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Home , Invariant (physics), Lorentz factor, Proper acceleration, Proper velocity

Proper- and f ≤ mα from the metric

P. Fraundorf∗ & Astronomy/CNS, U. Missouri-StL (63121), St. Louis, MO, USA (Dated: February 24, 2010) Proper and proper are relativity-smart concepts that, along with proper- , can (i) inject conceptual-clarity into low-speed intro-physics applications and (ii) inoc- ulate students against cognitive-dissonance when special and general relativistic applications are discussed. By calling the attention of introductory students to clock and free-float-frame perspectives from the start, the use of geometric (non-proper) in accelerated frames be- comes intuitive. Moreover, taking derivatives of event-coordinates with respect to proper-time τ via Minkowski’s flat-space version of Pythagoras theorem yields 3-vectors for both proper- velocity ~w = d~x/τ and proper-acceleration ~α that at any speed retain characteristics of their low-speed analogs. Dynamical quantities emerge on multiplying these derivatives by rest- m, which show that the scalar f dp/dt is less than or equal to mα with equality in the unidirectional case, long before students≡ are tasked with application of Lorentz transforms and Reimann geometry.

CONTENTS I. INTRODUCTION

I Introduction 1 This paper is about calling the attention of intro- A Short curriculum additions ...... 1 physics students to: (i) clocks attached to yardsticks, and 1 Clockswithyardsticks...... 1 (ii) the distinction between proper and geometric acceler- 2 Proper and geometric 2 ation. These simple actions can inject relativistic insight B Map-basedmotion...... 2 into Newtonian thought with everyday consequences, in- C Contentmodernization ...... 3 cluding a clearer awareness of the reaction gee- that presses down on freestyle skiers as they pass the bottom IIThetools 3 of the ramp leading to a jump. We further show that A Newtonianrecap...... 3 these items set the stage for descriptions of motion at B Thenewvariable ...... 3 “any speed” which are (a) accessible from the vantage C Theaddedequation...... 4 point of a single frame, and (b) patterned after descrip- D Takingderivatives ...... 4 tions that introductory physics teachers and students are already familiar with. III The 1st derivative and proper-velocity 5 A Speedtable...... 5 B Unidirectional velocity “addition” . . . 5 A. Short curriculum additions

IV The 2nd derivative and proper- This paper will recommend two minor curriculum ad- acceleration 6 ditions. A Inacceleratedframes ...... 6 B Athighspeeds...... 7 C Why is f ≤ mα?...... 7 1 Clocks with yardsticks 1 constantaccelerationexample . . . 8 2 electromagneticexample ...... 9 D Curvatureandequivalence ...... 9 Action: Point out to students that time t is gener- ally measured on clocks co-moving with the yardsticks of V Discussion and Conclusions 10 the reference used to measure vector- position ~r, and that clock motion can make a differ- Acknowledgments 10 ence. Rationale (not to be shared): The time and dis- References 10 tance between two events that take place at say ~r1, t1 and ~r2, t2 depends on the state of motion of the co-moving yardsticks (for measuring δr~ = ~r2 − ~r1) and where- possible synchronized clocks (for measuring δt = t2 − t1) ∗[email protected] that make up one’s extended . 2

TABLE I: Acceleration types and their connection to various forces. force geometric acceleration geometric acceleration category with points of action gone in free-float non-local effect normal string ⊕ ⊕ spring friction ⊕ ⊕ drag centripetal ⊕ ⊕ electromagnetic ⊕ ⊕ reaction “gees” centrifugal ⊕ ⊕ tidal Coriolis effect ⊕ ⊕

These δ-intervals obey Minkowski’s space-time version time if we invoke inertial forces to explain the geometric of Pythagoras’ theorem i.e. the flat-space metric equa- accelerations which occur in those frames. tion: The mathematics of geometric accelerations comes from the fact that in an object’s coordi- 2 2 (cδt) − δr~ · δr~ = (cδτ) . (1) nate acceleration (as distinct from its proper acceleration A) is equal to: where c is lightspeed and δt is the proper-time between events e.g. the time-elapsed on the isolated clock of a dU λ = Aλ − Γλ U µU ν , (3) flat-space traveler present at both events and traveling dτ µν with coordinate velocity ~v = δr/δt~ . As usual Pythagoras’ theorem in Cartesian coordinates gives us: where geometric accelerations are represented by the affine-connection term Γ on the right hand side, which δr~ · δr~ ≡ (δr)2 = (δx)2 + (δy)2 + (δz)2. (2) may be the sum of as many as sixteen separate velocity and position dependent terms. Coordinate acceleration The separation between events is time-like if (cδt)2 > 0, goes to zero whenever proper-acceleration is exactly can- null if (cδt)2 = 0, and space-like if (cδt)2 < 0. celed by that connection term[1], and thus when physical This observation contains virtually all of the physics and inertial forces add to zero. of , although fancier math is needed to draw out its more subtle consequences. B. Map-based motion

