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*Caution*: This
single-frame approach
provides students with
quantitative tools for dealing with time dilation, relativistic acceleration,
and even curved space-time. However, only a multi-frame approach can
fully protect students against everyday-language confusion regarding other
practical matters manifest at high speeds, like frame-dependent
simultaneity and length contraction. For students with only single-frame
perspectives on space-time, one bit of advice therefore is to *stick to
only distances measured using map-frame yardsticks*.

Begin with a "map frame", across which is distributed a
set of co-moving yardsticks and synchronized clocks. Simultaneity throughout
this page will be defined in the context of *this single frame of motion*.
Nonetheless, clocks carried by observers moving with respect to the map frame
(i.e. travelers) will keep their own time. Metric
equations are taylor-made* for this type
of **asymmetric** situation...

*Minkowski's space-time version of Pythagoras' Theorem* (the flat-space metric equation) for frame-invariant
proper (here "traveler") time is:

or`(`

*c**Traveler_Time_Elapsed)^2 =
(*c**Map_Time_Elapsed)^2 - Map_Distance_Traveled^2

*Useful velocity or rate measures* that arise because time's passage is
frame-variant (i.e. everyone's clocks tell a different
story) include...

or ```
Proper_Velocity =
Distance_Traveled/Traveler_Time_Elapsed
```

```
Coordinate_Velocity =
Distance_Traveled/Map_Time_Elapsed
```

```
Speed_of_Map_Time =
Map_Time_Elapsed/Traveler_Time_Elapsed
```

```
Rapidity =
ArcSinh[Proper_Velocity/
```

*c*] = ArcTanh[Coordinate_Velocity/*c*] =
(+/-)ArcCosh[Speed_of_Map_Time]

From the metric equation, one can obtain the last two equivalences above, as
well as the *Velocity Conversions* below:

or ```
Speed_of_Map_Time =
1/Sqrt[1-(Coordinate_Velocity/
```

*c*)^2] =
Sqrt[1+(Proper_Velocity/*c*)^2] = Cosh[Rapidity]

Concerning rates of velocity change, the metric equation
(along with the metric relation between
second proper-time derivatives of map-coordinates)
allows us to define frame-invariant proper (felt)
acceleration, coordinate (frame-variant)
acceleration *a*, and the relation between them as well:

or

```
Proper_Acceleration =
Speed_of_Map_Time^3*Coordinate_Acceleration
```

Lastly, the metric equation yields the *Motion Integrals* of
Constant Proper Acceleration:

or ```
Proper_Acceleration =
(Final_Proper_Velocity - Initial_Proper_Velocity)/Map_Time_Elapsed
```

`Proper_Acceleration = `

*c**(Final_Rapidity -
Initial_Rapidity)/Traveler_Time_Elapsed```
Proper_Acceleration =
```

*c*^2*(Final_Speed_of_Map_Time -
Initial_Speed_of_Map_Time)/Map_Distance_Traveled

These integrals can be combined to predict changes in any of these variables, in terms of changes in another. For example, a displacement-time equation for constant proper acceleration may be obtained by combining the Rapidity and Speed_of_Map_Time Integrals above:

On the right of the two equations above, you'll note that
these integrals reduce to the simpler Galilean integrals of constant
acceleration, when Coordinate_Velocity is much less than Light_Speed *c*.

Concerning insight into the causes of motion, experiment further
provides evidence for the below *dynamically conserved
quantities* which likewise depend on one's frame of reference:

or ```
Momentum =
Mass*Proper_Velocity; Mass_Energy =
Mass*
```

*c*^2*Speed_of_Map_Time

* no pun intended, even though the metric equation plays a central role in much of Edwin Taylor's recent work.

Reconcile yourself to simultaneity as manifest
to map-frame observers, specify alternate values for the three *green* variables below, and then click:

Equations for each of these quantities, in a given segment of the trip, are quite easy to obtain from the integrals of constant Proper_Acceleration given above. Thus with the right travel arrangements, you can find out how your younger sibling behaves when they find themselves senior to you by ten years! There is risk, of course, because if you don't like the way they behave, it may not be possible to go back to being the older of the two, unless you can persuade them to take a similar trip.

This page is http://www.umsl.edu/~fraundor/a1djs.html. Although there are many contributors, the person responsible for errors is P. Fraundorf. This site is hosted by the Department of Physics and Astronomy (and Center for Molecular Electronics) at UM-StL. For more on