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Traveler-point dynamics: A 3-vector bridge to curved Phil Fraundorf

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Phil Fraundorf. Traveler-point dynamics: A 3-vector bridge to curved spacetimes. 2021. ￿hal- 01503971v3￿

HAL Id: hal-01503971 https://hal.archives-ouvertes.fr/hal-01503971v3 Preprint submitted on 24 May 2021

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Traveler-point dynamics: A 3-vector bridge to curved spacetimes

P. Fraundorf∗ Physics & Astronomy/Center for Nanoscience, U. Missouri-StL (63121), St. Louis, MO, USA† (Dated: May 24, 2021) Locally-defined parameters chosen to be minimally frame-variant can be useful for describing mo- tion in accelerated frames and in curved spacetimes. In particular the metric-equation’s synchrony- free “traveler-point parameters”, namely scalar proper-time and the 3-vectors: proper- and proper-force, are useful in curved because extended arrays of synchronized clocks (e.g. for local measurement of the denominator in ∆x/∆t) may be hard to find. Combined with a recognition of improper (geometric) forces, these same parameters can better prepare intro-physics students for a world where Newton’s approach is a only useful approximation. This strategy also suggests an “accelerated-traveler perspective” on high speed travels that are exact at any speed in flat spacetime, and approximate in curved settings e.g. for computing gravitational trajectories at high speed.

CONTENTS proper velocity4, and 3-vector proper acceleration5 are inherently local to one traveler’s perspective, hence we I. introduction 1 refer to them as ”traveler-point” quantities. The advan- tage is that they are either frame-invariant (in the case II. 1 of proper time and the magnitude of proper-) or synchrony-free6 (i.e. do not require an extended net- III. 2 work of synchronized clocks which of course is not always available in curved space time and accelerated frames). IV.properacceleration 3 In a larger context, general-relativity revealed a cen- tury ago why we can get by with using Newton’s laws V.causesofmotion 4 in spacetime so curvy that it’s tough to jump higher than a meter. It’s because those laws work locally in all VI. discussion 6 frames of reference7,8, provided we recognize that motion is generally affected by both proper and “geometric” (i.e. Acknowledgments 6 -invisible connection-coefficient) forces. In that context, it’s probably time to give introductory References 6 students the good news. Here we discuss a way to do this without telling them to measure time and distance in e.g. A.possiblecoursenotes 7 kilograms, and without asking them to juggle more than 1. spacetime version of Pythagoras’ theorem 7 one concurrent defintion of simultaneity. 2. traveler-point 7 The metric equation of course avoids these things (cf. 3. dynamics in flat spacetime 8 Fig. 1) by specifying locally-defined frame invariants like 4. dynamics in curved spacetime 8 proper-time, and contenting itself with a single defini- tion of simultaneity i.e. that associated with the set of book-keeper (or map) coordinates that one chooses to de- I. INTRODUCTION scribe the spacetime metric. Hence the focus here is on a metric-first9, as distinct from transform-first10, strategy In an excellent 1968 article1, Robert W. Brehme dis- for describing motion in accelerated frames, in curved cussed the advantage of teaching relativity with four- spacetime, as well as at high speeds. vectors. This is not done for intro-physics courses. One reason may be that spacetime’s 4-vector symmetry is bro- ken locally into (3+1)D, so that e.g. engineers working II. PROPER TIME in curved spacetimes2 (including that here on earth) will naturally want to describe time-intervals in seconds and Proper-time is the time elapsed locally along the world- space-intervals e.g. in meters. In this note, we therefore line of any physical clock (broadly defined). It is also the explore in depth an idea about use of minimally frame- frame-invariant time-interval δτ that shows up in metric 2 variant scalars and 3-vectors, perhaps first hinted at in equations of the form (cδτ) = ΣµΣν gµν δXµδXν , which Percy W. Bridgman’s 1928 ruminations3 about the wis- in the flat-space Cartesian case takes the Pythagorean dom of defining 3-vector velocity using an ”extended” form (cδτ)2 = (cδt)2 − (δx)2 − (δy)2 − (δz)2, where book- time system rather than defining 3-vector velocity using keeper or map coordinates are found on the right-hand traveler proper time instead. side of the equation, and c is the “lightspeed” spacetime The problem with the latter, of course, is that mini- constant for converting e.g. seconds into meters. mally frame-variant quantities like proper time, 3-vector When simultaneity is defined by a flat-spacetime book- 2

