Elapsed Time

Let particle P be represented by some repetitive chain of events that are arranged in historical order

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2}, \sf{\Omega}_{3} \ \ldots \ \sf{\Omega}_{\it{i}} \ \ldots \ \sf{\Omega}_{\it{f}} \ \ldots \ \right)$

And let each orbital event $\sf{\Omega}$ be described by a time of occurrence $t$ so that P is also characterized by an ordered set of time coordinates. Consider a pair of events $\sf{\Omega}_{\it{i}}$ and $\sf{\Omega}_{\it{f}}$ where $\it{i} < \it{f}$. Since $\Psi$ is in historical order we call them the initial and final events of the pair. Definition: the elapsed time between these events is the difference in their time coordinates

$\Delta t \equiv t_{f} - t_{i}$

If P is isolated and the frame of reference is ideal then the mechanical energy and momentum do not change from event to event along the chain $\Psi$. For these conditions, the period $\hat{\tau}$ is constant too. But the time coordinate is defined by a sum of periods, so the elapsed time can be written as

$\Delta t = \delta _{t} ^{\, \sf{ P}} \delta _{t} ^{\, \sf{Earth}} \left(\, f-i \right) \hat{\tau}$

where $\delta _{ t}^{\, \sf{P}}$ notes P's temporal orientation and $\delta _{ t}^{\, \sf{Earth}}$ is the temporal orientation of the Earth. Since $\Psi$ is historically ordered, $\delta _{t} ^{\, \sf{ P}} = \delta _{t} ^{\, \sf{Earth}} = \pm 1$ and so

$\Delta t = \left(\, f-i \right) \hat{\tau}$

Sensory interpretation: The elapsed time during the experience of one orbital bundle of sensation $\sf{\Omega}$ is the period $\hat{\tau}$. Recall that this quantity can be understood as a fraction of a day. So all these ideas about time of occurence and elapsed time are based mostly on the reference sensation of seeing the Sun.
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