Real Time Relativity: exploration learning of special relativity

0.39 MB·English

by  C. M. Savage

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∗
“RealTimeRelativity”isacomputerprogramthatletsstudentsflyatrelativisticspeedsthougha
simulatedworldpopulatedwithplanets,clocks,andbuildings. Thecounterintuitiveandspectacular
opticaleffectsof relativity areprominent,while systematicexploration of thesimulation allows the
7 usertodiscoverrelativistic effectssuch aslength contraction andtherelativityof simultaneity. We
0 report on the physics and technology underpinning the simulation, and our experience using it for
0 teaching special relativity tofirst yearuniversity students.
2
PACSnumbers:
n
a
J
I. INTRODUCTION space and time. To overcome their non-relativistic pre-
8
conceptionsstudentsmustfirstrecognizethem,andthen
1
“RealTime Relativity”1 is a firstpersonpointof view confrontthem.3 TheRealTimeRelativitysimulationcan

] game-like computer simulation of a special relativistic aid this process.
h
world,whichallowstheusertomoveinthreedimensions In the next section we discuss some relevant physics
p
- amongst familiar objects, Fig. 1. In a first year univer- education research. Section III outlines the relativistic
d sityphysicscourseithasprovedcomplementarytoother opticsrequiredtounderstandthesimulation. SectionIV
e relativity instruction. briefly overviews the computer technology that is mak-
.
s Sincethereislittleopportunityforstudentstodirectly ing interactive simulations of realistic physics, such as
c
experience relativity, it is often perceived as abstract, Real Time Relativity, increasingly practical. Section V
i
s and they may find it hard to form an integrated rela- describes students’ experience of the simulation. Section
y
tivistic world-view. They find relativity interesting and VI shows how it provides fresh perspectives on physics
h
p exciting,butmaybe leftbemusedbythe chasmbetween suchastherelativityofsimultaneity. SectionVIIreports
[ the theoryandtheir everydayexperience.2,3,4 RealTime ourevaluationofitsuseinafirstyearphysicslaboratory.

Relativity can help bridge this chasm by making visual
1
observationsthe basis fromwhichthe theory is deduced.
v
0 In his original relativity paper Einstein discarded the II. EDUCATIONAL BACKGROUND
0 personal observer, who collects information from what
2 he sees, in favor of more abstract inertial observers who
There is substantial evidence for the value of ac-
1 use distributed arrays of rulers and conventionally syn-
0 chronizedclocks.5HoweverKomar6andothers7,8showed tive learning.9,10 Effective learning is stimulated by stu-
7 dentsparticipatingintheconstructionandapplicationof
thatspecialrelativitymaybeformulatedintermsofpos-
0 physics based world-views.11 A common factor in active
tulates about a personal observer’s visual observations.
/ learning is a cycle of developing, testing, and correcting
s ThisapproachtorelativityunderpinsouruseoftheReal
c understanding in a collaborative environment. Peer in-
Time Relativity simulation.
i struction is one way for this to occur in the classroom,12
s Studieshaveshownthatstudentsmayfailtolearnfun-
y while in the laboratory inquiry based approaches are
damentalconcepts,suchastherelativityofsimultaneity,
h known to be effective.11
p evenafter advancedinstruction.2,3,4 This is because spe-
Computer simulations may promote active learning in
: cial relativity contradicts some deeply held ideas about
v physics, especially where real laboratories are difficult
i to provide. However, research has shown that a testing
X
and development cycle is required to ensure good learn-
r
a ing outcomes.13,14 The effectiveness of simulations is re-
duced by poor interfaces,13 and by students’ lack of the
skillsrequiredtolearnfromthem.15 Theyalsoloseeffec-
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-
rectedtowardscleargoals,withanunderstandingoftheir
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 havebeen 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 largelyunexplored,as existing researchhas 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
byanexpertphysicist,ratherthanimmersivefirstperson standard configuration24 frames S and S′ are sufficient
simulations.13,14,15,16 forour purposes. We willuse “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
areaskingthe question: Canaspects ofspecialrelativity fcnc,x =γfw(nw,x−v/c),
be learntby explorationof the RealTime Relativity vir- f n =f n , f n =fn ,
c c,y w w,y c c,z w,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 remainingequationsexpressthe dependenceoftheprop-
the physics which determines what is seen in the sim- agationdirectionontherelativeframevelocity: aneffect
ulation. In our case this includes the finite speed of knownas“relativisticaberration”. IntheRealTimeRel-
light, the Doppler effect, and relativistic optical aber- ativity simulation let the world frame be that in which
ration. This preparationmight use conventionalinterac- 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
thescientificmethodisusedtodevelopunderstandingof byacamera. Werequirethefrequenciesandpropagation
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
understanding of time in special relativity is poor.2,3 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
cialrelativityattheundergraduatetograduatelevelsfail changessign. DividingthesecondandthirdofEqs.(2)by
to develop a functional understanding” .3 They identify the first and using nc,x = −cosθc and nw,x = −cosθw,
the reason for this as students misunderstanding funda- we get
mental ideas such as: the “time of an event, simultane-
cosθ = cosθw+v/c ,
ity, and reference frame”.3 They have developedinstruc- c 1+(v/c)cosθw
tional materials to address these problems. Mermin19 sinθ = sinθw . (3)
hasalsonotedthattraditionalrelativisticpedagogymay c γ(1+(v/c)cosθw)
makeincorrectassumptionsaboutstudents’priorknowl- Relativistic aberration is analogous to non-relativistic
edge. Real Time Relativity can address these problems, forms ofaberrationthat students may haveexperienced.
