PHSC 1053: Astronomy Relativity
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1 PHSC 1053: Astronomy Relativity
2 Postulates of Special Relativity The speed of light is constant in a vacuum and will be the same for ALL observers, independent of their motion relative to the each other and relative to the source of the light. This is the fastest that information can travel. There is no experiment yet performed that measures a vacuum speed of light (in any reference frame) that is not 3x10 8 m/s. Observers cannot detect their uniform motion except relative to other observers. The laws of physics must be the same for observers in any stationary or moving non-accelerated frame of reference.
3 Physics Revision Einstein s theory is not a modification to Newtonian physics. It is meant to REPLACE IT COMPLETELY. Before Newton: Galileo studied inertia, falling bodies and projectiles. He developed rules for describing motion. (free-fall, projectiles) Kepler studied and described planetary motion. (ellipses, P 2 = a 3 ) BUT neither explained the WHY. They lacked a synthesis, a complete unifying theory that tied things together. After Newton: Newton developed his laws of motion and the universal law of gravitation. All of the previous descriptions from Galileo, Kepler and others are explained by and can be derived from Newton s theories. In the same way, Newton s theories are explained by and can be derived from Einstein s Theory of Relativity.
4 Inertial Reference Frames An inertial frame of reference is at rest, OR moving at constant velocity in a straight line with respect to someone else s reference frame. Special Relativity deals with non-accelerating reference frames. General Relativity deals with accelerated reference frames.
5 Z u Relative Motion Z Y Y v X X u = arbitrary velocity of reference frame 1 (X, Y, Z) v = arbitrary velocity of reference frame 2 (X, Y, Z ) w = relative velocity between the two reference frames w = u + v
6 Velocity Addition Problem u = velocity of inertial reference frame #1 v = velocity of inertial reference frame #2 w = relative velocity between reference frame #1 and #2
7 Velocity Addition (w << c) w = u + v v u A person throws a rock at u = 60 km/hr while riding a truck traveling at v = 10 km/hr. What is the velocity of the rock as it strikes the idiot in the middle of the street?
8 Velocity Addition (w << c) w = u + v v u A person throws a rock at u = 60 km/hr while riding a truck traveling at v = 10 km/hr. What is the velocity of the rock as it strikes the idiot in the middle of the street? w = 70 km/hr Any idiot can calculate that!
9 Velocity Addition (w << c) u w = u - v v A person throws a rock at u = 60 km/hr while riding a truck traveling at v = 10 km/hr. What is the velocity of the rock as it strikes the idiot in the middle of the street?
10 Velocity Addition (w << c) u w = u - v v A person throws a rock at u = 60 km/hr while riding a truck traveling at v = 10 km/hr. What is the velocity of the rock as it strikes the idiot in the middle of the street? w = 50 km/hr
11 Velocity Addition (w ~ c) w = c + 2/3 c 2/3 c c c 2/3 c w = c - 2/3 c
12 Velocity Addition (w ~ c) w = c + 2/3 c 2/3 c c c 2/3 c w = c - 2/3 c Now What?! Must observe w = c... ALWAYS!
13 Relativistic Velocity Addition w = (u + v)/(1 + uv/c 2 ) Consistent with Newtonian velocity when uv << c 2, or when speeds are relatively SLOW. When uv << c 2 w = u + v When u and v are both the speed of light or close to it w = c CONCLUSION: Relativity is IMPORTANT only when speeds approach the speed of light in a vacuum.
14 Special Relativity Rules When viewed by the stationary observer Distances have their maximum lengths The observer at rest with respect to the object being measured, measures the proper length L o
15 Special Relativity Rules When viewed by the stationary observer Distances have their maximum lengths The observer at rest with respect to the object being measured, measures the proper length L o Masses have their minimum values The observer at rest with the mass measures the rest mass m o
16 Special Relativity Rules When viewed by the stationary observer Distances have their maximum lengths The observer at rest with respect to the object being measured, measures the proper length L o Masses have their minimum values The observer at rest with the mass measures the rest mass m o Clocks run the fastest (time required for an event is a minimum) The observer that measures an event from the same location measures the proper time t o
17 Transformations of length, mass and time Fundamental Units are not constant, but variables dependent upon the velocity through the relativity correction factor (γ). γ = 1/(1-β 2 ) 1/2 the relativistic correction factor, where β = w/c γ > 1.0 L = L o /γ L o is the proper length m = γ m o m o is the rest mass t = γ t o t o is the proper time
18 Relativity Relevance γ = 1/(1-β 2 ) 1/2 relativistic correction factor, where β = w/c w γ % Error between Newton and Einstein 0.1 c c c c c c c Error grows as the speed of light is approached Relativity is only important (noticeable) when traveling a significant fraction of the speed of light.