2 Proper and geometric accelerations Bell[2], Taylor & Wheeler[3, 4], Lagoute & Davost[5], Shurcliff[6], Moore[7], Dolby & Gull[8], Cook[1] and other Action: Point out that there are two kinds of acceler- experts in the use of ’s frame-invariants for ation: proper-accelerations caused by the tug of external education have encouraged introductions via the metric forces and geometric accelerations caused by choice of a equation, and/or from the vantage point of a single map- reference frame that’s not i.e. a local reference frame of co-moving yardsticks and synchronized clocks. coordinate-system that is not “in free-float”. We expand on that challenge here by showing how intro- In particular, proper-accelerations are felt through duction of one new variable (proper-time) via one added their points of action e.g. through forces on the bottom equation (the flat-space metric equation) sets the stage of your feet. On the other hand geometric accelerations for a logical narrative, patterned after that used for New- give rise to inertial forces that act on every ounce of an tonian dynamics, that yields all of special relativity (plus object’s being. They either vanish when seen from the useful steps toward and general relativ- vantage point of a local free-float frame, or give rise ity) from the vantage point of a single frame. to non-local force effects on your mass distribution Key to this process is use of several less-familiar ideas. that cannot be made to disappear, as summarized in the These include: attached table. (i) explicit mention of frame-relevance for variables e.g. Rationale (not to be shared): The assertion above by use of Shurcliff’s “eifo”-prefix (Effective for the Indi- contains the essence of general relativity’s equivalence cated Frame Only) for quantities whose value is impor- principle, which guarantees that Newton’s Laws can be tant to observers in an arbitrary frame but command no helpful locally in accelerated frames and curved space all-frame respect. 3

TABLE II: Cartesian 3-vectors at map-time t Newtonian dynamics. ⇒ component ~r velocity ~v acceleration ~a ~p net-force ΣF~ µ µ µ dRµ µ dUµ µ µ µ µ X R U dt A dt P mU ΣF mA 1 dx ≡ dvx≡ ≡ dpx ≡ X x dt vx dt ax mvx px dt fx 2 dy ≡ dvy ≡ ≡ dpy ≡ X y dt vy dt ay mvy py dt fy 3 dz ≡ dvz ≡ ≡ dpz ≡ X z dt vz dt az mvz pz dt fz µ ν 2 ≡2 ≡2 2≡ ≡ 2 gµν X X r v a p (ma)

(ii) proper-velocity ~w ≡ d~r/dτ, the distance per unit Given these tools, one takes derivatives with respect to traveler-time[9], retains many of the properties that or- map-time t so as to similarly define vector and dinary velocity loses at high speed. accelerations as shown in Table II. (iii) Proper-acceleration ~α, experienced relative to a The bottom row of quantities in this table are min- locally co-moving free-float-frame[3, 10], helps when ac- imally frame-dependent scalars only to the extent that celerating, speeding, and in curvy space-time. map-time t is frame-invariant. Newton had no way of (iv) How some of the space-like effect of sideways “felt” knowing that this condition is not met for frames mov- forces moves into the reference-frame’s time-domain at ing at sufficiently high speed with respect to one another. high speed, making dp/dt ≤ mα. Newton’s dynamical laws then emerge by multiplying velocity by frame- m to get momentum ~p, and noting that changes in this quantity require in- C. Content modernization teraction (namely an impulse-exchange) with the world around. This leads to: Hestenes[11], Taylor[12], Moore and Schroeder[13] d~p d~v among others, have discussed and helped to address Σif~i = = m = m~a (5) the challenge we face of modernizing content in spite dt dt of the fact that those who purchase textbooks like to along with the very powerful scalar work- integral teach it the way they learned it. In that context this (low-speed version): article encourages its readers to help take responsibil- ity for improvement of pedagogically-useful material in W 1 = ~a · ∆~x = v2 − v2 (6) joint-editing spaces on campus and on the web, includ- m 2 o ing Wikipedia’s pages on proper-velocity and proper-  acceleration. from which the principle of least-action[14] arises, along with Lagrangian/Hamiltonian dynamics and the Feyn- man propagator[15]. II. THE TOOLS As we see below, relativistic dynamics flows from this same sequence of steps simply by extending Pythagoras’ The conceptual tools needed are basically time- theorem to include time-elapsed on the clocks of a moving derivatives, frame-invariance, and mass. These are tools observer. that students of introductory physics may already be putting to use. B. The new variable

A. Newtonian recap The simplest way to prepare for what’s next is to ask students to specify which clock as well as which yardsticks Newtonian physics is naturally introduced by starting when they identify the reference frame they’ll use to de- with a Cartesian reference frame of co-moving yardsticks scribe motion. For the most part in Newtonian physics to measure map position coordinates ~r = {x,y,z} and co- class this means defining time t as the time-elapsed on moving synchronized clocks measuring map-time t with a set of synchronized clocks co-moving with those yard- which to describe motion. One also needs a notion of sticks. distance which is independent one’s choice of Cartesian By specifying the set of synchronized (or “map”) clocks axes. This comes from Pythagoras’ assertion that the used, course participants will get in the habit of mak- frame-invariant distance between two points in three- ing problems well-posed in terms of what we now know. dimensional space at time t is given by: This also sets the stage early-on for introduction of the one new variable needed for all of special rela- tivity, namely the frame-invariant time-elapsed (i.e. the (δr)2 = (δx)2 + (δy)2 + (δz)2. (4) proper-time τ) on the clocks of a moving object. This variable is given meaning through Minkowski’s extension of Pythagoras’ theorem. 4

