Introduction to Spacetime Geometry


 Luke Robbins
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1 Intrductin t Spacetime Gemetry Let s start with a review f a basic feature f Euclidean gemetry, the Pythagrean therem. In a twdimensinal crdinate system we can relate the length f a line segment t the crdinates f its endpints using the fllwing relatin. s x y Fr example, the length f the line segment shwn in Figure 1 is 10 m. s x y 7m1m 10m2m 6m 8m 10m 2 2 One f the prperties f the length f a line segment is that it s the same regardless f the crdinate system used t measure it, smething that wuld be difficult t demnstrate if we measured x and y in different units. In spacetime gemetry there s an analgus relatinship: t t x Fig 1: A line segment f length 10 m. where t is time, x is psitin, and Δt is called the prper time. But psitin and time must be measured in the same units, such as minutes, where a minute f distance is hw far light travels in a time f ne minute. Fr example, light takes 8 minutes t travel frm the sun t Earth, s the distance is 8 minutes. Nte that the speed f light c equals ne minute per minute. This equatin is valid nly in systems f units like this, where c = 1. Using these units, let s lk at an example. Suppse we have a rcket that mves alng the x axis. We are interested in tw events, the first event is when the rcket has a psitin f 3, and then later when it has a psitin f 6. We als nte that the first event ccurs at a time f 2 and the secnd ccurs at a time f 7. x We calculate the speed f the rcket: 0.6 t 72 5 And the prper time is t t x minutes. This means that nly 4 minutes f time passes fr peple abard the rcket! The result f a calculatin like this is s unusual that it causes us t scratch ur heads and wnder hw we can understand it. We have t use care in explaining what s happening. The axis we used t measure the rcket s psitin is at rest relative t us. Likewise the clck we used t measure time is als at rest relative t us. They frm what we call ur frame f reference. Nte that in ur frame f reference we are at rest. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 1
2 Figure 2 shws such a frame f reference. We imagine an bject such as a rcket mving alng the x axis. When the bject is lcated at sme particular psitin we call it an event. We use the clck t measure the time t f the event and the x axis t measure the psitin x f the event. We say that x and t are the crdinates f the event. Fig 2: Viewing a frame f reference used t measure psitin x and time t. The viewer is at rest relative t the frame f reference. The example we just lked at invlved tw events. In ur frame f reference the first event ccurred at psitin x = 3 and time t = 2, where we are measuring psitin in minutes and time in minutes. The secnd event ccurred at x = 6 and t = 7. Shwn in Figure 3 is a spacetime diagram f this situatin. T make clear that spacetime gemetry is nt like Euclidean gemetry, let s take anther lk at what s being shwn in Figure 3. Figure 4 shws the triangle we re using t d ur Fig 3: A prper time f 4 minutes. spacetime calculatin. Nte that it s a strange lking triangle because the hyptenuse is nt the lngest side! The type f thinking that we re used t ding just wn t help us ut here. The gemetry is nt Euclidean. We d nt use the Pythagrean therem and Euclidean gemetry. We use the definitin f prper time and spacetime gemetry. t t x Nte that peple abard the rcket have their frame f reference. It s a frame f reference in which they are at rest. It s the clck in their frame f reference that reads an elapsed time f 4 minutes between the events. Fig 4: A spacetime triangle. One f the things that makes this seem strange is that we are used t lking at things frm ur wn frame f reference, and thinking that what we measure wuld be the same when measured in every frame f reference. We culd instead think f ur planet Earth as ur rcket, traveling thrugh space at a speed f 0.6c relative t sme bserver. Suppse tw events ccur at the same psitin, and are separated by 4 minutes f time, in ur frame f reference. Thse same tw events are separated by 5 minutes f time in that bserver s frame f reference. Is there anther way f relating these clck readings that desn t invlve gemetry? Yes, it s the familiar frmula fr time dilatin. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 2
3 In ur example we have 0.6 and therefre t t S the time in ur frame f reference is t t Mun Decay: A Wrked Example Muns are particles that were first discvered in The mun is a much heavier cusin f the electrn, and it s unstable with a mean lifetime f 2.2 μs. Sme will live fr a lnger time, sme shrter, but n average, they live fr 2.2 μs befre decaying. Muns are created in Earth s upper atmsphere, and sme f them subsequently travel tward Earth s surface. Fr this example we ll suppse we have a single mun mving tward Earth s surface at a typical speed f c, and that it lives fr 2.2 μs befre decaying. T calculate the lifetime f this mun in ur frame f reference we first calculate the value f Giving us a lifetime f t t μs μs. Therefre the distance it travels in ur frame f reference is 8 m 6 s xvt s m. If we had instead used the prper time f 2.2 μs t d this calculatin we d have a distance traveled f nly 660 meters. Such a difference is s huge that it s easy t detect experimentally. Earth s atmsphere reaches t a height f mre than meters, s if the average mun decayed after traveling nly 660 meters we d expect very few t make it t Earth s surface. Abut 90 years has passed since the discvery f muns in ur atmsphere, but even then physicists knew that a large fractin f the muns created near the tp f the atmsphere wuld make it t Earth s surface. This is just ne f many experimental results and measurements that cnfirm the validity f Einstein s relativity. Let s lk at the spacetime gemetry. Remember that we need a system f units where c = 1, s we shall chse micrsecnds fr units f time and micrsecnds fr units f distance. A micrsecnd f 8 m 6 distance, the distance light travels in ne micrsecnd, is (310 s )(110 s) 300 m. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 3