FIG. 2. Proper-velocity addition for a sci-fi puzzler, in which FIG. 1. Even one extended-frame with synchronized clocks your starfleet battle-cruiser drops in from hyperspace heading may be more than we have in curved spacetime. away from a nearby , only to find the enemy ship heading off in another direction. keeper coordinate frame, time-elapsed on traveling clocks is always “dilated” (e.g. spread out), since from the ordinate velocity ~v ≡ d~x/dt it has no upper limit. Also metric equation the differential-aging or unlike coordinate velocity, its measurement does not de- γ ≡ dt/dτ ≥ 1. Expressions for this in terms of velocity pend on map-clock readings (synchronized or not) along will be discussed below. that . Proper-velocity is also represented by We won’t discuss the utility of proper-time τ in detail the space-like components of a traveling object’s velocity here because it is widely used now even by introductory 4-vector. textbook authors11. However it’s worth pointing out that An interesting thought problem for future engineers early treatments of were so focused on (as well as teachers) might be to ask if speed-limit signs, the equivalence between frames, each with their own ex- in a “Mr. Tompkins” world14 with a much slower light- tended set of synchronized clocks, that proper-time (if speed spacetime constant, would use coordinate-speed or discussed12) was generally a late-comer to the discussion, proper-speed. Four criteria to consider might involve the and of course used only for “rest frame” (i.e. unacceler- measure’s connection to , kinetic-energy, and ated) travelers10. to reaction-times (after the “warning ” arrive) for Moving yardsticks also have a “proper-length” associ- both the driver, and a pedestrian who is tempted to cross ated with them, which can be useful too. However this the street. is a “non-local” quantity not defined by the metric equa- The time-like component of that 4-vector is cγ, where tion, the concept of “rigid body” is only an approxima- γ ≡ dt/dτ is the also-useful (but not synchrony free) tion in spacetime, and itself involves Lorentz differential-aging or “speed of map-time” fac- three separate events as distinct from the two involved in tor mentioned above. In flat spacetime, from the met- time-dilation. Hence we don’t include proper-length in a ric equation it is easy to show that γ = w/v = collection of those traveler-point parameters which show p1 + (w/c)2 ≥ 1. promise of wide-ranging usefulness in accelerated frames In gravitationally-curved Schwarzschild spacetime and curved spacetime. with “far-coordinate” map distances and times, one gets instead from the metric for non-radial proper-velocity wφ 2 that γ ≡ dt/dτ = γrp1 + (wφ/c) ≥ 1, where the grav- III. PROPER VELOCITY itational time-dilation factor is γr ≡ 1/p1 − (rs/r) ≥ 1 and r ≥ rs is distance from the center of a M 13 Proper-velocity ~w ≡ d~x/dτ is the rate at which book- object whose Schwarzschild (event-horizon) radius rs = keeper or map 3-vector position ~x is changing per unit- 2GM/c2. Global positioning system satellites have to time elapsed locally on traveler clocks. Because it is consider both of these factors in compensating for the proportional to momentum (i.e. ~p = m ~w), unlike co- rate of time’s passage in their orbits around earth. 3

For unidirectional velocity addition, the advantage of a collider over an accelerator is easily seen because wAC = γABγBC (vAB +vBC ). In other words, the Lorentz-factors multiply even though the coordinate-velocites (which add) never exceed c. Proper velocity’s direct connection to momentum makes it more relevant to speed-limits e.g. in worlds with a reduced value for lightspeed, and its di- rect connection to traveler time makes it more relevant to passengers and crew when planning long high-speed trips. More generally, unlike coordinate-, proper ve- locities add vectorially provided that one rescales (in magnitude only) the “out of frame” component. In other words ~wAC = γAC~vAC = ( ~wAB )C + ~wBC . Here C’s view of the out-of-frame proper-velocity ( ~wAB)C is in the same FIG. 3. Accelerometer data from a phone dropped and caught direction as ~wAB but rescaled