Pedagogical Materials and Suggestions to Cure Misconceptions Connecting Special and General Relativity
aa r X i v : . [ phy s i c s . e d - ph ] J u l Pedagogical Materials and Suggestions to Cure Misconceptions Connecting Specialand General Relativity
R. A. Pepino ∗ and R. W. Mabile Florida Southern College, Lakeland, FL 33801
Many professional physicists do not fully understand the implications of the Einstein equivalenceprinciple of general relativity. Consequently, many are unaware of the fact that special relativityis fully capable of handling accelerated reference frames. We present results from our nationwidesurvey that confirm this is the case. We discuss possible origins of this misconception, then suggestnew materials for educators to use while discussing the classic twin paradox example. Afterwards,we review typical introductions to general relativity, clarify the equivalence principle, then suggestadditional material to be used when the Einstein equivalence principle is covered in an introductorycourse. All of our suggestions are straightforward enough to be administered to a sophomore-levelmodern physics class.
INTRODUCTION
We have found that it is fairly common for physicistswho do not specialize in general relativity (GR) to be-lieve that special relativity (SR) is incapable of model-ing dynamics within accelerated reference frames. Con-sequently, they may conclude that certain phenomena,such as time dilation due to acceleration, can only bedescribed with GR. The fact of the matter is: as longas spacetime if flat (Minkowskian), SR–not GR–is therelevant theory to use [1]. In
Gravitation , there is anentire chapter that discusses accelerated frames in spe-cial relativity [1]. In that chapter the authors state that“...special relativity was developed precisley to predictthe physics of accelerated objects.” Quoting
Spacetimeand Geometry : “The notion of acceleration in special rel-ativity has a bad reputation, for no good reason” [2]. Webelieve this misunderstanding is mainly caused by under-graduate students exposure to oversimplified discussionsof the twin paradox and the
Einstein equivalence principle (EEP)To confirm the existence of this misconception withinthe physics community, we developed a survey, whichwe emailed to prestigious institutions around the UnitedStates. This survey asked hundreds of physicists, fromgraduate students to professors, whether or not SR iscapable modeling time dilation due to acceleration.In this article, we review introductions of SR and GR insix popular undergraduate modern physics textbooks [3–8]. After pointing out misstatements in three of thesetextbooks, we discuss their presentations of the twinparadox and the Einstein equivalence principle. We thensuggest materials that can be adopted in the classroomfor these two topics that will help clarify subtle connec-tions between SR and GR.Specifically, with regards to the twin paradox, we en-courage the usage of lines of simultaneity in worldlineplots. These lines help demonstrate to students that notnot all of the time difference between frames occurs atthe turnaround point. We then introduce a straightfor- ward example that calculates time dilation due to theacceleration of the moving twin at the turnaround point.After students are introduced to the EEP, we suggestdiscussing the observations of time dilation accrued bya person in the presence of a gravitational field and an-other accelerating in free space by a third party observer.The third observer would prevent students from conclud-ing that there is a global equivalence between acceler-ation and gravity. In this context, geodesics , metrics ,Einstein’s summation convention and the Schwartzschildsolution can all be introduce into the classroom. Not onlywould the suggested materials help clarify the meaningand goals of GR, they would also introduce/reinforce con-cepts and tools widely used in other areas of physics. Oursuggestions are appropriate for sophomore-level physicscourses and above.Subjects like GR are esoteric, and most physicistsnever study the subject in great detail. When instruc-tors who are not familiar with the subject attempt toteach it in the classroom, they pass their own miscon-ceptions onto the next generation. Those students, inturn, pass their misconceptions onto the generation thatfollows them. This propagation of ignorance can be eas-ily broken at an early stage in a physics undergraduateprogram.
SURVEY QUESTIONS AND RESPONSES
The multiple choice question on our survey concerningtime dilation of accelerating systems was the following:
With regards to time dilation, what minimal levelof theory is needed to solve problems involving relativisticacceleration? • Special Relativity • Special relativity is not capable of dealing with ac-celeration. General relativity is required • A combination of both special and general relativ-ity are needed together • Cosmological relativity, because the boundary con-ditions of the universe are relevantIn addition to the multiple choice answer, we provideda space for free responses, which further confirmed thatmany were under the impression that SR was incapableof modeling dynamics within accelerated frames of refer-ence.Upon completion of this servey, some faculty and stu-dents disclosed their affiliated institutions. A list of theseinstitutions is provided below our conclusions section.Of the several hundred surveys sent, we received 57 to-tal responses from professionals: 38 from professors and19 from graduate and postdoctoral students. ExcludingGR specialists, reduces the number of responses to 33professors and 16 grad/post docs. Although this data setis not incredibly large, the results are still very alarming:
Ranking Total Participants Total correct % correct
Faculty 33 16 48%Students 16 5 31%TABLE I. Survey responses from faculty and Ph.D./Postdocstudents. This data excluded physicists specializing in thefield of GR. only 48% of professors and 31% of graduate and postdoc-toral students were aware of the fact that time dilationdue to relativitsic acceleration could be modeled solelywith SR. Of those who chose the wrong solution, 33% ofprofessors and 63% of students explicitly stated that onlyGR is capable of explaining time dilation for acceleratingsystems.
