CHAPTER 6 The Speial Theory of Relativity Units Galilean-Newtonian Relativity Postulates of the Speial Theory of Relativity Simultaneity Time Dilation and the Twin Paradox Length Contration Four-Dimensional Spae-Time Relativisti Momentum and Mass The Ultimate Speed E = m ; Mass and Energy Relativisti Addition of Veloities The Impat of Speial Relativity Basi Problems The speed of every partile in the universe always remains less than the speed of light Newtonian Mehanis is a limited theory It plaes no upper limit on speed It is ontrary to modern experimental results Newtonian Mehanis beomes a speialized ase of Einstein s Theory of Speial Relativity When speeds are muh less than the speed of light Foundation of Speial Relativity Reoniling of the measurements of two observers moving relative to eah other Normally observers measure different speeds for an objet Speial relativity relates two suh measurements Galilean Relativity Choose a frame of referene Neessary to desribe a physial event Aording to Galilean Relativity, the laws of mehanis are the same in all inertial frames of referene An inertial frame of referene is one in whih Newton s Laws are valid Objets subjeted to no fores will move in straight lines 1
Galilean Relativity Example A passenger in an airplane throws a ball straight up It appears to move in a vertial path This is the same motion as when the ball is thrown while at rest on the Earth The law of gravity and equations of motion under uniform aeleration are obeyed There is a stationary observer on the ground Views the path of the ball thrown to be a parabola The ball has a veloity to the right equal to the veloity of the plane Galilean Relativity Limitations The two observers disagree on the shape of the ball s path Both agree that the motion obeys the law of gravity and Newton s laws of motion Both agree on how long the ball was in the air Conlusion: There is no preferred frame of referene for desribing the laws of mehanis Galilean Relativity does not apply to experiments in eletriity, magnetism, optis, and other areas Results do not agree with experiments The observer should measure the speed of the pulse as v+ Atually measures the speed as
The Postulates of Speial Relativity The postulates of relativity as stated by Einstein: 1. Equivalene of Physial Laws The laws of physis are the same in all inertial frames of referene.. Constany of the Speed of Light The speed of light in a vauum, = 3.00 x 10 8 m/s, is the same in all inertial frames of referene, independent of the motion of the soure or the reeiver. The first postulate is ertainly reasonable; it would be hard to disover the laws of physis if it were not true! But why would the speed of light be onstant? It was thought that, like all other waves, light propagated as a disturbane in some medium, whih was alled the ether. The Earth s motion through the ether should be detetable by experiment. Experiments showed, however, no sign of the ether. Other experiments and measurements have been done, verifying that the speed of light is indeed onstant in all inertial frames of referene. With water waves, our measurement of the wave speed depends on our speed relative to the water: But with light, our measurements of its speed always give the same result: The fat that the speed of light is onstant also means that nothing an go faster than the speed of light it is the ultimate speed limit of the universe. Luminiferous Ether 19 th Century physiists ompared eletromagneti waves to mehanial waves Mehanial waves need a medium to support the disturbane The luminiferous ether was proposed as the medium required (and present) for light waves to propagate Present everywhere, even in empty spae Massless, but rigid medium Could have no effet on the motion of planets or other objets 3
Verifying the Luminiferous Ether Assoiated with an ether was an absolute frame where the laws of e & m take on their simplest form Sine the earth moves through the ether, there should be an ether wind blowing If v is the speed of the ether relative to the earth, the speed of light should have minimum (b) or maximum (a) value depending on its orientation to the wind Mihelson-Morley Experiment First performed in 1881 by Mihelson Repeated under various onditions by Mihelson and Morley Designed to detet small hanges in the speed of light By determining the veloity of the earth relative to the ether Used the Mihelson Interferometer Arm is aligned along the diretion of the earth s motion through spae The interferene pattern was observed while the interferometer was rotated through 90 The effet should have been to show small, but measurable, shifts in the fringe pattern Measurements failed to show any hange in the fringe pattern No fringe shift of the magnitude required was ever observed Light is now understood to be an eletromagneti wave, whih requires no medium for its propagation The idea of an ether was disarded The laws of eletriity and magnetism are the same in all inertial frames The addition laws for veloities were inorret 4
Albert Einstein 1879 1955 1905 published four papers on speial relativity 1916 published about General Relativity Searhed for a unified theory Never found one Einstein s Priniple of Relativity Resolves the ontradition between Galilean relativity and the fat that the speed of light is the same for all observers Postulates The Priniple of Relativity: All the laws of physis are the same in all inertial frames The onstany of the speed of light: the speed of light in a vauum has the same value in all inertial referene frames, regardless of the veloity of the observer or the veloity of the soure emitting the light This is a sweeping generalization of the priniple of Galilean relativity, whih refers only to the laws of mehanis The results of any kind of experiment performed in a laboratory at rest must be the same as when performed in a laboratory moving at a onstant speed past the first one No preferred inertial referene frame exists It is impossible to detet absolute motion The Constany of the Speed of Light Been onfirmed experimentally in many ways A diret demonstration involves measuring the speed of photons emitted by partiles traveling near the speed of light Confirms the speed of light to five signifiant figures Explains the null result of the Mihelson-Morley experiment Relative motion is unimportant when measuring the speed of light We must alter our ommon-sense notions of spae and time Consequenes of Speial Relativity Restriting the disussion to onepts of length, time, and simultaneity In relativisti mehanis There is no suh thing as absolute length There is no suh thing as absolute time Events at different loations that are observed to our simultaneously in one frame are not observed to be simultaneous in another frame moving uniformly past the first 5
