Student understanding of time in special relativity: simultaneity and reference frames

Rachel E. Scherr, Peter S. Shaffer, and Stamatis Vokos Department of Physics, University of Washington, Seattle, WA

This article reports on an investigation of student understanding of the concept of time in special relativity. A series of research tasks are discussed that illustrate, step-by-step, how student reasoning of fundamental concepts of relativity was probed. The results indicate that after standard instruction students at all academic levels have serious difficulties with the relativity of simultaneity and with the role of observers in inertial reference frames. Evidence is presented that suggests many students construct a conceptual framework in which the ideas of absolute simultaneity and the relativity of simultaneity harmoniously co-exist.

relativistic kinematics in the laboratory I. INTRODUCTION frame.1 We wanted to expand this research There is growing national interest in base to relativity in order to provide a guide increasing the exposure of students in for the development of instructional 2,3 introductory courses to modern physics materials by ourselves and others. topics, such as relativity. Proponents of This article reports on an investigation enlarging the scope of the curriculum argue of student understanding of time in special that in the beginning of the 21st century the relativity. A major purpose is to identify content of introductory classes should reflect and characterize the conceptual and some of the major intellectual breakthroughs reasoning difficulties that students at all of the 20th century. Others hold that the list levels encounter in their study of special of topics that must be covered is already too relativity. The emphasis is on the relativity daunting. Physics education research can of simultaneity and the role of reference play a pivotal role in this debate. Whether frames. We found that, after instruction, students first encounter modern physics many students are unable to determine the concepts at the introductory level or in time at which an event occurs, recognize the advanced courses, it is important to identify equivalence of observers at rest relative to what students can and cannot do after one another, and apply the definition of instruction and what steps can be taken to simultaneity. We illustrate the process help deepen their understanding of the through which we gradually obtained a material. In addition, analyzing the ways in detailed picture of student thinking by the which students undergo the transition design and successive refinement of a set of between understanding phenomena to which research tasks. they have immedia te access and understanding phenomena that lie outside II. PRIOR RESEARCH their everyday experience can help us There is currently only a small body of identify reasoning skills that are needed for research on student understanding of the study of advanced topics. relativity, mostly in Galilean contexts (in Over the last five years, the Physics particular, relative motion). In many cases, Education Group at the University of the research tasks used (e.g., multiple -choice Washington has been investigating student questions or single questions given to a understanding of key ideas in Galilean, small number of students) do not provide the special, and general relativistic kinematics. kinds of insights that are necessary to Extensive research has already been understand the prevalent modes of student conducted on student understanding of non- reasoning and to develop effective

Student understanding of time in special relativity Scherr, et al. 1 instructional strategies. We review salient III. FOCUS OF THE RESEARCH results from relevant investigations. A major goal of the investigation was to Galilean relativity determine the extent to which students, after Investigations of physics undergraduates instruction, are able to apply basic ideas of in India have identified the belief that special relativity to simple physical reference frames have limited physical situations. The context is one spatial extent. Studies by Panse et al. and Ramadas dimension. The concepts probed are et al. suggest that, for many students, frames summarized below. of reference have limited physical extent. The construct of a reference frame is at Students claim that a body can “emerge” the heart of relative motion. Most courses in from the reference frame of an object by special relativity begin with a discussion of leaving the vicinity of the obje ct (e.g., “a a reference frame as a system of observers ball can be thrown to ‘go outside’ a (or devices) by which the positions and 4 reference frame”). times of events are determined. An investigation by Saltiel and Understanding the concept of a reference Malgrange has identified difficulties with frame forms the foundation for relative motion among eleven-year-old- understanding any topic in special relativity. children and first- and fourth-year university It provides the basis for the determination of 5 students in France. The three groups all kinematical (and other physical) showed little difference in error rates to quantities and serves as the framework for written questions. Many students tended to relating measurements made by different identify an object’s motion as intrinsic, not a observers. The concept of a reference frame quantity that is measured relative to a presupposes an understanding of more basic reference frame. Students tended to make a measurement procedures. For clarity in the distinction between “real” motion, which discussion that follows, we review the basic has a dynamical cause, and “apparent” operational definitions associated with motion, which is “an optical illusion, devoid reference frames, i.e., the determination of of any physical reality.” the position and time of an event and the Special relativity conditions under which two events are Villani and Pacca have demonstrated treated as simultaneous.10 that university students’ reasoning in An event in special relativity is relativistic contexts is similar to that associated with a single location in space observed by Saltiel and Malgrange in and a single instant in time. The position of 6 Galilean contexts. A case study by Hewson an event is defined to be the coordinate label with a physics graduate student illustrated on a rigid ruler at the location of the event. the importance of “metaphysical beliefs” The ruler is envisioned to extend indefinitely (e.g., time is absolute) to his understanding from some chosen origin.11 The time of an 7 of special relativity. The student ni the event is most naturally defined as the study classified certain relativistic effects reading on a clock located at the event’s (including length contraction) as distortions position at the instant at which the event of perception. Posner et al. report similar occurs. The rulers and clocks used by any results in interviews with introductory observer are at rest relative to the observer. 8 students and their instructors. All observers in special relativity are O’Brien-Pride has conducted interviews assumed to be “intelligent observers” who and administered early versions of some of use synchronized clocks. To determine the the research tasks described here in which time of a distant event, an observer corrects university students appear to believe that the for the travel time of a signal originating at order of events depends on observer the event.12,13 Inertial observers at rest 9 location. Her preliminary results provided relative to one another determine the same impetus for the investigation detailed in this positions and times for events (and hence paper. the same relative ordering of events). Such

Student understanding of time in special relativity Scherr, et al. 2 observers are said to be in the same A. Student populations 14 reference frame. Most of the research was conducted at Events are defined to be simultaneous in the University of Washington in courses that a given frame if their corresponding time include special relativity. The study has readings are identical, according to the involved about 800 students from about 30 definition of the time of an event discussed sections of various courses. The populations above. A judgment of the simultaneity of include: non-physics students (in the widely separated events necessarily includes descriptive course for non-majors students); an appropriate definition of the time of a introductory students (in the introductory 15 distant event. The invariance of the speed calculus-based honors course and in the of light has an inevitable and unsettling sophomore-level course on modern physics); implication. Two events at different students in the junior-level courses on locations that occur at the same time in a electricity and magnetism and relativity and given frame are not simultaneous in any gravitation; and students in our upper- other frame. division course for prospective high school The relativity of simultaneity is among physics teachers.16 We also present results the key results of special relativity and one from physics graduate students at the that is particularly difficult to grasp, as University of Washington who participated evidenced by the numerous “paradoxes” that in interviews and others who were given a arise from it. It is not reasonable to expect written question on a graduate qualifying that students (even those facile with examination. In addition, the investigation mathematical formalism) will master all the includes students in the honors section of the intricacies of the counterintuitive results that calculus-based course at one of the other follow from the operational definition of research universities and advanced simultaneity. In this study, we considered a undergraduate students from the other meaningful understanding of relativistic collaborating university. We found that simultaneity to include the ability to identify student performance from all three relevant events, to determine the time at universities was similar. The results, which an event occurs (by correcting for therefore, have been combined.17 signal travel time), and to recognize that the time interval between two events is not B. Research methods invariant but depends on the reference The research was conducted through the frame. Other aspects of student ideas about analysis of student responses to written time and reference frames are not discussed questions and the analysis of interviews with (e.g., synchronization of clocks in relative individual students. The written questions motion, spatial measurements, differences were posed on course examinations and on between inertial and non-inertial frames). ungraded problems given during class.18 In some classes the questions were IV. OVERVIEW OF THE RESEARCH administered before instruction on the This investigation was conducted over relevant concepts; in other classes, the the last five years at the University of questions were administered after Washington and at two other large research instruction. Except where otherwise noted, universities. Fourteen instructors at the no class was given a question twice. University of Washington and one faculty The one-hour interviews were member at each of the other universities conducted with volunteers, who were have cooperated with the Physics Education primarily first-year physics graduate Group in this study. students. Several advanced undergraduate students and a few advanced graduate students also participated. In the interviews, we were able to probe student thinking in

