A traveler-centered intro to kinematics

P. Fraundorf

Physics & Astronomy/Center for NanoScience, U. Missouri-StL (63121)∗

(Dated: March 26, 2014)Treating time as a local variable permits robust approaches to kinematics that forego ques-

tions of extended-simultaneity, which because of their abstract nature might not be addressedexplicitly until a first relativity course and even then without considering the dependence ofclock-rates on position in a gravitational field. For example we here use synchrony-free “trav-eler kinematic” relations to construct a brief story for beginning students about: (a) time asa local quantity like position that depends on “which clock”, (b) coordinate-acceleration as anapproximation to the acceleration felt by a moving traveler, and (c) the geometric origin (hencemass-independence) of gravitational acceleration. The goal is to explicitly rule out global-timefor all from the start, so that it can be returned as a local approximation, while tantalizingstudents interested in the subject with more widely-applicable equations in range of their mathbackground.

PACS numbers: 05.70.Ce, 02.50.Wp, 75.10.Hk, 01.55.+b

CONTENTS

I introduction 1

II time as merely local 2

III kinematics teaser 3

IV sample extensions 3

A everyday problems . . . . . . . . . . . . 41 accelerated frames . . . . . . . . . . 42 gravitational time dilation . . . . . 5

B anyspeed problems . . . . . . . . . . . 51 colliders vs. accelerators . . . . . . 52 anyspeed road safety . . . . . . . . 63 single-frame time-dilation . . . . . . 74 accelerated roundtrips . . . . . . . . 75 adding 3-vector velocities . . . . . . 86 acceleration-related aging . . . . . . 9

V conclusions 9

Acknowledgments 10

A finding dt/dτ from the metric 10

B finding α from dt/dτ 10

C constant acceleration integrals 10

1 low-speed results . . . . . . . . . . . . . 102 any-speed results . . . . . . . . . . . . 11

D cool any-speed applications 12

1 proper-velocity at home . . . . . . . . . 12

E curving space-time 12

1 acceleration-related curvature . . . . . 122 gravity’s acceleration . . . . . . . . . . 13

References 13

I. INTRODUCTION

An emergent cross-disciplinary objective of Bayesianmodel-selection is the search for models that are leastsurprised by incoming data, because this goal at oncefocuses on both algorithmic simplicity and goodness offit1,2. The task of content-modernization in physics edu-cation research might, in theory, also be inspired by thisobjective. In that context this is a paper on kinematicscontent-modernization. The immediate goal is only tosuggest a minor shift in our introduction to the subject,with hopes that future experimental physics educationresearch can explore its impact on the physics experienceof future physics-majors, as well as on the much largerpopulation of future non-majors.

Because introductory physics sequences are oftentaught by multiple instructors at the same school, it isnot easy to adopt texts which involve radically differentapproaches. Hence progress in modernizing the approachhas been slow, even though the introductory series com-prises the last course contact that physics departmentshave with the lion’s share of the future consumers andvoters that they encounter. Since modern physics is in-creasingly relevant to everyday life, this is an increasinglyimportant opportunity missed.

The problem has been addressed variously in re-cent decades for example by Jonathan Reichert’s seven-chapter particle-physics introduction to unidirectionalmotion3, by Tom Moore’s Six Ideas text4 which has e.g.

2

at Washington University in St. Louis spawned a paral-lel introductory-physics sequence, and by Louis Bloom-field’s How Things Work text5 designed to look like acollection of everyday topics to the students while facultycan choose to see it as a standard-format two-semestersequence, with modern physics insights, squeezed into aone-semester course with few math requirements. Al-though these are important initiatives, the standard-format approach retains a significant market-share.

Given the standard-format approach and limited time,individual teachers and text authors may nonethelesswork to minimize the cognitive-dissonance between tra-ditional approaches and modern insights by integratingthose historical approaches into a modern context fromthe start. In statistical physics, for example, this mightinvolve starting from the statistical roots of concepts liketemperature and entropy6,7 i.e. the approximations un-derlying the ideal gas law and equipartition, as has beendone in senior undergrad texts for many decades8–11. Inquantum applications it might involve mention of “ex-plore all paths” logic12 and subsystem-correlations be-fore moving to energy-eigenvalue stationary states. Inmodern physics generally it might involve illustrations ofhow the value and application of a given concept (like po-sition in the subatomic world, or time on cosmic scales)can change significantly from one size-scale to the next13.

It is the much more modest objective of this paper tosuggest a few paragraphs to offer at the beginning of thesection on unidirectional kinematics. These paragraphswill introduce Galilean kinematics and Newtonian grav-ity as the approximations that they are, while placinga weak spot in everyone’s implicit assumption of globaltime. Thanks to synergy between relativistic insights atall levels, this opens the door to a more integrative per-spective on everyday topics like the perspective from ac-celerated frames, while offering to ambitious students arange of other types of problems with which they canplay from the vantage point of a single bookkeeper ormap frame14–16.

This paper therefore provides a “kinematics teaser” asone possible format for those opening paragraphs, fol-lowed by a survey of complementary problem-types thatmay pique student interest but which will be largely ig-nored. That’s because with or without a more modern-ized context, classes from most of today’s standardizedtexts have much business-as-usual to cover.

In particular here we argue that intro-physics studentsin an engineering-physics class might augment their in-troduction to unidirectional-kinematics with a technicalstory that’s in harmony with the general trend towardmetric-first approaches17–19. This might help nip the im-plicit Newtonian-assumption of universal time in the bud.It would also introduce: (i) momentum-proportionalproper-velocities20,21 that can be added vectorially at anyspeed22, and (ii) proper-acceleration23 in harmony withthe modern (equivalence-principle based) understandingof geometric-accelerations i.e. accelerations that arisefrom choice of a non free-float-frame coordinate-system.

In the process of putting into context theNewtonian-concepts of: reference-frame, elapsed-

time t, position x, velocity v, acceleration a,constant-acceleration-integral and gravitational-

acceleration g, this intro is designed to give students ataste of the more robust technical-concepts highlighted inbold below. If these engender critical-discussion (ratherthan a focus on intution-conflicts), all the better as suchconcepts might help inspire the empirical-scientist insidestudents even if they never take another physics course.

The few take-home equations from this introductionthat students will likely be asked to master in an introcourse will be highlighted in red. Hence you might simplyconsider these notes an alternate, but fun, intro to theGalilean kinematic relationships for tracking motion andits causes (including the assumption of global time) thatare often just presented or assumed fait accompli.