FLAWED ASSERTIONS IN TEXTBOOKS
While discussing the twin paradox and the EEP, wefound the following four inaccurate statements made instandard textbooks: • “The laws of special relativity apply only to iner-tial frames, those moving relative to one another atconstant velocity [4].” • “...Einstein’s 1905 [SR] theory applies only to non-accelerated motions... [GR] published in 1969 dealswith arbitrary motions, including accelerations...when we want to deal with accelerations in [SR], wehave to resort to ingenuity and approximations [7].” • “Special relativity is concerned only with inertialframes of reference... [GR] goes further by in-cluding the effects of accelerations on what we ob-serve [3].” • “...[GR] is more general than [SR] both because itincludes gravity and because it focuses on noniner-tial, as well as, inertial reference frames* [8].”Although the asterisk in Ref. [8] does lead to a footnotestating “Experts may object that even [SR] can handlenoninertial frames; nevertheless, its primary focus is in-ertial frames”. However, there is no ambiguity: experts will assert that accelerated frames can be treated withSR. TIME DILATION IN ACCELERATED FRAMES
The key to understanding time dilation between accel-erating and inertial reference frames is that for any in-stantaneous speed v of an accelerating frame S ′ (with re-spect to inertial frame S ), there exists an inertial Lorentztransformation connecting the two frames. This canbe understood through the invariance of worldline seg-ments. Assuming the spacetime metric signature to be(+ , − , − , − ): dW ′ = dW (1)= ⇒ cdτ ′ = p ( cdt ) − dx − dy − dz = ⇒ dτ ′ = r − v c dt for any given v where v = v x + v y + v z .Alternatively, this eqquation can be motivated by con-sidering the differential form of the Lorentz transforma-tions: dx = γ v ( dx ′ + vdt ′ ) (2) dt = γ v ( dt ′ + vc dx ′ ) , (3)where γ v = 1 / p − v /c is the Lorentz factor and c is the speed of light. The differential form eliminatesthe need to consider changing time and spatial originsbetween frames. Regardless how this discussion is moti-vated, for a particle stationary in S ′ , dx ′ = 0. This yieldsthe following relationship between t , the time measuredin S , to the proper time τ ′ of a particle at rest in S ′ dt = γ v dτ ′ , (4)which implies ∆ τ ′ = Z ∆ t r − v c dt. (5) ACCELERATION AND THE TWIN PARADOX
After undergraduates learn about time dilation, theyare typically exposed to the twin paradox. Throughoutthis article we will refer to the stationary twin as
Sara and the moving twin as
Mary . This paradox can be re-solved by noting the asymmetry between the worldlinetrajectories of the two twins: Mary has to reverse her di-rection while Sara remains stationary. In order to makethe discussion as clear as possible, standard discussionsinvolve three inertial (straight line) spacetime trajecto-ries: one for Sara, the other two for Mary.Some textbooks justify this inertial model by swappingMary from an inertial frame (moving away from Sara) toanother (that moves towards her) [3–6]. This argumentmight lead some students to incorrectly conclude that,since both twins will measure the other’s clock runningslower than theirs before the turnaround, 100% of thetime dilation is accrued at the turnaround point i.e. theframe swapping itself is responsible for the time differ-ence.
