Introducing Special Relativity Philip Freeman Roberta Tevlin
Planet Relativity Imagine a world where scientists have not realised that their planet is a sphere. • Aù • ùB Two scientists, Roberta and Philip, are observing two rockets, and measuring their heights above the horizon.
Planet Relativity • ùB • Aù • ùB • Aù • Aù • ùB Who’s right? Who cares?
Different directions for “UP” • The thing Roberta and Philip didn’t recognise is that they are both right, but they are talking about different things! • They have different directions for “up” and therefore disagree. • Each thinks the other has mixed together up/down with east/west
The relativity of “up” • The idea that “up” depends on the observer solves this sort of paradox, and many others(eg: why don’t people on the other side of the world fall off?) • The fact that up varies is an important clue as to the nature of space itself! • SO – how would we find out that up was different for different people?
Hints from balListics and Navigation • We might find hints in various fields, little inconsistencies and complications that add up after a while. • By adding rules and complexities we could make things consistent in one field or another, but we’d start to see contradictions between the fields.
100 years ago (OK, 106 years ago) • This is just exactly the situation 100+ years ago! • Physicists were starting to feel that they’d pretty much gotten everything sewn up and were near the end of physics (hmmm sound familiar?) • There were just a few little things, at the interfaces between fields, that were a bit messy.
Like what? • Mechanics • Electromagnetism • Thermodynamics Little contradiction to do with velocity of light in a vacuum Little contradiction to do with black body radiation (but that’s another story)
Contradiction? I don’t see no contradiction! • Mechanics: All motion is relative(laws of physics independent of constant velocity AKA Galilean Relativity) • Electromagnetism: The speed of light is absolute(can be calculated from electric and magnetic fields)
No big deal! We can fix this… • Speed of light must just refer to the speed of light compared to its medium. • No medium? There has to be a medium = ether. • But problems still pile up + can’t find the Earth’s motion through the ether.(Michaelson Morley – but actually not very significant in development of relativity)
Enter a wild card • Enter a young physicist, very irreverent, keeps getting in trouble for attitude problems… but with a gift. Guy name of ALBERT EINSTEIN. • Einstein’s gift can be seen as being able to say “Let’s just go with it and see what happens!” • What if BOTH statements are right?
Postulates of Special Relativity • All motion is relative= The laws of physics are the same in all inertial frames.= It is impossible to do an experiment in a closed laboratory which will measure that laboratories constant motion • The speed of light is absolute= The speed of light has the same value for all observers regardless of the relative motion of source or observer
So… what does that imply? • Light is constant so let’s use it to make measuring instruments, eg a Light Clock Tick! Tock! counter
What if I have a moving light clock? Your clock Myclock
Light in my clock goes farther! • The faster I go the further light travels in my clock. Your clock Myclock ct 3.0m 3.0m vt
Optional calculation: (helps make concrete) • Suppose I’m going at 70% the speed of light… Your clock Myclock How long does the light take to travel 3.0m in your clock (how long between tick-tocks?)ans: 10 ns = 2.1x108m/s ct 3.0m 3.0m vt
Light in my clock goes farther! • Suppose I’m going at 70% the speed of light… Your clock Myclock How long does it take light to travel the distance it has to in the moving clock? (how long do you measure between my tick-tocks?)(hint: express horizontal distance and diagonal in terms of time and use Pythagorean relation)ans: 14 ns = 2.1x108m/s ct 3.0m 3.0m vt
Moving Clocks run slow! • What do you measure? What do I measure? • My clock reports to me that 10 ns have passed, but YOU would say that actually it was 14 ns. My light clock is running slow. • That means ALL time measurements are slow too. My time is slowed down compared to yours • Why? Raise as many objections as you can to this result… eg: - maybe the light doesn’t go at an angle, it just misses the mirror?