TABLE III: Cartesian 4-vector relationships dynamics in (3+1)D spacetime. ⇒ component 4-position R 4-velocity V 4-acceleration A momENergy cP net 4-force ΣF µ µ µ dRµ µ dUµ µ µ µ µ X R U dτ A dτ cP cmU ΣF mA 0 dt ≡ c dγ ≡ 4 ~a ~v ≡2 1 dE ≡ f~ ~v X ct c cγ = 2 c = γ · γmc E γ · dτ ≡ 1 v dτ c ≡ c dτ ≡ c −( c ) 1 dx q dwx 2 4 ~a ~v vx dpx X x dτ wx = γvx dτ = γ ax + γ c· c cmwx cpx dτ γfx 2 dy ≡ dwy 2 4 ~a ~v vy ≡ dpy ≡ X y dτ wy = γvy dτ = γ ay + γ c· c cmwy cpy dτ γfy 3 dz ≡ dwz 2 4 ~a ~v vz ≡ dpz ≡ X z dτ wz = γvz dτ = γ az + γ c· c cmwz cpz dτ γfz µ ν 2 2 ≡ 2 2 ≡2 2 ≡ 2 gµν X X ρ (cτ) c α (mc ) (mα) ≡− − −

C. The added equation second proper-time τ derivatives of the displacement be- tween events, and then we multiply each by the traveling Minkowski’s space-time extension of Pythagoras’ the- object’s invariant mass m: orem, namely the flat-space metric equation, might then In addition to the now-familiar metric equation in- be described in Cartesian coordinates as: variant for 4-displacement R, this table yields frame- invariants for 4-velocity U, 4-acceleration A, momENergy (the energy-momentum 4-vector) cP, and net-4-force (the (cδτ)2 if δr < cδt (timelike) power-force 4-vector) ΣF . Key dynamical results that 2 2 follow directly are that relativistic momentum ~p = m ~w = (cδt) − (δr) = 0 if δr = cδt (null) (7) mγ~v, and relativistic energy E2 = (mc2)2 + (cp)2. −(δρ)2 if δr > cδt (spacelike)  Another more general result that emerges is the fact Just as Pythagoras’ theorem provides a frame-invariant that the magnitude-squared of a 4-vector has dif- measure of the distance between two points at a given ferent properties than the magnitude-squared of a 3- map-time t, this extension of that equation yields a mea- vector. First of all, the magnitude-squared can be pos- sure of the separation in space-time between two events itive (space-like) or negative (time-like). Similarly, the which is independent of reference-frame as well. Here absolute size of the magnitude also has different mean- events are specified by a map-time coordinate-value (in ing. distance units written as ct where c is the space-time For instance the absolute value of an object’s 4-velocity constant commonly called lightspeed) as well as Carte- is always c, whether it’s moving or not! Speeding up sian position values for {x,y,z}. changes its direction through space-time, but not the 4- In an introductory physics class, this equation might velocity magnitude itself. only be mentioned in passing as the reason one needs to Similarly the magnitude of an object’s four-vector ac- specify “which clock” when asking how much time has celeration does tell us about its felt-acceleration with elapsed. Once traveler-time has been defined in this way, respect to a local free-float frame. However it does it would let you later point to “anyspeed versions” e.g. of not give us directly the net sum of forces (i.e. rates the constant acceleration and force equations discussed of momentum-change) seen from a differently-moving here even if there is no time to cover them in class. frame, since dγ/dτ and the rate of an accelerated ob- In third-semester introductory and/or modern physics ject’s energy-change is only zero from the vantage point classes, this equation is a natural for exploring time- of a co-moving frame where γ bottoms out at the value dilation. To wit: If a car with a clock on its roof is 1. shown by video surveillance cameras to pass one bank Note that rates of energy-change are frame-dependent when it and the bank clock both read 2:02, and to pass a (and zero from the co-moving perspective) even in New- second bank at 2:06 when the car clock only reads 2:04, tonian physics. Surveys I’ve taken at national AAPT how fast was the car going? An even more interesting meetings suggest that this is not intuitively-obvious to if time-consuming prospect might be to start by giving many physics teachers, who may be used to thinking that students access to experimental data[16] on clock differ- dynamical properties (like force) are frame-independent. ences at high speed, modeling workshop style, to see what The following 4-vector facts should be easy to verify concepts they might use to quantify these effects. from Table III: The magnitude of an object’s 4-velocity (displacement’s proper-time derivative) is always time- like with magnitude c, while the magnitude of its 4- D. Taking derivatives acceleration is always space-like with magnitude equal to proper-acceleration α. Similarly the magnitude of an object’s momENergy is always time-like with mag- As we did above when recapping Newtonian dynamics, nitude mc2, while the magnitude of its net-4-force is al- it’s instructive to tabulate the derivatives of an event’s ways space-like with magnitude equal to the net-proper- coordinates {ct,x,y,z}. In Table III we take first and force mα. Note also that rates of momentum and energy change in a given frame are not frame-independent, and 5

FIG. 2: Undirectional velocity addition.

