Relativity Class 16 Prof J. Kenney June 18, 2018 boss
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
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.
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)
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.
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...
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!!
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!!
Does the train fit in the tunnel or doesn t it? A. Yes it fits B. No it doesn t fit C. Neither 1 nor 2 D. Both 1 and 2 E. My head hurts all answers sort-of correct
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!!
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
According to tunnel master at rest wrt tunnel: Train fits in tunnel GATES DO NOT SMASH INTO TRAIN & NO ONE DIES
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? A. Yes, but not all of them B. No, length contraction allows the train to fit between the gates C. No, since the gates don t close at the same time in their view D. No, because of time dilation E. Yes, but in a parallel universe an identical train survives, so it s OK
No... Everyone agrees that they do not die. But how?
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!!
What observers disagree on (these things are relative) Size of train & tunnel (space) When the gates close (& whether they close simultaneously) (time)
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
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
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 ] --- -------------- 0 1.0 0.01c 1.00005 0.1c 1.005 0.5c 1.15 0.9c 2.29 0.99c 7.09 0.999c 22.4 1.0c infinity
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
The Twin paradox
What "breaks the symmetry" between you and your twin, that allows the twin to be much younger than you on return? A. While the physiological effects on your bodies are different, you both have experienced the same passage of time B. Twin moved through space, whereas you did not C. Twin accelerated at start and end, whereas you did not D. Twin experienced different gravitational fields on journey. E. Twin lost track of time during journey
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!
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
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!
Motivation of Principle of Equivalence 1. You feel weightless if you are freefalling!
Can t distinguish between: Falling in gravity field Being at rest in zero gravity
Can t distinguish between: being at rest in Gravity field accelerating upwards
Motivation of Principle of Equivalence 2. All objects in gravitational field accelerate at same rate, independent of their mass
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
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)
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!
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!
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.]
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
Light beam as seen by elevator passenger
light beam passing through elevator which is accelerating upwards light path in accelerating elevator appears straight to someone not accelerating but curved to someone in accelerating elevator!
Principle of equivalence acceleration gravity Acceleration bends light -> gravity bends light
gravitational lensing (see animation ) or some other distant light source or some other foreground mass
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
gravitational lensing A remote light source as seen from an observer on Earth, while a mass which acts as gravitational lens (aqua), passes between Earth and the remote source. Without lens effect only the small circle placed in the center of the image would be visible. The white light creates two images or one circle of the same source. (wikipedia)
Curvature of space Gravitational bending of light suggests that we might think of space as being curved near masses
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
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
Gravitational lensing Light is deflected by curved space around masses Light paths of 2 photons which pass near a mass (deflector)
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
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
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
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
Tests of general relativity 1a. Deflection of starlight by masses: Shift of star position near limb of sun (1919)
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
Tests of general relativity 2. Advance of perihelion of Mercury s orbit
Tests of general relativity 3. Gravitational time delay: time passes more slowly in stronger gravitational field
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)
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)
Tests of GR 5. detection of gravitational waves from merging black holes! (2016) Nobel Prize 2017
Good books on black holes Kip Thorne Black Holes & Time Warps Mitch Begelman & Martin Rees Gravity s Fatal Attraction
Great NOVEL!! Alan Lightman Einstein s Dreams
Observing & Planetarium sessions might start this week! Wed night?? 7:30-9:30pm Only if clear! Cancellation decision by 6pm on website Must complete pre-assignment ticket before coming to observatory Further details by Wed
Spacetime tells matter how to move, matter tells spacetime how to curve
Gravity curves space around masses Low mass star High mass star Light is deflected by curved space around masses
Class 16 Prof J. Kenney June 18, 2018 Relativity