FIG. 1. (Color online) The standard twin paradox example.Worldline trajectories, observed in Sara’s frame, of Sara (reddashed) and Mary (blue solid) where Mary has an initial ve-locity of 0 . c , and travels to a distant planet 8 ly away. All of the referenced books dismiss the time dilationdue to acceleration at the turnaround point as negligiblecompared to that of the inertial dynamics [3–8]. Thisargument requires the magnitude of the acceleration of S ′ be very large compared to the frame’s inertial velocitymeasured by S . However, if Mary is to survive the trip,her acceleration needs to be reasonable. If this is the case,as we show below, the time dilation due to accelerationcannot be neglected. Suggested Additional Twin Paradox Material
Equal Time Plots
On order to help students understand that not all ofthe time dilation happens at the turnaround point, it ishelpful to quantify and discuss the lines of simultaneityconnecting Sara and Mary’s clocks as observed in Sara’sreference frame. This can be done by declaring t ′ in thestandard S → S ′ Lorentz transformation. This is pre-sented in FIG. 2 for Mary’s inertial speed is 0 . c as shetravels a distance of 8 ly, which are the numbers used forthe twin paradox example in Ref. [5]. FIG. 2. (Color online) The standard twin paradox example.Worldline trajectories in S of Sara (red dashed) and Mary(blue solid) for Mary having an initial velocity of 0 . c trav-eling to a distant planet 8 ly away. The lines of simultaneity(gray dot-dashed) for times 1, 3, 5, 7, 9 and 11 years in S ′ areincluded. Verification of Equation (5)
The standard inertial example of the twin paradox pro-vides the simplest mathematical computation for timedilation using Eq. (5): constant v implies∆ τ ′ = Z t r − v c dt = r − v c Z ∆ t dt = 1 γ v ∆ t (6)for each leg of the trip. This calculation also helps stu-dents see the connection between Eq. (5) and what theyare taught from the standard Lorentz transformations. Constant acceleration
Equation (5) can then be used with Mary undergoingarbitrary accelerations. Uniform acceleration in S or S ′ are intuitive cases.Uniform acceleration in S ′ is a well-studied problemdiscussed in several textbooks [2, 9–11]. Although thisproblem could lead to discussions clarifying the truemeaning of the EEP, math and analysis for this exam-ple might be difficult for early undergraduate students.The additional times acquired for the turnaround are notnegligible: assuming v i = 0 . c and g ′ = 9 . leadsto the additional measured time to be 2 .
59 years in S and 2 .
13 years in S ′ .Acceleration in S is a simpler scenario, which we intro-duce below. However, constant acceleration in S couldlead to Mary going faster than c , so one must designproblems such that the Lorentz factor remains real.We consider the following problem: at T i = 0, Sara ispassed by Mary, who is traveling in the positive directionwith initial speed v i while but undergoing a constant ac-celeration of − a in S . The worldline trajectories forthis problem are depicted in Fig. 3. According to Sara, FIG. 3. (Color online) Kinematic trajectories in S . Theworldline trajectories of Sara (red dashed line) and Mary (bluesolid curve) for Mary having an initial velocity of 0 . c and isundergoing a constant acceleration of a = 9 . . Mary’s trajectory is kinematic: v = v i − a t . In S , theadded time at the turnaround is ∆ t = 2 v i /a , while in S ′ , it is ∆ τ ′ = cg v i c r − (cid:16) v i c (cid:17) + arcsin (cid:16) v i c (cid:17) . (7)Taking v i = 0 . c and a = 9 . , the time measuredfor the turnaround is ∆ t ≃ .
55 years in for Sara and∆ τ ′ ≃ .
37 years for Mary. These times are not negligi-ble compared to 10 years. It should also be noted that,during this time interval, Mary experiences a maximumacceleration of 4 . g . TYPICAL INTRODUCTIONS OF GR
Undergraduate physics students might only be ex-posed to GR in their sophomore-level modern physicscourse. Coverage of GR at this level is very qualitative.This coverage typically starts with the EEP, which canbe stated as follows:An observer cannot perform any local experimentto determine whether they are being uniformly ac-celerated or stationary in the presence of a uniformgravitational field.Afterwards, some authors invoke the EEP to illus-trate the bending of a beam of light due to gravity bydrawing an equivalence to the path of light observed inan accelerating container [3–8]. However, some of thesebooks violate the local nature of the EEP[5, 7, 8] byillustrating the light source outside of the container.