II. TIME AS MERELY LOCAL

In the first part of the 20th century it was discoveredthat time is a local variable, linked to each clock’s loca-tion through a space-time version of Pythagoras’ theo-rem i.e. the local metric equation. Both height in theearth’s gravitational field, and clock-motion, affect therate at which time passes on a given clock. Both of theseeffects must, for example, be taken into account in thealgorithms used by handheld global positioning systems.

The fact that time is local to the clock that’s measuringit means that we should probably address the questionof extended-simultaneity (i.e. when an event happenedfrom your perspective if you weren’t present at the event)only as needed, and with suitable caution. Care is espe-cially needed for events separated by “space-like” inter-vals i.e. for which ∆x > c∆t where c is the space/timeconstant sometimes called “lightspeed”.

Recognizing that traveling clocks behave differentlyalso gives us a synchrony-free21 measure of speed withminimal frame-dependence, namely proper-velocity20

~w ≡ d~x/dτ = γ~v, which lets us think of momentum asa 3-vector proportional to velocity regardless of speed.Here τ is the frame-invariant proper-time elapsed on thetraveling object’s clock, Lorentz-factor γ ≡ dt/dτ , andas usual coordinate-velocity ~v = d~x/dt.

Recognition of the height-dependence of time as a kine-matic (i.e. metric-equation) effect, moreover, allows us toexplain the fact that free-falling objects are acceleratedby gravity at the same rate regardless of mass. Hencegravity is now seen as a geometric force instead of aproper one, which is only felt from the vantage point ofnon “free-float-frame” coordinate-systems like the shell-frame normally inhabited by dwellers on planet earth.

3

FIG. 1: The synchrony-free traveler-kinematic in (1+1)D.

III. KINEMATICS TEASER

The world is full of motion, but describing it (that’swhat kinematics does) requires two perspectives: (i) theperspective of that which is moving e.g. “the trav-eler”, and (ii) the reference or bookkeeper perspectivewhich ain’t moving e.g. “the map”. Thus at bare mini-mum we imagine a map-frame, defined by a coordinate-system of yardsticks say measuring map-position x withsynchronized-clocks fixed to those yardsticks measuringmap-time t, plus a traveler carrying her own clock thatmeasures traveler (or proper) time τ . A definition ofextended-simultaneity (i.e. not local to the traveler andher environs), where needed for problems addressed bythis approach, is provided by that synchronized array ofmap-clocks.

We will assume that map-clocks on earth can be syn-chronized (ignoring the fact that time’s rate of passageincreases with altitude), but let’s initially treat traveler-time τ as a local quantity that may or may not agreewith map-time t. The space-time version of Pythago-ras’ theorem says that in flat space-time, with lightspeedconstant c, the Lorentz-factor or “speed of map-time”is γ ≡ dt/dτ =

√

1 + (dx/dτ)2/c2. This indicates thatfor many engineering problems on earth (except e.g. forGPS and relativistic-accelerator engineering) we can ig-nore clock differences, provided we imagine further thatgravity arises not from variations in time’s passage asa function of height (i.e. from kinematics) but from adynamical force that acts on every ounce of an object’sbeing. In that case we can treat time as global, and imag-ine that accelerations all look the same to observers whoare not themselves being accelerated.

Before we take this leap, however, we might spenda paragraph describing kinematics in terms of traveler-centered variables (Fig. 1) that allow one to describemotion locally regardless of speed and/or space-time cur-vature. These variables are frame-invariant proper-

time τ on traveler clocks, synchrony-free proper-

velocity w ≡ dx/dτ defined in the map-frame, andthe frame-invariant proper-acceleration α experiencedby the traveler, which for unidirectional motion inflat space-time equals (1/γ)dw/dτ where dw/dτ is thebookkeeper-acceleration in proper units linked tomomentum-changes seen in the map frame. Acceleration

FIG. 2: Galileo’s approximation to the traveler-kinematic.

from the traveler perspective where the causes of motionare felt is key, because as Galileo and Newton demon-strated in the 17th century, those causes of motion areintimately connected to this second-derivative of positionas a function of time.

The relationships above allow us to write proper-acceleration as the proper-time derivative of hyperbolicvelocity-angle or rapidity η, defined by setting c sinh[η]equal to proper-velocity w in the acceleration direction.These relationships in turn simplify at low speeds (aslong as we can also treat space-time as flat) as shownat right below, because one can then approximate theproper-values for velocity and acceleration (Fig. 2) withcoordinate-values v ≡ dx/dt and a ≡ dv/dt.

proper-vel. c sinh[η] = dxdτ

proper-accel. α = cdηdτ

→v = dx

dt coord.-vel.a = dv

dt coord.-accel.(1)

As mentioned above, treating space-time as flat requiresthat our map frame be seen as a free-float-frame (i.e. oneexperiencing no net forces). Introductory courses there-fore concentrate on drawing out uses for the kinematicequations on the right hand side above. We provide theones on the left, to show that only a bit of added compli-cation will allow one to work in curved spacetimes andaccelerated frames where geometric-accelerations aswell as force-related proper-accelerations have an impacton the bookkeeper-accelerations observed.

IV. SAMPLE EXTENSIONS

The few-paragraph introduction to the “Galilean kine-matic approximation”, outlined in the teaser above, ismeant to serve as a self-sufficient lead-in on which no classtime at all need be spent. (Of course it might not hurtto mention that time t in the text might by default re-fer to time on clocks connected to one’s map-yardsticks.)The following extensions are therefore provided only toillustrate how the distinctions made in that introductionare “relativity smart”.

The example problems might also help empower fac-ulty/students in exploration of questions not covered in

4

FIG. 3: Proper (red) and geometric (blue) accelerations onleaving a stop-sign.

standard courses but within range of available math e.g.to the extent that time and motivation outside of classis available. Experimental physics education research, ofcourse, is still needed to uncover the problems that fac-ulty and students at various levels will encounter by suchexplorations. This in turn, will be important to textauthors downstream to determine how minor changeslike this might help to constructively evolve the standardcourse.

A. everyday problems

We first look at problems that a traveler-kinematicsintroduction might help one address in the relatively low-speed activities characteristic of everyday life on earth.

