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Real Time Relativity: Exploration Learning of Special Relativity Real Time Relativity: exploration learning of special relativity C. M. Savage, A. Searle, and L. McCalman Department of Physics, Australian National University, ACT 0200, Australia∗ “Real Time Relativity” is a computer program that lets students fly at relativistic speeds though a simulated world populated with planets, clocks, and buildings. The counterintuitive and spectacular optical effects of relativity are prominent, while systematic exploration of the simulation allows the user to discover relativistic effects such as length contraction and the relativity of simultaneity. We report on the physics and technology underpinning the simulation, and our experience using it for teaching special relativity to first year university students. PACS numbers: I. INTRODUCTION space and time. To overcome their non-relativistic pre- conceptions students must first recognize them, and then 3 “Real Time Relativity”1 is a first person point of view confront them. The Real Time Relativity simulation can game-like computer simulation of a special relativistic aid this process. world, which allows the user to move in three dimensions In the next section we discuss some relevant physics amongst familiar objects, Fig. 1. In a first year univer- education research. Section III outlines the relativistic sity physics course it has proved complementary to other optics required to understand the simulation. Section IV relativity instruction. briefly overviews the computer technology that is mak- Since there is little opportunity for students to directly ing interactive simulations of realistic physics, such as experience relativity, it is often perceived as abstract, Real Time Relativity, increasingly practical. Section V and they may find it hard to form an integrated rela- describes students’ experience of the simulation. Section tivistic world-view. They find relativity interesting and VI shows how it provides fresh perspectives on physics exciting, but may be left bemused by the chasm between such as the relativity of simultaneity. Section VII reports the theory and their everyday experience.2,3,4 Real Time our evaluation of its use in a first year physics laboratory. Relativity can help bridge this chasm by making visual observations the basis from which the theory is deduced. In his original relativity paper Einstein discarded the II. EDUCATIONAL BACKGROUND personal observer, who collects information from what he sees, in favor of more abstract inertial observers who There is substantial evidence for the value of ac- use distributed arrays of rulers and conventionally syn- 9,10 chronized clocks.5 However Komar6 and others7,8 showed tive learning. Effective learning is stimulated by stu- dents participating in the construction and application of that special relativity may be formulated in terms of pos- 11 tulates about a personal observer’s visual observations. physics based world-views. A common factor in active This approach to relativity underpins our use of the Real learning is a cycle of developing, testing, and correcting Time Relativity simulation. understanding in a collaborative environment. Peer in- struction is one way for this to occur in the classroom,12 Studies have shown that students may fail to learn fun- damental concepts, such as the relativity of simultaneity, while in the laboratory inquiry based approaches are known to be effective.11 even after advanced instruction.2,3,4 This is because spe- cial relativity contradicts some deeply held ideas about Computer simulations may promote active learning in physics, especially where real laboratories are difficult to provide. However, research has shown that a testing and development cycle is required to ensure good learn- ing outcomes.13,14 The effectiveness of simulations is re- arXiv:physics/0701200v1 [physics.ed-ph] 18 Jan 2007 duced by poor interfaces,13 and by students’ lack of the skills required to learn from them.15 They also lose effec- tiveness if the exploration is not conducted according to the scientific method.16 However, when such issues are addressed the results can be spectacular.14 Computer simulations are most effective when di- rected towards clear goals, with an understanding of their strengths and limitations.14,16 They can provide an ad- FIG. 1: Screenshot from Real Time Relativity. The speed ditional active learning mode, and address broad goals of the camera relative to the objects is v = 0.9682c. The such as “thinking like a physicist”.17 However, learning Doppler and headlight effects have been turned off. to use software increases cognitive load, lessening capac- 2 ity to learn other new material.15 The value for physics unit vector in the propagation direction. From the 4- teaching of first person simulations, such as Real Time frequency components in a particular frame, the compo- Relativity, is largely unexplored, as existing research has nents in any other frame may be found using a Lorentz