Student understanding of time in special relativity Scherr, et al. 3 greater depth than is possible with short erupted first since its signal travels farther in written questions. The interviews were order to reach the ground-based observer at audiotaped or videotaped and transcribed for the same time as the signal from Mt. later analysis. Rainier. A correct answer can also be obtained C. Research tasks using the Lorentz transformation for time: 2 2 2 -1 2 To obtain an increasingly deeper dt¢ = g (dt - vdx / c ), g = (1 - v c ) . In understanding of how students apply the this context, dt¢ = tH¢ - t R¢ and dt = tH - t R concepts of simultaneity and reference are the elapsed times between the eruptions frame, we used variants of three questions: at Hood and Rainier in the spacecraft frame and the ground frame respectively, is the the Spacecraft question (four versions), the v velocity of the spacecraft relative to the Explosions question, and the Seismologist ground, and dx = xH - xR is the spatial question. All involve two observers with a coordinate separation between the eruptions given relative motion. Students are told the in the ground frame.23 Taking the positive time ordering of the events for one observer direction to be directed from Rainier to and asked about the time ordering of the Hood, then > 0 and . Since 19 v dx > 0 events for the second observer. These dt = 0 (simultaneous eruptions in the questions and their solutions are described ground frame), then dt ¢ <0 . below. 2. Explosions question 1. Spacecraft question The Explosions question is the converse Results from four versions of the of the Spacecraft question. Students are told Spacecraft question are discussed in this that two events occur at different times in a paper. All involve two volcanoes, given frame and are asked if there is another Mt. Rainier and Mt. Hood, that erupt frame in which the events are simultaneous. simultaneously according to an observer at In the Explosions question, an explosion rest on the ground, midway between the occurs at each end of a landing strip with volcanoes.20 A spacecraft moves at a given proper length of 3000 m. In the frame of an relativistic velocity from Mt. Rainier to engineer at rest on the strip, the explosion at Mt. Hood.21 Students are asked questions the right end occurs a time c dt = 1200 m that probe their beliefs about the order of the after the explosion on the left end. (In some eruptions in the moving frame. variations, students were given a time of A correct answer to all versions can be dt = (1200 m)/c = 4 ms. The conversion obtained by qualitative or quantitative seemed to present no difficulty.) Students reasoning or from a spacetime diagram. The are asked whether there is a frame in which following is an example of a qualitative the explosions are simultaneous, and if so, to argument that we would have accepted as determine the relative velocity of that frame. 22 correct. In the spacecraft frame, light from A correct answer can be found through the two eruptions moves outward at the use of the Lorentz transformations.24 The speed of light in spherical wavefronts from spatial separation between the explosions two points that are stationary. In that frame, (dx) is 3000 m and the time separation (c the observer on the ground, who receives dt) is 1200 m. Thus the time duration both signals simultaneously, is moving between the explosions (c dt´) is zero in a backward (i.e., in the direction of an arrow frame that moves from left to right with pointing from the front of the spacecraft speed 0.4c. toward the rear). According to the spacecraft observer, the ground-based 3. Seismologist question observer is closer to the center of the signal The Seismologist question probes from Mt. Rainier at the instant that observer student understanding of reference frames receives both signals. The spacecraft and simultaneity within a single reference observer therefore concludes that Mt. Hood frame. The context is similar to that of the

Student understanding of time in special relativity Scherr, et al. 4 Spacecraft question: two volcanoes, A. Failure to recognize spontaneously Mt. Rainier and Mt. Hood, suddenly erupt that simultaneity is relative and a seismologist at rest midway between We refer to the first version of the them sees the eruptions at the same instant. Spacecraft question as undirected. We were The Seismologist question differs from the interested in finding out whether or not Spacecraft question in that the second students would recognize, without observer (the “assistant”) is not moving, but prompting, that simultaneity is relative and, remains at rest relative to the ground at the if not, the degree of prompting that is base of Mt. Rainier. Students are asked necessary for them to apply the relativity of whether Mt. Rainier erupts before, after, or simultaneity. at the same instant as Mt. Hood in the reference frame of the assistant. Spacecraft Question: Undirected version To answer the Seismologist question The students were asked to draw correctly, students must be able to apply the spacetime diagrams for both the ground and definition of simultaneity and understand the spacecraft frames. They were told to show role of a reference frame in establishing a the volcanoes, the spacecraft, and the common time coordinate for observers at eruption events. They were not asked rest relative to one another. Since the explicitly whether the eruption events are seismologist and the assistant are in the simultaneous in the spacecraft frame. same reference frame, they obtain the same Rather, we inferred their ideas indirectly answer for the order of the eruptions. Since from their diagrams. the seismologist is equidistant from the We administered this undirected version mountains, the signal travel times are the as an interview task to 7 graduate students same. Therefore, the eruptions occurred at and later to 20 advanced undergraduate the same time in the frame of the students enrolled in a course in special and seismologist and the assistant. general relativity. All the graduate students Commentary had had undergraduate instruction in special At each stage of our study, we tried to relativity and were studying relativistic determine whether student responses truly kinematics, dynamics, and electromagnetism reflected their understanding of the material. in their graduate-level electricity and For instance, we wanted to determine the magnetism course at the time of the extent to which specific difficulties are interviews. The undergraduates had linguistic or conceptual and the extent to completed instruction on the relativity of which mistaken beliefs are easily addressed simultaneity. All had worked with or deeply held. To this end, we continually spacetime diagrams in their current or refined the research tasks. Results from previous courses. earlier tasks guided us in designing new All the graduate students correctly drew questions that would enable us to probe the worldlines of each object in the student thinking more thoroughly. spacetime diagram for each frame. However, only one recognized V. PRELIMINARY INVESTIGATION spontaneously the relativity of simultaneity OF STUDENT UNDERSTANDING in this context. All the others indicated on OF THE CONCEPTS OF their spacetime diagrams and in their verbal SIMULTANTEITY AND explanations that the two eruptions had REFERENCE FRAMES identical vertical (time) coordinates in the Our preliminary investigation of student ground frame and identical time coordinates understanding of special relativity was based in the spacecraft frame. [See Fig. 1 for on two versions of the Spacecraft question: examples of correct and incorrect spacetime an undirected and a directed version. The diagrams.] The results from the two versions and the results from each are undergraduate students were similar. All described below. drew spacetime diagrams that included the