1 accelerated frames

From the vantage point of any reference frame (ac-celerated or not) in any spacetime (flat and/or curved),general relativity tells us in four-vector terms24 thatmomentum & energy changes per unit mass i.e. 4-

vector bookkeeper-accelerations dUλ/dτ arise di-rectly from some combination of (i) 4-vector proper-

accelerations Aλ caused by forces acting on a travelingobject and (ii) frame-dependent 4-vector geometric-

accelerations −ΓλµνUµUν that act on every ounce of

that object’s being.From the vantage point only of unaccelerated or “in-

ertial” reference frames, Newton’s laws further tell usthat three-vector momentum changes per unit mass i.e.coordinate-accelerations d~v/dt of objects traveling atspeeds small compared to “lightspeed” are more simplycaused by the net-force

∑

F per unit mass acting on thatobject, provided that we also think of gravity as a forceitself (albeit one which acts on every ounce of our object’sbeing).

Beyond this, general relativity’s equivalenceprinciple19,25 gives us the good news that New-ton’s second law works well locally at low speedswhether or not your frame is accelerated, as long as

geometric-accelerations other than gravity may also betreated as proper-accelerations due to forces that acton every ounce of an object’s being. In other words,questions about inertial forces in an accelerated frame,e.g. of your car when you step on the gas or take a curveat constant speed, can be treated in this context.

Example Problem IV.A.1a: You are passenger ina car accelerating at 0.5 gee from rest at a stop sign. Ifyour bookkeeper-acceleration (in the accelerated frameof your car) is zero, then what backwards geometric-acceleration on every ounce of your being appears (inthat accelerated frame) to be canceling out the forwardproper-acceleration caused by the horizontal force on youfrom the back of your seat?

Example Problem IV.A.1b: You are in a car go-ing at constant speed v around a leftward curve of radiusR. In order to hold your bookkeeper-acceleration (in theaccelerated frame of your car) to zero, what rightwardgeometric-acceleration on every ounce of your being mustappear (in that accelerated frame) to cancel out the cen-tripetal proper-acceleration caused by the leftward forceof your car seat on you? Also in which direction wouldthe resulting torque appear to operate, if your center ofmass is located above your point of contact with the seat?

Example Problem IV.A.1c: You are pilot of aninverted aircraft going 200 mph at the top of a half-mileradius loop-the-loop. What net geometric-accelerationtoward the cupholder is the coffee feeling in your now-inverted cup?

In most books, of course, only gravity’s geometric-acceleration will be treated as due to a force that actson every ounce of an object. Other local effects ofgeometric-acceleration are avoided by applying Newton’sLaws only in inertial frames that are not experiencingmotion-related geometric-acceleration.

For shell frame observers experiencing gravitational ac-celeration g, we might amend Newton’s 2nd law for aframe undergoing longitudinal (tangential) and/or trans-verse (radial) acceleration ~aframe under non-relativisticconditions to read something like:

N∑

i=1

~Fi + m~g − m~aframe ≃d~v

dt(2)

Here the proper (force-related) and geometric (kine-matic) accelerations are on the left side of the equation,while the bookkeeper-acceleration (e.g. of a piece of iceon the floor of an accelerated vehicle) is on the right. Asyou can see, the effect of gravity m~g enters this equationin the same way whether it is conceptualized as the resultof a force, or the result of a geometric acceleration thatacts on every ounce of an object’s being.

Thus introducing gravity as an effect due to spacetimecurvature, that may be locally-approximated as a 1/r2

force, gives students heads-up at the outset that the dif-ferential passage of time is involved, that applied scienceis all about strategic approximation, and that Newton’sLaws may with caution prove useful even in accelerated

5

FIG. 4: Traveler-kinematics in a Schwarzschild shell-frame.

frames.

2 gravitational time dilation

The traveler kinematic generalizes nicely to curvedspacetime as well as accelerated frames, as shown in the(1+1)D Schwarzschild metric example in Fig. 4. Asin the accelerated-frame case, more care must be takenthan in the Galilean-approximation since geometric-accelerations of only local utility are likely to be presenthere as well.

In particular the stationary (i.e. zero proper-velocity)Lorentz-factor in the far-time Schwarzschild metric,namely dt/dτ = 1/

√

1 − 2GM/(c2r) ≃ 1 + (1/c2)GM/r,is directly linked to the gravitational potential-energy perunit mass (GM/r) that we need to escape the planet. Asdiscussed later, such differential-aging effects also occurfor accelerated-frames in flat space-time.

Thus when a dropped object accelerates downward itis confirmation that objects closer to the earth’s centerare aging more slowly than objects further away. Thisdifferential-aging must be taken directly into accountwhen estimating your location with a global positioningsystem, because GPS satellites operate at altitudes wherethe effect of gravitational time-dilation is significant.

Example Problem IV.A.2a: If you spend one quar-ter of your first decade standing up as a 1.8 meter talladult, and very little of that time standing on your headto cancel the effect, about how much older is your headthan your feet?

Example Problem IV.A.2b: If the officialformation-date for the earth was 4.5 billion years ago,how much older is the earth’s surface, than its center,today?

Note that these problems involve small differences be-tween relatively large numbers, so they might have to bedone with very high precision calculations. For example,clocks 1.8 meters above the earth’s surface only pick upabout 1.96 additional attoseconds per second of elapsedtime at the earth’s surface.

The moving Schwarzschild Lorentz-factor dt/dτ =Sqrt[1 + (w/c)2]/Sqrt[1 − 2GM/(c2r)], which is a prod-uct of stationary and motion-related terms, can of coursealso be used by students to examine time-dilation effects

FIG. 5: Time-dilation effects in satellite global-positioning.

associated with satellite orbits. One must of course con-sider time at two radii (e.g. on earth and in orbit) in-stead of Schwarzschild far-time t per se, in which case onefinds that low-earth orbit effects are dominated by satel-lite motion while high earth-orbit effects are dominatedby gravitational time-dilation, as shown in Fig. 5.

B. anyspeed problems

In addition to working in accelerated frames andcurved spacetime, the traveler-kinematic approach worksregardless of speed. As a result, the short introductionabove opens to the door to calculations involving veloc-ity and acceleration at super as well as at sub relativisticspeeds i.e. for proper velocities (and momenta per unitmass) well above and well below 1 lightyear per traveleryear. Some examples of this are provided here.

