Class 16. Prof J. Kenney October 31, Relativity

Transcription

1 Class 16 Prof J. Kenney October 31, 2016 Relativity

2 Length contraction (moving sticks are shorter) A measuring stick at rest has a length Lo. When it is propelled at velocity v, it has a shorter length according to an observer not moving with it

3 Time dilation (moving clocks run slow) Suppose you are at rest on earth Your friend is flying past you at high speed v According to you, the interval between your heartbeats is t 0 = 1 second. Your friend measures the interval between her heartbeats to be the same t 0 = 1 second.

4 Time dilation (moving clocks run slow) But YOU measure the interval between HER heartbeats to be a longer time t 0 If v = 0.99c, t = 7t o = 7 sec According to you, your moving friend is aging more slowly! But according to her, YOU are aging more slowly! (in this case the effect is symmetric)

5 Key points of Special Relativity Motions affect our measurements of distance and time Space is not fixed and uniform Time does not pass at uniform rate Space & Time cannot be thought of as separate entities. They are intrinsically linked! So introduce linked concept spacetime.

6 Train in the tunnel paradox Train and tunnel at rest According to observer at rest: Train 1,000 ft long Tunnel 800 ft long So train at rest does not fit in tunnel...

7 Train now moving at 0.8c According to observer at rest wrt tunnel: Train 600 ft long (length contraction) Tunnel 800 ft long So train DOES fit in tunnel, according to observer at rest wrt tunnel!!

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9 Train now moving at 0.8c According to observer on train: Train 1,000 ft long Tunnel 480 ft long (length contraction) So train DOES NOT fit in tunnel, according to observer on train!!

10 Does the train fit in the tunnel or doesn t it? 1. Yes it fits 2. No it doesn t fit 3. Neither 1 nor 2 4. Both 1 and 2 5. My head hurts

11 Does it or doesn t it fit into the tunnel...and does it matter? Add something to the situation so that it DOES matter so that people could die, depending on the answer!!

12 Tunnel master controls iron gates at the 2 ends of the tunnel When the train is in the tunnel (according to the tunnel master, at rest wrt tunnel), he slams the 2 iron gates shut at the same time, then quickly opens them

13 According to tunnel master at rest wrt tunnel: Train fits in tunnel GATES DO NOT SMASH INTO TRAIN & NO ONE DIES

14 What do people on the train observe? For them, the train does not fit into the tunnel... What happens when the iron gates close? DO THEY DIE? 1. Yes, but not all of them 2. No, length contraction allows the train to fit between the gates 3. No, since the gates don t close at the same time in their view 4. No, because of time dilation 5. Yes, but in a parallel universe an identical train survives, so it s OK

15 No... Everyone agrees that they do not die. But how?

16 For observers on the train, the gates do not shut at the same time... The front gate shuts just before the front of the train reaches it, then it opens and the front of the train goes through. THEN A BIT LATER the rear gates shuts, just after the rear of the train passes. So, the train doesn t fit in the tunnel but no one dies because the gates do not close simultaneously!!

17 What observers disagree on (these things are relative) Size of train & tunnel (space) When the gates close (& whether they close simultaneously) (time) What observers agree on: events & their outcome: (events: something that occurs at a location in space at a point in time) Front gate misses front of train Back gate misses back of train

18 simultaneity Observers don t agree on whether things are simultaneous, i.e. whether spatially separate events happen at the same time! For 1 observer, 2 events might be simultaneous, but for another observer moving wrt the first, the 2 events are NOT simultaneous

19 Why does relativity seem strange to us? Need to be moving at very high speeds (v>0.1c) before effects are noticeable v 1/ [1-v 2 /c 2 ] c c c c c c c infinity

20 Why does relativity seem strange to us? Fast car 200 m.p.h 3x10-7 c Fast plane 2000 m.p.h 3x10-6 c Space shuttle 17,500 m.p.h 3x10-5 c Orbital velocity 67,000 m.p.h 1x10-4 c of earth We live in a low velocity environment. We can safely use Newtonian mechanics at these speeds

21 The Twin paradox

22 What "breaks the symmetry" between you and your twin, that allows the twin to be much younger than you on return? 1. While the physiological effects on your bodies are different, you both have experienced the same passage of time 2. Twin moved through space, whereas you did not 3. Twin accelerated at start and end, whereas you did not 4. Twin experienced different gravitational fields on journey. 5. Twin lost track of time during journey

23 The answer is : acceleration is involved! Special Relativity doesn t cover acceleration, it only covers constant velocities. (so there is no paradox no logical inconsistency) For acceleration, we need General Relativity. And it turns out, acceleration is intimately related to gravity!

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25 Basic Postulates of General Relativity 1. Relativity (or Special Relativity) a. laws of physics same in all non-accelerating reference frames b. Speed of light constant 2. Equivalence Principle there is no way to locally distinguish between gravity and acceleration they are equivalent!

