The Nature of Space and Time-Special Relativity. READ: Chapter 7

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1 HW The Nature of Space and Time-Special Relativity READ: Chapter 7 HW #1 Chapter 7: (HW#1) Questions 1, Problems (HW#2)1, (HW#3)3, (HW#4)4, (HW#5)6, (HW#6)7, (HW#7) Computer problem 7.1 (HW#8Additional task: Substitute the Lorentz equations 7.10 into R.2 (see lecture notes) to see if you Get R.1 which would demonstrate the transformation is the proper one for c to be constant in all frames moving at v!

2 Review of Basic Physics for space science 160 x The average velocity is rate at which the displacement occurs xavg, It s a vector..ie it has direction t t average speed is a scalar quantity same units as velocity d v total distance / total time: avg t Instantaneous Speed v=dv/dt :the magnitude of the instantaneous velocity(a vector) Basic Units MKS or (SI) (meter kilogram sec) =meter/sec =m/s other (km/hr); English(E) (mi/hr) ft/s Basic Units of text are cgs (cm gram sec) = centimeter/sec = cm/s I will use cgs that follows along with some MKS examples Particle under constant velocity vf =vi=v or d = xf xi = v t where t is time traveled (tf-ti) Or xf=xi+vt vx vxf vxi axavg, Average Acceleration is the rate of change of the velocity t tf ti 2 v dv dx Acceleration =instantaneous acceleration = in the x direction for example is a lim x x In general a= dv/dt basic units are cm/s/s =cm/s 2 other (SI)\m/s 2 (E) ft/s 2 x t 0 2 t dt dt BASIC EQUATIONS OF MOTION UNDER CONSTANT ACCELERATION ARE 1. Given dv/dt =a -> dv= a dt integrate( t =0 to t v= v 0 to v) Gives v = v 0 + at 2. Given FROM 1 THAT v= dx/dt =v 0 + at integrate( t t =0 to t x= x 0 to x) Gives x=x 0 +v 0 t +1/2 a t 2 3, Variations for problem solving it is useful to eliminate a by substituting for a from 1 a =(v-v 0 ) /t into 2 Gives: x=x 0 +1/2(v 0 + v)t TRY THIS!!!! Hand in for extra credit show all steps v x f x i 4. Also for problem solving eliminate t in 2 by substituting for t from 1 ie t=(v -v 0 )/ a into 2 Gives: v 2 = v a (x-x 0 ) TRY THIS!!!! Hand in for extra credit show all steps

3 Galileo Galilei ;Italian physicist and astronomer; Formulated laws of motion for objects in free fall; Supported heliocentric universe Some info from Wikipedia..note the links a paper on Galileo would be EXTRA CREDIT! Galileo Galilei - Wikipedia, the free encyclopedia Played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism. Galileo has been called the "father of modern observational astronomy",the "father of modern physics", [ the "father of science", [ and "the Father of Modern Science". Galileo later defended his views in Dialogue Concerning the Two Chief World Systems, which appeared to attack Pope Urban VIII and thus alienated him and the Jesuits, who had both supported Galileo up until this point. He was tried by the Inquisition, found "vehemently suspect of heresy", forced to recant, and spent the rest of his life under house arrest.it was while Galileo was under house arrest that he wrote one of his finest works, Two New Sciences, in which he summarised the work he had done some forty years earlier, on the two sciences now called kinematics and strength of materials. Galileo found that the acceleration of all objects in free fall is directed downward, regardless of the initial motion and disregarding air resistance t he magnitude of free fall acceleration is a= g = 9.80 m/s 2 (SI) = 980 c/s 2 NOTE: magnitude means just value not direction! g decreases with increasing altitude g varies with latitude The 9.80 m/s 2 is the average at the Earth s surface The symbol g is normally used for the acceleration due to gravity ie a=g=-9.80 m/s 2 on average -Not to be confused with g for grams TODAY we know the nominal "average" value at the Earth's surface, known as standard gravity is, by definition, in SI; m/s 2 ( ft/s 2 ). Thus, approximately g=9.81 m/s 2 or 32.2 ft/s 2