1 colliders vs. accelerators

Although v ≪ c, w ≪ c and K ≪ mc2 are all naturalinequalities that define the sub-relativistic regime, wherefor example δt ≃ δt, coordinate-velocity v inequalities arenot useful compared to the proper-velocity and kinetic-energy inequalities w ≫ c and K ≫ mc2 for defininga supra-relativistic regime, where for example K ≃ pc.Moreover, a proper-velocity of just one lightyear per trav-eler year i.e. w ≃ c is a natural dividing point betweenthose two limiting cases.

Relativistic-particle land-speed records also becomemore interesting in lightyears per traveler year (w in

6

FIG. 6: The proper-velocity advantage of colliders due toLorentz-factor multiplication.

[ly/ty]), than in lightyears per map year (v in [ly/y]). Forexample, a 45 GeV electron accelerated in 1989 by theLarge Electron-Positron Collider (LEP) at Cern wouldhave had a coordinate-speed v of only about sixty fourtrillionths shy of 1 [ly/y]. On the other hand, its proper-speed would have been around 88,000 [ly/ty], a numbermuch more useful for comparisons from one run to thenext.

To see this, note that proper-velocity ~w ≡ d~x/dτ =γv can be written as Lorentz-factor γ ≡ dt/dτ timescoordinate-velocity ~v ≡ d~x/dt. For unidirectional mo-tion, proper-velocities simply add by the symmetric re-lation:

wAC ≡ γACvAC = γABγBC(vAB + vBC) (3)

i.e. the coordinate-velocities add while the Lorentz-factors multiply. Reporting accelerator speeds in properunits (e.g. of 1 [ly/ty]) better connects high-speed num-bers to properties (like momentum and energy) familiarat low speeds, and it makes the numbers more impressiveas well. It also illustrates the collider advantage muchmore clearly.

Example Problem IV.B.1a: What’s the properland-speed-record for protons accelerated in the labora-tory on earth? How about for electrons, or for iron nu-clei?

Example Problem IV.B.1b: If two of the pro-tons mentioned in the previous problems were put ontoa head-on collision-course, what would be the relativeproper-speed of collision, obtained at likely less thantwice the cost of running one of those protons into astationary target?

Voters enjoying land-speed records in [ly/ty] might re-

FIG. 7: Momentum (blue), energy (red) plus driver (yellow)and pedestrian (green) reaction times as a function of proper(left) or coordinate (right) speed.

ally enjoy hearing about the collider advantage. As seein Fig. 6, accelerating two 45 GeV electrons and collid-ing them takes the proper land-speed for a single elec-tron of 88,000 [ly/ty] up to a relative collider speed ofwAC ≃ 88, 0002(1+1) ∼ 1.55×1010 [ly/ty]. This is quitea bargain over one accelerator, for something like twicethe cost.

Of course proper speed is not the speed experienced bythe couch potato watching the particle go by, but it is alsoperhaps the measure of speed which is least dependenton the existence of a set of synchronized bookkeeper ormap-clocks. Given the local nature of time, as discussedbelow, this may be a non-trivial advantage.

2 anyspeed road safety

Unlike coordinate-velocity dx/dt, proper-velocity isproportional to momentum at any speed, has no upperlimit, and only makes local assertions about the readingon one clock (the traveler clock). One might howeverwonder about its relative utility. At high speeds (or in adream-world of George Gamow’s Mr. Tompkins26 wherelightspeed was 55 miles per hour) would highway speed-limits be more usefully expressed in proper or coordinatevelocity units?

Example Problem IV.B.2a: On a car trip with 300map-miles remaining, is it proper-velocity or coordinate-velocity which determines most directly how much longeryou have to remain in the vehicle? Is it proper-velocity orcoordinate-velocity which determines most directly whattime you can be expected to arrive on map-clocks?

Example Problem IV.B.2b: What upper limiton vehicle momentum, vehicle kinetic energy, driverreaction-time for obstacle avoidance, and pedestrianreaction-time for oncoming-vehicle avoidance, is providedby a proper-speed limit of w/c in the range of positivereal numbers? How would these relationships change fora coordinate-speed limit of v/c instead?

Example Problem IV.B.2c: Your 1000 kg space-roadster accelerates from rest to 0.707 lightspeed i.e.

7

FIG. 8: Single-frame time-dilation puzzler.

0.707 [lightseconds/map-second]. The kinetic energy ac-quired on this “jump to lightspeed” is equivalent to howmany gallons of gasoline, each of which contains 100 mil-lion Joules of energy available to do work?

3 single-frame time-dilation

As long as only one map-frame is required, solvingtime-dilation problems may be much easier using proper-velocity in the traveler-kinematic, than using coordinate-velocity, since the former relates directly to the rela-tion between map-distance covered in a given amountof traveler-time:

Example Problem IV.B.3a: Muons with a lifetimeof ∆τ = 50 [micro-seconds] are created at ∆x = 100 [km]altitude, in collisions between air molecules and high-energy cosmic rays. Since these muons routinely makeit to sea level for detection before their on-board clockscause them to decay, how fast (earth distance traveledper unit earth time) must they be moving?

Example Problem IV.B.3b: The timer carried by ahigh speed airtrack glider registers only one third of theelapsed time recorded by gate clocks, on the trip betweena pair of synchronized gates fixed to the airtrack. Howfast was the glider traveling?

Example Problem IV.B.3c: The highly-accuratetimer carried by a fast-moving vehicle registers only 99percent of the elapsed-time recorded by synchronizedbank clocks that it passes. How fast was the vehicle trav-eling?

The caveat is that length-contraction always requirestwo frames of synchronized clocks, so careful treatmentof length-contraction effects should likely await the in-troduction of Lorentz transforms. We do attempt to ad-dress some questions of extended radar-time simultaneity

FIG. 9: An accelerated roundtrip puzzler.

without them,later on in this paper.

4 accelerated roundtrips

Given that acceleration is not always discussed muchin special-relativity texts30, one might imagine that equa-tions for any-speed acceleration are irrelevant to every-day life. On the contrary thanks to proper-acceleration’sframe-invariance and general relativity’s equivalence-principle, which allows Newton’s laws to work locally inaccelerated (non-free-float) frames with help from non-proper geometric (affine-connection) forces that act onevery ounce of an object’s being, proper-acceleration al-lows one to explain the difference between gravity andmost other intro-physics forces22 from the first day ofclass.