Suggested Additional GR Material
The term local in the EEP is very important: acceler-ation is not globally equivalent to gravity. A third partyobserver will measure a difference between the time evo-lution of one person stationary in a uniform gravitationalfield and another undergoing uniform acceleration in freespace. The rate of time dilation for the stationary indiv-itual is constant, while acceleration of the other implies γ v increases with time. Time dilation for the stationaryobserver in an uniform gravitational field can be approx-imated using the Schwartzschild metric provided below. Suggested Introductory GR Material
In practice, GR involves setting up Einstein’s fieldequations for a particular scenario and then solving forthe spacetime metric g µν . g µν determines how time andlengths are related to displacements [11]. Consequently, g µν quantifies the curvature of spacetime.The field equations are tensor equations which, in theircompact form, are R µν − Rg µν = 8 πGc T µν (8)where R µν is the Ricci tensor , R is its contraction, G isNewton’s gravitational constant, c is the speed of light, T µν is the energy-momentum tensor . The subindices µ and ν each range from 0 to 3. After defining T µν fora given situation, one solves for g µν in Eq. (8). In thisform, Eq. (8) looks deceptively simple. Once unpacked,it yields a set of nonlinear, coupled, partial differentialequations.References [3–8] all mention that massive ob-jects/energy generate curvature of spacetime. However,none of these mainstream textbooks either present thefield equations, or mention how the curvature of space-time is quantified. Discussions of the field equations, andhow to interpret the metrics they yield, could inspire stu-dents to learn more about general relativity.A geodesic is the ‘shortest path’ to get from one pointin an arbitrary space to another. Common examples in-clude straight lines in Cartesian space (Eq. (10)), or mo-tion on the surface of a sphere. In the context of GR,the metric relates the geodesic to the chosen coordinatesystem. Below are the metric-geodesic relationships forCartesian space ( δ µν ), Minkowski ( η µν ) and general ( g µν )spacetime coordinates. ds = δ jk dx j dx k = ds x + ds y + ds z (9) dW = η µν dx µ dx ν = dt − ds x − ds y − ds z (10) dW = g µν dx µ dx ν (11)where ds is the Cartesian length and dW is, once again,the worldline segment for an arbitrary spacetime. Herethe Einstein summation convention is being used and { j, k } ∈ { , , } .Metrics, geodesics and the Einstein summation con-vention show up in areas of physics ranging from classicalmechanics to quantum field theory. This makes introduc-ing undergraduates to these topics worthwhile. Addition-ally, this coverage would add a quantitative element todiscussions to lessons introducing GR.With metrics and geodesics introduced, analyses canbe performed on famous solutions to the Einstein fieldequations such as the Schwartzchild metric: dW = (cid:18) − GMr (cid:19) dt − (cid:18) − GMr (cid:19) − dr − r d Ω (12)where G is the Gravitational constant, M is the massof a spherical planet, r is the distance away from thecenter of the planet and Ω is the solid angle. This is thesolution for a stationary spherical mass. As stated above,this metric allows students to calculate time dilation dueto gravitational fields as well as provide a quantitativeintroduction to black holes. CONCLUSIONS
In this article, we have pointed out misstatements re-garding SR and GR in widely-used modern physics text-books. We have discussed the merits of covering exam-ples of accelerated reference dynamics in SR to under-graduate physics courses. In addition to clarifying as-pects of the twin paradox, they can help eliminate com-mon misconceptions among physicists that SR is inca-pable of determining time dilation due to acceleration. We have provided a simple model of how accelerationcan be taught in the classroom. We discussed how linesof simultaneity can clarify lessons on time dilation in twinparadox example. We suggested that, while introducingthe EEP, a third party observer would help clarify itstrue meaning, bute also help eliminate the confusion be-tween SR and GR. Finally, we suggested adding basiccoverage of Einstein’s field equations, metrics, geodesicsand the Einstein summation convention into the class-room. Such lessons not only inspire students to learnmore about GR, they add more of a quantitative compo-nent to the discussions. The mathematical tools acquiredin these lessons will also be useful for students when theytake more advanced courses.
PARTICIPATING INSTITUTIONS
We received surveys from several institutions. Ofthe responses received, the participants who disclosedtheir affiliation included the following institutions: Cal-tech, Harvard University, Iowa State University, MIT,Montana State University, New Mexico State University,North Dakota State University, Noth Carolina State Uni-versity, Notre Dame, Texas Tech University, Tufts Uni-versity, University of Alabama, University of Illinois, Uni-versity of Maryland, University of Minnesota, Universityof Nebraska, University of North Carolina, University ofPittsburgh, University of South Carolina, University ofTennessee, University of Texas at Austin and from Wash-inton University, St Louis.
ACKNOWLEDGMENTS
We would like to thank S. Carroll and A. Hamiltonfor their clarifying discussions on the fundamentals ofgeneral relativity. ∗ rpepino@flsouthern.edu[1] Charles W Misner, Kip S Thorne, and John ArchibaldWheeler. Gravitation . Princeton University Press, 2017.[2] Sean M Carroll.
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