4 Applying the constant accelration equations to motion in the y direction see the image on the left, so for this image the equations can be written As below. Still 1 dimension but y is the direction here! Initial in now i rather than 0 f for final etc. see equivalent 1 D equations in red. v yf = v yi +a yt v = v 0 + at y f =y i + ½ (v yi + v yf )t : x=x 0 +1/2(v 0 + v)t y f = y i + v yi t + ½ a y t 2 x=x 0 +v 0 t +1/2 a t 2 v yf2 = v yi2 + 2a y (y f -y i ) : v 2 =v a (x-x 0 ) Some books and the PROF prefers to use these with a y =-g-> a=-g and thus use the following equations for clarity and simplicity in the y direction (can you use the result below to get the numerical values in the figure to the left? Try it EXTRA CREDIT HAND IN SHOW WORK!) v = v 0 - gt y=y 0 +1/2(v 0 + v)t y=y 0 +v 0 t -1/2 g t 2 v 2 =v g (y-y 0 )

5 Simple drill and practice EXTRA CREDIT Be sure to organize what you know and what you are looking for to help your logic. 1. Kinematics in the vertical direction. You shoot an arrow straight up. It leaves your bow at 160 km/hr. (watch units) a. How far up does it go before it starts down.? b. How long does it take to get to the top of its motion? 2. A baseball is hit so that it travels straight upward. A fan observers it take 3.00 s To reach its maximum height. Find a. The ball s initial velocity?.hint. What is the velocity at the top of the motion? b. The height it reaches?

6 Sir Isaac Newton, LAWS OF MOTION AND GRAVITY Isaac Newton - Wikipedia, the free encyclopedia , Formulated basic laws of mechanics, Discovered Law of Universal Gravitation Invented form of calculus, Many observations dealing with light and optics PUT THE CUTS ON THE SIDE OF COINS? WHY? Forces are what cause any change in the velocity of an object Newton s definition Force =dp/dt p = mv defined at the momentum of mass m A force is that which causes an acceleration Ie if m is constant than F=dp/dt=d(mv)/dt=mdv/dt And dv/dt =a so F=ma in this case or a=f/m also called Newton s 2 nd law of Motion UNITS OF F (SI) = ma = 1kg x 1 m/s 2 =1 Newton cgs: F= 1 gm x 1 cm/s 2 = 1 dyne English F= 1 Ib = 1 slug x 1ft/s 2 NOTE: 1 Newton = 10 5 dynes and 1 lb ~= 4 Newtons or 1N ~=1/4 lb Newtons Third law..action reaction F F Newtons Universal Law of Gravity is that two bodies pull on each other equal and opposite with the force being equal to F =-Gm 2 1 m 2 /r

7 Work-> Energy - Non Relativistic W = F r cos F F. r = change in energy of system )=Fd unit wise The units of work and thus energy is (SI) joule (J) or (cgs) erg or Foot-pound in English from W= FD for ( SI)1 joule = 1 newton x 1 meter or J = N m In cgs 1 erg=1 dyne x 1 cm.. Or 1 joule = 10 7 erg x f x f W F d x m a d x x i x i The Work-Kinetic Energy Theorem states W = K f K i = K ->> Hence K=1/2 mv 2 is a a natural for kinetic energy expression. Of mass m moving at speed v. Work-Potential Energy Theorem state W = U f U i = - U for conservative forces which means -> Or formally x f W C F x x d x U i F x d U d x W v f v i m v d v 1 1 W m v m v 2 2 W K K K net f i 2 2 f i Using the Work Energy concept we derive Potential energies for two important force, Universal Gravitaion-> F = -Gm 1 m 2 /r 2 and electrical (Coluomb force) F = kq 1 q 2 /r 2 ( q the electric charge on a mass) W = F G dr = - Gm 1 m 2 1/r 2 dr = -Gm 1 m 2 (1/r f -1/r i ) or we can say at a point r, U G = Gm 1 m 2 /r W = F e dr = kq 1 q 2 1/r 2 dr = kq 1 q 2 (1/r f -1/r i ) or we can say at a point r, U e =kq 1 q 2 /r Or potential energies for these forces go as 1/r UNITS are energy or Joules, or ergs etc Note from above since these F s are conservative that F = -du/dr with U G = Gm 1 m 2 /r U e =kq 1 q 2 /r we get back the the expresson of the 1/r 2 forces by taking the derivatives of the potentials!