Student understanding of time in special relativity Scherr, et al. 5 t t´ t´

Spacecraft Spacecraft Spacecraft

Eruption Eruption Eruption Eruption Eruption Eruption x x´ x´

Mt. Rainier Mt. Hood Mt. Rainier Mt. Rainier Mt. Hood Mt. Hood

(a) (b) (c)

Figure 1: Spacetime diagrams for the first (undirected) version of the Spacecraft question. (a) Correct diagram for the ground frame. (b) Correct diagram for the spacecraft frame. (c) Typical incorrect diagram for the spacecraft frame drawn by students. correct features except that about 85% Spacecraft Question: Directed version denoted the eruption events as simultaneous in both frames. Some of the difficulty may In the directed version of the Spacecraft have been related to a lack of facility in question, students are asked explicitly relating spacetime diagrams for two frames. whether, in the reference frame of the Nonetheless, the majority of the students spacecraft, Mt. Rainier erupts before, after, failed to recognize spontaneously that or at the same time as Mt. Hood. They are simultaneity is relative and to draw their also asked to find the time between the diagrams appropriately. eruptions. The relativity of simultaneity is arguably We administered this directed version of the central result of relativistic kinematics the Spacecraft question to 49 students on an and a key consequence of the Lorentz examination in an introductory honors transformations. The fact that advanced calculus-based physics class. The students students do not apply this idea had completed the study of the relevant spontaneously is a matter of concern. On material. All students answered that the the other hand, the fact that the time order of eruptions are not simultaneous in the events is not the same in all frames is among spacecraft frame. (Very few students had the most counterintuitive ideas in special done so on the undirected version.) relativity. We wondered whether students However, only about 45% gave the correct might apply the simultaneity of relativity if time ordering (Hood erupts first in the prompted explicitly to do so. spacecraft frame) with correct reasoning.25 About 25% of the students gave the correct B. Failure to apply spontaneously the time order of events but used incomplete formalism of a reference frame in reasoning to support their answers. About determining the time of an event 25% of the class gave the reverse time We decided to develop a new version of ordering. the Spacecraft question that would be more Essentially all of the students who gave directive than the first. This version was a correct answer with incomplete reasoning written in collaboration with the course answered in a similar way. The responses instructor. The terminology was identic al to below are typical. that used in class. “Mt. Hood erupted first because the spacecraft is moving towards it, so the wavefront of the eruption

Student understanding of time in special relativity Scherr, et al. 6 of Mt. Hood will reach the craft One student claimed necessary information first.” (introductory student) was missing from the problem statement.

“Hood first, because the “It would depend on where the spacecraft will encounter those spacecraft was when the first wavefronts first.” (introductory explosion occurs. If it is close student) enough to Hood that the distance between the ship and Hood plus We categorize the reasoning given by the distance the ship travels while these students as incomplete since these the light is en route, then it sees students describe only the order in which the Hood explode first.” signals from the distant events reach the (introductory student) spacecraft. They make no explicit mention of the relative velocity or the relativity of The students quoted above all failed to simultaneity. The sequence in which the treat the spacecraft observer as signals are received does not provide enough representative of a class of observers, all information to determine the time ordering moving with the same velocity. They of the eruption events. Different choices of seemed to interpret the time of an event as observer location result in different an observer- (not frame-) dependent reception orders and some students might quantity. These results suggest that have obtained the correct answer by a students, on their own, fail to apply the fortuitous choice of observer location (e.g., formalism of reference frames (i.e., a system half-way between the two volcanoes). We of clocks and rods) in defining the time of started to suspect the presence of incorrect an event. ideas when we realized that most of the Commentary remaining students (about 25%) seemed to The versions of the Spacecraft question have made a different assumption about the used in the preliminary investigation are spacecraft location. These students obtained similar to many end-of-chapter questions on the opposite answer for the time ordering of relativistic simultaneity. Students are told the events but gave explanations similar to that two events are simultaneous in one those illustrated above. The following frame (S) and asked about the time order of response was typical. the events in another frame (S¢) that moves “The observer will witness Mt. relative to the first with a given velocity. Rainier erupting first because they In the context of the Spacecraft are directly over Rainier when the question, we found that many students fail to explosion happens, so the light apply spontaneously the relativity of from the explosion has less simultaneity. When prompted to think distance to travel than the light explicitly about the order of the events, from Mt. Hood’s explosion.” essentially all students state that the events (introductory student) are not simultaneous in the spacecraft frame. However, most reason incorrectly. They Several students regarded both the tend to focus on the relative position of the velocity and position of the spacecraft as spacecraft and volcanoes and fail to important in the time ordering of the recognize that the relative velocity eruption events. However, like the students determines the time order of the eruptions in above, they focused on the reception of the the spacecraft frame. light signals by the observer. They did not The results from the undirected and treat relative motion as the determining directed versions of the Spacecraft question feature of a reference frame but as a factor suggested the presence of serious conceptual that complicates the calculation of the time and reasoning difficulties with basic at which the observer receives the signals. concepts in special relativity. We

Student understanding of time in special relativity Scherr, et al. 7 considered, however, the possibility that student responses did not reflect what Mt Rainier and Mt. Hood, which are 300 km students actually thought. We needed to apart in their rest frame, suddenly erupt at the same time in the reference frame of a probe more deeply into the nature of their seismologist at rest in a laboratory midway conceptions of time, simultaneity, and between the volcanoes. A fast spacecraft reference frames. flying with constant speed v = 0.8c from Rainier toward Hood is directly over Mt. VI. DETAILED INVESTIGATION OF Rainier when it erupts. STUDENT UNDERSTANDING OF THE CONCEPTS OF TIME, Let Event 1 be “Mt Rainier erupts,” and SIMULTANEITY, AND Event 2 be “Mt. Hood erupts.” REFERENCE FRAMES In the reference frame of the spacecraft, This section is divided into three inter- does Event 1 occur before, after, or at the related and inter-dependent parts. Part A same time as Event 2? Explain your deals primarily with a prevalent and reasoning. persistent student interpretation of simultaneity that is observer-dependent. Figure 2: Location-specific version Many students believe that the time order of of the Spacecraft question. distant events is determined by the time order in which signals from the events are thought the observer’s location affects the perceived by an observer. Part B presents time ordering of the events. By describing evidence that many students have a deeply the eruptions as events, we tried to make held underlying belief that simultaneity is clear to students that the time interval of absolute. Part B also describes how students interest is not that between the reception of often attempt to reconcile these two the light signals by the moving observer, but contradictory beliefs with each other and rather that between the emission of the with what they have been taught about the signals by the volcanoes. relativity of simultaneity. Student Spacecraft question: Location-specific interpretations and beliefs about version simultaneity described in Parts A and B have direct implications on student In the third (location-specific) version of understanding of the role of an observer in a the Spacecraft question, students are told given frame. Part C is devoted to an that the spacecraft, which is flying from exploration of student beliefs about the Mt. Rainier to Mt. Hood, is over Mt. Rainier concept of a reference frame. at the instant Mt. Rainier erupts. The eruption events, which are simultaneous in A. Belief that events are simultaneous if the ground frame, are explicitly labeled as an observer receives signals from the Event 1 (Mt. Rainier erupts) and Event 2 events at the same instant (Mt. Hood erupts). Students are to We decided that the Spacecraft question determine whether, in the reference frame of would be a better probe of student thinking the spacecraft, Event 1 occurs before, after, about simultaneity if two changes were or at the same time as Event 2. [See Fig. 2.] made: (1) specifying not only the velocity Table I summarizes the results of the but also a location for the moving observer location-specific Spacecraft question, which and (2) rewording the question to describe was administered as an ungraded written the eruptions as spacetime events. By question to non-physics students, choosing the observer location introductory students, and advanced appropriately, we sought to distinguish undergraduate students. The question was between students who obtained a correct also given as an interview task to advanced answer for correct reasons and students who undergraduates and physics graduate students. In most cases, the question was

Student understanding of time in special relativity Scherr, et al. 8 Table I. Results from the third (location-specific) version of the Spacecraft question. Students are told the eruptions are simultaneous for one observer and are given the location of an observer in a second frame moving relative to the first. They are asked for the order of the eruptions in the frame of the second observer. [Percentages have been rounded to the nearest 5%.]