Thus thanks to the metric-equation’s assignment of a frame-invariant traveler or proper-time τ to the displace- ment between events in context of a single map-frame of co-moving yardsticks and synchronized clocks, proper- FIG. 1: Log-log plot of γ, v , and η vs. w . c c velocity becomes one of three related derivatives in spe- cial relativity (coordinate velocity ~v, proper-velocity ~w, and γ) that describe an object’s rate of therefore that frame-variant force components defined di- travel. For unidirectional motion, in units of lightspeed c rectly in terms of momentum and energy change come in each of these is also simply related to a traveling object’s handy along with net-proper-force at high speeds. hyperbolic velocity angle or η by

III. THE 1ST DERIVATIVE AND w v η = sinh−1[ ] = tanh−1[ ]= ± cosh−1[γ] (10) PROPER-VELOCITY c c . Proper-velocity, the distance traveled per unit time elapsed on the clocks of a traveling object, equals co- ordinate velocity at low speeds. Proper velocity at high A. Speed table speeds, moreover, retains many of the properties that coordinate-velocity loses. Table IV illustrates how the proper-velocity of wo ≡ c For example proper-velocity equals momentum per or “one map-lightyear per traveler-year” is a natural unit mass at any speed, and therefore has no upper limit. benchmark for the transition from sub-relativistic to At high speeds, as shown in Figure 1, it is proportional super-relativistic motion. to an object’s energy as well. Note from Figure 1 that velocity angle η and proper- The full table of four-vector relationships above is not velocity w run from 0 to infinity and track coordinate- needed to introduce proper-velocity. For example simply velocity when w ≪ c. On the other hand when w ≫ taking the derivative of the metric equation itself with re- c, proper-velocity tracks Lorentz factor γ while velocity spect to τ, to describe the timelike worldline angle η is logarithmic and hence increases much more of a traveler, yields: slowly. dt 2 dr 2 c − = c2 (8) dτ dτ B. Unidirectional velocity “addition”     and therefore For unidirectional motion, at low speeds the velocity dt 1 w 2 v13 of object 1 from the point of view of oncoming object γ ≡ = = 1+ , (9) 3 might be described as the sum of the velocity v12 of dτ v 2 r c 1 − c   object 1 with respect to lab frame 2 plus the velocity q v23 of the lab frame 2 with respect to object 3, i.e. as where coordinate-velocity ~v ≡ d~r/dt and proper-velocity ~w ≡ d~r/dτ = γ~v. 6

TABLE IV: Benchmark values around the relativistic slope-change in KE vs. momentum. Parameter Coordinate-velocity v Velocity-angle η Proper-velocity w Lorentz factor γ Condition [ly/map-year] [hyperbolic-radians] [ly/trav-year] [map-yr/trav-yr] traveler stopped 0 0 0 1 1 1 1+√5 1 √5 momentum 2 mc √5 0.447 ln[ 2 ] 0.481 2 2 1.118 ≃ ≃ 1 1 1 ≃1 − − 1 e 1 1 e 2 e 2 e 2 +e 2 rapidity ı-radian − 0.462 − 0.521 1.128 2 e+1 ≃ 2 2 ≃ 2 ≃ coord-vel v = 1 c 1 1 ln[3] 0.549 1 0.577 2 1.155 2 2 2 ≃ √3 ≃ √3 ≃ 1 √ √ momentum mc √2 0.707 ln[1 + 2] 0.881 1 2 1.414 2 ≃ ≃ ≃ e 1 e 1/e e+1/e − rapidity 1 ı-radian e2+1− 0.761 1 2 1.175 2 1.543 2 √3 ≃ ≃ ≃ mc 2 0.866 ln[√3 + 2] 1.317 √3 1.732 2 2 ≃ √ ≃ ≃ √ momentum 2mc √5 0.894 ln[2 + 5] 1.444 2 5 2.236 4 ≃ ≃ 2 2 2 ≃2 e 1 e 1/e e +1/e rapidity 2 ı-radian 4− 0.964 2 − 3.627 3.762 e +1 ≃ 2 ≃ 2 ≃ coord-vel v = c 1 ∞ ∞ ∞