in magnitude by a factor three times. During the free-fall segments, the accelerometer 2 reading drops to zero because is a geometric force of (γBC + (γAB − 1) ~wBC · ~wAB/wAB) ≥ 0. Hence vector diagrams, and even the original low-speed equation for (like centrifugal) which acts on every ounce of the phone’s velocity-addition (almost), survive intact when proper- structure, and is hence not detected. The positive spikes occur velocity is used. when the fall is arrested as the falling phone is caught (also by hand) before it hit the floor. This would open the door to new settings for those rela- tive motion problems (that students dislike so much). For example, suppose that a starfleet battle-cruiser drops out proper-, like that due to the upward force of hyperspace in the orbital plane of a ringworld, travel- of your hand when holding the phone in place before the ing at a proper-velocity of 1[ lightyear/traveler-year] ra- drop, nothing is detected when the phone is released even dially away from the ringworld’s star. An enemy cruiser through the net acceleration 4-vector in that case has a drops out of hyperspace nearby at the same time, trav- vertically downward component of g. eling 1[ly/ty] in the rotation-direction of the ringworld’s Thus in curved spacetimes and accelerated frames, un- orbit, and in a direction perpendicular to the starfleet derstanding motion requires a recognition of geometric cruiser’s radial-trajectory. What is the proper-velocity (connection-coefficient) effects which thankfully (follow- (magnitude in [ly/ty] and direction) of the enemy cruiser ing Newton) can often be approximated locally as gravi- relative to the starfleet ship? The vector solution to this tational and inertial forces. The bottom line is that in flat is illustrated in the Fig. 2. spacetime motion is explained by the forces that cause , but as soon as we move to curved spacetimes and accelerated frames (even here on earth) IV. PROPER ACCELERATION a second kind of cause comes into play. As we’ll see, these geometric effects act on every ounce of their target, The felt or proper-acceleration 3-vector ~α is defined vanish locally from the perspective of a comoving free- as the acceleration felt by (broadly de- float frame, and are therefore undetected by on-board fined) that are moving with an object15 relative to the accelerometers. tangent free-float (or as a generalization of Returning to flat spacetime for the moment, let’s also “inertial”) frame. In flat spacetime its magnitude is consider the equations of constant proper-acceleration the frame-invariant magnitude of the net-acceleration 4- (cf. Fig. 4). In flat spacetime, it is proper and not co- λ 2 λ λ δU δ X ordinate acceleration that can be (and normally is) held vector A ≡ δτ ≡ δτ 2 . This four-vector has a null time-component in the frame of the accelerated traveler, constant. from whose “traveler-point” perspective its 3-vector di- If we stick to (1+1)D i.e. unidirectional motion, con- rection is defined. stant proper acceleration yields some wonderfully simple integrals of motion, namely α = ∆w/∆t = c∆η/∆τ = In flat spacetime, this 3-vector is simply related to 2 δ ~w c ∆γ/∆x, where “hyperbolic velocity angle” or rapid- the rate of proper-velocity change through ~α = δt || ~w + −1 −1 δ ~w ity η = sinh [w/c] = tanh [v/c]. These are ap- γ δt ⊥ ~w, which in the unidirectional case reduces to ~α = proximated by the familiar introl-physics relationships δ ~w 1 2 δt . This link between proper acceleration and proper ve- a = ∆v/∆t = 2 ∆(v )/∆x at low speed, but also al- locity will be important when we look at traveler-point low beginning students to explore interstellar constant dynamics in the next section. proper-acceleration round-trip problems almost as easily To illustrate what happens in curved spacetime (and as one does low-speed constant acceleration problems. accelerated frames), try running an accelerometer app on In (3+1)D the integrals are also analytical, but messy your phone and performing a hold, drop, catch sequence and perhaps best expressed16 in terms of a characteristic as shown in Fig. 3. Because accelerometers only detect time τo equal to p(γo + 1)/2c/α, which connects to a 4

FIG. 4. Metric-1st equations of (1+1)D motion. The dot over the equals sign means this works only for unidirectional motion. The Lorentz transform section at far right is the only place which requires more than one map (or bookkeeper) reference frame of yardsticks and synchronized clocks.