19 Proper Length L = L o /γ L o is the proper length Proper length is measured by the observer at rest in relation to the object being measured. All other lengths which are not at rest with respect to the observer must be transformed into his reference frame with γ. Length Contraction: moving lengths are contracted.
20 Proper Time t = γ t o t o is the proper time Proper time is measured by the observer that views events at the same location. All other events which are not at the same location with respect to the observer must be transformed into his reference frame with γ. Time Dilation: moving clocks run slow.
21 Doctor, Doctor A doctor tells me I have a heart condition and will only live 2.0µs. I decide to party a little before my untimely death and rocket dive off a cliff on your property traveling at 0.99c oward the ground. Everyone needs a little thrill, or at least to go out in style. I want to enjoy every last moment, so I ask you if I need a parachute. You say the cliff is 1000 meters high, so d = v t d = 0.99c x 2.0 µs = 600 meters. You say NO, you ll be dead before impact, you should only travel 600 meters in 2µs, enjoy yourself.
22 Doctor, Doctor You said the cliff was 1000 meters high, and I could only travel 600 meters. I jump and I hit the ground and writhe in pain before I die. I am very rightfully upset, albeit dead anyway. 1. You say I lived longer than I said I would. 2. I say you can t make a measure heights of cliff s very well. WHO IS RIGHT?
23 Time Dilation fix 1. You say I lived longer than I said I would. You transform my lifetime from my frame to your reference frame. Moving Clocks Run Slow from your perspective. γ = 1/( ) 1/2 = 7.09 t = γ t o = 7.09 (2µs) = µs How long I live in your frame. Now, d = 0.99c x 14.18µs = 4200 meters. You say I should have gotten a parachute.
24 Length Contraction fix 2. I say you can t make a fundamental cliff measurement. The cliff rushes past me at 0.99c. I transform your measurement of the cliff to my reference frame. Moving Lengths Are Contracted in my reference frame. γ = 1/( ) 1/2 = 7.09 L = L o /γ = 1000/7.09 = 141 meters Cliff height in my frame. Now, d = 0.99c x 2 µs = 600 meters I say I needed a parachute, we both agree, physics works!
25 Problem I fly over a fast food joint in my spaceship at 0.4c. You call me on my cell phone and say you are at a 6 foot table having a 1/4 pound burger that takes you 5 minutes to eat. What do I say?
26 Solution I fly over a fast food joint in my spaceship at 0.4c. You call me on my cell phone and say you are at a 6 foot table eating a 1/4 pound burger that takes you 5 minutes. What do I say? γ = 1/( ) 1/2 = 1.09 You measure both the proper time and the proper length, L o, t o and the rest mass in this case. I say, t = γ t o = 1.09 x 5 = 5.45 minutes L = L o /γ = 6/1.09 = 5.5 foot table m = γ m o = 1.09 x 0.25 = 0.27 pound burger
27 Twin Paradox A trip is taken by one identical twin to alpha centauri in a rocket ship traveling at w = 0.95c. Alpha centauri is 4.3 LY distant. What will be the difference in ages between the twin that stayed on Earth and the twin taking the trip?
28 Twin Paradox (time dilation) A trip is taken by one identical twin to alpha centauri in a rocket ship traveling at w = 0.95c. Alpha centauri is 4.3 LY distant. What will be the difference in ages between the twin that stayed on Earth and the twin taking the trip? 1. Identify the two events of interest for the time interval. 2. Determine the reference frame in which the events occur at the same location. An observer at rest in this frame measures the proper time.