Thus the local metric equation allows one to addressacceleration problems, as well as those involving constantvelocity. These may be quite relevant, for example, tothe question of long-distance space-travel. Since humansare not adapted to accelerations above 1-gee but can takeadvantage of a 1-gee acceleration to both minimize travel-time and keep our bodies in shape, let’s focus specificallyon 1-gee constant proper-acceleration roundtrips as theideal if we can find a way to deal with the problems offuel mass and high-speed collisions.

A simple model for “one-gee” round-trips might besomeone doing jumping-jacks. Technically of course, onearth at least, these take place in a non-free-float framein which launch is accomplished with help of a proper-acceleration while the return trip is accomplished withhelp of a geometric-force that acts on every ounce of one’sbeing.

The good news for interstellar roundtrips, if you simplyput in the numbers, is that how far one can go in a givenamount of elapsed traveler-time exceeds the distance one

8

can go in a Galilean world (without a finite value forspace-time constant c) by a ridiculous amount. In otherwords, relativity opens up rather than closes down possi-bilities for interstellar travel in terms of time-elapsed ontraveler-clocks31. It’s the couch-potatoes at home thatrelativity hurts, not the travelers themselves.

For instance, a 57-year 1-gee roundtrip usingthe low-speed (non-relativistic) equations for constantcoordinate-acceleration above would at most allow oneto go about 200 lightyears and back. The same 57-yeartrip using the relativistic equations for constant proper-acceleration would take you all the way to Andromedagalaxy 2 million lightyears away and back.

Example problem IV.B.4a: How much traveler-time would elapse on a 1-gee constant proper-accelerationroundtrip to and from Sirius if it were 8.6 lightyears awayfrom Earth? How much map time would elapse on thesame trip? How much on-board fuel would be requiredat the start of each one-way leg of the trip, assuming thatthe fuel could be efficiently converted to photons directedin the desired thrust direction.

Example problem IV.B.4b: An enemy spaceshipwith its FTL-drive disabled drops out of hyperspace inthe neighborhood of a ringworld habitat, traveling at 1ly/ty radially away from the ringworld’s star. A starfleetbattle-cruiser capable of continuous 1 gee acceleration atrest nearby takes up the chase. How much cruiser timeand enemy time elapses before the cruiser can catch up tothe enemy? What are the ringworld coordinates and timefor that encounter? What is the relative proper-speed ofthe two ships when that encounter occurs?

Example problem IV.B.4c: What’s the travelertime-elapsed on a 1-gee constant proper-accelerationround-trip to Andromeda galaxy 2, 538, 000 lightyearsaway?

The bad news is that carrying on-board fuel (evenone-way) on these trips will make trips just to nearbystars difficult, and the thrust-profile for constant proper-acceleration very heavily front-loaded22. Extended timesat 1-gee acceleration of course might make collisionswith dust particles (or even hydrogen atoms) at ambi-ent speeds a non-trivial problem as well.

5 adding 3-vector velocities

The traveler-kinematic also extrapolates nicely into adescription of (3+1)D accelerated motion, as illustratedin Fig. 10. We’ve already discussed some advantagesof unidirectional proper-velocity addition, in compari-son e.g. to the addition of relative coordinate-velocitiesat high speed. We here explore the familiar “tail-to-nose” addtion of three-vector velocities, and in particularapplication of the dreaded “relative-velocity equation”~vAC = ~vAB + ~vBC .

Coordinate-velocity 3-vectors only add at speeds smallcompared to that of light. However, regardless of speed

FIG. 10: The synchrony-free traveler-kinematic in (3+1)D.

FIG. 11: Tail to nose proper-velocity addition.

local proper-velocities ~w ≡ d~x/dt do add naturally as 3-vectors, provided that we rescale the magnitude of theany “out-of-frame” vectors. As a result the familiarthree-vector relative-velocity equation from introductoryphysics continues to work for proper-velocities regardlessof speed, i.e.

~wAC = (~wAB)C + ~wBC (4)

where Lorentz-factor γ ≡ dt/dτ =√

1 + (w/c)2, withthe caveat that the magnitude-only of wAB is rescaledinto frame C using:

(~wAB)C ≡

(

~wAB · ~wBC

(1 + γAB)c2+ γBC

)

~wAB (5)

.This makes it possible to use familiar tools for address-

ing problems like the following:Example Problem IV.B.5a: A starfleet battle-

cruiser drops out of hyperspace in the orbital plane ofa ringworld, traveling at 1 [ly/ty] radially away fromthe ringworld’s star. An enemy cruiser drops out of hy-perspace nearby at the same time, traveling 1 [ly/ty] inthe rotation-direction of the ringworld’s orbit, and in adirection perpendicular to the starfleet cruiser’s radial-trajectory. What is the proper-velocity (magnitude and

9

FIG. 12: In special-relativity with radar-time simultaneity27,acceleration curves flat spacetime.

direction) of the enemy cruiser relative to the starfleetship?

6 acceleration-related aging

Although beyond the scope of an introductory physicsclass, the curvature of spacetime coordinates for an ac-celerated frame in flat-spacetime can be visualized viathat accelerated traveler’s radar-time simultaneity ρ− cτisocontours, as shown Fig. 12. This allows one to ex-amine differential-aging from the traveler’s perspective,associated with event-points on arbitrary world lines inthis space.

The map-frame ct versus x radar-time simultaneityplot in the figure at right shows how acceleration, inthis case of a 1-gee proper-acceleration round-trip lasting4 traveler-years, distorts distances (blue vertical mesh-lines) and simultaneity (blue horizontal mesh-lines) ex-perienced by that accelerated observer. For objects thatare extended along the line of their acceleration, thesedistortions in space and time will occur even across anaccelerated-objects own length.

For example, in addition to the metric-equationsmotion-related time-dilation in which δttraveler/δtmap =√

1 − (v/c)2, for accelerated objects of length Lin the direction of proper-acceleration α, one findsan acceleration-related time-dilation of the form:δtrailing/δtleading = e−αL/c2

. Here the leading-edge of theobject is in the direction of the acceleration α, not nec-essarily in the direction of travel.

Example Problem IV.B.6a: Imagine that standingon the earth’s surface at radius R is like undergoing con-stant proper-acceleration for an extended period of time.What estimate would this give for the differential agingbetween your head and your feet as a 1.8 meter tall adult,if you spent 25 percent of your life standing during thelast 10 years?