hyperbolic velocity angle τ/τo. Here γo is the “aging energy change are frame-variant, the only hand to go up factor at turnaround”, which for unidirectional motion in the audience was that of Edwin F. Taylor (that year’s yields the (1+1)D discussed above. Oersted Medal winner). In curved spacetime and accelerated frames, the move from 4-vector to local (3+1)D dynamics is normally done V. CAUSES OF MOTION by the “geometric force” approximation, in which motion is seen as caused by adding geometric to proper forces in the equations above to give Σξ~ +Σζ~ ≃ m~α and δE = Traveler-point dynamics has good news and bad news (Σξ~+Σζ~) · δ~x. The geometric forces (ζ) are unfelt by on- for beginners. The good news is that Newton’s laws can board accelerometers, act on every ounce of an object’s often be made to work locally in curved spacetimes and mass, and vanish when seen from a local free-float frame. accelerated frames. Gravity itself is an example of that. These relations are inspired by the general-relativistic Also, “felt” or proper acceleration is something that all 4-vector equation of motion, written in force-units as observers should able to agree on. mDU λ/dτ − mΓλ U µU ν = mdU λ/dτ, which may be de- The bad news is that even in flat spacetime, rates of µν scribed as “net proper 4-vector force” + “net geometric momentum change are frame-variant if more than one 4-vector force” = mass times the “net 4-vector acceler- direction is involved. As a result, Newton’s laws generally ation”. As usual here m is frame-invariant mass, upper ~ break into two parts, with net felt force obeying Σξ = case D denotes the of 4-velocity com- m~α while net frame-variant force obeys Σf~ = δ~p/δt = λ λ ponent U , Γµν denotes one of 4 × 4 × 4 = 64 connection mδ ~w/δt. That’s because the effect of each felt force ξ coefficients defined in terms of metric tensor derivatives, ~ λ µ ν on the observed rate of momentum change, namely f = and repeated indices in products like Γµν U U are im- ~ ~ ξ|| ~w +(1/γ)ξ⊥ ~w, depends on that object’s proper velocity plicitly summed over all (spatial and temporal) values of ~w. those indices. The idea that rates of momentum change depend on The traveler-point approach expresses this in terms of one’s may seem odd, but rates of en- a locally-useful set of 3-vector and scalar relations, like ergy change (δE/δt =Σξ~ · ~v → Σf~ · ~v) are frame-variant Σξ~ +Σζ~ = m~α and δE = (Σξ~ +Σζ~) · δ~x. The scalar even at low speeds. This in spite of the fact that when part connects to energy conservation, while the vector I asked an AAPT audience in 1998 for a show of hands part connects to momentum conservation locally through by those who thought that classical rates of an object’s the flat spacetime connection to frame-variant forces f~ 5

FIG. 5. Two views of proper (red) and geometric (dark blue or brown) forces in some everyday settings. discussed above. Both the “geometric force” approxima- tion coefficients (8 of 40 with independent values) for a tion which gives rise to the ζ~’ vectors, and the connec- Schwarzschild object of mass M and “far-coordinate” ra- tion of ”felt plus geometric” forces to rates of momentum dius r, to get Σζ~ = −GMmr/rˆ 2. If we wish to support change, rely on the general assertion that spacetime is the object in place so that net acceleration 3-vector ~α is ”locally flat” even in curved spacetimes and accelerated zero, then we require an E-field directed in the radial di- frames. The locality requirement, of course, may be more rection for which upward proper force qE′ = GMm/r2. severe as spacetime curvature increases. One nice thing about discussing this force-balance in For example, if we imagine a stationary charge q held traveler-point terms is that all observers, those using far- in place by some combination of electromagnetic forces in time in a Schwarzschild potential as well as e.g. those a