29 Twin Paradox The events are, departure from the Earth and arrival at alpha centauri. Upon departure, the Earth is just outside the rocket door. Upon arrival at alpha centauri, it is just outside the door. The passenger of the rocket views the events in the same place and therefore measures the proper time. The twin on Earth views the events at different places. He started here, then the ship blasted and his twin left. Arrival is there. Light travel time to alpha centauri is 4.3 LY. The rocket is traveling at 0.95 the speed of light c. Therefore, the twin on earth calculates the time taken as t = d/w t = 4.3 years/0.95 = 4.5 years
30 Time is transformed by t = γ t o Twin Paradox The twin on the rocket measures t o and the earth bound twin measures t. Therefore, t o = t/γ γ = 1/( ) 1/2 = 3.2 t o = 4.5 years / 3.2 t o = 1.4 years Earth twin is 3.1 years older than rocket twin!
31 Twin Paradox (Length Contraction) A trip is taken by one identical twin to alpha centauri in a rocket ship traveling at w = 0.95c. Alpha centauri is 4.3 LY distant. What will be the difference in ages between the twin that stayed on earth and the twin taking the trip? 1. Identify the length of interest. 2. Determine the reference frame in which the observer is at rest with respect to this length. An observer at rest in this frame measures the proper length.
32 Twin Paradox Length is transformed by L =L o /γ Rocket twin says clock is OK, earth flew away at 0.95c. The twin on the rocket measures L and the earth bound twin measures the proper length L o (distance to alpha centauri). Therefore, L = L o /γ γ = 1/( ) 1/2 = 3.2 L = 4.3 LY / 3.2 L = 1.34 LY The rocket twin claims to have only traveled 1.34 LY. Traveling at 0.95c, 1.34 LY takes t = 1.34/0.95 or 1.4 years. Age difference is still 3.1 years (= )
33 Equivalence of Gravity and Acceleration
34 General Relativity: Equivalence Principle There is no way to tell the difference between gravity and an accelerated reference frame. Observers can not distinguish locally between inertial forces due to acceleration and uniform gravitational forces due to the presence of a massive body.
35 Who is accelerating? Acceleration Is Relative
36 Objects are accelerated at a rate equal to g near the Earth s surface. Gravity m a = m GM/d 2 a ~ M/d 2 g = G M earth /R earth 2
37 Equivalence Principle Gravity is indistinguishable from accelerated reference frames
38 The Accelerating Lab In Space Projectiles in an accelerated lab will behave like projectiles on the surface of the Earth as long as a = g
39 The Accelerating Lab In Space Projectiles will behave like projectiles on the surface of the Earth as long as a = g
40 Imagine a photon, entering a peephole in the lab on the left. LASER Light Beams LARGE Acceleration The laboratory is accelerating very very rapidly. To an observer, the beam must appear to be deflected because of the motion of the lab.
41 Laser Light Deflection LASER Even this experiment must not be able to distinguish between acceleration and gravity. Earth s Surface CONCLUSION: Light is bent by gravity!
42 Space-Time Curvature Light travels the shortest distance between two points in the local curved space-time.
43 Bending of Star Light
44 Solar Eclipse Observations Theoretical deflections and actual 1919 solar eclipse data from the Eddington expedition showing expected and actual deflections of starlight in the direction of the sun.
45 Galaxy Cluster Abell 2218 Gravitational Lenses
46 Abell 383 Gravitational Lenses
47 Geodesic? Shortest Distance Between Two Points Depends Upon Geometry. For a Plane, it is along a line. For a Sphere, it is the arc of a great circle.
48 Newton versus Einstein The geometry you know is valid when drawn on a flat surface. The rules change if the surface is not flat.
49 Space-time Space-time can have three possible geometries: flat the rules of Euclidean geometry apply spherical parallel lines eventually meet saddle-shaped parallel lines eventually diverge Space-time may have different geometries in different places. If space-time is curved, then no line can be perfectly straight. Since being in free-fall is equivalent to traveling at constant velocity (i.e. a straight line) objects experiencing weightlessness must be traveling along the straightest possible line in space-time (traveling along a geodesic) which may be curved Objects in orbit are weightless. the shapes & speeds of their orbits can reveal the geometry of space-time these same orbits are determined by gravity
50 Newton versus Einstein Newton: Force tells mass how to accelerate F = m a Mass tells gravity how to exert a force F = m G M/D 2 Planets orbit the sun as a consequence of a balance between flying off in a straight line at constant velocity and gravity accelerating them at a right angle towards a center of mass. Einstein: Curved space-time tells mass-energy how to move Mass-energy tells space-time how to curve E = m c 2 Planets move without any forces upon them, traveling inertially along the local curved space-time created by the mass of the sun and other bodies.