ND v13 = v12 + v23. This Galilean addition rule is noted in IV. THE 2 DERIVATIVE AND Fig. 2 by the red outline. PROPER-ACCELERATION At coordinate-speeds approaching c, coordinate- velocity deviates from this simple addition rule in that Proper-acceleration is the physical acceleration expe- (hyperbolic velocity angle boosts) add in- rienced by an object relative to a locally co-moving stead of velocities, i.e. η13 = η12 + η23. As a re- free-float frame. This may be quite different than the sult coordinate-velocity follows the green checkerboard in coordinate-acceleration seen in a separate reference frame the figure, while proper-velocity (that transparent sheet) (inertial or not) of co-moving yardsticks and synchro- goes through the roof. nized clocks. For highly relativistic objects (i.e. with momentum The proper-acceleration 3-vector, combined with a per unit mass much larger than lightspeed) the result null time-component, yields the object’s four-acceleration of the coordinate-velocity expression (green mesh in the (as measured by the object itself) which makes figure at right) familiar from most textbooks is rather proper-acceleration’s magnitude Lorentz-invariant. Thus uninteresting since the coordinate-velocities all peak out proper-acceleration comes in handy: (i) with accelerated at c, i.e. (c + c)/(1 + 1) ⇒ c. coordinate systems, (ii) at relativistic speeds, and (iii) in For relative proper-velocity, the result is: curved spacetime. Proper-acceleration reduces to coordinate-acceleration in an inertial coordinate system in flat spacetime (i.e. w13 = γ13v13 = γ12γ23 (v12 + v23) . (11) in the absence of gravity), provided the magnitude of the object’s proper-velocity (momentum per unit mass) This expression shows how the momentum per unit is much less than the c. Here we focus mass as well as the map-distance traveled per unit trav- on situations where proper-acceleration and coordinate- eler time of object 1, as seen in the frame of oncoming acceleration are not always the same. particle 3, goes as the sum of the coordinate-velocities the product of the gamma (energy) factors. This is denoted by the sky-ward climbing radial mesh in the A. In accelerated frames figure. The proper-velocity equation is especially important in The intro-physics caution about using Newton’s Laws high energy physics, because colliders enable one to ex- only in un-accelerated or inertial frames is a strategy plore proper-speed and energy ranges much higher than for avoiding confusion by non-proper (i.e. “geometric”) accessible with fixed-target collisions. For instance each coordinate-accelerations like those experienced e.g by of two (traveling with frames 1 and 3) in a head- passengers in vehicles accelerated from a stop-sign or go- on collision traveling in the lab frame (2) at γ mc2 = 12 ing around a curve. As indicated below, the equivalence 45[GeV] or w = w = γv ≃ 88, 000[lightseconds 12 23 principle of general relativity in fact shows that Newton’s per traveler second] would see the other coming to- laws work locally in any coordinate system provided one ward them at v ≃ c and w = 88, 0002(1 + 1) ≃ 13 13 invokes a suitable set of “geometric” or affine-connection 1.55 × 1010[lightseconds per traveler second] or γ mc2 ≃ 13 accelerations/forces. 7.9[PeV]. From the target’s point of view, that’s quite an This result validates the intro-physics practice of treat- increase in both energy and momentum per unit mass! ing shell-frames on planet earth as free-float-frames so that gravity becomes an external force. It also validates 7 local reference to other “fictitious” forces like Coriolis and centrifugal. The rule of thumb in each case is that non-proper or “geometric” accelerations act on every ounce of an object’s mass, and vanish if one describes the motion from the perspective of a free-float-frame moving with our accelerated object. Thus the concept of proper- acceleration, as the physical-acceleration experienced by an object with reference to a locally-comoving free-float- frame, is already an important introductory physics tool.

B. At high speeds

The time derivatives of proper-velocity in Table III show that momentum change (hence force) has an energy-change component in the velocity direction, pro- FIG. 3: Proper-acceleration components for v = 0.65c. portional to ~v · ~a, as well as the expected acceleration component in the direction of ~a||~α. Hence acceleration and force are no longer in the same direction. Thus if our observer is moving at fixed velocity v⊥ However the last column in that table suggests a fairly “perpendicular” to the path of a uniformly-accelerated simple dependence of proper-acceleration’s magnitude α traveler, derivatives with respect to map-time t can on force f~ and velocity ~v|| ~w vectors. In terms of compo- be written in terms of the frame-invariant proper- nents of velocity ~v transverse (t) and longitudinal (ℓ) to acceleration α as: the force vector f~, direct substitution gives: dw α dw dγ v k = and ⊥ = c ⊥ , (17) 2 dt γ dt dt c vℓ 2 ⊥ mα 1 − c wt   = = 1+ ≡ γt ≥ 1. (12) v v 2 where f u 1 −  r c u c   t dγ v  c = αγ k . (18) As you can see the equality obtains in the high-speed dt ⊥ c unidirectional, as well as the low-speed, limits.   Since the angle between force and acceleration varies, Here wk ≡ dxk/dt = γvk is the component of proper- a more detailed analysis is needed in order to compare velocity parallel to the proper-acceleration direction, 2 their directions. One might begin by writing the Lorentz while γ⊥ ≡ 1/ 1 − (v⊥/c) and v⊥ ≡ dx⊥/dt = w⊥/γ factor in terms of velocities parallel and perpendicular to are respectively the Lorentz-factor and the (unchanging) the acceleration direction, i.e. component ofp coordinate-velocity perpendicular to the same. 1 dγ v dv γ = ⇒ c = γ3 k k . (13) From this one can write the integrals of constant 2 2 vk v⊥ dt c dt proper-acceleration in (3+1)D spacetime as: 1 − c − c   q 2   ∆wk ∆ηk c ∆γ From this it’s relatively easy to calculate the proper- α = γ⊥ = c = (19) velocity derivatives: ∆t ∆τ γ⊥ ∆xk