Schwarzschild (non-spinning spherical mass) potential, passing by in a star ship accelerating at 1 gee, will be λ β talking about the same thing in terms of time-intervals the net proper force is the Lorentz 4-vector qFβ U = and forces experienced by the traveler, albeit with posi- qγ{E~ ·~v/c, E~ +~v×B~ }. In traveler-point form17 for charge tion coordinates of their own choosing. ~ ~ ′ ~ ~ ~ q this becomes Σξ = qE = q(E|| ~w+γE⊥ ~w+ ~w×B), where The net proper-force that our cell-phones measure is E~ and B~ are the bookkeeper-frame field vectors, and E~ ′ frame-invariant in magnitude as is proper time, with a is only the electric field seen by charge q. 3-vector direction presumably defined in traveler terms. The magnetic field B~ ′ seen by the charge has no effect This makes it useful18 for comparing electromagnetic on its motion, so that in traveler-point terms such prob- forces in different frames (e.g. to see how magnetic at- lems become purely electrostatic even in curved space- traction becomes purely electrostatic), and makes curved time! As an aside in this context, the above relations may spacetime (like that we experience on earth’s surface as be used to show how frame-variant proper forces can in illustrated in Fig. 5) a bit simpler to understand than general be decomposed into “static” and “kinetic” com- we might have imagined in a world where time passes ~ 3 ~ ~ ~ 1 ~ ponents, fs = γ m~a = ξ|| ~w + γξ⊥ ~w and fk = ( γ − γ)ξ⊥ ~w, differently according to your location as well as your rate which are analog to the electrostatic and magnetic forces of travel. (respectively) associated with electromagnetic fields. Thus Einstein’s , far from invalidat- The net geometric force, on the other hand, is ob- ing Newton, revealed that the classical laws work locally tained by summing the 13 out of 64 non-zero connec- in all frames (including accelerated-frames in curved- 6

multaneity defined by a free-float bookkeeper frame. In curved spacetimes the best bet for this reference frame may be the tangent free-float9 frame (or Fermi-normal8) , like for instance the “shell frame” that we use for our inertial reference here on earth. This traveler-point perspective perhaps does not do much for rigorous simplification of “geometric-force” analysis in curved spacetime, because e.g. the geodesic equation (i.e. without proper-forces) for arbitrary coordinate-systems is not reducible to purely 3-vector form. However, the electromagnetic example discussed above would make it quite straightforward e.g. to define λ µ ν a “gravitational proper-force” of the form mGµν U U . This proper-force approximation for including gravito- FIG. 6. Years elapsed in proper time (blue) and map time magnetic effects at any speed might, for example, come (red) on a 16+12+20 = 48 lightyear 1-gee acceleration round in handy for making 3-vector simulators of interstellar trip lasting about 17.2 traveler years and 53.7 map years. travel, e.g. using the celestiaproject.net database, which would work pretty well in the vicinity of main-sequence spacetime) provided that, in addition to proper-forces, and planets, as well as between stars. we recognize geometric (i.e. connection-coefficient) forces In that case, interesting extreme-gravity effects might like gravity and inertial-forces (acceleration “gees”, cen- have to be patched into the algorithms after the fact, trifugal, etc.) that “act on every ounce” of an object’s be- e.g. with help from a mean-field estimate of gravitational ing. These highly frame-dependent geometric forces (esp. dt/dτ. Regardless of whether one uses geometric-force 21 when the metric is diagonal) link to differential-aging gravity or its proper-force approximation, metric-first (γ ≡ dt/dτ) with “well-depth” as well as kinetic ener- simulations of accelerated motion between and around gies (e.g. in rotating habitats, accelerating spaceships19, stars