How much are things slowed?Let’s calculate symbolically. Your clocktime = T Myclocktime=t Pythagorian Relation: (vt)2 + (cT)2 = (ct)2c2T2 = c2t2 – v2t2 T2 = t2 – (v2/c2)t2 T2 = (1 - v2/c2) t2 \\ ct L = cT L vt
Moving clocks are slow by a factor of • The moving clock is slowed by a factor of
It’s about time, it’s about space • If time is changed then space must be affected too, otherwise we wouldn’t agree about the speed of light! • If I measure a shorter time than you then I must measure the distances as shorter too. • You see my moving clocks slow (by ) • I see your moving metre sticks shrunk (by )
What do I see looking at you? • You see my clock run slow, including my watch running slow, my heart beating slow, my thoughts going slow… my life is just slowed down by this factor. • If I try to talk to you it wiilllbeeeeveeeryslooow… • What will I see if I look at you? What will I hear if you talk to me? • Discuss and be prepared to describe what I’ll see/hear!
But wait a moment! • You saw me moving, and therefore MY clock was slow. Myclock Your clock
Who’s moving though? My clock Your clock • But what do Isee? • I’m the one who’s at rest, YOU are moving!YOUR clock is the one that is slow!
Whose clock is slow? • ùB • Aù • I think YOUR clock is running slow. • YOU think MY clock is running slow. • Does this sound familiar? • Aù • ùB
Different directions • Philip thought Roberta’s metre stick was too short, and Roberta thought Philip’s was. • This was because their metre sticks were pointed in different directions. They had different directions for “up”. • The same is true for the observers comparing clocks. We each think the other’s clock is slow becausewe have different directions for TIME!
How can time have a direction?? Grandma’s house LRR house • A spacetime diagram: time “World Lines” Red Riding Hood Big Bad Wolf space
The time axis • Time axis = same point at different times 3) Yep, right here time 2) Still here 1) I’m here space
Moving at constant velocity • Moving observer’s world line (axis) is Tilted. 3) Haven’t budged Time axis points in a different direction in spacetime! time 2) Still here What is all time for one observer is partly time and partly space to another! 1) I’m here space
BACK TO Planet Relativity for a moment: • Aù • ùB “Invariant Distance” Philip and Roberta measure different up/down and east/west distances. But it they put their x and y values together they will find an invariant distance.
But nothing is ever simple! • It takes Roberta and Philip a while to figure out this invariant because they have always measured vertical height in feet and east/west distance in kilometers. • Eventually they do find a constant that combines the units, so they can convert to common units: C=3280.84 ft/km
Vertizontal • What Philip and Roberta have discovered is that vertical and horizontal are not different things, but are ONE unified thing, “vertizontal” which can be seen from different angles. • What is purely vertical to one observer is partly vertical and partly horizontal for the other, and vice versa. Hence all the ruler arguments!
Spacetime • You guessed it! Einstein (and others, esp. Poncaire and Minkowski) realised that space and time are not separate things, but the combo was the real thing: spacetime • We also have the quaint habit of using different units for space and time. We need to know how to convert, ie how many metres are there in a second? (wanna guess?)
Converting seconds to metres • Conversion factor c = 3.0 × 108 m/s • Exercise: How old are you in metres? Why so MANY metres/second?
Rotation Vertizontal: The different directions of ‘up’ were because of a regular ‘circular’ rotation: Spacetime:The different directions of time are a rotation too, but not a circular one. They are a hyperbolic rotation: y t x x (eqn of circle) (eqn of hyperbola)
Let’s keep this simple! • Postulates: • Laws of physics don’t depend on constant motion • Speed of light is the same for all observers • Results: • Moving clocks run slow (by a factor of ) • Moving metre sticks shrink (by a factor of ) • Moving masses are ‘increased’ (mass-energy) • Spacetime: • Space and time are one thing, motion is changing direction of time
Where to slip this in (being subversive again) • Kinematics? • Waves? Einstein Simplified • E/M? • Other?