Written question Interview task

Introductory students Non-physics Advanced (Honors calculus- Advanced undergraduates students undergraduates based physics) and graduate students Before After Before After Before After instruction † instruction instruction instruction instruction instruction ‡

(N = 23) (N = 16) (N = 67) (N = 73) (N = 20) (N = 93) (N = 11) % % % % % % %

(N) (N) (N) (N) (N) (N) (N) Correct answers (Hood erupts first) 15% 15% 5% 10% 15% 25% 25% with correct reasoning * (3) (2) (3) (8) (3) (24) (3) or incomplete reasoning Simultaneous eruptions 45% 30% 20% 5% 25% 20% 0 (reasoning consistent with being (10) (5) (12) (5) (5) (19) (0) based on absolute simultaneity) Rainier erupts first (reasoning consistent with being 35% 45% 70% 75% 45% 40% 55%

based on the times at which signals (8) (7) (46) (55) (9) (39) (6) are received by the observer) Other 10% 15% 10% 5% 15% 10% 20% (e.g., student stated not enough (2) (2) (6) (5) (3) (11) (2) information given)

* Some students gave a correct answer with reasoning that was incomplete, but not incorrect. Although it was not possible to tell whether they were correct in their reasoning, in this article the responses are treated as correct. † These students had received research-based instruction on reference frames in Galilean relativity in which they had developed a definition for the time of an event. We have evidence that this special instruction may have been responsible for the low percentage of students answering that Rainier erupts first. The students had had no instruction in special relativity. ‡ One of the five classes of advanced undergraduates who were given the location-specific version of the Spacecraft question had previously been given a variant of the question as a pretest. The results were comparable to the other four classes.

posed after lecture instruction on the The majority of students in every relativity of simultaneity. As can be seen student group answered incorrectly. from the table, prior instruction had little Analysis of student responses revealed two effect on student performance.26 related modes of incorrect reasoning. In every student group, fewer than 25% 1. Tendency to associate the time of of the students gave a correct response an event with the time at which an (ignoring reasoning). Of those who gave the observer receives a signal from the correct order, most used complete reasoning. event Only a few used incomplete reasoning similar to that in the preliminary Despite having been told explicitly that investigation (e.g., “Hood erupts first since the events of interest are Event 1 (Mt. the spacecraft is flying towards Hood.”). Rainier erupts) and Event 2 (Mt. Hood erupts), many students attributed the time of

Student understanding of time in special relativity Scherr, et al. 9 each eruption to the time at which a given distant events are simultaneous is, therefore, observer sees the eruption. Both the judged by many students only on the basis advanced students in the interviews and the of the time order of the received signals. introductory students on the written The difficulties seem to be intimately tied to problems made this error. The following their ideas of reference frames. statements are typical. The tendency of students to interpret simultaneity in terms of signal reception “The spacecraft is near Rainier, so had, thus far, prevented us from determining he gets the signal about the same whether or not the students recognized that time Rainier erupts. So the the events themselves are not simultaneous spacecraft pilot would say Rainier in all frames. We thought, however, that the erupts before Hood.” (graduate failure to use the emission events might be student) easy to correct with upper-level “Mt. Rainier erupts first because undergraduate students and graduate the light from Mt. Hood takes students. We wondered whether students time to reach the spaceship.” would apply the relativity of simultaneity (introductory student) properly if, during interviews, it were pointed out to them that the reception events 2. Tendency to regard the observer are not the ones to consider. as dependent only on his or her personal sensory experiences B. Belief that simultaneity is absolute The failure to distinguish between the As described earlier, we found that time of an event and the time at which an many students fail to apply spontaneously observer sees that event occurring did not the idea of relativity of simultaneity after seem to be a superficial error but seemed to instruction. On the other hand, we also found that many students appear to believe have deep roots. Many students failed to recognize that an observer is not isolated but in a type of relativity of simultaneity that we has access to information provided by other would term excessive. Students often claim observers in his or her frame. that observers at different locations determine different time orderings for events “For example if he looks at [the based on the reception of signals from the volcano, it] looks peaceful. There events. We now present evidence that these is nothing going on. So he would apparently contradictory beliefs (that say it hasn’t erupted. I would say, simultaneity is absolute, and that to state that something happened simultaneity is “excessively” relative) often you have to have any evidence, coexist harmoniously. Our results suggest and he hasn’t got any evidence that many students believe that simultaneity that something happened. So it is relative only in the limited sense that hasn’t happened.” (graduate signals from events arrive at different student) observers at different times – and that Commentary fundamentally, simultaneity is absolute. The third, location-specific version of Spacecraft question: Explicit version the Spacecraft question provided insights into student thinking about simultaneity that We wanted to ensure that students were the first two versions had not. The not hindered by semantic misinterpretations responses to the written questions and of technical terms such as “intelligent observer” and “reference frame.” In an interviews show that many students very strongly associate the time of an event with effort to remove possible ambiguity from the the time at which an observer receives a task and to probe even more explicitly than signal from the event. Whether or not before, we designed a fourth version of the Spacecraft question. We refer to it as the

Student understanding of time in special relativity Scherr, et al. 10 In this problem, all events and motions occur along a single line in space. Non-inertial effects on the surface of the Earth may be neglected. Two volcanoes, Mt. Rainier and Mt. Hood, are 300 km apart in their rest frame. Each erupts suddenly in a burst of light. A seismologist at rest in a laboratory midway between the volcanoes receives the light signals from the volcanoes at the same time. The seismologist’s assistant is at rest in a lab at the base of Mt. Rainier. Define Event 1 to be “Mt. Rainier erupts,” and Event 2 to be “Mt. Hood erupts.” A fast spacecraft flies past Mt. Rainier toward Mt. Hood with constant velocity v = 0.8c relative to the ground (g = 5/3). At the instant Mt. Rainier erupts, the spacecraft is directly above it and so the spacecraft pilot receives the light from Mt. Rainier instantaneously. All observers are intelligent observers, i.e., they correct for signal travel time to determine the time of events in their reference frame. Each observer has synchronized clocks with all other observers in his or her reference frame. For each intelligent observer below, does Event 1 occur before, after, or at the same time as Event 2? Explain. • Seismologist • Seismologist’s assistant • Spacecraft pilot