8 Energy is conserved in an Isolated System For an isolated system, E mech = 0 E mech = K + U This is conservation of energy for an isolated system with no nonconservative forces acting If nonconservative forces are acting, some energy is transformed into internal energy like friction->heat Conservation of Energy becomes E system = 0 ( example book falling) The changes in energy E system = 0 Or K + U=0 K=- U Ie. K f -K i = -(U f U i ) can be written out and rearranged K f + U f = K i + U i Remember, this applies only to a system in which conservative forces act Or 1/2mv f2 +mgh f =1/2mgv i2 +mgh i where mgh is just W = change in potential near earth or Fd =mgh Example is for Free fall see -> WHAT IS the expression v f here-> Analyze Apply Conservation of Energy K f + U gf = K i + U gi K i = 0, the ball is dropped Solving for v f v v 2g h y Can you do this last algebra? No? Then try it as a practice of algebraic manipulation.its its easy! f 2 i

9 Our last definition of physics concepts is = Instantaneous Power Power is the time rate of energy transfer The instantaneous power is defined as Or simple E/t for Unit anaylsis de dt Using work as the energy transfer method, this can also be written as Or W/t for unit analysis or also the average power of a system avg W t UNIT of Power in the SI units Watt is a Joule/sec called? Answer WATT! 1 watt=1 joule/sec =W Does an erg/sec have a name????? A unit of power in the US Customary system is horsepower W/t= Fd/t a horse can lift 550 ibs in one sec Or 1hp = 550ft lbs /sec or 1 hp = 746 W Units of power can also be used to express units of work or energy as from your electric company Since P =E/t than E =Pt or what you pay the bill for. E= 1 kwh = Pt= (1000 W)(3600 s) = 3.6 x10 6 J This is the end of the physics review for Ast 160

10 Einstein and Relativity Chapter 7 Einstein ( ) noticed that Newton s laws of motion are only correct in the limit of low velocities, much less than the speed of light. Theory of Special Relativity Also, revised understanding of gravity Theory of General Relativity

11 The Speed of Light We start S. Relativity with a qualitative approach Special Relativity becomes important in systems which are moving on the order of the speed of light The speed of light is c= 300,000 km/s =3X10 8 m/s is the ultimate fast : Is exactly 299,792,458 m/s (how can they know this is the exact speed?) EXTRA CREDIT? ~1 foot per nanosecond ~1 million times the speed of sound. ~ Around the earth 7.8 times in a second Earth to sun in 8.3 min. Galileo was the first person to propose that the speed of light be measured with a lantern relay. His experiment was tried shortly after his death. In 1676 Ole Roemer first determined d the speed of light (how can this be done with 17 th cent equipment. EXTRA CREDIT?

12 The Speed of Light In 1873, Maxwell first understood that light was an electromagnetic wave!. It was the understanding of the nature of EM radiation which h first led to a conceptual problem that t required relativity as a solution. According to his equations, a pulse of light emitted from a source at rest would spread out at velocity c in all directions. But what would happen if the pulse was emitted from a source that was moving? This possibility confused physicists until 1905.