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
visualpositionofahighflyingaircraftandthatindicated
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
thatstraight-linesareseenaseither linesorcirculararcs
Some ofthe basicphysicsofrelativisticoptics,namely
in other frames. He also showed that a sphere, which
the Doppler effect andaberration,wasdiscussed by Ein-
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 haveanellipticaloutline),willcontinuetohaveacircular
Terrell,21 and Weisskopf22 showed that relativity gives a outline after aberration, and hence continue to look like
a sphere. These effects are immediately apparent in the
rich and unexpected visual environment.
Real Time Relativity simulation, Fig. 2.
Inthissectionwesummarizerelativisticopticsusing4-
Thenon-relativisticDopplereffectmayalsobefamiliar
vectors,becausethatishowitisimplementedintheReal
TimeRelativityprogram(seeSectionIV). Rindler23pro- to some students. This, together with the analogy to
non-relativistic aberration, emphasize the closer relation
videsamorecompleteintroduction,bothwithandwith-
of relativistic optics to direct experience than the usual
out using 4-vectors.
space-time approachto special relativity.
A plane light wave is described by its 4-frequency F,
which has components23 A convenient form of the Doppler effect follows from
the first two of Eqs. (2) after eliminating n ,
w,x
F =f[1,~n], (1)
1−v2/c2
f = p f =Df , (4)
where f is the frequency and ~n = (nx,ny,nz) is the c 1−(v/c)cosθc w w
3
the photons, and the change in photon flux due to the
combined effects of time dilation and the observer’s mo-
tion,whichisanadditionalmanifestationoftheDoppler
effect. IntermsoftheDopplerfactorinEq.(4)thesecon-
tribute factors to the change in intensity of D2, D, and
D respectively, for a combined intensity change factor of
D4. However for common detectors, such as the eye or
a CCD camera, it is the photon number flux P that is
detected, and this changes by a factor of D3, since the
energy change per photon is irrelevant,
P =D3P . (6)
B A
IV. TECHNOLOGY
Computers can generate images incorporating
special,28 and general,29 relativistic optics. By the early
1990s it was possible to interactively render simple
objects, such as cubes.30 The highest quality images
were generated by the ray-tracing method, which is
capable of producing photo-realistic images.31 However
ray-tracing is currently too slow for interactive simula-
FIG. 2: Screenshots from Real Time Relativity. Top frame:
tions,althoughindividualimagescanbe strungtogether
at rest in the world frame. Bottom frame: v = 0.9682c,
to make movies.18
corresponding to γ = 4. The Doppler and headlight effects
have been turned off. In the world frame the camera is in The development of the programmable graphics
front of the position in thetop frame. processor32 has made it possible to render complex rel-
ativistic scenes in real time. The first such systems ap-
pear to have been developed by D. Weiskopf33 and M.
where this equation defines the “Doppler factor” D. For Bochers34withinthephysicseducationgroupattheUni-
v/c ≪ 1 the denominator is the familiar non-relativistic versity of Tu¨bingen.35 This group has focussed on us-
wave compression or expansion. For waves incoming at ing relativity visualization for science communication.36
θ =π/2radianstotherelativemotion,thedenominator Our Real Time Relativity simulation is similar to these,
c
is one and the observed frequency is less than the world is freely available, and is being developed as an Open
frequency, at which they were emitted. This means that SourceprojectundertheLesserGeneralPublicLicense.1
thetimebetweenwavecrests,theperiod,islonger;which The screen image displayed by Real Time Relativ-
isexactlytheeffectoftimedilation,ifthewavecrestsare ity is created using the computer graphics technique
regarded as a clock. knownasenvironmentmapping,whichrendersthethree-
The effect of aberrationon small angles may be found dimensional virtual world onto a two-dimensional cube
by taking differentialsof the inverseLorentztransforma- map. A cube map may be visualised as the 4π ster-
tion of the 4-frequency.25 This yields radian view-field mapped onto the interior surface of a
cube centered on a camera, representing the user’s field
dθ =D−1dθ . (5) ofview. Infact,thecubemapisadatastructureinwhich
C W
the image pixels are addressed by line of sight direction,
HencesmallanglestransformbytheinverseDopplerfac- rather than by spatial position. The relativistically cor-
tor. In particular, for objects directly ahead, so that rect scene is produced by transforming the cube map.