Figure 3: Explicit version of the Spacecraft question.

explicit Spacecraft version because it makes travel time. When students used technical explicit the correction for signal travel time terms such as “reference frame,” the employed by intelligent observers. [See Fig. interviewer probed their understanding of 3.] the term. If a student’s interpretation We initially administered the question as differed from the conventional an interview task so that we could carefully interpretation, the interviewer attempted to observe and correct, as necessary, the way in correct the student, and asked the student to which students interpreted the question. We reconsider his or her response in light of the probed both qualitative and quantitative accepted interpretation. reasoning. The interviews were conducted The results from the interviews and the with 2 advanced undergraduates and 5 written doctoral examination were similar. graduate students. The question was In the interviews, a correct answer was subsequently given to 23 graduate students given by 1 of the 2 advanced as part of a written qualifying examination undergraduates, and 2 of the 5 graduate for doctoral candidacy. students. On the written question, 7 of the In the explicit version of the Spacecraft 23 graduate students (30%) answered question, students are told that “observers correctly. The quotes given in the are intelligent observers, i.e., they correct for discussion below are from the interviews signal travel time in order to determine the since the interview format allowed us to time of events in their reference frame. probe student reasoning in greater detail Each observer has clocks that are than is possible in a written question. synchronized with those of all other 1. Tendency to regard the relativity observers in his or her reference frame.” In of simultaneity as an artifact of the course of the interview, students were signal travel time reminded repeatedly to consider all observers as making corrections for signal During the interview, many students seemed to resist thinking about simultaneity

Student understanding of time in special relativity Scherr, et al. 11 in terms of emission, rather than reception, spacecraft frame? Considered of the signals. As the interviews progressed, separately from the time of we realized that part of the difficulty was the light hit[ting] the that they believed strongly that, in every spacecraft. reference frame, the two events occurred at the same instant. Many seemed to treat the S: I’m not really sure how to do non-simultaneity of the reception of the that. It would seem to me that signals as a way of reconciling this belief just logically, it doesn’t with what they thought they had learned matter, it would still go about the relativity of simultaneity. Rainier, Hood. Four of the seven advanced undergraduate and graduate students who I: You’re speculating, if I can responded to the explicit Spacecraft paraphrase you, that after the question clearly articulated the idea that the spacecraft [observer] made order of events in the spacecraft frame is corrections for the fact that it determined by the order in which the signals took the Hood signal some from the events arrive at the spacecraft. time to get to him, after he (About 60% of the graduate students did so made those corrections he on the qualifying examination.) Their would wind up concluding explanations were similar to those quoted that Rainier went first? previously in response to the other versions of the Spacecraft question. When reminded S: If he did everything right then to consider the spacecraft observer as he would have appropriately making corrections for the signal travel come up with the amount of time, all of these interview subjects claimed time the signal would have that after making such corrections, the traveled, the distance between intelligent observer in the spacecraft would the two mountains would determine the eruptions to have been probably also have been simultaneous. different to him, and so he’d have to account for that too, “Using her correction, assuming but once he made all of those she’s intelligent…I mean if she accounting things then he measured the effect would have to say that yes, relativistically, she would measure they went up at the same time. them happening at the same time if she subtracted the time she I: Oh, at the same time. I calculated.” (graduate student) thought you said Rainier went first. “If we are in relative motion we will measure different distances S: No, no, no, he would see and so on but if we are all Rainier go up first, but he intelligent observers we will all would eventually after doing figure out that the events were all of the math would agree simultaneous in our rods-and- with [two observers at rest on clocks reference frame.” (graduate the ground], that they went up student) at the same time. (advanced An advanced undergraduate reached a undergraduate student) similar conclusion after some clarification: These students appeared to believe that I: Can you tell which order the events simultaneous in the ground frame eruptions occur in, in the would be simultaneous according to an observer in any reference frame who made

Student understanding of time in special relativity Scherr, et al. 12 appropriate corrections for signal travel S: Relativity of simultaneity is time. In the following example a graduate this local thing. It’s not the student compared the spacecraft observer to rods and clocks thing, because an observer on the ground under the if we are intelligent, we spacecraft, at Mt. Rainier: correct for that. It’s this thing that if I see them at different “There is no real difference times, they occurred at between the spacecraft and the different times in my [observer on the ground under the reference frame. (graduate spacecraft]. Because I said the student) signals reach [that observer] at different times, but he can Responses such as these seemed to determine at which times the indicate that the belief that simultaneity is signals were emitted. …So I can absolute is deeply held.27 Any appearance do the same thing on the to the contrary is only that – a visual spacecraft: I might see the signals appearance due to differences in the at different times but I can figure reception of signals from the events. This out that they happened at the same belief was typical among students who time.” (graduate student) asserted that the order of events in the spacecraft frame is the order in which Such responses indicated students’ signals arrive at the spacecraft. belief that simultaneity is absolute. Yet students, in their discussion of relativistic I: This thing about events that effects, appeared to have heard the idea of are simultaneous in one the relativity of simultaneity. How could reference frame, not being these incompatible ideas coexist so openly? simultaneous in another The resolution to this apparent contradiction reference frame? Do you came from in-depth probing. have a sense of where that comes from? S: There are really two separate kinds of reference frames. S: Light has a finite speed, so it’s There is the kind of reference going to take some time for frames with all those rods and the information to travel from clocks extending to infinity, point A to point B wherever like in [the textbook]. But in the observer is. This is a practice, nothing happens pretty good example actually. except right where you are. One observer is right between So really, your reference the mountains and he sees frame is something you carry them at the same time, the around with you … other observer is not and so he sees them at different times. I: There is this thing about (graduate student) simultaneity being relative, about events that are 2. Tendency to regard the Lorentz simultaneous in my reference transformation for time as frame not necessarily being correcting for signal travel time simultaneous in another reference frame. Which kind In deriving the relativity of simultaneity, of reference frame does that many instructors invoke the Lorentz refer to? transformation for time. We found that conceptual difficulties with reference frames and the time of an event can prevent students from interpreting appropriately the