using simultaneity defined by bookkeeper far-time, and gravity on earth8) that go something like (γ −1)mc2. but which are predicted on actions taken by a single trav- eler on their proper-time clock (cf. Fig. 6), will likely become quite robust in the days ahead. VI. DISCUSSION In summary, traveler-point parameters can help open the door: (a) to introductory students wanting to explore The foregoing, summarized in Fig. 4, is targeted to- the consequences of life in accelerated frames and curved ward university physics teachers, but uses concepts which spacetime here on earth, as well as (b) to engineers want- may be unfamiliar to some. In that context, in Ap- ing to apply Newton’s laws locally in environments where pendix A we provide notes that may provide shortcuts high speeds and differential aging must be taken into ac- for intro-physics “relativity sections”, and we are work- count. ing on a one-class intro to Newtonian variables as useful approximations20. The traveler-point proper coordinate perspective on the importance of specifying “which clock”, and on the ACKNOWLEDGMENTS distinction between proper and geometric forces with help from your cell-phone, can be helpful for students at Thanks to: Matt Wentzel-Long for his ideas, the late the very beginning of their engineering physics education. Bill Shurcliff (1909-2006) for his counsel on “minimally- It also provides “Newtonian-like” tools for the 3-vector variant” approaches, Phil Asquith for calling the Bridg- addition of proper-velocities and momenta, as well as for man, Bondi & Brehme papers to our attention, plus D. proper-force/acceleration analysis at any speed using si- G. Messerschmitt & D. Neustadter for their interest.

∗ [email protected] the IEEE 105, 1511–1573 (2017). † also Physics, Washington University (63110), St. Louis, 3 P. W. Bridgman, The Logic of Modern Physics (The MO, USA Macmillan Company, New York, NY, 1928). 1 Robert W. Brehme, “The advantage of teaching relativity 4 Hermann Bondi, Relativity and Common Sense (Double- with four-vectors,” American Journal of Physics 36, 896– day/Dover, New York NY, 1964/1980). 901 (1968). 5 A Einstein and N Rosen, “The particle problem in the 2 David G. Messerschmitt, “Relativistic timekeeping, mo- general ,” Phys. Rev. 48, 73–77 (1935). tion, and gravity in distributed systems,” Proceedings of 6 W. A. Shurcliff, “Special relativity: The central ideas,” 7

(1996), 19 Appleton St, MA 02138. 7 Charles W. Misner, Kip S. Thorne, and , Gravitation (W. H. Freeman, San Francisco, 1973). 8 James B. Hartle, Gravity (Addison Wesley Longman, 2002). 9 E. Taylor and J. A. Wheeler, Exploring black holes, 1st ed. (Addison Wesley Longman, 2001). 10 A. P. French, Special Relativity (W. W. Norton, New York, New York, 1968). 11 David Halliday, Robert Resnick, and Jearl Walker, Fun- damentals of Physics, 5th ed. (Wiley, New York, 1997). 12 , Relativity: The special and the general theory, a popular exposition (Methuen and Company, 1920, 1961). 13 Francis W. Sears and Robert W. Brehme, Introduction to the theory of relativity (Addison-Wesley, NY, New York, 1968) section 7-3. 14 George Gamow, Mr. Tompkins in paperback (Cambridge University Press, 1996) illustrated by the author and John Hookham. 15 E. Taylor and J. A. Wheeler, Spacetime physics, 1st ed. (W. FIG. 7. Pythagorean frame-invariance of the hypotenuse 2 2 2 H. Freeman, San Francisco, 1963) contains some material (in black), from the metric equation (δh) = (δx) +(δy) not found in the 2nd edition. for seven different Cartesian coordinate systems, whose unit- 16 P. Fraundorf and Matt Wentzel-Long, “Parameterizing the vectors are shown in color. Although none of these coor- proper-acceleration 3-vector,” HAL-02087094 (2019). dinate systems agrees on the coordinates of the black line’s 17 John David Jackson, Classical Electrodynamics, 3rd ed. endpoints, all agree that the length of the hypotenuse is 5. (John Wiley and Sons, 1999). 