θ = 0, and for v/c ≪ 1, the inverse Doppler factor is Each camera image pixel is formed by light incident
C
approximately D ≈ 1−v/c, and objects angular sizes from a particular direction; that is, with a specific prop-
are shrunk. While for objects behind, D ≈ 1+v/c and agationdirectionn~ inthecameraframe. Therelativistic
c
objects are expanded. physicsproblemistofindthecorrespondingpropagation
Perhapsthemostsubtleoftherelativisticopticseffects vector n~ in the world frame in which the cube map is
w
istheheadlighteffect. Indeed,acompletediscussionwas constructed. This vector then addresses the pixel on the
not given until 1979, by McKinley.26,27 It refers to the cube map that is mapped to the camera pixel. The re-
increased intensity of light coming from objects we are sulting camera image is displayed on the screen.
moving towards. The intensity decreases for objects we A plane light wave is represented by the relativis-
are moving away from. Three things combine to pro- tic 4-frequency, Eq. (1). The propagation direction
duce these intensity changes: the change in angular size in the world frame is found by the inverse Lorentz
of the emitting region, the Doppler change in energy of transformation25 of this 4-vector from the camera frame
4
into the world frame. This is implemented as a four- tails of any particular graphics processing unit. Conse-
dimensional matrix multiplication of the 4-frequency. quently, it is only available on Windows computer sys-
The transformation matrix is calculated before each tems. DirectX 9 includes the High Level Shader Lan-
frameisrendered,usingthecurrentcameravelocity,and guage in which the pixel shader controlling the graphics
isthenappliedtoa4-frequencyconstructedforeachcam- processing unit is written.
era pixel. This has a spatial component equal to the Graphics processing units have been increasing in
pixel’s imaging direction and the time component set to processing power more rapidly than central processing
one. The spatialpartofthe transformed4-frequencyad- units.32 This is driven by the demand for parallel com-
dresses the cube map pixel that is then rendered to the puting from the gaming community. For example, the
screen. Xbox 360 graphics processing unit has forty-eight 32-
The Graphics Processing Units (GPUs) on computer bit processors running at 500 MHz, each capable of a
video cards are designed to do 4-vector matrix algebra floating point 4-vectoroperationper cycle, giving nearly
efficientlyandinparallel. Thismakesitpossibleforsim- 100GigaFlops,comparedtoperhapsafewGigaFlopsfor
ulations such as Real Time Relativity, and that due to a central processing unit.39 The main limitation is that
Bochers,34,36 to perform the Lorentz transformations in graphics processing units do data-parallel computing, in
real time. The 4-vectors that graphics processing units whichthesameoperationisrepeatedoneachelementofa
normallyworkwithspecifythex,yandzcoordinatesofa dataarray. Nevertheless,computationalscientistsarede-
vertex and a fourth w component that facilitates certain veloping algorithms that harness their processing power
non-linear transformations (such as translation and per- for tasks such as solving partial differential equations.32
spectiveprojection),orspecifythered,green,bluecolour The Folding@Home distributed computing project has
and alpha (transparency) of a (texture) pixel. Since the a client available which runs their molecular dynamics
processing of different vertices or pixels is usually inde- calculationongraphicsprocessingunits, increasingcom-
pendent, the operations can be performed in parallel. putationalpowerbyabouttwentytimespercomputer.40
The Doppler shift factor D is givenby the ratio of the These developments may have an impact on the kinds
time components of the 4-frequencies in the camera and of physics teaching simulations that are possible in the
world frames, Eq. (4). However, to determine the effect future.
of the Doppler shift on a general colour requires the en-
tire intensity spectrum. But in current graphics systems
the spectrum is specified at just three frequencies; red, V. THE REAL TIME RELATIVITY
green, and blue. Hence interpolation is used to gener- SIMULATION
ate the spectrum. This simple approach, together with
the lack of any infrared or ultraviolet spectra, prevents
In this section we introduce the Real Time Relativ-
a true representation of Doppler shifted colors, and is a
ity simulation as experienced by students in the first
significantlimitationof the currentversionofRealTime
year course for physics majors at The Australian Na-
Relativity. In particular, stars do not maintain a black- tionalUniversity.41Itwasusedinathreehourlaboratory
body spectrum.26,37,38
structuredtoencourageexploration,whilerequiringthat
The headlight effect, Eq. (6), is implemented by mul- certain measurements be made and compared to theory.