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14 In Water Things Look Like This WAVE IN WATER IS expanding AT SPEED c IN THE WATER A boat (Z) moving through water (v b ) will measure forward going waves (A) as going slow v<c IE v= c-v b and dbackwards going waves (B) as going fast v>c IE v= c+v b and at (C) v=c Z A C B

15 What is the velocity of light ( the arrows) as measured from the moving earth? We expect if light moves in medium called the ether, that A beam would be >c B beam would be <c and C beam would =c analogous to the boat in water case. A C B

16 Michelson-Morley Morley Experiment Albert Michelson and Edward Morley were two American physicists working at Case Western Reserve University in Cleveland They constructed a device which compared the velocity of light traveling in different directions (1887). They found, much to their surprise that t the speed of light was identical in all directions! c m / This is strange???? HOW COULD THAT BE? s

17 Michelson-Morley eso o eyexperiment e t(cont.) t) If the aether theory were correct, light would thus move more slowly against the aether wind and more quickly downwind. The Michelson-Morley apparatus should easily be able to detect this difference. In fact, the result was the exact opposite: light always moves at the same speed regardless of the velocity of the source or the observer or the direction that the light is moving!

18 With light, things look like this: A person on a cart moving at half the speed of light will see light moving at c. A person watching on the ground will see that same light moving at the same speed, whether the light came from a stationary or moving source

19 Something must be changing and effecting the measurement made By all three observers at different speed coming up with The same answer for c?

20 So how is this possible?? In the 18 years after the Michelson-Morley experiment, the smartest people in the world attempted to explain it away In particular C.F. FitzGerald and H.A. Lorentz constructed a mathematical formulation (called the Lorentz transformation) which seemed to explain things but no one could figure out which it all meant. In 1905, Albert Einstein proposed the theory of Special Relativity which showed that the only way to explain the experimental result is to suppose that space and time as seen by one observer are distorted when observed by another observer (in such a way as to keep c invariant) invariant means constant..non changing!

21 Welcome to The Strange World of Albert Einstein Some of the consequences of Special relativity are: Events which are simultaneous to a stationary observer are not simultaneous to a moving observer. Nothing can move faster than c, the speed of light in vacuum. A stationary observer will see a moving clock running slow. A moving object will be contracted along its direction of motion. Mass can be shown to be a frozen form of energy according to the relation E=mc².

22 Events In physics jargon, the word event has about the same meaning as it s everyday usage. An event occurs at a specific location space at a specific moment in time:

23 Reference Frames A reference frame is a means of describing the location of an event in space and time. To construct a reference frame, lay out a bunch of rulers and synchronized clocks You can then describe an event by where it occurs according to the rulers and when it occurs according to the clocks.

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27 Lorentz Transformation As we shall see, space and time are not absolute as in Newtonian physics and everyday y experience. The Mathematical relation between the description of two different observers is called the Lorentz transformation. Some phenomena which follow from the Lorentz transformation are: Relativity of Simultaneous events Time Dilation Length Contraction

28 Reference Frames (cont.) What is the relation between the description of an event in a moving reference frame and a stationary one? To answer this question, we need to use the two principles of relativity ***

29 The First Principle of Relativity An inertial frame is one which moves through space at a constant velocity The first principle of relativity is: The laws of physics are identical in all inertial frames of reference. For example, if you are in a closed box moving through space at a constant velocity, there is no experiment you can do to determine how fast you are going In fact the idea of an observer being in motion g with respect to space has no meaning.

30 The Second Principle of Relativity The second principle of relativity is a departure from Classical Physics: The speed of light in vacuum has the same value, C, in all inertial frames regardless of the source of the light and the direction it moves. This is what the MM experiment shows. The speed of light is therefore very special This principle is not obvious in everyday experience since things around us move much slower than c. In fact, the effects of relativity it only become apparent at high velocities

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32 What Happens to Simultaneous Events? Are events which are simultaneous to one observer also simultaneous to another observer? We can use the principles of relativity to answer this question. Imagine a train moving at half the speed of light

33 View from the Train

34 The View From The Ground

35 Simultaneous Events Thus two events which are simultaneous to the observer on the train are not simultaneous to an observer on the ground The rearwards event happens first according to the stationary observer The stationary ti observer will therefore see a clock at the rear of the train ahead of the clock at the front of the train

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38 Time Dilation Let us now consider the relation between time as measured by moving and stationary observers. To measure time let us use a light clock where each tick is the time it takes for a pulse of light to move a given distance.