concerned simulations of models, such as might be used transformation. The transformation between the usual by an expert physicist, rather than immersive first person standard configuration24 frames S and S′ are sufficient simulations.13,14,15,16 for our purposes. We will use “world” (w) and “camera” Real Time Relativity differs from other physics simu- (c) to refer to the frames S and S′, lations in providing a realistic, explorable environment. f = γf (1 − n v/c), In the context of a first year university physics class, we c w w,x are asking the question: Can aspects of special relativity fcnc,x = γfw(nw,x − v/c), be learnt by exploration of the Real Time Relativity vir- fcnc,y = fwnw,y, fcnc,z = fnw,z, tual world? Many students are comfortable interactively γ = (1 − v2/c2)−1/2, (2) discovering the rules of virtual worlds; perhaps they can use this experience for discovering the rules of physics? where v is the component of the relative velocity of the Successful learning from simulations is more likely frames along the positive xw axis of the world frame. if students are suitably prepared and guided.11,13,15 The first equation expresses the Doppler effect, and the Preparation should develop a basic understanding of remaining equations express the dependence of the prop- the physics which determines what is seen in the sim- agation direction on the relative frame velocity: an effect ulation. In our case this includes the finite speed of known as “relativistic aberration”. In the Real Time Rel- light, the Doppler effect, and relativistic optical aber- ativity simulation let the world frame be that in which ration. This preparation might use conventional interac- the objects are at rest, and the camera frame be the tive multimedia.18 Preparation should also include how user’s instantaneous rest frame, as we represent the user the scientific method is used to develop understanding of by a camera. We require the frequencies and propagation novel phenomena. directions in the camera frame, fc and ~nc. Scherr, Shaffer, and Vokos have found that students’ Since ~nc is a unit vector, its x-component, nc,x is the 2,3 understanding of time in special relativity is poor. cosine of the angle θc between the light ray and the They conclude that “... many students who study spe- xc axis: if the ray is coming towards the observer nc,x cial relativity at the undergraduate to graduate levels fail changes sign. Dividing the second and third of Eqs. (2) by 3 to develop a functional understanding” . They identify the first and using nc,x = − cos θc and nw,x = − cos θw, the reason for this as students misunderstanding funda- we get mental ideas such as: the “time of an event, simultane- cos θw+v/c 3 cos θ = , ity, and reference frame”. They have developed instruc- c 1+(v/c) cos θw tional materials to address these problems. Mermin19 sin θw sin θc = γ(1+(v/c) cos θw) . (3) has also noted that traditional relativistic pedagogy may make incorrect assumptions about students’ prior knowl- Relativistic aberration is analogous to non-relativistic edge. Real Time Relativity can address these problems, forms of aberration that students may have experienced. as fundamental ideas, such as the time of an event, have For example, the dependence of the angle of falling rain intuitive operational meanings. on an observer’s velocity, or the difference between the visual position of a high flying aircraft and that indicated by its sound. This understanding may be made quanti- III. RELATIVISTIC OPTICS tative using the relativistic velocity addition formulae.23 Penrose20 showed that relativistic aberration implies Some of the basic physics of relativistic optics, namely that straight-lines are seen as either lines or circular arcs the Doppler effect and aberration, was discussed by Ein- in other frames. He also showed that a sphere, which stein in his first relativity paper.5 However it was not always has a circular outline (unlike a circle which may until about 1960 that the pioneering work of Penrose,20 have an elliptical outline), will continue to have a circular Terrell,21 and Weisskopf22 showed that relativity gives a outline after aberration, and hence continue to look like rich and unexpected visual environment. a sphere. These effects are immediately apparent in the In this section we summarize relativistic optics using 4- Real Time Relativity simulation, Fig. 2. vectors, because that is how it is implemented in the Real The non-relativistic Doppler effect may also be familiar Time Relativity program (see Section IV). Rindler23 pro- to some students. This, together with the analogy to vides a more complete introduction, both with and with- non-relativistic aberration, emphasize the closer relation out using 4-vectors. of relativistic optics to direct experience than the usual A plane light wave is described by its 4-frequency F , space-time approach to special relativity. which has components23 A convenient form of the Doppler effect follows from the first two of Eqs.
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