26 Motivation of Principle of Equivalence 1. You feel weightless if you are freefalling!

27 Can t distinguish between: Falling in gravity field Being at rest in zero gravity

28 Can t distinguish between: being at rest in Gravity field accelerating upwards

29 Motivation of Principle of Equivalence 2. All objects in gravitational field accelerate at same rate, independent of their mass Newton s explanation of this: Equation for dynamics: F = ma acceleration depends on force and mass, so you might expect different accelerations for different masses Equation for gravity: F grav = GMm/R 2 equating these yields: F = F grav ma = GMm/R 2 a = GM/R 2 = g (little g is gravitational acceleration at earth s surface)

30 Motivation of Principle of Equivalence 2. All objects in gravitational field accelerate at same rate, independent of their mass Newton s explanation of this: Equation for dynamics: F = ma acceleration depends on force and mass, so you might expect different accelerations for different masses Equation for gravity: F grav = GMm/R 2 equating these yields: F = F grav ma = GMm/R 2 a = GM/R 2 = g so mass m of object cancels out! acceleration (Motion) of object in gravity field doesn t depend on its mass!

31 Motivation of Principle of Equivalence 2. All objects in gravitational field accelerate at same rate, independent of their mass Newton s explanation of this: Equation for dynamics: F = ma acceleration depends on force and mass, so you might expect different accelerations for different masses Equation for gravity: F grav = GMm/R 2 equating these yields: F = F grav ma = GMm/R 2 a = GM/R 2 = g so mass m of object cancels out! this is some guy s attempt to explain observed fact! acceleration (Motion) of object in gravity field doesn t depend on its mass! this is observed fact!

32 Acceleration (Motion) of object in gravity field doesn t depend on its mass! So Newton came up with a pair of equations which each contain the mass m, but just so happen to have m cancel out when they are combined, so that the prediction agrees with the observations How convenient! [ Newton may have thought this was interesting or curious, but he didn t pursue it.] But Einstein thought this was fishy and amazing. So amazing that it deserved to be a postulate which transcends Newton s equations. This postulate is the of Principle of Equivalence

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35 Light beam as seen by elevator passenger

36 Principle of equivalence acceleration gravity Acceleration bends light -> gravity bends light

37 gravitational lensing (see animation )

38 Gravitational lenses Arcs formed by distorted images of background galaxies lens is large galaxy cluster Nearly complete ring formed by distorted image of background galaxy lens is large elliptical galaxy

39 A light source passes behind a gravitational lens (point mass placed in the center of the image). The aqua circle is a source as it would be seen if there was no lens; white spots are the multiple images of the source. (wikipedia)

40 Curvature of space Gravitational bending of light suggests that we might think of space as being curved near masses

41 Curvature of space the reason light & other things follow a curved trajectory near a mass is that space is curved near the mass, and things simply follow their natural trajectory in this curved space

42 Curvature of space Imagine 3D space as 2D rubber sheet with mass as heavy ball bearings causing indentations in sheet Rolling balls on curved sheet follow trajectories & orbits similar to those of things in curved space

43 Gravitational lensing Light is deflected by curved space around masses Light paths of 2 photons which pass near a mass (deflector)

44 Curvature of space What determines trajectory or orbit: a. How sheet (or space) is curved by the masses bigger masses make big indentations in sheet (or space) b. Speed of thing DOESN T depend on mass of thing that is orbiting! This is why concept of curved space is a good one

45 Newton vs. Einstein on gravitational lensing Newton: can t explain this! Trajectory of any mass is bent from straight line by force of gravity Light has no mass, shouldn t be affected by force of gravity Einstein: can explain this! Light travels in straight line (shortest distance between 2 points) Space(time) itself is distorted (curved) by presence of matter

46 Basic ideas of General Relativity Mass of object alters properties of space & time around it Gravity causes space to become curved and time to slow down Eliminates idea of force of gravity, instead, gravity is curved spacetime

47 Tests of general relativity 1a. Deflection of starlight by masses: Shift of star position near limb of sun (1919)

48 Tests of general relativity 1b. Deflection of starlight by masses: Gravitational lenses Nearly complete ring formed by distorted image of background galaxy lens is large elliptical galaxy Arcs formed by distorted images of background galaxies lens is large galaxy cluster

49 Tests of general relativity 2. Advance of perihelion of Mercury s orbit

50 Tests of general relativity 3. Gravitational time delay: time passes more slowly in stronger gravitational field

51 Tests of general relativity 3. Gravitational time delay: radio signals from Viking Spacecraft on Mars were delayed as they passed by the Sun This is NOT because speed of light changes but because time passes more slowly in gravitational fields (gravitational time dilation)

52 Tests of general relativity 4. gravitational redshift observed from dense astronomical objects (white dwarfs, neutron stars, black holes) Gravitational time delay -> gravitational redshift Think of the frequency of an electromagnetic wave as a clock, which slows down in a gravitational field (slower frequency -> redshift)

53 Tests of general relativity 5. detection of gravitational waves from merging black holes! (2016)