tiplying eachpixelcolorvectorby the third powerofthe Studentswereprovidedwithamanualgivingbackground
Doppler shift factor D. There are significant limitations informationandaskingbothqualitativeandquantitative
on how the resulting large intensity range is rendered to questions. Many students completed the laboratory be-
the screen by current hardware. fore they attended the relativistic optics lecture. Stu-
The graphics processing unit does the Lorentz trans- dents worked in groups of two or three, and discussion
formations as well as its usual graphics work. First, a was encouraged. Preparation included answering simple
non-relativistic three-dimensional scene is rendered to a pre-lab questions which were assessed at the beginning
cube map, then relativistic transformations are applied of the laboratory.
to it. To generate a frame, the 4-frequency associated Aninitialproblemoforientationwithinrelativisticop-
with eachcamerapixelis inverseLorentztransformedto tics simulations arises because the speed of light is very
findthecorrespondingworldframecubemappixel. This large in everyday terms. This means that either the ob-
is then Doppler and intensity shifted, also by the graph- jects in the simulationmust be very large,roughly light-
ics processing unit. An 800 by 600 window has 480,000 seconds, or the speed of light must be artificially slow,
pixels, so displaying 50 frames per second requires 24 as in Gamow’s Mr. Tompkins story.42 In the interest of
million pixel transformations per second, which is well realism, we have taken the former view, which allows
within the capabilities of inexpensive graphics process- us to include realistic astronomical objects such as the
ing units. Consequently, it is the conventional graphics Earth,which is 0.042light-seconds in diameter. The top
processinggeneratingthe cube mapthatlimits the over- frame of Fig. 2 shows a screen from Real Time Relativ-
all performance, not the relativistic calculations. ity. TheEarthisvisible,asistheSunbehindit.43 These
RealTime Relativity isprogrammedusing Microsoft’s objects set the scale of the simulated world. Other ob-
DirectX 9 interface, so that it is independent of the de- jects, such as the columns, have been chosen for their
5
they see a rainbow effect, as for directions towards that
of motion, θ < θ , the color is blue shifted, while for
0
directionsawayfromthe directionofmotion,θ >θ , the
0
color is red shifted. Fig. 3 shows the red shifting of a
bluecylinderthroughgreenandredastheviewingangle
increases.
If students already understand the non-relativistic
Doppler effect, they may be guided to discover the rel-
ativistic version. In particular, it is possible to deduce
timedilationfromtheobservationthatthereisreddening
when viewing perpendicular to the direction of motion,
Eq. (4).
FIG. 3: Screenshot from Real Time Relativity showing how
the Doppler effect depends on the view angle. The camera The bottom frame of Fig. 2 shows the scene with the
is looking perpendicular to the direction of motion, which is camera travelling down the row of columns with a speed
fromrighttoleftatv=0.5c,atastippledbluecylinder. The corresponding to γ = 4. The Doppler and headlight ef-
Doppler factor is approximately one, D ≈1, in the direction fects have been turned off. The circular curvature of the
of the left edge of the image. The rest of the cylinder is red nearest columns is due to relativistic aberration, as dis-
shifted through green and red. The headlight effect has been cussed in section III. The curvature of the more distant
turned off.
columns is barely noticeable. However, they are shrunk
by approximately the inverse Doppler factor D−1, ac-
familiar shapes, although they would be absurdly large cording to Eq. (5). The camera field of view covers a
if they existed in the real world. Familiar objects aid in wide field in the world frame: the hoops and cubes on
the recognition of the distortions caused by relativistic theedgesoftheimagearebehindthecameraintheworld
aberration. frame.
Students start by accelerating from rest down the row Aberrationmaybeunderstoodasaconsequenceofthe
of columns shown in Fig. 2. At first it seems that they finite speed of light. The key idea is that the light that
aremovingbackwards.44 Thisis completelycounterintu- reaches the camera at a particular instant was reflected
itive and prompts them to question what they see: the by objects at different times. The light from closer ob-
exploration has begun! The effect is due to relativistic jects wasreflectedlater thanthat fromfar awayobjects.
