t_1 + t_2) = \text{addTime}(x, \text{diff}(x, (\text{toTime}(t_0, x))^{-1}(t_2), t_0), (\text{toTime}(t_0, x))^{-1}(t_1))$$ where $(\text{toTime}(t_0, x))^{-1}$ is the inverse of the homeomorphism that maps points in the time manifold to their representations as durations relative to $t_0$ in the unit $x$, $\text{diff}$ is the signed difference between two points in time, and $\text{addTime}$ is the addition of a duration to a time point.

Let $t_0 \in \text{TimeTransMan}$ be a choice of origin and $x \in \text{TimeUnit}$ be a choice of time unit. For any two time points $t_1, t_2 \in \text{TimeTransMan}$, the signed difference between $t_2$ and $t_1$ in the units of $x$ is equal to the difference between their representations in the space $\text{Time}$ (relative to origin $t_0$ and unit $x$): $$\text{diff}(x, t_2, t_1) = \text{toTime}(t_0, x, t_2) - \text{toTime}(t_0, x, t_1)$$ where the subtraction on the right-hand side is performed in the space $\text{Time}$.

Let $t_0 \in \text{TimeTransMan}$ be a chosen origin and $x \in \text{TimeUnit}$ be a choice of time unit. Let $\text{toTime}(t_0, x) : \text{TimeTransMan} \to \text{Time}$ be the homeomorphism that maps a point in the time manifold to its coordinate representation relative to $t_0$ and $x$. For any point in time $t \in \text{TimeTransMan}$, the following identity holds: $$\text{toTime}(t_0, x)(\text{neg}(t_0, t)) = -\text{toTime}(t_0, x)(t)$$ where $\text{neg}(t_0, t)$ is the point in the time manifold obtained by reflecting $t$ across the origin $t_0$, and the negation on the right-hand side is the additive inverse in the space $\text{Time}$.

Let $t_0 \in \text{TimeTransMan}$ be a choice of origin and $x \in \text{TimeUnit}$ be a choice of time unit. For any duration $r \in \text{Time}$, the point in the time manifold corresponding to the negative duration $-r$ (via the inverse of the homeomorphism $\text{toTime}(t_0, x)$) is equal to the negation of the point corresponding to duration $r$ around the origin $t_0$. That is: $$(\text{toTime}(t_0, x))^{-1}(-r) = \text{neg}(t_0, (\text{toTime}(t_0, x))^{-1}(r))$$ where $\text{neg}(t_0, t)$ is the reflection of the time point $t$ across the reference point $t_0$.

For any choice of origin $t_0 \in \text{TimeTransMan}$, time unit $x \in \text{TimeUnit}$, and durations $r_1, r_2 \in \text{Time}$, let $f: \text{TimeTransMan} \cong \text{Time}$ be the homeomorphism $\text{toTime}(t_0, x)$ that identifies points in the time manifold with durations relative to the origin. The inverse of this homeomorphism, $f^{-1}$, satisfies the following identity for the difference of two durations: $$f^{-1}(r_1 - r_2) = \text{addTime}(x, \text{diff}(x, t_0, f^{-1}(r_2)), f^{-1}(r_1))$$ where $\text{addTime}(x, r, t)$ denotes the addition of duration $r$ to time point $t$ in unit $x$, and $\text{diff}(x, t_A, t_B)$ is the signed difference between time points $t_A$ and $t_B$ in unit $x$.