Space-Time Diagrams: Visualizing Special Relativity

1 Page 1 of 4 Space-Time diagrams : Visualizing Special RelativityProf. Steuard Keck Science Center, The Claremont CollegesA Space-Time diagram shows the history of objects moving through space (usually in justone dimension). A specific point on a Space-Time diagram is called an event. To make a Space-Time diagram, take many snapshots of the objects over time and setthem on top of each other. Lines in the diagram are like contrails through time . Given a diagram, to take a snapshot find the positions of each object on the appropriate time slice (the dashed lines below).A time slice labels a set of events thatthe observer considers will draw time slices at equally spacedintervals so that each slice corresponds to one tick of the observer s (yr)x (ly)c = 1 ly/yr12345678910012345601234560 Space-Time diagramSnapshots ( time slices)123456012345601234560123456012345 60x (ly)x (ly)x (ly)x (ly)x (ly)x (ly)In these diagrams : The closer an object s path is to vertical, the slower it is moving:slope = 1/v.

2 A 45 angle corresponds to the speed of light: we will always measure time in years (yr)and space in lightyears (ly) (or in seconds and light-seconds, etc.).Thus,c= 1 few examples:t (yr)x (ly)1234560123450 Elastic collisiont (yr)x (ly)1234560123450 Totally inelastic collisiont (yr)x (ly)1234560123450 There and back againPage 2 of 4 Thus far, this is just another way to visualize motion: it applies just as well to classicalphysics as it does to Relativity . Relativity adds two new ingredients: Nothing can travel at more than a 45 angle from vertical (even for a moment):v < c. Different observers see different time slices!In particular, the time slices seen by the observer drawing the diagram (the observer atrest ) are the same horizontal lines as before.

3 But the time slices seen by an observertraveling at speedvrelative to the diagram s reference frame have the following properties(recall that =1/q1 v2c2, so for example,v=12cgives = ): The time slicestilt up toward 45 asvincreases: they haveslopev/c2=v 1 long as we drawcat 45 , this means thata moving observer s time slices tiltup by the same angle that her path tilts downin our diagram.(Technically,we are graphing the Lorentz transformation equations, plugging in various values fort .) time slices getsqueezed togetherasvincreases. (Comparev=12cand45cbelow.)A moving observer s time slices spaced t apart are separated vertically on the diagramby t ( forv=12c, 1 yr time slices intersect thet-axis every yr). A moving observer s clock ticks occur when her time slices intersect herpathon the diagram.

4 Thus, an observer at rest sees the moving clock tickinglessoften: theintersectionshave larger vertical spacing, every t= t (so the movingobserver s t = 1 yr time slice intersects her path at diagram time t= yr).t (yr)x (ly)123450123401 yr2 yr3 yr4 yr = v = (yr)x (ly)123450123401 yr2 yr3 yr = = :Above, the diagram observer clearly sees moving clocks running slow: for each 1 yr intersection along the moving path, his dotted horizontal time slices come more than1 yr apart. For example, in thev=12cdiagram the moving observer s 2 yr clock tick isdelayed untilt= yr (and whenv=45c, it is delayed even more: untilt= yr).But the moving observer would say the same thing about the diagram observer! In thev=12cdiagram, the diagram observer s clock tick atx= 0 andt= 2 yr falls between themoving observer s 2 yr and 3 yr time slices.

5 (In fact, it is at exactly t = yr.) Bothobservers see the other clock running slow by the same amount. The page about the twin paradox will give you some idea of why this isn t a 3 of 4In Relativity (including Space-Time diagrams ), time and space are mixed together in twoNot-Quite-Pythagorean Theorems for Space-Time ( the Space-Time interval ): The time t experienced along an observer s path obeys (c t )2= (c t)2 ( x)2. A length x measured along an observer s time slice obeys ( x )2= ( x)2 (c t) are equivalent to t= t andL=L (just plug in x=v t, etc.).WARNING!Never measure lengths at an angle on a spacetime diagram with a ruler: lines onpaper obey theordinaryPythagorean theorem, not these Special Space-Time (s)x (ls)12345012340 Measuring length5 x tL = x L = xmoving framediagram frame(different x!)