39 Time Dilation (cont.) Now let us imagine a train passing a stationary observer where each observer has an identical light clock. The observer on the train observes his light clock working normally each microsecond the clock advances one unit as the light goes back and forth:

40 Time Dilation (cont.) Now what does the Compared to a stationary stationary observer see? observer, the light beam travels quite far. Thus each tick of the moving clock corresponds to many ticks of the stationary clock

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42 Time Dilation Let us now consider the relation between time as measured by moving and stationary observers. To measure time let us use a light clock where each tick is the time it takes for a pulse of light to move a given distance.

43 Time Dilation (cont.) Now let us imagine a train passing a stationary observer where each observer has an identical light clock. The observer on the train observes his light clock working normally each microsecond the clock advances one unit as the light goes back and forth:

44 Time Dilation (cont.) Now what does the Compared to a stationary stationary observer see? observer, the light beam travels quite far. Thus each tick of the moving clock corresponds to many ticks of the stationary clock

45 So How Much Does The Moving Clock Run Slow? Let t 0 be the time it takes for one tick according to someone on the train and t be the time according to some one on the ground. From what we just discussed t>t 0 but by how much? The factor ( ) quantifies the amount of time dilation at a give velocity.

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47 The Factor Gamma Thus, the time recorded on the moving clock, is related to the time that the t 0 stationary clock records according: t0 t 2 1 ( v / c) For simplicity we write the relation as: 1 t t0 where 1 ( v / c ) is the time dilation factor. 2

48 Some Time Dilation Factors Velocity Gamma Space Shuttle 5000m/s Earth in Orbit 30000m/s c 0.01c c c c 0.8c c c c c c 70.7

49 Time Dilation (cont.) For example, suppose that a rocket ship is moving through space at a speed of 0.8c. According to an observer on earth 1.67 years pass for each year that passes for the rocket man, because for this velocity gamma=1.67 But wait a second! According to the person on the rocket ship, the earth-man is moving at 0.8c. The rocket man will therefore observe the earth clock as running slow! Each sees the other s clock as running slow. HOW CAN THIS BE!!!!!

50 FitzGerald Length Contraction Just as relativity tells us that different observers will experience time differently, the same is also true of length. In fact, a stationary observer will observe a moving object shortened by a factor of which is the same as the time dilation factor. Thus, if L is the length of an object as seen by a stationary observer and L 0 is the length in the moving frame then: L L 0 /

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53 Why Length Contraction Suppose that t a rocket moves from the Sun to the Earth at v=0.95c ( =3.2). According to an observer from Earth, the trip takes 500s. As seen by earthbound Ship covers 150,000,000, km in 500 s observer By time dilation, only 500s/3.2=156s pass on the ship. The crew observes the Earth coming at them at 0.95c This means that t the sun-earth distance according to the crew must be reduced by 3.2! Earth covers 47,000,000 km in 156 s As seen by crew member observer

54 The Twin Paradox To bring this issue into focus, consider the following story: Jane and Sally are identical twins. When they are both age 35, Sally travels in a rocket to a star 20 light years away at v=0.99c and the returns to Earth. The trip takes 40 years according to Jane and when Sally gets back, Jane has aged 40 years and is now 75 years old. Since gamma=7.09, Sally has aged only 5 years 8 months and is therefore only 40 years and 8 months old. Yet according to the above, when Sally was moving, she would see Jane s clock as running slow. How is this possible???