4 First, we calculate that x t Finally, we verify that the definitin f prper time is satisfied: t t x Fig 5: Spacetime diagram fr flight f mun. Figure 5 shws the spacetime diagram fr the flight f the mun. Nte that the tw legs f the right triangle are almst identical in length (because the mun s speed is very near the speed f light) and that again the hyptenuse is nt the lngest side. The Twin Paradx: A Wrked Example Tw twins, Tess and Sam, test ut the validity f relativity by ding an experiment. Sam stays at hme while Tess travels. Tess spends ne hur traveling at a speed f 0.6c, then turns arund and heads back hme at the same speed. When she gets back 2 hurs have elapsed n her clck, but 2.5 hurs have elapsed n Sam s clck. Scratching their heads, they try t use the definitin f prper time t figure ut the discrepancy in their clck readings. T d their spacetime calculatins they decide t measure time in minutes and distance in minutes. A 8 m 10 minute f distance is hw far light travels in a minute f time: (310 s )(60s) m. First, we lk at things frm Sam s frame f reference. He imagines an x axis with the rigin at his huse n planet Earth. It stretches ut int space fr several minutes f distance. Each minute representing the distance a light beam wuld travel in ne minute f time. Sam realizes that half f 2.5 hurs is 75 minutes f time. He uses this t calculate the distance t the turn arund pint: x t (0.6)(75) 45. S there are tw events, Event 1 is when Tess leaves, and Event 2 is when Tess reaches the turn arund pint. In Sam s frame f reference they ccur at x = 1 and x = 45, respectively. Fig 6: Event 1 is the departure, ccurs at x = 1. Event 2 is the turn arund, ccurs at x = 45. Nw let s lk at things frm Tess s frame f reference. These tw events ccur at the same place in her frame f reference, s the amunt f time that elapses between them is a prper time. The relatinship between her crdinates and his is given by 2 t t x Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 4
5 Figure 7 is the spacetime diagram Sam drew f the utbund half f Tess s trip. We can verify the calculatin f the elapsed time using the time dilatin frmula. Recalling that = 1.25 when = 0.6. t t S, frm Sam s perspective Tess takes 60 minutes t g ut. And then she wuld take 60 minutes t cme back, fr a ttal time f 2 hurs. But fr Sam it s 75 minutes f waiting fr Tess t g ut, 75 minutes fr her return, fr a ttal f 2.5 hurs! Fig 7: Sam s spacetime diagram f Tess s utbund trip. The questin arises, and this is the apparent paradx, why can t we lk at things frm Tess s frame f reference, and cnclude the reverse? In ther wrds, Tess imagines an x axis stretching ut frm her space ship. In her frame f reference she stays at the rigin, but she sees Sam as mving away frm her alng that x axis until he gets t the turnarund pint. Then it wuld be her that experiences 75 minutes f time while he experiences 60. In fact, that is a perfectly valid way t lk at it! Figure 6 culd just as well be Tess s spacetime diagram f Sam s utbund jurney. It seems strange t think f Sam as ging anywhere when he stays at hme, but frm Tess s pint f view that s exactly what she sees Sam ding. Yu culd imagine a trip like this ccurring at a much faster speed and fr a much lnger time, s the difference is mre prnunced. The traveling twin culd be gne fr 50 years, and be nly 5 years lder upn return. The stay at hme twin is 50 years lder and is nw a grandfather. It seems perfectly reasnable t ask why the reverse desn t happen s that the traveling twin is 50 years lder and the stay at hme twin is nly 5 years lder. The reslutin f the paradx lies in lking at the cmplete spacetime diagram f bth the utbund and the return legs f the jurney, as shwn in Figure 9. This has t be a diagram f Tess s jurney frm Sam s perspective, it cannt be a diagram f Sam s jurney. The reasn is the kink at the half way pint where Tess turns arund and heads back hme. Such a change in directin is smething that nly Tess experiences. Even if she had her eyes clsed she wuld ntice the acceleratin frm turning arund. She can feel it. Sam experiences n such thing at the half way pint. He culd be relaxing in a chair, eyes clsed r pen, and he d feel nthing. N acceleratin. It s nt he wh turns arund, it s she. That s what breaks the symmetry and lets us realize that the scenari where the stay at hme twin ages mre is the crrect ne. Fig 9: Sam s spacetime diagram f Tess s cmplete trip. A mre in depth analysis is required t wrk things ut frm Tess s tw different frames f reference, the ne she s in during the utbund half, and the ne she switches t fr the return half. Ding ne f these mre cmplete analyses gives f curse the same result. Tess experiences a prper time f 2 hurs, Sam experiences a prper time f 2.5 hurs. The difference in their ages is a difference in prper times. Nte that this is nt the same thing as the difference between a prper time and a dilated time. A difference in prper times is smething all bservers will agree n, whereas the difference between an bserved dilated time and a prper time depends n the bserver s frame f reference. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 5
6 Exercises 1. A subatmic particle lives fr 7 micrsecnds befre decaying, as measured in its wn frame f reference. (a) Calculate its lifetime as measured by an bserver with a relative speed f 0.96c. (b) Sketch a spacetime diagram f the particle s mtin frm the frame f reference f the bserver. Draw the diagram t scale. Label the axes with numbers. Label the length f the lines drawn n the diagram. 2. One twin leaves hme and travels in a straight line at a speed f 0.96c fr 7 years, as measured in his wn frame f reference. He then turns arund and cmes back hme at the same speed, having aged 14 years during the trip. (a) His twin sister, wh stayed at hme, ages hw much during her brther s absence? (b) Sketch a spacetime diagram f the traveling twin s mtin frm the frame f reference f the stay at hme twin. Draw the diagram t scale. Label the axes with numbers. Label the lengths f the lines drawn n the diagram. Intrductin t Spacetime H Trivilin, Cllege f the Mainland, December 2015 Page 6
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