2 −1 dw dv dγ dv v where ηk ≡ sinh [wk/c]. These reduce to the unidirec- k = γ k + v = γ3 k 1 − ⊥ (14) dt dt k dt dt c2 tional proper-acceleration equations   ∆w ∆η ∆γ and α = = c = c2 (20) ∆t ∆τ ∆x dw⊥ dγ 3 dvk v⊥ vk = v⊥ = γ . (15) when v = 0 ⇒ γ = 1, and to the Newtonian equa- dt dt dt c c ⊥ ⊥   tions for coordinate-acceleration with respect to an iner- Converting these to derivatives with respect to proper- tial frame in three-dimensions when v ≪ c. time on multiplication by γ ≡ dt/dτ, one can then form the invariant 4-vector sum α2 to get: C. Why is f mα? ≤ 2 2 2 3 dwk dw⊥ dγ γ dvk α = + − c = . (16) From the foregoing analysis one can also express New- s dτ dτ dτ γ⊥ dt       ton’s 2nd Law in (3+1)D spacetime for a single (frame- 8

TABLE V: Various quantities ranked according to their frame-(in)dependence. quantity frame 4-vector effective for the invariant component indicated frame only (eifo) time proper (clock-frame) map-time time ∆τ ∆t = γ∆τ length proper (rest-frame) map-distance contracted length ∆ρ ∆r length ∆ρ/γ speed of map-time Lorentz factor γ dt/dτ and total energy E = γmc2≡ 3-vector velocity proper-velocity ~w d~r/dτ coordinate-velocity and momentum momentum ~p =≡m ~w ~v d~r/dt = ~w/γ ≡ 3-vector acceleration proper-acceleration α coordinate-acceleration and proper-force proper-force Σ ~ m~α ~a d~v/dt =(γ /γ3)~α F ≡ ≡ ⊥ rate of momentum d~p/dτ net-force

change Σf~ d~p/dt = (Σ /γt)ˆiℓ ≡ F rate of energy dE/dτ power change dE/dt = Σf~ ~v · variant) force or net-force by writing[17]: other hand the proper force were independent of vx, e.g. in the case of a constant proper-acceleration rocket-ship, d~p mα then the reference-frame force peaks when vx = 0. f~ ≡ = ˆiℓ (21) dt γt In both cases one can think of this as the force dilu- tion (or “eifo-weakening” ala Shurcliff) that takes place where the unit vector longitudinal to the frame-variant as changes in proper-velocity (hence momentum) are lost force direction is: to the reference-frame’s time-domain because of side- ways motion. Since this stems from the frame-invariance γt v⊥ vk ˆiℓ ≡ ˆik + γtγ⊥ ˆi⊥. (22) of proper-acceleration, it’s really just another metric- γ⊥ c c     based kinematic-effect like time-dilation (eifo-slowing) Here the gamma-value transverse to the frame-variant and length-contraction (eifo-shortening). force direction can be written:

2 wt 1 1 constant acceleration example γt ≡ 1+ = . (23) c 2 v 2 r 1 + γ⊥v⊥ k   γ⊥ c c As a function of traveler or proper time τ, for con- r stant proper-acceleration in the x direction from rest at    k Conversely for any observed rate of momentum-change the origin we might write the traveler’s map time and µ (or combination of frame-variant forces) one can asso- position coordinates, or 4-vector displacement R , as: ciate 3-vectors with the scalar frame-invariants proper- c2 ατ acceleration α and proper-force F ≡ mα. These obey: R0 = ct = γ sinh . (26) α ⊥ c  h i F~ ≡ m~α = fγtˆik (24) 2 1 c ατ where the unit vector in the proper-acceleration direction R = xk = cosh − 1 ≥ 0. (27) is: α c  h i  ˆ γt ˆ 1 wt wℓ ˆ 2 ik ≡ iℓ + it (25) 2 c γ⊥v⊥ ατ γ⊥ γ⊥γt c c R = x = sinh . (28)     ⊥ α c c  h i and subscript t denotes quantities transverse to the Taking derivatives and calculating the magnitude of frame-variant force. d~p/dt ≡ md~w/dt, one gets: The magnitude of the proper force is greater than or equal to the force seen in an observer-frame, i.e. F ≡ dp v 2 ατ 2 mα ≥ f ≡ dp/dt, with equality at any speed as long as f ≡ = mα 1 − ⊥ 1 − tanh . (29) the accelerated object’s velocity transverse to the force dt s c c    h i  (i.e. wt) is zero. As we show below the proper-force Fz = mα on a As predicted, f ≤ mα. charge moving sideways at speed vx with respect to some protons at rest is least when vx = 0, since the reference- frame force f ≡ dp/dt is independent of vx. If on the 9