Student understanding of time in special relativity Scherr, et al. 13 terms in the Lorentz transformations. In proportional since the travel time for the particular, many students appeared to signal from Hood is reduced as the speed of believe that the Lorentz transformations the spacecraft in the ground frame is constitute a correction for signal travel time: increased. S: In the reference frame of the 3. Tendency to treat simultaneity as spacecraft, does Event 1 independent of relative motion occur before, after, or at the Some students stated explicitly that same time as Event 2? … relativity of simultaneity is not directly Before. Even though the related to relative motion. Even graduate spacecraft is traveling very students expressed this idea. As in the fast, I would say that it’s preliminary investigation, some students right next to Mt. Rainier so appeared to believe that relative motion does it’s going to see Mt. Rainier play a role in the timing of events in the go off, and even though it’s spacecraft frame – but only to the extent that traveling towards Mt. Hood it influences the reception of signals by the and the light from Mt. Hood spacecraft. is traveling towards it, it will Commentary still take some amount of We have found that students often time for the information of incorporate the relativity of simultaneity into Mt. Hood exploding to reach their own conceptual framework in a way it. that allows them to continue to believe in absolute simultaneity. They do so by I: Okay, so it’ll see Mt. Rainier treating the time of an event as the instant at first. In the spacecraft’s which that event is seen to occur by an reference frame, after he observer and attributing the relativity of makes any corrections for simultaneity to signal travel time. Such signal travel time that might incorrect beliefs can insulate students from be appropriate – gaining an understanding of the relativity of simultaneity as a consequence of the S: Are we including Lorentz invariance of the speed of light. Instructors transformations in that? and textbooks often admonish students to (advanced undergraduate distinguish between the corrections of a student) finite signal travel time and the inevitability The “desynchronization term” in the of the relativity of simultaneity.29 Lorentz transformation for time (– vdx/c2) Apparently, such admonitions are presented particular difficulty for students.28 insufficient. Several cited that term in support of the idea that the time of an event is influenced by the C. Belief that every observer constitutes position of an observer relative to the event: a distinct reference frame The Spacecraft question discussed thus “[This term] is the correction for far had originally been designed to probe the travel time of the light. The student understanding of simultaneity. The time I have to wait in my frame to results suggested, however, that we needed see one event and then the next to investigate more deeply their one.” (graduate student) understanding of reference frames. We therefore developed other questions to probe The student above went on to express student beliefs about simultaneity, reference his confusion about the fact that the term is frames, and the role of observers. Results proportional to the speed of the spacecraft. from two of these, the Explosions question He thought it should be inversely

Student understanding of time in special relativity Scherr, et al. 14 and the Seismologist question, are described “If we travel at a speed in which below. [one] side is the chasing side, it will be ahead by a certain time. Explosions question Now if we set it to be In the Explosions question, students are cdt = 1200 m ahead, and we stand given the time interval between two non- where the engineer is standing, simultaneous events in one frame and asked we’ll see the explosions at the whether there is second frame in which the same time. So, you must be events are simultaneous. As in the first two traveling at 0.4c, and must be at versions of the Spacecraft question, no the point where the engineer is mention is made of a specific observer in the standing.” (introductory student – second frame. However, students often italics added for emphasis) raised the issue of the observer location spontaneously. The student seems to believe that the The question has been given in written explosions are simultaneous only for one form to introductory students in sophomore observer in the ‘moving’ frame (the observer modern physics courses and to students in a who sees them simultaneously). Another junior-level course on electricity in introductory student answered similarly. magnetism. In both cases, the question was administered on an examination after “There is no ‘frame’ that you will always see them [the two instruction in special relativity. We have explosions] at the same instant, also asked the Explosions question during but there is a position. We can be interviews with 5 advanced undergraduate [a certain number of] meters from and 12 graduate students. the right [end] and see them at the The percentage of correct responses on the written question was 40%. In the same time.” (introductory student) interviews, 3 of the 5 undergraduates and 7 The student seems to have interpreted of the 12 graduate students answered the question “Is there a frame in which the correctly. Many students attempted to solve events are simultaneous?” to mean “Is there the questions mathematically and it was an observer who sees the events at the same often difficult to characterize their errors. time?” This student apparently believes that However, the responses of students at all a set of observers at rest relative to one levels often reflected the following another would not agree that the explosions difficulty. are simultaneous since such observers would not all receive light from the two explosions 1. Tendency to treat observers at the at the same instant. The student was unable same location as being in the same to apply the idea of a reference frame as a reference frame, independent of system for measuring the time of events. relative motion The following student correctly Seismologist question determined the relative speed of the frame in In the Seismologist question, students which the two explosions are simultaneous. are asked about the relative ordering of two [The student uses the rule developed in class events for a seismologist and an assistant at that clocks in moving frames that follow (or rest relative to one another. The question “chase”) other clocks read earlier times.] was designed to probe whether students However, the student places an additional would incorrectly treat simultaneity as constraint on the solution by specifying a relative, even for two observers in the same location for the observer such that the reference frame. observer would see the explosions The question has been administered to simultaneously. introductory and advanced students both as an examination question and as an ungraded

Student understanding of time in special relativity Scherr, et al. 15 Table I. Results from the third (location-specific) version of the Spacecraft question. Students are told the eruptions are simultaneous for one observer and are given the location of an observer in a second frame moving relative to the first. They are asked for the order of the eruptions in the frame of the second observer. [Percentages have been rounded to the nearest 5%.]

Written question Interview task

Introductory students Advanced Non - physics (Honors calculus- Advanced undergraduates undergraduates students based physics) and graduate students During After Before After Before After instruction instruction instruction instruction instruction instruction ( N = 40) ( N = 26) ( N = 88) ( N = 79) ( N = 48) ( N = 63) ( N = 17) % % % % % % % (N) (N) (N) (N) (N) (N) Correct answers 20% 15% 10% 30% 40% 40% 60% (simultaneous eruptions) (19) (6) (3) (25) (20) (24) (10) regardless of reasoning 60% Rainier erupts first 65% 75% 65% 55% 50% 40% (49) (25) (20) (57) (26) (33) (7) Other ( e.g., Hood erupts first, student 25% 10% 15% 5% 5% 10% 0

stated not enough information (9) (3) (12) (5) (2) (6) (0) given)

written question. It has also been given first, and until he sees its light, during interviews to advanced effectively it hasn’t erupted yet.” undergraduates and graduate students. (introductory student) Relatively few students at any level answered correctly about the time order of “Reference frames, they’re events in the frame of the assistant. Only dependent on position, so that’s between 15% and 40% of the students why [the seismologist] and [the answered correctly on the written questions. assistant] measure two different As shown in Table II, even advanced things. I think of them being in students had difficulty answering this different reference frames simply question correctly. Many of the errors because some people would say indicated the following tendency. ‘Well, if they’re in the same 2. Tendency to treat observers at rest reference frame everything should relative to one another as being in happen the same.’ And it doesn’t separate reference frames happen the same, because they Essentially all of the students who measure two different times. answered the Seismologist question That’s why I am tempted to say incorrectly stated that Mt. Rainier erupts although they’re both at rest with first in the frame of the assistant. Some of respect to each other, they’re in these students were explicit about their different reference frames.” interpretation of the term “reference frame.” (graduate student)

“Assuming the assistant is his “… ‘in the reference frame of the reference frame, Rainier will erupt assistant’ means you’re sitting first because he will see its light there and waiting for events to