This calculation, using Dolby and Gull’s radar-timemetric, gives essentially the same differential-aging asdoes the Schwarzschild metric for this “stationary-acceleration” problem, with a simpler approximation. Infact, in this case one gets simply an increased tick rateof 109 attoseconds per second for each meter of height.This of course is only a local calculation, in comparisonto the Schwarzschild one, since the acceleration due togravity falls off with height h above the surface unlessh ≪ R.

However this sort of calculation is not very useful forspaceship problems. That’s because a ”rigid spaceship”at 1-gee acceleration will to first order force its leadingand trailing portions into a Rindler coordinate system,whose length and clock rates will remain constant al-though trailing parts of the ship will in effect experienceslightly greater proper-acceleration during acceleration-phase of the trip.

For the 1-gee proper-acceleration of a standing humanin the vertical direction, this differential-aging betweenhead and foot on the order of 1− 2× 10−16. This meansthat if you stand up (or sit tall) for a sizeable frac-tion of your lifetime, your head may be a few-hundrednanoseconds older than your feet. This is a small ef-fect for humans, but as discussed earlier its quite sig-nificant for global-positioning satellites for which nano-second timing-errors give rise to macroscopic errors inposition.

The differential-aging of accelerated objects is linkedto the gravitational time-dilation experienced by shell-frame objects in a planet’s gravitational field, because aproper-acceleration is needed to keep objects in such afield from following a rain-frame trajectory.

V. CONCLUSIONS

To recap, we suggest here a few paragraphs of introduc-tion about: (i) time as a local variable for describing mo-tion regardless of speed using a traveler-kinematic involv-ing only frame-invariant or synchrony-free variables, and(ii) the Galilean-kinematic approximation, which is thesubject of most courses because it works so well for engi-neering problems on earth. We further provide exampleproblems for applying the traveler-kinematic which aremathematically accessible to most introductory physicsstudents, should they be tempted to explore them fur-ther outside the bounds of a traditional class. Appen-dices are provided as background on the metric equa-tions, Lorentz-factor calculations, and constant accelera-tion integrals that underlie these well-known results.

10

Concerning wider applications for the traveler kine-matic, four-vectors are of course written in the traveler-kinematic i.e. in terms of derivatives with respectto proper-time. Moreover the free-particle Lagrangianin curved space-time, when parameterized in terms ofproper-time, is simply −mc2. This yields the most el-egant and comprehensive prediction of free-particle mo-tion yet: In the absence of proper-acceleration, objectsmove so as to maximize aging28,29 i.e. elapsed proper-time.

More generally of course the metric-equation strategyused above for relating proper-acceleration to bookkeeperacceleration works in any curved space-time or acceler-ated frame. Of course relation between the two involvesa covariant-derivative with connection-coefficients, whosemulti-component sums are beyond the interest of mostundergraduates.

Nonetheless, a closer look at everyday curvatures (likethe gravity around a planet) may be of interest to some.Questions to explore there, for instance, might include:(a) how satellite orbits can be predicted by choosing thepath of maximal aging, or (b) how the Schwarschild-metric yields within one expression the equations for bothshell-frame gravity and accelerated-frame centrifugal-force.

ACKNOWLEDGMENTS

I would like to thank Roger Hill, Bill Shurcliff, andEdwin Taylor for their enthusiasm about new ways tolook at old things.

APPENDIX A: FINDING dt/dτ FROM THE

METRIC

Here we outline the basis relationships using physicalunits for the benefit of folks interested in some concretecalculations.

The metric equations discussed here are the Minkowskimetric, the Schwarzschild metric, and the flat-spaceradar-time constant proper-acceleration metric. TheLorentz factor defined as dt/dτ for these is simply a rear-rangement of the metric equation itself: Divide throughby cδτ , and solve for dt/dτ by moving terms around andtaking the square root.

For the (1+1)D Minkowski (flat-space) metric

(cδτ)2 = (cδt)2 − (δx)2 (A1)

, this yields the special-relativistic Lorentz factor γ ≡dt/dτ =

√

1 + (w/c)2 = 1/√

1 − (v/c)2 in terms ofproper-velocity w ≡ dx/dτ or coordinate-velocity v ≡dx/dt.

For the (1+1)D Schwarzschild (gravitational) metricwith only radial motion, we start with:

(cδτ)2 = (1 −ro

rfar

)(cδtfar)2 −

(δrfar)2

(1 − ro

rfar

). (A2)

On earth’s surface the metric equation doesn’t differmuch from the Minkowski case since ro/rfar ≃ 1.39117×10−9, given that event-horizon radius ro = 2GM/c2 ≃8.87 [mm], where G is the universal gravitation constantand M the earth’s mass. Nonetheless, the effects of grav-ity are far from negligible!

The Lorentz-factor in this case becomes the prod-uct of a motion-related and a radius-dependent term,namely γ ≡ dtfar/dτ =

√

1 + (w/c)2/√

1 − ro/rfar,where bookkeeper coordinates tfar and rfar represent “far-coordinates” i.e. values determined using the synchro-nized clocks and yardsticks of observers in flat space faraway from our gravitational object.

Finally the (1+1)D accelerated-frame radar-time met-ric, by comparison, looks like

(cdτ)2 = e2αxα

c2 (c2δt2α − δx2

α) (A3)

. We will only be considering objects which are a fixedradar-time distance xα from our accelerated traveler (andhence in this specific sense “stationary”), the Lorentz

factor of interest is γ ≡ δtα/dτ = e−2xαα/c2

.

APPENDIX B: FINDING α FROM dt/dτ

Finding a relation between the frame-invariant magni-tude of a travelers proper-acceleration α and parameterslike velocity and position with help from the Lorentz-factor calculated from the metric is relatively easy withthe Minkowski (flat-space) metric. This is becauseproper-velocity and the Lorentz-factor make up compo-nents of the 4-vector velocity whose proper-time deriva-tive is the 4-vector proper-acceleration.

The problem is more complicated in curved space-time and in accelerated frames because of need for con-nection (i.e. geometric acceleration) terms in the co-variant derivative that describes the 4-vector proper-acceleration. Here we consider only two special cases:Low speed radial motion relative to the shell-frame ofa gravitating sphere, and travelers whose radar-distancefrom an accelerating map-frame is fixed.