2 electromagnetic example Hence the net upward Coulomb’s Law force from the metric equation becomes: A fixed charge Q acts on a moving charge q via the ra- dial frame-variant force f = kqQ/r2. Here we choose the 1 1 1 rest frame of the source charge so that the field is sta- fz = Fz = q(Eo)γx + q(−Eo)  (34) tionary. The “frame-invariant” but velocity-dependent γx γx γx  proton F  proper-force F ≡ mα on any moving charge from above  F  is then: | {z }  so that | {z } 2 qQ wφ F~ = k 1+ ˆi , (30) 1 v 2 r2 c k f = qE 1 − = qE x . (35) r z o γ2 o c    x  where   If we define magnetic field using Biot-Savart as w 2 w φ ˆ φ wr ˆ 1+ c ir + c c iφ ˆ k Ids k λvxds vxEo ik ≡ . (31) By ≡ = = , (36)  2 2 2 2 2 2 2 2 wφ wφ wr c r c r c 1+ c + c c r then this analysis yields the equation:     This velocity dependence of the proper force in (3+1)D space-time gives rise to relativistic effects for almost all fz = qvxBy. (37) forces, and to magnetism from Coulomb’s Law. To illustrate this, consider the simpler special case D. Curvature and equivalence when the force is perpendicular to the line of a test parti- cle’s velocity v⊥ = vφ. Then the proper-force F is simply related to the frame-variant force f (defined as usual in In the language of general relativity, the components terms of momentum and energy transfer) by: of an object’s acceleration four-vector A (whose magni- tude is proper acceleration) are related to elements of the 2 four-velocity via a D with respect to ~ qQ wφ ˆ F = k 2 1+ ik = fγ⊥r.ˆ (32) proper time τ: r r c   λ λ This lets one use and the frame-invariance λ DU dU λ µ ν A = = +Γ µν U U . (38) of proper force in a single frame, instead of length con- dτ dτ traction and two-frames ala Purcell[18], to illustrate the Here U is the object’s four-velocity, and Γ represents connection between relativity and magnetism. For ex- the coordinate system’s 64 connection coefficients or ample consider a horizontal and electrically-neutral wire Christoffel symbols. Note that the Greek subscripts take element ds with −λ Coulomb of free electrons per unit on four possible values, namely 0 for the time-axis and length moving at speed v to the right. Their excess x 1-3 for spatial coordinate axes, and that repeated indices charge is cancelled out by the same number of bound pro- are used to indicate summation over all values of that tons per unit length, standing still. The current in the index. Trajectories with zero proper acceleration are re- wire (defined as moving in the -x direction) is I = λv . x ferred to as or free-float trajectories. For a stationary positive charge q a distance r Thus coordinate acceleration goes to zero whenever above the wire, γ = γ is 1 and the net upward (repul- ⊥ x proper-acceleration is exactly canceled by the connection sive) force is: (or “geometric acceleration”) term on the far right[1]. Caution: This term may be a sum of as many as sixteen fz = Fz = q(Eo) + q(−Eo) = 0 (33) separate velocity and position dependent terms, since the proton F electron F repeated indices µ and ν are by convention summed over | {z } | {z } 2 all pairs of their four allowed values. where the base electric field magnitude Eo ≡ kλds/r . The above equation also offers some perspective on However if the same positive charge q is now forces and the . Consider “local” moving to the right with those electrons, the proper- book-keeper coordinates[4] for the metric (e.g. a local force exerted on our test charge by the stationary posi- Lorentz tetrad[19] like that on which global positioning tive charges remains in the direction of their separation systems provide information) to describe time in seconds, but increases in magnitude by that factor of γx. What and space in distance units along perpendicular axes. If the moving electrons do to the test particle as it matches we multiply the above equation by the traveling object’s speed with them would be less clear, except that by sym- rest mass m, and divide by Lorentz factor γ ≡ dt/dτ, metry the frame-invariant proper-force in that case must the spacelike components express the rate of momentum decrease by the same factor of γx since the test particle change for that object from the perspective of the coor- now moves with them rather than the protons. dinates used to describe the metric. 10