Student understanding of time in special relativity Scherr, et al. 16 happen, and you record them I: So if he made measurements when you see them, and that’s of [times and distances], and when you mark them down so he knew the speed of the that’s when they happen.” signal and so on, what would (graduate student) he conclude about the eruption times? … In the view of the student above, “the reference frame of the assistant” consists of S: For the assistant Mt. Rainier a single observer (the assistant). Many would erupt before Mt. Hood. students treated a reference frame as being This is what I would say. local to the position of an observer.30 In one version of the Seismologist I: In the assistant’s reference question, students are asked explicitly about frame Rainier would erupt both (1) the order in which the light signals first? from the eruptions reach the assistant and (2) the order of the eruption events in the S: Yeah, definitely. reference frame of the assistant. In their responses, some students stated explicitly I: And what if the assistant that the questions were identical. made measurements, and had assistants of his own, “If by his reference frame, you whatever measurements he mean, ‘When did he see it?’ it needed to make to reach a would be before.” conclusion about the timing of Despite the fact that both observers are the eruptions? After all those in the same frame, some students referred measurements, the assistant explicitly to the relativity of simultaneity or would say, “In my reference to the lack of synchronization of clocks. frame, Mt. Rainier erupted first”? “Mt. Rainier erupts first in the assistant’s frame according to the S: Yes. (graduate student) relativity of simultaneity.” Unlike some of the introductory students, nearly all of the advanced students “According to the assistant distinguished clearly between emission Mt. Rainier erupted first because events (the eruptions) and reception events the two clocks (eruptions) are (the arrival of the light from an eruption at a unsynchronized according to him particular observer). They also recognized and the light from Rainier is that the travel time of light is relevant. closer.” (advanced undergraduate However, nearly all of these students student) indicated a belief that such corrections were The belief that each observer constitutes not appropriate for observers to make in a separate reference frame was common and attempting to determine the time of an event seems to be quite strongly held. The in their reference frames. During the dialogue below provides an example. interviews, many expressed the view that “reference frame” describes what an S: The assistant is at Mt. Rainier, observer perceives at a particular location. and Mt. Rainier erupts, and One student went so far as to express the Mt. Hood erupts at the same belief that the time ordering of events in an time. But he can’t see it the observer’s frame depends on which signals same time, because if you that observer is able to detect: look at this mountain you always see in the past.

Student understanding of time in special relativity Scherr, et al. 17 “Within normal human ability to Mt. Rainier, with my fancy comprehend time, I would say that Rolex. And you determine the eruptions are going to be at the that at exactly noon, you took same time. But if he’s blind, he’s your vitamins. And suppose I going to hear Rainier for sure go wanted to figure out what off before Hood. And so he’s time you took your vitamins. going to say that Rainier went off before Hood because it’s going to S: If you’re looking at me take much longer for the sound through a telescope you will from Hood to get there.” (graduate look at my clock and it will student) say 12, but you will look at your clock and it will say 12 The student quoted above was also plus ? t. asked to sketch pictures of the explosions as they happen in the seismologists’ reference I: So in my reference frame, did frame. She responded, “Which you take your vitamins at 12? seismologist? The one at the base of the Or at 12 plus ? t? mountain, or the one in the middle?” At the interviewer’s cautious response of “Both,” S: Twelve plus ? t. (advanced she sketched two sets of pictures – one for undergraduate student) the seismologist in the middle, and one for the assistant at Mt. Rainier. [See Fig. 4.] Commentary She explicitly indicated that the eruptions When students remarked that the were simultaneous according to one assistant can only know what happens at his observer but not the other. own location, they were indicating that they Some students argued that observers at had not understood the basic concept of a rest disagree on the time of an event even reference frame. They had not recognized when the signal from that event explicitly that intelligent observers at rest with respect contains the time of that event. The to one another can communicate about the following exchange is illustrative. spacetime coordinates of events at each location. Instead, these students seemed to I: Let’s say … you are in the think that each observer is restricted to the middle, with your fancy Rolex information that he or she can obtain [watch], and I’m at directly.

"Seismologist's frame" "Assistant's frame"

Rainier Hood

Rainier Hood

Rainier Hood

Figure 4: Student response to the Seismologist question. The student has indicated that the eruptions are simultaneous in the reference frame of the seismologist (in the middle), but that in the reference frame of the assistant (at Mt. Rainier), Mt. Rainier erupts first.

Student understanding of time in special relativity Scherr, et al. 18 VII. CONCLUSION concepts. Formulating an appropriate This investigation has identified measurement procedure for the time of an widespread difficulties that students have event involves recognizing the inherently with the definition of the time of an event local nature of measurement, applying a and the role of intelligent observers. After well-defined measurement procedure in a instruction, more than 2/3 of physics given reference frame, and understanding undergraduates and 1/3 of graduate students the relationship between measurements in physics are unable to apply the construct made by different observers. These ideas of a reference frame in determining whether are crucial in contexts ranging from the or not two events are simultaneous. Many rolling of a steel ball on a level track to the students interpret the phrase “relativity of motion of objects in the vicinity of massive simultaneity” as implying that the stars. This investigation documents simultaneity of events is determined by an prevalent modes of reasoning with these observer on the basis of the reception of fundamental concepts as a first step toward light signals. They often attribute the making special relativity meaningful to relativity of simultaneity to the difference in students. signal travel time for different observers. In this way, they reconcile statements of the ACKNOWLEDGMENTS relativity of simultaneity with a belief in The investigation described in this paper absolute simultaneity and fail to confront the has been a collaborative effort by many startling ideas of special relativity. members of the Physics Education Group, Experienced instructors know that present and past. We are extremely grateful students often have trouble relating to Lillian C. McDermott for her intellectual measurements made by observers in contributions in helping synthesize and different reference frames. It is not clarify the issues addressed in the paper and surprising that students, even at advanced her help in the writing. Bradley S. Ambrose levels, do not fully understand the and Andrew Boudreaux played significant implications of the invariance of the speed roles in the initial stages of the research. of light. What is surprising is that most Special thanks are also due to Paula R.L. students apparently fail to recognize even Heron. Arnold Arons and Edwin F. Taylor the basic issues that are being addressed. gave us invaluable insights into the Students at all levels have significant treatment of the relativity of simultaneity. difficulties with the ideas that form the Also deeply appreciated is the ongoing foundations of the concept of a reference cooperation of the colleagues in whose frame. In particular, many students do not physics classes the written problems have think of a reference frame as a system of been administered, especially James observers that determine the same time for Bardeen and E. Norval Fortson. We would any given event. Such difficulties appear to also like to thank Tevian Dray and Greg impede not only their understanding of the Kilcup. The authors gratefully acknowledge relativity of simultaneity, but also their the support of the National Science ability to apply correctly the Lorentz Foundation through Grants DUE 9354501 transformations. and DUE 9727648. Special relativity offers instructors an opportunity to channel student interest in 1 For an extensive bibliography, see the relevant modern physics into a challenging sections in L.C. McDermott and E.F. Redish, intellectual experience. For most people, the “Resource Letter: PER-1: Physics Education implications of special relativity are in Research,” Am. J. Phys. 67, 755-767 (1999). strong conflict with their intuition. For 2 For an article that illustrates how research can students to recognize the conflict and guide the development of curriculum, see L.C. appreciate its resolution, they need to have a McDermott, Millikan Award Lecture, “What functional understanding of some very basic