APPENDIX C: CONSTANT ACCELERATION

INTEGRALS

For a change of pace from most texts, let’s discuss theassumptions needed for a computer to derive the equa-tions of constant acceleration. For equations that workat any speed, we’ll also give you some practice treatingtime as a local instead of as a global variable i.e. as avalue connected to readings on a specific clock.

1. low-speed results

If we define coordinate-acceleration a ≡ dv/dt andcoordinate-velocity v ≡ dx/dt where x and t are map-

11

position and map-time, respectively, then holding con-stant the coordinate-acceleration a (which is not theacceleration felt by our traveler at high speeds) allowsone to derive the v ≪ c low-speed constant coordinate-acceleration equations familiar from intro-physics textsfor coordinate-velocity v[t] and map-position x[t]. Canyou do it?

The following is what Mathematica needs to pull it off:

FullSimplify[DSolve[{v[t] == x’[t],a == v’[t],

x[0] == xo,v[0] == vo

},{x[t], v[t]},t]

].

Here we’ve added the intial (t = 0) boundary-conditions by defining xo as initial map-position andvo as initial coordinate-velocity to eliminate the twoconstants of integration. Mathematica’s output is:

{ {v[t] -> a t + vo,

x[t] -> (a t∧2)/2 + t vo + xo} } .

Thus the equations for constant coordinate-acceleration in one direction may be written:

v = vo + at (C1)

x = xo + vot +1

2at2 = xo +

v2 − v2o

2a, (C2)

where as usual ∆f ≡ ffinal − finitial for any time-varying quantity f . Here the first equation tells us howcoordinate-velocity v changes with elapsed map-time t,while the second tells us how map-position x changeswith map-time t as well as with state-of-motion (thework-energy equation) since work is W ≃ ma∆x andkinetic energy is K ≃ 1

2mv2.

In terms of increments instead of differentials for con-stant unidirectional acceleration, we can therefore alsowrite: a = ∆v/∆t = 1

2∆[v2]/∆x.

2. any-speed results

For equations that work at any speed, we begin bytreating the “proper” time τ on the clocks of a trav-eler as a local-variable, whose value we’d like to figureout relative to the local value of the traveler’s positionx and time t on the yardsticks and synchronized clocksof a reference map-frame. The space-time Pythagorean

theorem or “metric-equation” for flat space-time, namely(cδτ)2 = (cδt)2 − (δx2) with “lightspeed” constant c, al-lows us to define minimally frame-variant “proper” valuesfor the velocity and acceleration, as well as for the time,experienced by our traveler.

In particular proper-velocity w (map-distance x perunit proper-time τ) is just w ≡ dx/dτ ≡ c sinh[η],where η is referred to as hyperbolic velocity-angle orrapidity. The frame-invariant proper-acceleration αfelt by a traveler equals this “length-contracted” proper-velocity derivative (1/γ)d2x/dτ2 = cdη/dt i.e. constant ctimes the traveler-time τ derivative of rapidity η. Hold-ing α fixed thus allows one to derive constant proper-acceleration equations that work at any speed for map-position x[τ ] and proper-velocity w[τ ].

In the above discussion we are using proper-time τ lo-

cal to the traveler’s clocks as the independent variable,so as to avoid thinking of time as a global variable.Note that unlike coordinate-velocity v ≡ dx/dt, proper-velocity w ≡ dx/dτ always equals momentum per unitmass and has no upper limit. Can you figure how map-position x and proper-velocity w = c sinh[η] depend ontraveler-time τ , given this information?

The following is what Mathematica needs to pull it off:

FullSimplify[DSolve[{c Sinh[eta[tau]] == x’[tau],

alpha == c eta’[tau],x[0] == xo,

eta[0] == etao},{x[tau], eta[tau]},{tau}]].

As before we specify two initial (τ = 0) conditions, inthis case for initial map-position xo and initial rapidityor hyperbolic velocity-angle ηo. The result is:

{ {eta[tau] -> etao + (alpha tau)/c,

x[tau] -> (alpha xo + c∧2 (-Cosh[etao] +Cosh[etao + (alpha tau)/c]))/alpha

} } .

Note that we also get these bonus relationships:Traveler-speed on the map can be expressed in sev-eral ways, including: Lorentz-factor γ ≡ dt/dτ =

cosh[η] =√

1 + (w/c)2 = 1/√

1 − (v/c)2, wherecoordinate-velocity v ≡ w/γ = c tanh[η]. For incre-mental changes when proper-acceleration is constant andall motion is along that direction we can also writeα = ∆w/∆t = c∆η/∆τ = c2∆γ/∆x = γ3a.

All of the foregoing assertions are local to the trav-

eler’s position in the map-frame of yardsticks andsynchronized clocks. If we use those synchronized clocksto define simulaneity between separated events, the above

12

also tells us about traveler motion from the per-

spective of stationary observers anywhere on the

map. Hence these equations are spectacular for explor-ing constant-acceleration round-trips between stars.

Thus the equations, analogous to the Newtonian ones,for unidirectional constant proper-acceleration at anyspeed might be written:

w = c sinh[ατ

c+ ηo

]

= wo + α

∫ τ

0

γ[τ ′]dτ ′ = wo + α∆t,

(C3)

x = xo +c2

α

(

cosh[ατ

c+ ηo

]

− cosh [ηo])

= xo +c2

α∆γ,

(C4)where again ∆f ≡ ffinal − finitial, and ηo ≡ sinh−1[wo/c].Here the first equation tells us how proper-velocity wchanges with elapsed traveler-time τ and map-time ∆t,while the second tells us how map-position x changeswith traveler-time τ as well as with state-of-motion (thework-energy equation) since work is mα∆x and change-in kinetic energy is ∆K = mc2∆γ given that K =(γ−1)mc2. These results generalize nicely to the (3+1)Dcase (Fig. 10) and to curved spacetime (Fig. 4), althoughmore care must be taken than in the Galilean approxi-mation since 3-vector magnitudes show more dependenceon observer frame.

At low speeds (w ≪ c) of course, map and travelerclock times go at the same rate i.e. dt ≃ dτ , the velocity-parameters (coordinate, proper and angle) are essentiallythe same i.e. v ≃ w ≃ cη, coordinate and proper accel-eration are about equal i.e. a ≃ α, and the any-speedequations reduce to the low-speed ones discussed above.

