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A simple model for “one-gee” round-trips might be someon
Thus the local metric-equation allows one to address acceleration problems, as well as those involving constant velocity. These may be quite relevant, for example, to the question of long-distance space-travel. Since humans are not adapted to accelerations above 1-gee but can take advantage of a 1-gee acceleration to both minimize travel time and keep our bodies in shape, let’s focus specifically on 1-gee constant proper-acceleration roundtrips as the ideal if we can find a way to deal with the problems of fuel mass and high-speed collisions.
Given that acceleration is not always discussed much in special-relativity texts (cf. French 1968), one might imagine that equations for any-speed acceleration are irrelevant to everyday life. On the contrary thanks to proper-acceleration’s frame-invariance and general relativity’s equivalence principle, which allows Newton’s laws to work locally in accelerated (non-free-float) frames with help from non-proper geometric (affine-connection) forces that act on every ounce of an object’s being, proper-acceleration allows one to explain the difference between gravity and most other intro-physics forces from the first day of class.
e doing jumping-jacks. Technically of course, on earth at least, these take place in a non-free-float frame in which launch is accomplished with help of a proper acceleration (floor against feet) while the return trip is accomplished with help of a geometric-force (gravity) that acts on every ounce of one’s being. The good news for interstellar roundtrips, if you simply put in the numbers, is that how far one can go in a given amount of elapsed traveler-time exceeds the distance one can go in a Galilean world (without a finite value for space-time constant c) by a ridiculous amount. In other words, relativity opens up rather than closes down possibilities for interstellar travel in terms of time-elapsed on traveler-clocks (Lagoute 1995). It’s the couch-potatoes at home that relativity hurts, not the travelers themselves.
For instance, a 57-year 1-gee roundtrip using the low-speed (non-relativistic) equations for constant coordinate-acceleration above would at most allow one to go about 200 lightyears and back. The same 57-year trip using the relativistic equations for constant proper-acceleration would take you all the way to Andromeda galaxy 2 million lightyears away and back. A Wolfram computable document format calculator for plotting proper-time and map-time elapsed, and for listing minimum one-way launchmass-to-payload ratios, during constant proper-acceleration roundtrips to and from stars within 100 lightyears of earth is illustrated in the bottom figure at right.
Example problem 4.2.4a: How much traveler-time would elapse on a 1-gee constant proper-acceleration roundtrip to and from Sirius if it were 8.6 lightyears away from Earth? How much map time would elapse on the same trip? How much on-board fuel would be required at the start of each one-way leg of the trip, assuming that the fuel could be efficiently converted to photons directed in the desired thrust direction.
Example problem 4.2.4b: An enemy spaceship with its FTL-drive disabled drops out of hyperspace in the neighborhood of a ringworld habitat, traveling at 1 ly/ty radially away from the ringworld's star. A starfleet battle-cruiser capable of continuous 1 gee acceleration at rest nearby takes up the chase. How much cruiser time and enemy time elapses before the cruiser can catch up to the enemy? What are the ringworld coordinates and time for that encounter? What is the relative proper-speed of the two ships when that encounter occurs?
Example problem 4.2.4c: What's the traveler time-elapsed on a 1-gee constant proper-acceleration round-trip to Andromeda galaxy 2,538,000 lightyears away?
The bad news is that carrying on-board fuel (even one-way) on these trips will make trips just to nearby stars difficult, and the thrust-profile for constant proper-acceleration very heavily front-loaded. Extended times at 1-gee acceleration of course might make collisions with dust particles (or even hydrogen atoms) at ambient speeds a non-trivial problem as well.
Example problem 4.2.4d (with sample solution): Consider an interstellar spaceship which undergoes constant proper acceleration of 1 "gee" (≈ 1[ly/y2]) from rest over a distance of 2[ly] i.e. halfway to Proxima Centauri from Sol. What's the proper-velocity and γ = E/mc^2 at that point? How long on ship clocks (e.g. in years) did this take? How long on map clocks?
General equations for (1+1)D constant proper acceleration might be summarized as:
Using these, a sample solution might take the form:
Part 2 of the problem is more challenging: At this halfway point to proxima centauri, our shuttle gets the message that a space emergency requires flyby of a point 0.2 lightyears away (using galactic map coordinates), with thruster pointed in a direction (to our traveler) perpendicular to the line between earth and centauri. Instead of rotating the thruster by 180 degrees to begin the deceleration, they then rotate it by the required 90 degrees to make the detour. How long will it take to get there? Refocusing on their original destination will of course be a natural (and more challenging) followup to this question.
The general equations for (3+1) constant proper acceleration needed now might be written as:
In this case, a sample solution might take the form:
Related references
Anthony P. French (1968) Special relativity (W. W. Norton, NY) page 154: "...acceleration is a quantity of limited and questionable value in special relativity."
C. Lagoute and E. Davoust (1995) "The intersellar traveler", Am. J. Phys. 63, 221.
P. Fraundorf (2016/2017) Traveler-point dynamics, hal-01503971 working draft.
P. Fraundorf (2016) "The proper-force 3-vector", HAL-01344268 current pdf.
21 Jan 2015 update: A conceptual (no equations) paper on this topic may be found in draft form here. Suggestions to make it better, e.g. up on our traveler kinematic google-group, are invited. /pf
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