This in turn can be broken down into parts due to Lorentz transforms[7] follows via a single-frame narrative proper and geometric components of acceleration and patterned after that used already to introduce Newtonian force. If we further multiply the time-like component by dynamics. lightspeed c, and define coordinate velocity as ~v ≡ d~x/dt, The frame-invariants that emerge for describing mo- we get an expression for rate of energy change as well: tion, namely lightspeed c and proper-acceleration α, are scalars even though proper-acceleration in the free-float dE d~p = ~v · (timelike) (39) frame co-moving with an accelerated traveler can be bro- dt dt ken nicely into three space-like components as well. Since they are true frame-invariants, these scalars are useful in and curved-spacetime too. Similarly, as William Shurcliff put d~p it in a private communication, proper-velocity is speed =Σf~ +Σf~ = m( ~a + ~a ) (spacelike). (40) dt o g o g “defined in a manner which requires no synchrony”. No wonder it equals momentum per unit mass, and repre- Here ao is an acceleration term (not necessarily a phys- sents three of four components of a 4-vector whose time- ical acceleration) due to proper forces and ag is, by de- like component is cγ. fault, a geometric acceleration term that we see applied The metric equation is sometimes mentioned at the end to the object because of our coordinate system choice. of introductory sections on relativity. It’s argued here At low speeds these accelerations combine to generate a that, with help from proper-time, proper-velocity, and coordinate acceleration like ~a = d2~x/dt2, while for uni- proper-acceleration, the metric equation might instead directional motion at any speed ao’s magnitude is that be used from the start. of proper acceleration α as above. In general expressing these accelerations and forces can be complicated. Nonetheless if we use this breakdown to describe the ACKNOWLEDGMENTS connection coefficient (Γ) term above in terms of geomet- ric forces, then the motion of objects from the point of Thanks are due to: Roger Hill for some lovely course view of any coordinate system (at least at low speeds) can notes, Bill Shurcliff for his counsel on minimally-variant be seen as locally Newtonian. This is already common approaches and on ways to cause cognitive dissonance in practice e.g. with gravity and . relativity texts, as well as John Mallinckrodt, Eric Man- dell, and Edwin Taylor for their ideas and enthusiasm. V. DISCUSSION AND CONCLUSIONS REFERENCES Einstein’s writings on special relativity, which served to guide early texts, were inspired by the frame- [1] R. J. Cook, Am. J. Phys. 72, 214 (2004). invariance of lightspeed. Thus extended frames in rel- [2] J. S. Bell, in The speakable and unspeakable in quan- ative motion were front and center. However just as Ein- tum (Cambridge University Press, 1987). stein’s impression of Minkowski’s insight as ¨uberfl¨ussige [3] E. Taylor and J. A. Wheeler, Spacetime physics (W. gelehrsamkeit[20] (superfluous erudition) was eclipsed by H. Freeman, San Francisco, 1963), 1st ed., chapter its adoption as key to a geometric understanding of grav- 1 Exercise 51 page 97-98 ”Clock paradox III” on itation, so recognition of privileged frames (e.g. the proper-acceleration is not found in the 2nd edition. proper-time frame of a moving clock, and the rest-frame [4] E. Taylor and J. A. Wheeler, Exploring black holes of a moving yardstick) is becoming more and more com- (Addison Wesley Longman, 2001), 1st ed. mon in modern texts. [5] C. Lagoute and E. Davoust, Am. J. Phys. 63, 221 The late William A. Shurcliff’s passion for clear and (1995). minimally-variant descriptions, like that of the metric [6] W. A. Shurcliff, Special relativity: The central ideas equation with its “moving clock time dτ”, can be used to (1996), 19 Appleton St, Cambridge MA 02138; In- take this trend further as outlined in Table V. But text- troduces the “eifo” prefix (effective for the indicated books alone (as time has shown) cannot lead the way to frame only) in Chapter 8, uses for proper-speed in simpler and clearer presentations. Shared materials in Chapter 23. joint-editing space, however, can give energetic teachers [7] T. Moore, Six Ideas that Shaped Physics (McGraw- grass-roots input into the evolution of content. Responsi- Hill, 1998). bility for participation in the process, of course, lies with [8] C. E. Dolby and S. F. Gull, Am. J. Phys. 69(12), the reader. 1256 (2001). In summary, the behavior of spacetime at high speeds [9] Sears and Brehme, Introduction to the theory of rel- does not have to be introduced by expanding the dreaded ativity (Addison-Wesley, New York, 1968), proper- “relative motion section” of an introductory physics velocity is introduced in Section 7-3. course to include Lorentz transforms. An alternative is to introduce the traveler-time variable τ via the metric equation. Given this, all of special relativity including 11

[10] A. J. Mallinckrodt, What happens when at > c? [17] P. Fraundorf, Some minimally-variant map- (1999), presented at the Summer 1999 AAPT meet- based rules of motion at any speed (1997), ing in San Antonio, Texas. arXiv:physics/9704018. [11] D. Hestenes, Am. J. Phys. 71, 105 (2003). [18] E. M. Purcell, Electricity and Magnetism (McGraw- [12] E. F. Taylor, Am. J. Phys. 66, 369 (1998), the Oer- Hill, 1965). sted Medal lecture. [19] C. W. Misner, K. S. Thorne, and J. A. Wheeler, [13] T. A. Moore and D. V. Schroeder, Am. J. Phys. Gravitation (W. H. Freeman, San Francisco, 1973). 65(1), 26 (1997). [20] A. Pais, Subtle is the Lord...: The science and life [14] T. A. Moore, Am. J. Phys. 72(4), 522 (2004). of Albert Einstein (Oxford University Press, Oxford, [15] E. F. Taylor, S. Vokos, J. M. O’Meara, and N. S. 1982), cf. the private communication with Valentin Thornber, Computers in Physics 12, 190 (1998). Bargmann on page 152. [16] P. Fraundorf, An experience model for anyspeed mo- tion (2001), arXiv:physics/0109030.