aberration. An important way that motion is sensed is This is irrelevant when the camera is at rest relative to
bythechangeinangularsizeasourdistancetotheobject theobjects,butwhenitismoving,thepositionoftheob-
changes. Normally,as we approachanobjectits angular jects in the camera frame depends on time. For a large
size increases, roughly proportionate with the distance. object this means that the parts nearer the camera re-
In contrast, the decrease in angular size due to relativis- flected the received light later than the further parts. If
tic aberration, Eq. (5), is approximately proportional to the camerais movingtowardsthe object,atasignificant
1−(v/c)cosθ , for v/c ≪ 1, and occurs before the dis- fractionofthe speed of light, the near parts reflectwhen
c
tance hastime to change. Thereforethe initialview isof they are significantly closer and hence look bigger than
objectsshrinking,andthisisinterpretedasthemmoving thefarpartswhichreflectedwhentheywerefurtheraway
away, and hence as backwards movement of the viewer. and hence looked smaller. If we are moving directly to-
Astheycontinuetoaccelerate,nearbyobjectseventually wards the middle of an object the net result is that the
passby,andtheperceptionofforwardmotionisrestored. middlelooksfatterthantheends,seeFig.4. Iftheobject
is off to one side it is curved into a circular arc.
Colors change due the Doppler effect Eq. (4), but the Currently, Real Time Relativity is limited to all ob-
headlight effect quickly saturates the scene with bright jects being at rest in the world frame. This means that
light, dominating all other effects, due to its dependence relativistic dynamics is not within its capabilities.
on the third power of the Doppler factor, Eq. (6). Con-
sequently, it is useful to be able to turn it off. Although
this goes against the principle of making the simulation VI. THE RELATIVITY OF SIMULTANEITY
as realistic as possible, it is difficult to see some other
effects if it is left on.
The relativity of simultaneity has been identified as a
The Doppler effect depends on the viewing angle,
particularly difficult concept for students to learn from
Eq. (4). There is a particular angle to the direction of passive instruction.2,3,4 In order that students might ac-
motion θ for which there is no effect, since the Doppler
0 tively discover the relativity of simultaneity for them-
factor D =1 when
selves, the Real Time Relativity simulation includes
cosθ =(c/v)(1−γ−1). (7) clocks in the world frame. Even when the camera is
0
at rest in the world frame, clocks at different distances
◦
For v = 0.5c this angle is θ = 1.3 radians (74 ). If a from the camera are seen to read different times due to
0
student looks at at a pure colored object at this angle the light propagation delay, see the top frame of Fig. 5.
6
FIG. 4: Screenshot from Real Time Relativity showing rela-
tivistic aberration.The camera is moving towards the center
of the column with v =0.9682c, so that γ =4. The Doppler
and headlight effects are off.
Studentsgenerallyhavenodifficultyrecognizinganduti-
lizing this fact.2,3,4 Note that clocks the same distance FIG. 5: Screenshots from Real Time Relativity explaining
from the camera read the same time. the relativity of simultaneity. Top frame. The effect of light
propagation delay on observed clocks. The camera is at rest
The middle frame shows the same view of the clocks,
relative to the clocks, which are lined up perpendicular to
but with the camera moving with v = 0.5c parallel to
the line of sight to the central clock. The clocks are 5 light-
the clocks from left to right. The camera is looking per-
secondsapartandreadseconds. Middleframe. Thecamerais
pendicular to its direction of motion. Note that the eye
movingfrom left torightparallel totheclockswithv=0.5c.
gets confusingcues fromthis image,asthe clocksarero- Theperpendiculardistancetotheclocksisthesameasinthe
tatedasifwe werelookingatthemfrominfront,butwe topframe(about31light-seconds). Themajorcontributorto
are not. This effect is a result of relativistic aberration the different clock readings is the relativity of simultaneity.
knownas“Terrellrotation”.21 Lengthcontractionbythe However, light delay causes clocks to the left to differ more
factor γ−1 =0.87 is also apparent. fromthecentralclockthanthosetotheright. Bottomframe.
Thecamerahasbeenbroughttorestimmediatelyaftertaking
The relativity of simultaneity is apparent from the
themiddleframe,althoughsometimethenelapsedbeforethe
readingsontheclocksinthemiddle frameofFig.5. The
image wastaken. Intheclocks’ rest framethedifferent clock
right-most clock is ahead of the left-most by 10 seconds.
readings are entirely a consequence of the light propagation
This cannot be explained by light delay in the camera
delay. The field of view is thesame in each frame.
frame, as the observed time difference is too large, and
the times increase from left to right. However it is ex-
plained by light delay in the clocks’frame. Students can
mation; in particular, the times read by the clocks when
see this by immediately stopping the camera relative to
they were emitted. The different times of the different
the clocks. Due to relativistic aberrationthey mustthen
clocksis understoodby Bobasaresultofthe lightprop-
look back to see the clocks: this view is shown in the
agation delay over the different distances to the clocks.
bottom frame of Fig. 5. In the clocks’ rest frame the
However,the clockswereatapproximatelythe same dis-
camera is not opposite the clocks, but is to their right.
tance from Alice when they emitted the light, so she re-
From this perspective it is clear why the clocks read as
quiresanotherexplanation. Thisisanewphysicaleffect:
they do: the left-most clock is furthest and reads earli-
the relativity of simultaneity. The relativity postulate
est, while the right-most is closest and reads latest. The
ensures that what is true for these clocks is true for any
time difference between them is exactly that seen by the
clocks, and hence for time itself. A complete discussion
moving camera.
is given in Appendix A.