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57 Twin Paradox Another way of thinking about the situation is as follows: If two observers move past each other, each sees the other s clock as moving slow. The apparent problem is resolved by the change in time with position. In the case of the twin paradox, there is not a symmetric relation between the two twins. The earthbound twin was in an inertial frame the whole time The traveling twin underwent an acceleration when she turned around and came back. This breaks the symmetry between the two

58 The Concept of Space-time Recall that an event takes place at a specific point in space at a specific time. We can therefore think of an event as a point in space-time. It is conventional to display time as a vertical axis and space as the horizontal axis.

59 Space-Time Diagrams Every event can be represented as a point in space-time An object is represented by a line through space-time known as it s world line If we label l the axes in natural units, light moves on lines at a 45º angle

60 An Object standing still Time (in seconds) Apieceof light An Object Moving The light cone Position (in lt-seconds)

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63 The second principle of relativity implies that you can never catch up to a piece of light, therefore you cannot accelerate through the light barrier If there did exist a magic bullet that could travel faster than light, it would imply that t you could travel or at least send information back in time Thus an event can only effect what lies in its future light cone and can only be effected by events in its past light cone The Moving finger writes; and, having writ, Moves on: nor all thy piety nor wit Shall lure it back to cancel half a line, Nor all thy tears wash out a word of it. -Omar Khayyam

64 A trip to the Stars Consider a space ship which accelerates at 1g for the first half of the trip decelerates at 1g for the second half of the trip At this acceleration one can achieve speed near the speed of light in about a year At 1 year of acceleration v=0.761 c I f t ithi th lif ti f th In fact, within the life time of the crew, one could reach the edges of the universe!!!

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66 Acceleration/Deceleration = 1 g Time (in years) Deceleration Distance (ly) Ship time (y) Turnaround Distant Star 30,000 2,000, Acceleration Position (in lt-years)

67 Energy Since the speed of light is the ultimate speed limit If you accelerate an object towards c, it s velocity gets closer to c but never reaches it The amount of energy required to do this is thus greater than ½mv² 2 In fact K ( 1) mc Einstein realized that to have a meaningful definition of Energy which is connected to the geometry of spacetime it is necessary to assign an energy E =mc² to an object at rest. 0 Thus, the total energy of an object including its rest energy and kinetic energy is E rel E0 K mc 2

68 Two Postulates Leading to Special Relativity (1) we now REHASH!!!! 1. Observers can never detect their uniform motion, except relative to other objects. This is equivalent to: The laws of physics are the same for all y observers, no matter what their motion, as long as they are not accelerated.

69 The latter was discovered by Einstein when he considered the equations That are used for light and other electromagnetic phenomena (x-rays, uv, ir, micro Gamma, radio, etc). These equations are known as Maxwell s equations. Maxwell s equations predict that E&M energies are waves that vary sinusoidally in space And time! The equations are normally written as follows involving the Electric vector E And the magnetic vector B

70 Einstein s though experiment (Gedanke!) is illustrated in your text Fig 7.1 Basically he reasoned that if someone could move at the speed of light then The wave would appear to be stationary in time. However, the mathematical solutions of Maxwell s equations do not permit this case. Ie all solutions are Sinusoidal in space and time. Moving observer sees( NOT POSSIBLE!) FROM FIG 7.1 t Stationary observer sees Hence, he chose to reason it is impossible to move at an arbitrary speed relative to an E&M wave and this must mean that the speed of light is a Constant in all moving frames!

71 Two Postulates Leading to Special Relativity (2) 2. The velocity of light, c, is constant and will be the same for all observers, independent of their motion relative to the light source.