18 P. Fraundorf, “The proper-force 3 vector,” HAL-01344268 (2016). 19 The quantity τ is the proper-time elapsed on the clocks Wolfgang Rindler, Relativity: Special, General, and Cos- of a traveling observer whose map-position x may be mological (Oxford University Press, 2001). 20 written as a function of map-time t. As usual c is the Matt Wentzel-Long and P. Fraundorf, “A class period on spacetime-smart 3-vectors with familiar approximates,” spacetime constant (literally the number of meters in a HAL-02196765 (2019). second) which is traditionally referred to as lightspeed 21 Stephen S. Bell, “A numerical solution of the relativistic because it equals the in a vacuum. Kepler problem,” 9, 281–285 (1995). The term on the left in the metric equation is referred to as a frame-invariant. Just like a given hypotenuse (cf. Fig. 7) can be expressed in terms of a bunch of Appendix A: possible course notes different xy coordinate systems, all of which agree on its length, so a given proper-time interval e.g. on a traveling The sections to follow might be dropped into the in- object’s clock, can be expressed in terms of many differ- troductory physics schedule as the usual “relativity sec- ent “bookkeeper” reference-frames, all of which will also tion”, near the end of the course, or may be rearranged agree on its duration even if they can’t agree on which of for piecemeal discussion earlier so that the Newtonian two spatially-separated events happened first. models, that students will be working with, are framed This equation is seriously powerful. As Einstein illus- as approximations from the start. trated, if one tweaks the “unit” coefficients of the terms on the right by only “one part per billion”, we find our- selves in a gravitational field like that on earth where a 1. spacetime version of Pythagoras’ theorem fall of only a few meters can do you in.

Time is local to a given clock, and simultaneity is determined by your choice of reference frame. Al- 2. traveler-point kinematics though Maxwell’s equations on electromagnetism were “informed” to this reality in the mid 1800’s, humans re- From the foregoing, it is easy to define a proper- ally didn’t start to get the picture until the early 1900’s. velocity ~w ≡ d~x/dτ = γ~v, where ~v ≡ d~x/dt is But how might one deal with this quantitatively? coordinate-velocity as usual, and speed of map- Start with the (1+1)D flat-space metric equation, time or “differential-aging factor” γ ≡ dt/dτ = namely (cδτ)2 = (cδt)2 − (δx)2 where x and t are po- p1 + (w/c)2 = 1/p1 − (v/c)2 ≥ 1. This last relation sition and time coordinates associated with your refer- tells us that when simultaneity is defined by a network of ence “map-frame” of yardsticks and synchronized clocks. synchronized map clocks, a moving traveler’s clock will 8 always run slow. These relationships follow directly from by a traveler. Under a constant net proper-force, we can the metric equation itself. therefore expect constant net-acceleration. Thus having a new time-variable τ also gives us some At high speeds the constant proper-force equations are new ways to measure rate of travel. Proper-velocity, as messier because of that pesky γ in equations like ∆w = we’ll see, has no upper limit and is related to conserved- ∆(γv) = αo∆t. The low speed approximation (namely quantity momentum by the simple vector relation ~p = ∆v = a∆t) is therefore a bit simpler to deal with, and m ~w where m is our traveler’s frame-invariant rest-mass. works fine for speeds well below c. The upper limit of c ≥ dx/dt on coordinate-velocity re- 4. dynamics in curved spacetime sults simply from the fact that momentum (and dx/dτ) have to be finite. Speed of map-time γ, on the other In both curved spacetime and in accelerated frames, total energy 2 hand, relates to by E = γmc , and to Newton’s equations still work (at