PHYS 113: Relativity ISpacetime DiagramsWe keep seeing the word “relativity” appear in our discussion. To the person on the street,relativity (normally associated with Einstein) elicits the statement “everything is r elative,”implying that we cannot know anything obj ectively. In fact, b oth Einstein and Galileo (whowe will also discuss) would have emphatically disagreed with this assessment. We can saythings objectively – it’s just that part of our objective statement needs to include informationabout how to translate what I see, to what you see.Our standard piece of equipment for this experiment will be a train moving to the right atsome known speed, v. You will be on that train, while I will be on the platform.Prof. Goldberg’s (unprimed)framev−vYour (primed) frameNote that the train is moving at all times. The “Special” in special relativity comes fromthe fact that there are no a ccelerations.Of course, you also know from your own experience that from your vantage, it appears thatI am moving backwards at sp eed, v.In general, when we talk about different perspectives, we’ll refer t o the “unprimed” (meaning,in this case, my) or “primed” (meaning your) frame. Fo r example, if I saw a puppy roll over10 meters in front of me, I’d say, x = 10m. However, if you also observed the same “event,”but it was behind you 5m, we’d say, x′= −5m, which is just a shorthand for saying that theposition of the puppy rolling over is -5m from the perspective of the person in the train.I used the word “event” in the previous paragraph. For us, there are only events, a nd eventshave a time, and a position. Thus, “The party begins at 9:00pm at the funky disco” indicatesan event.We can imagine a particularly simple 1-d universe full of events, and plot it up on a “space-time” diagram.Relativity I: Galilean Relativity– 1PHYS 113: Relativity ITimeSpaceStationaryObserverObserver Travelingat 1/2 cLight Detectedand ReflectedLight Detected3 "Events" in spacetimeLight Beam SentIn general we tend to scale these things so that 1 unit in the x-direction corresponds to, say,1 light-second, and 1 unit in the y-direction corresponds to a second, and thus, light travelsat 45 degree ang les (or -45 deg. if it’s traveling to the left). Someone sitting still will movevertically through time (but not through space). And, of course, since no-one can travelfaster than light, your “world line” can’t ever be shallower than 45 degrees.On the space-time diagram, I’ve drawn a light-beam fired from a ship moving half the speedof light toward an observer at rest (Event 1). The beam is detected by the stationary observer(event 2), and re-deflected. Finally, the the lightbeam is detected by the moving observer(event 3).For each observer, each event has a definite time and position. I have drawn things from theperspective of the stationary observer, since from your own perspective it always seems asthough you are sitting still.Galilean RelativityAll of this is just notation leading us up to relativity. And when you think relativity, you nodoubt think of Galileo. Well, probably not. But in reality, Galileo’s relativity is the one inour everyday experience.Time: For Galileo, time was an absolute, and thus, all clocks ran at the same rate. So, eventhough it is legitimate (say if you and I are in different time-zones) f or us to measure anevent at different “times” – say t1= 0, and t′1= 3600s (if you measure a particular event asbeing one hour later than may because you never reset your watch fro m daylight savings),it is t rue that our watches run at the same rate. Thus, if we measure a second event, and Imeasure it at t2= 10s, you’ll measure it as t′= 3610s.Relativity I: Galilean Relativity– 2PHYS 113: Relativity IAccording to Galileo, For both of us:Galileo : ∆t = t2− t1= ∆t′= t′2− t′1Space: With space it’s a different matter. Consider that for a moving observer, obj ects,and hence events, will look like they’re in different positions at different t imes. Thus, if aparticular event appears to me a t some position, x, you’ll observe it asx′= x − vtwhich should accord with your intuition. If you don’t believe me, think about how a station-ary observer appears as you pull away from the station. As more time passes, his positionlooks more and more negative.Finally, we can relate the velocity of an observed object. Consider that velocity is simplydefined as:u ≡∆x∆t(1)where we simply observe a moving object at two different times. Using the relations:∆x′= ∆x − v∆t (2)∆t′= ∆t (3)simple algebra yields:u′= u − v (4)In other words, if you were on a train going 100mph, and I fired a bullet parallel to the trainat 150mph, you would observe the bullet to be moving at 50 mph.Finally, note that these equations are completely symmetric. We could just as easily say:∆x = ∆x′+ v∆t′∆t = ∆t′u = u′+ vAnd thus, there’s no way from Newton physics to tell who is moving and who is staying still.According to Newton, the o nly thing that matters are accelerations, and consider that asystem which obeys Newton’s laws in one frame will also satisfy them in any other inertialframe.No problem, right?Wrong. Wait until next time.Relativity I: Galilean Relativity–