72 CONSIDER A LIGHT CLOCK (SEC 7.2) TIME DILATION TWO MIRROR LIGHT BOUNCES BACK AND FORTH STATIONARY clock L ROUND TRIP TIME t o = 2l/c 7.1 Moving Light clock Velocity=v t on moving Frame of Reference c is same! L?= (L 2 + (vt/2) 2 ) 1/2 vt? vt/2 L Time t for round trip must be = (from rate x time =distance) time=distance/rate or t= 2(L 2 + (vt/2) 2 ) 1/2 c Solve for t (squaring then use 7.1 to eliminate L ) t o we get the time dilation formula t= Ie. Moving clock appears to be slower (1- (v/c) 2 ) 1/2 The time in the rest frame is called the proper time

73 Special Relativity time dilation. t=t o /(1 [v/c] 2 ) 1/2 = t 0 eq. 7.2 (where = 1/(1 [v/c] 2 ) 1/2 is the sometimes called the Lorentz factor. Ex. v=0.5c t=1.547 t o ie v=0 t=t o or the moving time interval is greater than when object is at rest Longer interval -> Move fast age slower! Example 7.1 How fast must a particle be traveling to live ten times as Long as the same particle at rest? SOLVE NOW! ANS: t=to/(1 [v/c] 2 ) 1/2 -> 10 = 1(1-(v/c) 2 ) 1/2 -> 100=1/(1-(v/c) 2 ) -> (/) (v/c) 2 =0.99 or v= 0.995c

74 Other Effects of Special Relativity Length contraction: Length scales on a rapidly moving object appear shortened in direction of the motion. L=L o / Mass increases: m= m o The energy of a body at rest is not 0. Instead, we find E 0 = m o c 2

75 since m= m o or t = t o Rapid Rise to Infinity! see fig 7.5! or t/t o Relativistic mass increase with speed

76 Basic Doppler effect illustrated t Relativistic Doppler effect Source S moving with velocity v emits N waves in time t measured by receiver at R S v R Wave move at c Frequency (emitted by source is =N/t) So in time t S moves vt and wave moves ct wavelength measured by R will be = (ct -vt ) /N vt ct (ct -vt ) Frequency seen by R is =c/ =c/ (ct -vt ) /N=c/(c-v) N/t Non relativistic =c/(c-v) 1-v/c) 1 ie t=t ie /(1-v/c) Relativity t = t or =c/ =c/ (ct -vt ) /N=c/(c-v) N/t =1/(1-v/c) N/ t Or =1/(1-v/c) see algebra breakdown to Get equation 7.5 that follows

77 Relativistic Doppler effect Lengths and times are different for moving observers hence this effects wavelengths And frequencies when sources or observers are moving. Section 7.4 goes through some derivations the key results are Moving source or Moving observer at velocity v frequency becomes = (1+v/c) and for velocities close to c v/c 1/2 = 1-v/c Example 7.2 Find the wavelength we would observe the H line if it is Emitted by an object moving away with v=0.3 c? SOLVE NOW! ANS =1.36 x nm nm Extra credit: show for frequency that the relativistic doppler effect becomes the classical doppler effect for small v!

78 Space Time section 7.5 If we take into account space and time changes then we get the full Lorentz transformations between moving and stationary coordinate systems: for motion,v, along an x axis we get (introducing i v/c) /)the moving system set t (Check out also the reverse 7.9) ct = ct - x) x = (x ct) y =y y z =z

79 Einstein applied the Lorentz transformation to show truly that the speed of light Is constant in all frames. In fact the Lorentz transformation guarantees it! As follows: ASSUME A PULSE OF LIGHT IS SPREADING OUT FROM A POINT BY EINSTEINS POSTULATE A SPHERE OF LIGHT EMERGING FROM THE POINT SHOULD BE MOVING A THE SAME SPEED MEASURED IN A STATIONARY FRAME OR MOVING FRAME IN STATIONARY FRAME THE RADIUS OF THE SPHERE AT A TIME t =ct In the moving frame(velocity v) the radius at a time t (time measured by Moving observer) = ct ie c is the same in both! Equation of this expanding sphere in stationary frame is r x 2 + y 2 + z 2 =r 2 = (ct) 2 =c 2 t 2 equation R.1 Equation of this expanding sphere in the moving frame Velocity is v is x 2 + y 2 + z 2 =r 2 = (ct ) 2 =c 2 t 2 equation R.2 How can this be..it must be that coordinates in each frame are different and Moving coordinates have changed..the Lorentz transformations are the changes!. r Home Work! Substitute the Lorentz equations 7.10 into R.2 to see if you Get R.1 which would demonstrate the transformation is the proper one for c to be constant in all frames moving at v! similar to discussion of 7.12!

