Physlib.SpaceAndTime.SpaceTime.TimeSlice

Let $d$ be a natural number representing the spatial dimension, and let $M$ be a normed real vector space. Given a speed of light $c$ and a function $f: \text{SpaceTime } d \to M$, if $f$ is $n$-times continuously differentiable ($C^n$), then its uncurried time-slice representation—the function mapping the pair of time and space $(t, x) \in \text{Time} \times \text{Space } d$ to $M$—is also $n$-times continuously differentiable.

Let $d$ be a natural number representing the spatial dimension and $M$ be a real normed space. Given a speed of light $c$, let $f: \text{SpaceTime } d \to M$ be a function. If $f$ is differentiable over $\mathbb{R}$, then its uncurried time-sliced representation $F: \text{Time} \times \text{Space } d \to M$ is also differentiable over $\mathbb{R}$.

Let $d$ be a natural number representing the spatial dimension, and let $M$ be a real vector space. For a given speed of light $c$, the operator $\text{timeSliceLinearEquiv}_c$ is a linear equivalence that transforms a function on spacetime $f : \text{SpaceTime } d \to M$ into its representation as a function of time and space. This theorem states that the application of this linear equivalence to $f$ is equal to the direct time-slicing of $f$: $$\text{timeSliceLinearEquiv}_c(f) = \text{timeSlice}_c(f)$$ where $\text{timeSlice}_c(f)$ is the function of type $\text{Time} \to \text{Space } d \to M$ obtained by decomposing the spacetime coordinates.

Let $d$ be the spatial dimension and $M$ be a real vector space. For a given speed of light $c$, let $\text{timeSlice}_c$ be the operation that transforms a function defined on spacetime into a function from $\text{Time}$ to $\text{Space } d$. Let $\text{timeSliceLinearEquiv}_c$ be the linear equivalence representing this same transformation. For any function $f: \text{Time} \to \text{Space } d \to M$, the application of the inverse of the linear equivalence to $f$ is equal to the application of the inverse of the time-slice map to $f$: $$(\text{timeSliceLinearEquiv}_c)^{-1}(f) = (\text{timeSlice}_c)^{-1}(f)$$

Let $d$ be the spatial dimension and $M$ be a real normed space. Given a speed of light $c$, let $f$ be an $M$-valued distribution on $\text{SpaceTime } d$ (a continuous linear map from $\mathcal{S}(\text{SpaceTime } d, \mathbb{R})$ to $M$). For any Schwartz test function $\kappa \in \mathcal{S}(\text{Time} \times \text{Space } d, \mathbb{R})$, the action of the time-sliced distribution $\text{distTimeSlice}_c f$ on $\kappa$ is given by: $$(\text{distTimeSlice}_c f)(\kappa) = f(\kappa \circ \Phi_c)$$ where $\Phi_c : \text{SpaceTime } d \to \text{Time} \times \text{Space } d$ is the continuous linear equivalence that identifies spacetime coordinates with time and space components according to $c$.

Let $c$ be the speed of light. Let $M$ be a real normed space and $d$ be a natural number representing the spatial dimension. Given an $M$-valued distribution $f$ on the product space $\text{Time} \times \text{Space}_d$ and a Schwartz test function $\kappa \in \mathcal{S}(\text{SpaceTime}_d, \mathbb{R})$, the action of the inverse time-slice operator $(\text{distTimeSlice } c)^{-1}$ on $f$ evaluated at $\kappa$ is given by: $$((\text{distTimeSlice } c)^{-1} f)(\kappa) = f(\kappa \circ \Phi_c^{-1}),$$ where $\Phi_c^{-1}: \text{Time} \times \text{Space}_d \to \text{SpaceTime}_d$ is the inverse of the coordinate transformation map $\Phi_c: \text{SpaceTime}_d \to \text{Time} \times \text{Space}_d$ defined by the speed of light $c$.