Student understanding of time in special relativity Scherr, et al. 19

we teach and what is learned – Closing the Press, Middletown, CT, 1962) and gap,” Am. J. Phys. 59, 301-315 (1991). A.B. Arons, A guide to introductory physics 3 For examp les of curriculum that have been teaching (Wiley, New York, NY, 1990). developed through an iterative process of 11 The definition of a global coordinate system research, curriculum development, and breaks down in non-inertial frames and in instruction, see L.C. McDermott, P.S. Shaffer, general relativity. The restriction of an inertial and the Physics Education Group at the frame to a finite extent in both space and time University of Washington, Tutorials in is a refinement not usually encountered in Introductory Physics, Preliminary Edition courses in special relativity. (Prentice Hall, Upper Saddle River, NJ, 1998); 12 L.C. McDermott and the Physics Education One method of synchronizing clocks in special Group at the University of Washington, relativity includes sending the reading on one Physics by Inquiry (Wiley, New York, NY, clock to a clock at another location by means 1996), Vols. I and II. of some signal. The second clock is 4 synchronized with the first by setting it to read S. Panse, J. Ramadas, and A. Kumar, the time sent from the first clock plus the “Alternative conceptions in Galilean relativity: signal travel time. The use of light signals for Frames of reference,” Int. J. Sci. Educ. 16, 63- the synchronization of clocks is customary but 82 (1994); J. Ramadas, S. Barve, and A. not necessary. See, for instance, the first book Kumar, “Alternative conceptions in Galilean in Ref. 10. relativity: Inertial and non-inertial observers,” 13 Int. J. Sci. Educ. 18, 615-629 (1996). Although it is possible to define the time of an 5 event as the time at which an observer sees the E. Saltiel and J.L. Malgrange, “‘Spontaneous’ event, the time then depends on observer ways of reasoning in elementary kinematics,” location. Einstein, for instance, considered Eur. J. Phys. 1, 73-80 (1980). and rejected such a definition. See A. 6 A. Villani and J.L.A. Pacca, “Students’ Einstein, “On the electrodynamics of moving spontaneous ideas about the speed of light,” bodies,” in The principle of relativity: A Int. J. Sci. Educ. 9, 55-66 (1987) and collection of original memoirs on the special “Spontaneous reasoning of graduate students,” and general theory of relativity (Dover, New Int. J. Sci. Educ. 12, 589-600 (1990). York, NY, 1952). 14 7 P.W. Hewson, “A case study of conceptual Some authors define the term “observer” to change in special relativity: The influence of indicate the full set of measuring devices and prior knowledge in learning,” Eur. J. Sci. procedures that comprise a reference frame. Educ. 4, 61-76 (1982). For an example of this approach, see E.F. Taylor and J.A. Wheeler, Spacetime Physics 8 G.J. Posner, K.A. Strike, P.W. Hewson, and (W.H. Freeman, New York, NY, 1992), pp. W.A. Gertzog, “Accommodation of a 39. scientific conception: Toward a theory of 15 conceptual change,” Sci. Ed. 66, 211-227 The process by which “time is spread over (1982). space” is described meticulously in the first book in Ref. 10. 9 T.E. O’Brien-Pride, “An investigation of 16 student difficulties with two dimensions, two- For a description of the course for high school body systems, and relativity in introductory teachers, see L.C. McDermott, “A perspective mechanics,” Ph.D. dissertation, Department of on teacher preparation in physics and other Physics, University of Washington, 1997 sciences: The need for special courses for (unpublished). teachers,” Am. J. Phys. 58, 734-742 (1990). 17 10 That these are concepts that require definition The results from the various classes at the is itself a new idea to many students. For University of Washington were consistent insightful discussions of these definitions and within statistical fluctuations. The results the pedagogical concerns that they raise, see, from the other universities were within the for instance, P.W. Bridgman, A sophisticate’s same range. For the purposes of this article, primer of relativity (Wesleyan University

Student understanding of time in special relativity Scherr, et al. 20

the results from corresponding classes have 25 We have evidence that the students in this been combined. class performed as well as they did on the directed version of the Spacecraft question as 18 The problems are “ungraded” in the sense that a result of special instruction that they had they are not graded for correctness. Rather, received. On the basis of the research students receive credit for giving responses described in this paper, we have been that reflect an attempt to answer the questions. developing instructional materials to address Student performance on the examination the specific difficulties that we have identified. questions and the ungraded problems was very This class had used preliminary versions of similar. This is consistent with the results of these instructional materials. As is other investigations conducted by our group, demonstrated in section VI, other classes after in which we have found that students take the standard instruction did not do as well on ungraded questions seriously. similar questions. 19 Each research question was posed in at least 26 This finding is consistent with our experience two ways, with different context and slight that the study of advanced material does not changes in wording. In general, we found that necessarily deepen conceptual understanding. such changes had little effect on student See, for example, S. Vokos, P.S. Shaffer, B.S. performance. Therefore, in this paper we Ambrose, and L.C. McDermott, “Student present only a representative description or understanding of the wave nature of matter: example of each question. Diffraction and interference of particles,” 20 The fact that the curved surface of a Phys. Educ. Res., Am. J. Phys. Suppl. 68, S42- gravitating, rotating Earth is not an inertial S51 (July 2000); B.S. Ambrose, P.S. Shaffer, frame did not elicit student concern. In one R.N. Steinberg, and L.C. McDermott, “An version of the question, students were told to investigation of student understanding of neglect non-inertial effects. In another, the single-slit diffraction and double-slit context was set in deep space. Neither interference,” Am. J. Phys. 67, 146-155 statement seemed to change student responses. (1999); K. Wosilait, P.R.L. Heron, P.S. 21 Shaffer, and L.C. McDermott, “Development In keeping with standard practice in one- of a research-based tutorial on light and dimensional problems on special relativity, shadow,” ibid. 66, 906-913 (1999); T. O’Brien students were told in all questions that all Pride, S. Vokos, and L.C. McDermott, “The motions were to be considered as occurring challenge of matching learning assessments to along a single line in space. No student teaching goals: An example from the work- seemed to have had difficulty neglecting the energy and impulse-momentum theorems,” vertical dimension. ibid. 66, 147-157 (1998); L.C. McDermott, 22 The prototype of such a qualitative analysis is P.S. Shaffer, and M.D. Somers, “Research as a that of the classic train paradox often used to guide for teaching introductory mechanics: An develop the relativity of simultaneity. See, for illustration in the context of the Atwood’s instance, P.A. Tipler and R.A. Llewellyn, machine,” ibid. 62, 46-55 (1994). Modern Physics (W.H. Freeman, New York, 27 The belief in absolute simultaneity may be NY, 1999). related to the strong belief of students in a 23 We use the notation dt instead of ? t, etc., to try preferred reference frame, which has been to minimize confusion between the difference documented in the context of Galilean between two quantities and a change in a relativity. See Refs. 5, 6, and 9. quantity. We are grateful to Eric Mazur for a 28 This term is related to the amount of time that discussion on this point. two specific synchronized clocks in the 24 Since the events are separated by a spacelike spacecraft frame are measured to be out of interval (c2dt2 – dx2 < 0), students can predict synchronization by observers in the ground the existence of a frame in which the events frame. are simultaneous without use of the Lorentz 29 See, for instance, D. Griffiths, Introduction to transformations. Electrodynamics (Prentice Hall, Upper Saddle

Student understanding of time in special relativity Scherr, et al. 21

River, NJ 1989), p. 452. This widely used text states explicitly that the relativity of simultaneity is “a genuine discrepancy between measurements made by competent observers in relative motion, not a simple mistake arising from a failure to account for the travel time of light signals.” 30 The belief that each observer constitutes a distinct reference frame is similar to the belief documented in Galilean contexts in Ref. 4. The authors describe a tendency of students to treat the ext ent of an observer’s reference frame as limited to the physical object on which the observer is located (e.g., the deck of a boat).

Student understanding of time in special relativity Scherr, et al. 22