Let us restate the argumentin terms of two photogra-
phers: Alice is moving relative to the clocks, and Bob is
stationaryrelativetotheclocks. BothAliceandBobtake
VII. LABORATORY EVALUATION
photographs of the clocks at an event “CLICK”, chosen
sothatAlice,inherownframe,isapproximatelyequidis-
tant from the locations of the clocks when they emitted Real Time Relativity was incorporated in a first
the photographed light.45 Both Alice and Bob are sam- year laboratory session at The Australian National
pling the set of photons originating from the clocks and University.41 Thecourseincludedninelecturesandthree
present at CLICK. These photons carry the same infor- tutorials on special relativity. One lecture was devoted
7
to relativistic optics. Theflawsinthesimulationhadabiggernegativeimpact
The content of the laboratory has been indicated in on the students than expected. Students sometimes at-
section V. Its effectiveness was assessed in three ways. tributedtheir lackofunderstandingofthe physicsofthe
First,studentscompletedquestionnairesbeforeandafter simulationto a “bug”,evenwhen there wasnone,rather
the laboratory. Second, one of the authors was present than to their need to develop better understanding.13
asanobserverineachlaboratory,recordinghowstudents OurexperiencesuggeststhattheRealTimeRelativity
interactedwiththesimulation. Alaboratorydemonstra- simulation canstimulate discoverylearning,and provide
torwasalsopresent. Third,studentsrecordedtheirwork complementary learning opportunities to those provided
in laboratory log books which were assessed. by lectures and problem solving tutorials. However, re-
The pre-laboratory surveys indicated that students alizing its full value will require further cycles of testing
usually had prior knowledge of relativity and were eager and development. Next time we use it, we shall require
to learn more. However they tended to perceive it as an students to “play” with the simulation as part of the
abstract subject. The post-laboratory surveys indicated pre-laboratory preparation, so that they have some fa-
that students felt they had learnt about relativity from miliarity with the controls and with the peculiarities of
the simulation, andthat it hadstimulated their interest. navigation in a relativistic world. We shall also provide
Some students reported that the “concrete” or “visual” moreopportunityforopenendedexploration,asthisap-
nature of the simulation was helpful:46 pears to be its strength.
“Real Time Relativity is very useful - many
people are visual learners.” VIII. CONCLUSION
However, students often reported that the laboratory
manual was too prescriptive and did not allow them to Real Time Relativity is an immersive physics simu-
adequately pursue their own investigations. This criti- lation of a kind that is becoming increasingly accessi-
cism focussed on the quantitative exercises: ble due to the improving cost effectiveness of computer
technology. It givesstudents the opportunity to discover
“Why arewe forcingequations fromthe sim- and confront their misconceptions about relativity, and
ulation?” to construct resolutions.
Our experience with Real Time Relativity suggests
There were also many complaints about the difficulty of
that it provides new perspectives on special relativity.
using the program,and the inadequate time available to
Thismaybeparticularlyvaluabletostudentswhoprefer
develop proficiency with navigation through the virtual
the concrete over the abstract. Important physics, such
world:
as the relativity of simultaneity, can be introduced with
minimal mathematics. This may broaden the group of
“Thecontrolswerereally,reallyhardtouse.”
students who can learn relativity. However, the educa-
The laboratory observer enabled a testing and refine- tional value of first person simulations, like Real Time
mentcycle. We identified problems,andcorrectedthem, Relativity, is an interesting area for further physics edu-
before the next student group took the laboratory. In cation research.
particular, students often tried to push simulations to
the limits to see what happened, behavior noted by the
APPENDIX A: THE RELATIVITY OF
University of Colorado Physics Education Technology
group.13,14 Ifasimulationdoesnotrespondsensibly,stu- SIMULTANEITY
dentscanloseconfidenceinitsreliability. Observerswere
able to monitor what engaged students, and what frus- In this appendix we expand on the explanation of the
trated them. The most engaging aspect was the explo- relativity of simultaneity in terms of light delays that
ration of a novel and open ended world. Amongst the was introduced in section VI. It uses the aberrationand
morefrustratingthingswerethesimulation’scontrolsnot length contraction formulae. In the context of discovery
behaving in ways students considered natural. learning with the Real Time Relativity simulation each
The log books completed during the laboratory did of these formulae may, in principle, be deduced from ob-
not capture the excitement that was observed in work- servations. Along the way we also deduce time dilation.