Let $M$ be a real normed space and $d$ be the spatial dimension. For a given speed of light $c$, let $f$ be an $M$-valued distribution on $\text{SpaceTime } d$. Let $\text{distTimeSlice}_c$ be the operator that transforms distributions on spacetime into distributions on the product space $\text{Time} \times \text{Space } d$. Let $\partial_0$ denote the distributional partial derivative with respect to the zeroth (temporal) coordinate index in spacetime, and let $\partial_t$ denote the physical time derivative operator on $\text{Time} \times \text{Space } d$. The theorem states that the time slice of the coordinate derivative is proportional to the physical time derivative of the time slice: $$\text{distTimeSlice}_c (\partial_0 f) = \frac{1}{c} \frac{\partial}{\partial t} (\text{distTimeSlice}_c f)$$

Let $M$ be a real normed space and $d$ be the spatial dimension. For a given speed of light $c$, let $f$ be an $M$-valued distribution on the product space $\text{Time} \times \text{Space } d$. Let $(\text{distTimeSlice } c)^{-1}$ denote the inverse time-slice operator that maps distributions on $\text{Time} \times \text{Space } d$ back to distributions on $\text{SpaceTime } d$. Let $\partial_0$ denote the distributional partial derivative with respect to the zeroth (temporal) coordinate index in spacetime, and let $\partial_t$ denote the physical time derivative operator on $\text{Time} \times \text{Space } d$. The theorem states that: $$\partial_0 ((\text{distTimeSlice } c)^{-1} f) = \frac{1}{c} (\text{distTimeSlice } c)^{-1} \left( \frac{\partial f}{\partial t} \right)$$

Let $M$ be a real normed space and $d$ be the spatial dimension. For a given speed of light $c$, let $f$ be an $M$-valued distribution on the product space $\text{Time} \times \text{Space } d$. Let $(\text{distTimeSlice } c)^{-1}$ denote the inverse time-slice operator that maps distributions on $\text{Time} \times \text{Space } d$ back to distributions on $\text{SpaceTime } d$. Let $\partial_t f$ (or `Space.distTimeDeriv f`) denote the physical time derivative of the distribution on the product space, and let $\partial_0$ denote the distributional partial derivative with respect to the zeroth (temporal) coordinate index in spacetime. The theorem states that the inverse time-slice of the temporal derivative is equal to $c$ times the coordinate derivative of the inverse time-slice: $$(\text{distTimeSlice } c)^{-1} \left( \frac{\partial f}{\partial t} \right) = c \cdot \partial_0 ((\text{distTimeSlice } c)^{-1} f)$$

Let $d$ be the spatial dimension, $M$ be a real normed vector space, and $c$ be the speed of light. Let $f: \text{SpaceTime } d \to d[\mathbb{R}] M$ be an $M$-valued distribution on spacetime. For any spatial index $i \in \{0, \dots, d-1\}$, the time slice of the $i$-th spatial partial derivative of $f$ is equal to the $i$-th spatial partial derivative of the time-sliced distribution $f$. That is, $$\text{distTimeSlice}_c \left( \frac{\partial f}{\partial x_i} \right) = \frac{\partial}{\partial x_i} (\text{distTimeSlice}_c f)$$ where on the left-hand side $\frac{\partial}{\partial x_i}$ is the distributional derivative on $\text{SpaceTime } d$, and on the right-hand side it is the distributional derivative on $\text{Time} \times \text{Space } d$.

Let $d$ be the spatial dimension, $M$ be a real normed vector space, and $c$ be the speed of light. Let $f: (\text{Time} \times \text{Space } d) \to d[\mathbb{R}] M$ be an $M$-valued distribution. For any spatial index $i \in \{0, \dots, d-1\}$, the distributional spatial derivative of the inverse time-sliced distribution is equal to the inverse time-slice of the spatial derivative of $f$. That is, $$\partial_i (\text{distTimeSlice}_c^{-1} f) = \text{distTimeSlice}_c^{-1} \left( \frac{\partial f}{\partial x_i} \right)$$ where on the left-hand side $\partial_i$ is the distributional derivative on $\text{SpaceTime } d$, and on the right-hand side $\frac{\partial}{\partial x_i}$ is the distributional derivative on $\text{Time} \times \text{Space } d$.