6 T (yr)x (ly)12345012301 yr2 yr3 yr4 yr = v = cTo findthe length of an object, an observer mustmea-sure between points on the SAME time slice, as shown.(Why? At right, if you located the tail atx= 0 whent= 0and then located the nose atx= ls ( light-seconds ) whent= 5 s, you shouldn t conclude that the ship is ls long!)Motion in the xdirection:The observer s path tiltsdown to the left toward 45 and her time slices tilt up, as shownat right.(Noteworthy aside: This would be the diagram drawn bythemovingobserver in thev=12cdiagram on the previous her, the formerly at rest observer is moving to the left withvelocityv= 12c. The original observer s formerly horizontal timeslices are tilted in this reference frame.)Multiple moving objects:The diagrams below showthree observers with different velocities.

7 In both, the timeslices shown are those for the middle observer: we changefrom the left observer s perspective to his. After the change,the middle observer s time slices become horizontal and thevelocities{v0= 0,v,u}change to{v 0,v = 0,u }as cannot simply add and subtract velocitieswhen changing reference framesexcept in a couple ofspecial cases. Instead, use the following result (derived bymeasuring a moving object s position along tilted time slicesusing the methods above):u = (u v)/(1 uvc2).t (yr)x (ly)123450123405v = cu = ct (yr)123451230x (ly)123v0 = cu = cChanging to another reference frame 1 yr2 yr3 yrv = 0v0 = 0 Page 4 of 4 The Twin Paradox in a Space-Time Diagramt (yr)x (ly)123456789100123401 yr2 yr3 yr4 yr5 yr6 yr7 yr8 yr = v = cTwin Paradox Our textbook describes a 20 year old mannamed Speedo who takes a trip in a space -ship while his identical twin brother Goslostays on Earth.

8 If Speedo travels atv=12cfor five years (as measured on Earth) andthen quickly turns around and comes homeat the same rate, his path on Goslo s Space-Time diagram is as shown (Goslo s path isstraight up thetaxis). The diagram at rightshows Speedo s time slices (dashed lines) atone year intervals (by his measurement). Italso includes extra (dotted line) time slicesat (Speedo s) quarter-year intervals betweenhis years 4 and 5 as he turns first important observation is thatwhen Speedo gets home, Goslo has aged 10years (he is now 30), but Speedo has onlyaged years (he is not yet even 29).As long as Speedo moves at constantspeed (away from Earth or toward it), he andGoslo each see the other as aging less quickly.

9 (When Speedo celebrates his 24th birthdayhe thinks that Goslo is still only , butGoslo thinks the very same thing on his own24th birthday.)But that leads to the second importantpoint: the difference between motion in aninertial frame (without acceleration) and mo-tion that includes acceleration. During thebrief period aroundt= 5 yr on Earth (oraroundt = yr on Speedo s spaceship) when Speedo reverses direction, the angles of histime slices change because his velocity is changing. Because those angles change while he isfar away from Earth, he sees Goslo age very quickly during the brief time it takes him toturn around (by an extra years). The faster he changes direction, the faster he sees thoseextra years pass back on Earth.

10 Meanwhile, Goslo sees no such extra time pass for Speedobecause Goslo s time slices never change their this reason, we could not draw a correct Space-Time diagram based on Speedo as theobserver at rest. Special Relativity is only valid when the observer drawing thespace- time diagram moves with constant a side note, it might be possible to draw a similar kind of Space-Time diagram fromSpeedo s perspective using techniques fromgeneralrelativity, but that s beyond the scope ofthis class. Doing so would probably require us to draw the picture on a curved surface ratherthan on a flat sheet of paper: essentially, Speedo would move straight along the time axis atx= 0, but Goslo s path through time would have to pass over a hill, making it longer.