ing laboratory groups. However, successful quantitative WewillrefertoFig.6,whichshowsschematicdiagrams
measurements were generally made: for example, of the of the scenario shown in Figs. 5. At event CLICK both
Doppler effect and of length contraction as a function of AliceandBobtakephotographsoftheclocks. Wechoose
speed. CLICKto be the co-incidentoriginsofAlice’s andBob’s
Our experience confirmed the importance of develop- rest frames, which we assume to be in standard configu-
ing educational software through a testing and refine- ration with relative velocity v. Therefore CLICK occurs
mentcycle.13,14 Students usedthe simulationinwayswe at times t = t = 0. In the notation of section III Al-
A B
had not anticipated, and had different ideas to the au- ice’s frame is the camera frame, and Bob’s frame is the
thors about what constituted a natural user interface. world frame.
8
gationtimefromC toBob,thetimeonC ’sphotograph
1 1
will be that it read at time t = −d /c = −γd/c. This
B B
differs from the time deduced by Alice by the time dila-
tion factor γ. Thus we obtain time dilation fromaberra-
tion.
However the focus here is on the relativity of simul-
taneity. The path length difference ∆d between the
B
paths from clocks C and C may be approximated by
1 2
a method familiar from diffraction theory. We drop a
perpendicular to C from the line between clock C and
FIG.6: Schematicdiagramsfortherelativityofsimultaneity. 2 1
BothpanelsrefertothetimeofeventCLICK,indicatedby*s, Bob. Thedistancealongthislinefromtheperpendicular
whenthephotographsaretaken. ThelinesfromclocksC1and to C1 is the approximate path length difference. Using
C2 toAliceandBobarethepathstakenbythelightforming the corresponding right-angle triangle with hypotenuse
thephotographsintheirrespectiveframes. (a)Alice’sframe. L and angle π−θ we have
B B
(b) Bob’s frame in which theclocks are at rest.
∆d =L cos(π−θ )=−L cosθ =L (v/c), (A3)
B B B B B B
Fig. 6(a) shows the light paths taken from the clocks where we again used the aberration formulae, Eqs. (3),
C and C to Alice, for whom they are moving from with θ = π/2, to find cosθ = −v/c. The correspond-
1 2 A B
right to left with speed v. She looks perpendicular to ing light propagation time difference, ∆t = L v/c2, is
B B
thedirectionofrelativemotiontoseethem,atθ =π/2, the time difference between the clocks in Bob’s photo-
A
andinfersthatwastheirdirectionwhentheyemittedthe graph. However, it is also the time difference between
lightshe images. Letthe perpendiculardistance toclock the clocks in Alice’s photograph, since both images are
C be d, and the distance between the clocks, in Alice’s made from the same group of photons; those present at
1
frame, be L . Due to the light propagation delay, the event CLICK.
A
time on the photograph of clock C will be that it read We can express this time difference in terms of Alice’s
1
at time t = −d/c. The path length difference between quantities by using the length contractionformula L =
A B
the paths from clocks C and C is γL ,
2 1 A
∆dA =qd2+L2A−d≈L2A/(2d), (A1) ∆tB =(γLA)(v/c2)=γ(LAv/c2), (A4)
where we have assumed L ≪ d and Taylor expanded which is precisely the term responsible for the relativity
A A
of simultaneity in the inverse Lorentz transformation,
the square root to first order. The corresponding light
propagationtimedifferencecanbemadearbitrarilysmall
∆t =γ(∆t +∆x v/c2). (A5)
by making L a sufficiently small fraction of d45. B A A
A
Fig. 6(b) shows the light paths taken from the clocks
Thus we have shown how the relativity of simultaneity
to Bob, who is at rest relative to them. He looks back
can be understood in terms of light propagation delays,
at the angle θ to photograph them. Let the distance
B and be deduced from direct observations of clocks.
to clock C be d . Since lengths perpendicular to the
1 B
relative motion are invariant this is given in terms of d
by
ACKNOWLEDGMENTS
d sin(π−θ )=d sinθ =d⇒d =γd, (A2)
B B B B B
TheauthorsacknowledgetheassistanceofDr.K.Wil-
where we used the aberration formulae Eqs. (3), with son with the implementation and evaluation of the Real
θ = π/2, to find sinθ = γ−1. Due to the light propa- Time Relativity laboratory.