51 Ant Experiments attempting to measure the curvature of their universe may involve measurements of circles. The circumference of a circle should be 2πL if they live in a 2-D world as they assume. They of course actually measure 2πr. R and r are a part of the 3-D sphere that the ants are not aware of. BUT... Measuring Curvature radius = L R r 2πr
52 The circumference of circles as measured, first grows with radius L, then shrinks as L becomes greater than 1/4 the circumference of their world. Measuring Curvature radius = L R r 2πr By measuring the discrepancy between 2πr/L and 2π, they can obtain information about the curvature of their world (R).
53 Expansion Confusion One might be puzzled by a world of finite area and no edges. Especially so if you are the 2-D ant on the 3-D sphere. If our world is expanding, the area is increasing, where s that extra area come from!? Their perception of expansion is that it is real motion along the surface of their world. Those are the only 2 dimensions they are aware of and they can conceive only those two dimensions when they are told the world is expanding. The expansion of their world takes place because the surface is carried in time to a different location in the third unobservable dimension. The radius of the sphere increases, thereby increasing their surface area as if by magic.
54 Surveying the number counts of galaxies at larger and larger distance, can yield information about the curvature of the universe. Galaxy Counts
55 Total Energy A projectile has an amount of kinetic energy given by K.E. = 1/2 m V 2 It has a potential energy relative to earth's surface of P.E. = - mgm/r The total energy is conserved so that Total Energy = K.E. + P.E. Total Energy = 1/2 m V 2 - mgm/r In order to escape earth's gravity you have to have Total Energy > 0, so the critical point is when the Total Energy = 0.
56 Escape Velocity The Escape Velocity is the velocity an object (of ANY mass) must have in order to leave the gravitational field of a massive body. It depends on the total energy an object has. There is an escape velocity from the Earth's surface, from the Sun's surface, even from the solar system, and it depends on how massive that body is and how far you are away from the body.
57 Energy & Orbits e >> 0 e > 0 e = 0 Conic Section Velocity Total Energy Orbit Hyperbola V > V esc > 0 Unbound Parabola V = V esc = 0 1 pass only Ellipse V < V esc < 0 a(1-e), a(1+e) Circle V = V circ Minimum a = radius
58 Energy For the Universe, the density is the key. It either expands forever, or eventually stops, expanding and re-collapses
59 What have we learned? What is relative about the theory of relativity? The theory is based on the idea that all motion is relative. That is, there is no correct answer to the question of who or what is really moving in the universe, so motion can be described only for one object relative to another. What is absolute according to the theory of relativity? (1) The laws of nature are the same for everyone, and (2) the speed of light is the same for everyone. How are paradoxes useful to understanding relativity? Because paradoxes seem to violate common sense or to be selfcontradictory, many of the ideas of relativity can best be understood by confronting the paradoxes and finding their underlying resolutions.
60 What have we learned? What do we mean by a reference frame in relativity? Two (or more) objects share the same reference frame if they are not moving relative to each other. In that case, the objects will experience the passage of time and measurements of distance and mass in the same way. Time, distance, and mass will be different for objects in different reference frames. Why can t you reach the speed of light? Light always travels at the same speed, so your own light (light that you emit or reflect) is always moving ahead of you at the speed of light. All other observers will also see your light moving at the speed of light and because it is moving ahead of you, the observers will always conclude that you are moving slower than the speed of light.
61 What have we learned? How are time, space, and mass different for a moving object than for an object at rest? If you observe an object moving by you at high speed, you ll find that its time is running slower than yours, its length is shorter than its length when at rest, and its mass is greater than its mass when at rest. Will observers in different reference frames agree when events happen at the same time? They will not agree unless both events also occur in the same place. In general, when an observer in one reference frame sees two events happen simultaneously, observers in other reference frames will claim that one event preceded the other.
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