least locally) provided 2 by K = (γ − 1)mc . that we recognize the existence of non-proper or geomet- At this point, a variety of famiilar time-dilation and ric forces, like gravity as well as inertial forces that arise relativistic energy/momentum topics might be covered as in accelerated frames. Newton’s law for causes of motion examples. If there is added time, one path to take is “in the neighborhood of a traveling object” then takes to introduce length-contraction, velocity addition, and the 3-vector form ΣF~ +ΣF~ = m~α, where ~α is the net- Doppler effect with or without Lorentz transforms, his- o g acceleration actually observed by the traveler. torical notes on lightspeed measurement, etc. In what follows, we instead push the alignment with traveler-point The reason that your cell-phone’s accelerometer can’t concepts and the Newtonian treatment of kinematics and see gravity (or ), even when gravity is mechanics a bit further. causing a net-acceleration downward, is that these are ~ It is also conceptually interesting to note that proper- geometric forces. Geometric forces (ΣFg), which act on acceleration, in turn, is simply the vector-acceleration “every ounce” of an object, result from being in an ac- detected by a cell-phone accelerometer in the traveler’s celerated frame or in curved spacetime. Accelerometers ~ pocket. This quantity, after a couple of proper-time can only detect the result of net proper-forces (ΣFo), i.e. derivatives, pops up on the left side of the metric equa- one’s proper-acceleration ~αo. tion as a frame-invariant as did “hypotenuse” and proper- It’s traditional to approximate gravity on the surface of time. the earth as simply a proper-force that is proportional to For unidirectional motion in flat-spacetime (i.e. us- mass m, for which F~ = m~g where ~g is a downward vector ing map-coordinates in an inertial frame), proper- with a magnitude of about 9.8 [m/s2]. This works quite 3 2 2 acceleration ~αo = γ ~a, where ~a ≡ d ~x/dt is the usual well for most applications. However unlike geometric- coordinate-acceleration. These relations also yield a forces, proper-forces are not associated with positional few simple integrals for “constant” proper-acceleration, time-dilation like that which must be figured into GPS namely α = ∆w/∆t = c∆η/∆τ = c2∆γ/∆x. system calculations. That 2nd equality involves “hyperbolic velocity-angle” Just as kinetic energy in flat spacetime is related via −1 −1 or rapidity η = sinh [w/c] = tanh [v/c], so that (dt/dτ −1)mc2 to the faster passage of map-time (t) with γ = cosh[η]. The first and third equalities reduce to the respect to traveler time (τ) when simultaneity is defined familiar conceptual-physics relationships a = ∆v/∆t = by the map-frame, so is the position-dependent potential- 1 2 2 ∆(v )/∆x at low speed. However they allow begin- energy well-depth of some geometric-forces in accelerat- ning students to explore interstellar constant proper- ing frames and curved spacetime. Thus (dt/dτ − 1)mc2 acceleration round-trip problems, almost as easily as they also describes the potential-energy well-depth for mass- do problems involving projectile trajectories on earth. m travelers located: (i) at radius r from the axis in a habitat rotating with ω, which in the 1 2 2 low-speed limit reduces to the classical value 2 mω r ; 3. dynamics in flat spacetime (ii) a distance L behind the leading edge of a spaceship undergoing constant proper acceleration α, which in the The net proper-force may in general be written as small L limit reduces to the classical value mαL; and ΣF~o ≡ m~αo. In flat (and unaccelerated) spacetime (iii) on the radius R surface of a mass M planet, which coordinate-systems, all forces are proper, and proper- in the R ≫ 2GM/c2 limit reduces to the classical value acceleration ~αo equals the net-acceleration ~α observed of GMm/R where G is the gravitational constant.