Anyspeed kinematics from the metric equation

UM-StL Physics & Astronomy, St. Louis MO 63121, Copyright (1999) P. Fraundorf

For those of you interested, assume that (by experiment or the grapevine) you've gotten wind of the flat-space metric equation (Minkowski's space-time version of Pythagoras' theorem) that relates traveler time τ, to map time t and map distance x on a reference coordinate system (the "map") with both standardized yardsticks and synchronized clocks, via:

(c dτ)2 = (c dt)2 - (dx)2, where c is the speed of light.

This equation describes a quantity dτ that has proven by experiment to be a frame-invariant measure of distance between events (i.e. the same to all observers) along a path or "world line" in space-time. We write it in terms of infinitesimal increments, so that it can be applied to world lines (e.g. traveler trajectories) of arbitrary shape.

Two velocities and a 3rd speed are suggested by its three variables. To be specific, call v = dx/dt the "coordinate velocity" (e.g. in lightyears per map-year"), γ = dt/dτ the "speed of map time" (e.g. in map-years per traveler year), and w = dx/dτ= γv the "proper velocity" (e.g. in lightyears per traveler year). Then see if you can answer the question, how can one express γ and w in terms of v? Hint: A bit of algebra shows that γ is near one when v < c (lightspeed), and that w goes to infinity as v approaches c. Hence making v approach c makes map clocks tick infinitely faster than traveler clocks. No wonder it is difficult to do! Also note that γ is greater than one if the traveler is moving. This means that the time on traveler clocks appears dilated (spread out over a longer period) to map-frame observers.

By the way, these quantities γ and w are very important in the physics of motion at any speed because of the conservation laws for energy and momentum, and the fact that map-time speed γ equals total energy E divided by mc2, and proper velocity w equals momentum p per unit mass m.

The next question: Define a = dv/dt as the "coordinate acceleration", α = dw/dt as the "proper acceleration in one spatial dimension", and ask how we can express α in terms of a and one of the speeds (e.g. γ for example). If α is constant, what happens to a at high speeds? Hint: The proportionality between these two accelerations is γ to some integer power...

Finally, and this is a bit more challenging, imagine planning a business that would take customers on a space cruise involving constant "1-gee" proper acceleration, bringing them back 100 years in the future (e.g. think about what their investments might be worth)! How long by their clocks would this cruise last, and how far would it take them? Hint: To get at this, you may need to determine from the equations above both the proper time, and the map-distance, integrals of constant proper acceleration α. These are analogous to the familiar constant coordinate acceleration integrals that are so useful at low speeds: a = dv/dt = d(v2)/(2dx). In fact, when motion is uni-directional the constant proper acceleration integral for map-time at any speed is almost the same: it's just α= dw/dt.

Let me know what you find! Cheers. /pf :)

Postscript: Note that we avoided discussion of traveler distances above. That is because such discussion would involve multiple frames, which are messy although occasionally quite useful to consider. Hence if you are a glutton for such punishment, assume that in describing two events in spacetime (separated by dt and dx) that the same events in a moving frame have linearly related separation, i.e.

dt' = A dt + B dx, dx' = C dt + D dx.

This linearity is needed to make sure an object moving at constant speed in one frame is also moving at constant speed in the other.

Using the metric equation above and three special cases (e.g. dx = c dt, dx = - c dt, and dx' = 0 -> dx = vreldt where vrel is the velocity of the primed frame with respect to the unprimed one), you can show that A = D = γrel, B = -wrel /c2, and C = - wrel. This is the Lorentz transform, which among other things shows that a ruler at rest in the primed frame of length dx' presents to unprimed frame observers a contracted instantaneous length (i.e. dx = dx'/γrel when dt=0). This transform also yields a velocity subtraction rule: wrel = γ γ' (v - v'). Change the minus to a plus, and it tells us how relative velocities add.

Note therefore that to calculate the relative proper velocity between frames, one should subtract or add coordinate velocities (v's) but multiply map-time speeds (γ's)! This has very practical consequences. For example, we can accelerate electrons on earth only to about 70GeV or γ = E/mc2 = 70GeV/511keV = 140000. However, if we run two such high speed particles into each other, wrel= g2(v+v) = (140000)2(2c) so that wrel = 4x1010 c. Thus colliders let us go from w = γ v = 140000c to wrel= 4x1010c. This is a 280,000-fold improvement, for 2 times the cost! The record for relative velocities is thus well over 1010 [lightyears per traveler year], much better than the 140,000 [lightyear per traveler year] land-speed record for particles with respect to the earth itself.

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