80 Space-time and the 4 dimensional world The previous fact shows that the Lorentz transformations do not change the radius Of the expanding light sphere from a moving or stationary perspective. This fact is Said to be an invariant property. Ie. The radius of the light sphere is invariant Between stationary and moving frames of reference. The Lorentz transformations act like a rotation of an axis, if you plot just the x axis And ct time axis they Look like (fig 7.8) we use ct as the time axis (unit is length). ct ct = (ct - x) v=c We note that this space-time Diagram shows that the transformation Of coordinate axis appears to rotate x Them to the special line when v=c! x = (x ct) x Check out the discussion that shows that length is invariant under a rotation Page 133. Which is analogous to our showing the radius of the light sphere is Invariant.

81 Space-time and the 4 dimensional world Physicist developed a way of seeing events in our 4 dimensional world by Viewing it on a diagram known as the Space time diagram which in a 2 D rep is Similar to the last diagram We define in this 2D rep ( s) 2 =(c t) 2 ( x) 2 ct B Timelike events Called the space time interval And as in our previous light sphere Lightlike events(v=c) Calculation is invariant under the A C Spacelike events Lorentz transformation x The diagram defines three times of space time intervals If you are a photon or an expanding sphere of light (1D here) starting ti At the origin you follow the red Lightlike line. s=0! ie change in x And change in c-time equal each other! If you have an event at the origin moving less than light speed than you have a timelike event..(event starting at A can in the future reach B!) What about C? s>0 in Timelike events s<0 like C are called spacelike if we start at the origin we would have to move s<0 like C are called spacelike if we start at the origin we would have to move Faster than light to get to C. Study the three dimensional case in sec 7.5.1

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84 ENERGY IN SPECIAL RELATIVITY NON-RELATIVITY CALLED NEWTONIAN OR CLASSICAL PHYSICS.. A particle mass m, moving at a velocity has energy called KINECTIC ENERGY, E k = ½ mv 2 (UNITS joules) the particle carries an impact capability called Momentum, p, given as p=mv.( UNITS kg-m/sec) Note momentum has a direction or it is a 3D vector and has x,y and z components (p x,p y,p z) just like a point is space needs three values, x,y and z. SPECIAL RELATIVITY Herr Doctor Explains -- The total Energy of a particle with relativistic mass m= m o moving with velocity v is E= where m o and E o =m o c 2 the rest mass and rest energy of a particle Or E= m o c 2 The relativistic momentum p= m o v and has components (p x,p y,p z ) ( like in classical theory since it has direction) The total Energy in relativity includes the energy of motion and the particles enormous rest Mass energy m 2 o c, hence, subtracting the rest energy should give us the energy of motion ie. KINECTIC ENERGY, E k = m o c 2 -m o c 2 =( m o c 2 In fact, if v is small that the v/c is small we can use the expansion when x is small ie (1-x) n =1-nx to show that the relativistic Kinetic Energy becomes The classical Kinetic energy for low velocities, v. ie -(v/c) 2 ) 1/2 =1-1/2 v 2 /c 2 Hence, k = (1-1/2 v 2 /c 2-1)m o c 2 =1/2 m o v 2 =classical expression! See section on how Energy and Momentum is set up in SPACE_TIME!

85 HW The Nature of Space and Time-Special Relativity READ: Chapter 7 HW #1 Chapter 7: (HW#1) Questions 1, Problems (HW#2)1, (HW#3)3, (HW#4)4, (HW#5)6, (HW#6)7, (HW#7) Computer problem 7.1 (HW#8Additional task: Substitute the Lorentz equations 7.10 into R.2 (see lecture notes) to see if you Get R.1 which would demonstrate the transformation is the proper one for c to be constant in all frames moving at v!