A B
∗
Electronic address: [email protected] ofchangingdeeplyheldstudentbeliefsabouttherelativity
1 The Real Time Relativity web site contains informa- of simultaneity”, Am.J. Phys. 70, 1238-1248 (2002).
tion about the program and a downloadable installer: 4 R. E. Scherr, “An investigation of student un-
hhttp://www.anu.edu.au/Physics/Savage/RTRi. derstanding of basic concepts in special relativ-
2 R. E. Scherr, P. S. Shaffer, and S. Vokos, “Student un- ity”, Ph.D. dissertation, Department of Physics,
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69, S24-S35 (2001). index.htmli.
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Annalen der Physik 17, 891-921 (1905). Reprinted in En- ofthestarfieldfromarelativisticspaceship”,Am.J.Phys.
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(1988). 31 Ping-Kang Hsiung , R. H. Thibadeau, and M. Wu, “T-
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755-767 (1999). 32 J.D. Owens et al., “A Survey of General-
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64-74 (1998). hhttp://graphics.idav.ucdavis.edu/publications/i.
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Hall, Englewood Cliffs, NJ, 1997). 34 M. Bochers, “Interactive and stereoscopic visu-
13 W.K.Adams,S.Reid,R.LeMaster,S.B.McKagan,K.K. alization in special relativity”, Ph.D. disserta-
Perkins and C. E. Wieman, “A Study of Interface Design tion, University of Tu¨bingen 2005 (unpublished);
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14 C. Wieman and K.Perkins, “A powerful tool for teaching 36 D. Weiskopf et al., “Explanatory and illustrative visual-
science”, Nature Physics 2, 290-292 (2006). ization of special and general relativity”, IEEE transac-
15 S.Yeo,R.Loss,M.Zadnik,A.Harrison,andD.Treagust, tionsonVisualizationandComputerGraphics12,522-534
“What do students really learn from interactive multime- (2006).
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(2004). thecosmic background radiation and its rest frame”, Am.
16 R. N. Steinberg, “Computers in teaching science: to sim- J. Phys. 58, 942-945 (1990).
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Suppl.68, S37-S41 (2000). the visual appearance of a large sphere revisited”, Am. J.
17 A. Van Heuvelen, “Learning to think like a physicist: a Phys. 68, 56-60 (2000).
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Phys. 59, 891-897 (1991). ture”, IEEE Micro 26, 25-37 (2006).
18 ThroughEinstein’sEyes,amultimediawebsiteaboutrel- 40 Folding@home project web site:
ativisticoptics.AvailableonCDfromtheauthors,andon- hhttp://folding.stanford.edu/i.
line at: hhttp://www.anu.edu.au/Physics/Savage/TEE/i. 41 PHYS1201. Lectures and tutorials were 50 minutes each.
19 N.D.Mermin,It’sabout time(PrincetonUniversityPress, There were 74 studentsin the class.
Princeton, 2005). 42 G. Gamow, Mr Tompkins in Paperback (Cambridge U.P.,
20 R.Penrose,“Theapparentshapeofarelativistically mov- Cambridge, England, 1965). Note that the figures do not
ing sphere”, Proc. Camb. Phil. Soc. 55, 137-139 (1959). show what would actually be seen, as this was not known
21 J. Terrell, “Invisibility oftheLorentzcontraction”, Physi- at thetime thestory was written.
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22 V.F.Weisskopf,“Thevisualappearanceofrapidlymoving diameter is 4.7 light seconds) the distance between them
objects”, Physics Today 13, 24-27 (Sept.,1960). is not to scale.
23 W. Rindler, Relativity: special, general, and cosmological 44 The best way to experience this is to run the simulation.
(Oxford UniversityPress, USA,2006). However, a movie of the sequence discussed in the text is
24 Standard configuration of frames S and S′ has theS′ ori- available from the Real Time Relativity web site.1
gin traveling with speed v along the positive x axis of S. 45 Choosing the perpendicular distance to the clocks suffi-
The frames’ origins are coincident at t = t′ = 0, and the cientlylarge,comparedtothedistancebetweentheclocks,
corresponding coordinate axes are parallel. makes this approximation as good as required. It is only
25 TheinverseLorentztransformationsofthe4-frequencyare within this approximation that pure length contraction
obtainedfromEqs.(2)byswappingthewandclabelsand and relativity of simultaneity are observed.
changing the sign of v. 46 Quotationsaretakenfromthepost-laboratoryevaluations
26 J.M. McKinley and P. Doherty, “Relativistic transforma- completed by students.
tions of light power”, Am.J. Phys.47, 602-605 (1979).
27 J.M.McKinley,“Insearchofthestarbow: theappearance