APPENDIX D: COOL ANY-SPEED

APPLICATIONS

Invariant proper-time τ is already finding its way intointro-physics and special-relativity books, e.g. to recog-nize that the frame a clock resides in is special when thetopic of time elapsed on that clock comes up. Proper-velocity w and proper-acceleration α for a traveler areless consistently mentioned, but also have uses that maybe of interest to introductory physics teachers. In thissection we discuss a few of the possibilities.

1. proper-velocity at home

In the multi-directional case, moreover, a 3-vectorequation very similar to the low-speed velocity equationcan be written, namely ~wAC = (~wAB)C + ~wBC . Theonly complication is that frame C’s view of the out-of-frame proper-velocity (~wAB)C is in the direction of~wAB ≡ γAB~vAB but must be rescaled in magnitude22

before the addition will work.

Another fun, but less practical topic, is that of rela-tivistic traffic safety which includes games in Mr. Tomp-kins style universes26 where e.g. lightspeed is 55 mph.In such a universe, would interstate highways still needspeed-limit signs? The answer is yes, and it would more-over be a proper and not a coordinate velocity limit.

To see this, simply consider (one at a time)which velocity-measure best reflects maximum possi-ble collision-damage in terms of vehicle (i) momentumand (ii) kinetic energy, and which measure best re-flects minimum chance for collision-avoidance in termsof (iii) driver and (iv) pedestrian reaction-time. At allspeeds, both vehicle momentum and kinetic energy scalenicely with proper-velocity while their dependence oncoordinate-velocity goes through the roof as v → c. Sim-ilarly driver reaction-time decreases, as does pedestrianreaction-time after the warning photon arrives, in a com-plementary way with increasing proper-velocity but notwith coordinate-velocity22.

Thus in our c = 55 mph universe, limiting travelers tospeeds of less than 55 map-miles per traveler-hour makesmore sense than limiting them to less than 55 or even54.991 map-miles per map-hour. Not only would rais-ing the limit to 60 mph remain a viable option, but as anadded bonus the speedometer-reading for proper-velocitydivided into destination distance directly answers thequestion that kids in the back seat are asking i.e. “Howlong (to me) before we get there?”

APPENDIX E: CURVING SPACE-TIME

Our analysis in the first section, of time as local to theclocks used to measure it, was in part to distance our-selves as much as possible from a discussion of extendedsimultaneity. In this section, for dealing with acceler-ated travelers we choose a radar-time model for extendedsimultaneity27 (instead of the tangent free-float-framemodel) in order to show students how proper-accelerationcurves space-time for the traveler all by itself. This modelin hand, the door my open a bit wider to experimen-tation by interested students with simple gravitationalmetrics in the spirit of Taylor and Wheeler’s “Explor-ing Black Holes” text18, whose pre-publication draft-titlewas “Scouting Black Holes: Exploring General Relativitywith Calculus” likely in part to inspire a closer look byintro-physics students.

In this note, we don’t have the opportunity to de-velop the equations to treat curved space in detail. In-stead, therefore, we focus on visualizations and on a fewbottom-line relationships that might pique a student’sinterest.

1. acceleration-related curvature

The map-frame ct versus x radar-time simultaneityplot in Fig. 12 shows how acceleration, in this case of

13

a 1-gee proper-acceleration round-trip lasting 4 traveler-years, distorts distances (blue vertical mesh-lines) andsimultaneity (blue horizontal mesh-lines) experienced bythat accelerated observer. For objects that are extendedalong the line of their acceleration, these distortions inspace and time will occur even across an accelerated-object’s own length.

For example, in addition to the metric-equation’smotion-related time-dilation in which:

δτtraveler = δtmap

√

1 −(v

c

)2

, (E1)

for accelerated objects of length L in the direction ofproper-acceleration α, one finds an acceleration-relatedtime-dilation of the form:

δτtrailing ≃ δτleading

√

1 −2αL

c2. (E2)

Here the leading-edge of the object is in the directionof the acceleration α, not necessarily in the direction oftravel.

For the 1-gee proper-acceleration of a standing humanin the vertical direction, this differential-aging betweenhead and foot becomes dtfoot/dthead ≃ 1 − 2 × 10−16.This means that if you stand up (or sit tall) for a size-able fraction of your lifetime, your head may be a

few-hundred nanoseconds older than your feet.This is a small effect for humans, but as discussed below(and illustrated in Fig. 5) it’s quite significant for global-positioning satellites for which nano-second timing-errorsgive rise to macroscopic errors in position.

2. gravity’s acceleration

Einstein’s general-relativity shows how a gravitational-acceleration that is the same for all masses can be seento result from a mass-related static-distortion of space-time. This can be described most simply with a modifiedmetric-equation of the form:

(cδτradius-r)2 = (1 −

ro

r)(cδtfar)

2 −(δxfar)

2

(1 − ro

r ). (E3)

On earth’s surface the metric equation doesn’t changeby much since ro/r ≃ 1.39117× 10−9, given that event-

horizon radius ro = 2GM/c2, where G is the universalgravitation constant and M the earth’s mass.

Nonetheless, this modified-metric gives rise to grav-

ity’s geometric-acceleration of GM/r2 that at earth’ssurface becomes g ≃ 9.8 meters per second-squared,which must be countered by an upward proper-force ofmg (as shown later in this course) to keep a shell-frame

observer’s radius fixed in the neighborhood of a planet.That’s because shell-frames (of fixed radius) are not free-float-frames. Around gravitational objects, free-float-frames are sometimes called rain-frames instead.

The space-time curvature associated with gravity’sgeometric-acceleration also distorts space and time. Oneresult of this is the gravitational time-dilation of global-positioning-system (GPS) satellites, as well as of yourhead, relative to your boots on the ground.

As with the previous two expressions for differential-aging, this dilation is also linked to an expression involv-ing

√

1 − 2energy/mc2, namely:

δτradius-r ≃ δtfar

√

1 −2GM

rc2, (E4)

where potential-energy per unit mass at radius r (alsoto be shown later in the course) is GM/r. This furthermeans that if clocks at the earth’s center and surfacebegan ticking together on the day when earth’s formationfrom the solar-nebula was complete, since then time-

elapsed at earth’s center is about a year less than

on the surface. Such differential-aging effects are evenmore severe with extremely dense objects, like neutron

stars and the event-horizons of black holes.

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