Simultaneity In Speial Relativity, Einstein abandoned the assumption of simultaneity Thought experiment to show this A boxar moves with uniform veloity Two lightning bolts strike the ends The lightning bolts leave marks (A and B ) on the ar and (A and B) on the ground Two observers are present: O in the boxar and O on the ground Simultaneity Thought Experiment Set-up Observer O is midway between the points of lightning strikes on the ground, A and B Observer O is midway between the points of lightning strikes on the boxar, A and B The light signals reah observer O at the same time He onludes the light has traveled at the same speed over equal distanes Observer O onludes the lightning bolts ourred simultaneously By the time the light has reahed observer O, observer O has moved The light from B has already moved by the observer, but the light from A has not yet reahed him The two observers must find that light travels at the same speed Observer O onludes the lightning struk the front of the boxar before it struk the bak (they were not simultaneous events) Simultaneity Thought Experiment, Summary Two events that are simultaneous in one referene frame are in general not simultaneous in a seond referene frame moving relative to the first 6
That is, simultaneity is not an absolute onept, but rather one that depends on the state of motion of the observer In the thought experiment, both observers are orret, beause there is no preferred inertial referene frame Time Dilation The vehile is moving to the right with speed v A mirror is fixed to the eiling of the vehile An observer, O, at rest in this system holds a laser a distane d below the mirror The laser emits a pulse of light direted at the mirror (event 1) and the pulse arrives bak after being refleted (event ) Time Dilation, Moving Observer Observer O arries a lok She uses it to measure the time between the events (Δt p ) The p stands for proper She observes the events to our at the same plae Δt p = distane/speed = (d)/ Time Dilation, Stationary Observer Observer O is a stationary observer on the earth He observes the mirror and O to move with speed v By the time the light from the laser reahes the mirror, the mirror has moved to the right The light must travel farther with respet to O than with respet to O Time Dilation, Observations Both observers must measure the speed of the light to be The light travels farther for O The time interval, Δt, for O is longer than the time interval for O, Δt p 7
tp t t 1 v where 1 1v Time Dilation, Time Comparisons Observer O measures a longer time interval than observer O Time Dilation, Summary The time interval Δt between two events measured by an observer moving with respet to a lok is longer than the time interval Δt p between the same two events measured by an observer at rest with respet to the lok A lok moving past an observer at speed v runs more slowly than an idential lok at rest with respet to the observer by a fator of -1 Identifying Proper Time The time interval Δt p is alled the proper time p The proper time is the time interval between events as measured by an observer who sees the events our at the same position You must be able to orretly identify the observer who measures the proper time interval Alternate Views The view of O that O is really the one moving with speed v to the left and O s lok is running more slowly is just as valid as O s view that O was moving The priniple of relativity requires that the views of the two observers in uniform relative motion must be equally valid and apable of being heked experimentally Time Dilation Generalization All physial proesses slow down relative to a lok when those proesses our in a frame moving with respet to the lok These proesses an be hemial and biologial as well as physial Time dilation is a very real phenomena that has been verified by various experiments 8
Time Dilation Verifiation Muon Deays Muons are unstable partiles that have the same harge as an eletron, but a mass 07 times more than an eletron Muons have a half-life of Δt p =.µs when measured in a referene frame at rest with respet to them (a) Relative to an observer on earth, muons should have a lifetime of Δt p (b) A CERN experiment measured lifetimes in agreement with the preditions of relativity Example 1: A ar traveling 100 km/h overs a ertain distane in 10.0 s aording to the driver s wath. What does an observer at rest on Earth measure for the time interval? to 10.00s 10.00s t 15 v 7.8 / 1 (8.59x10 ) 1 m s 1 8 3.00x10 m / s A alulator will give an answer of 10.00s. A omputer will give 4x10 14 s more than 10.00 s. Example : The period of a pendulum is measured to be 3.00 s in the inertial frame of the pendulum. What is the period as measured by an observer moving at a speed of 0.950 with respet to the pendulum? t p 3.00s t 9.61s 1 v / (0.950 ) 1 The moving observer onsiders the pendulum to be moving, and moving loks are observed to run more slowly. The Twin Paradox The Situation A thought experiment involving a set of twins, Speedo and Goslo Speedo travels to Planet X, 0 light years from earth His ship travels at 0.95 After reahing planet X, he immediately returns to earth at the same speed When Speedo returns, he has aged 13 years, but Goslo has aged 4 years 9
The Twins Perspetives Goslo s perspetive is that he was at rest while Speedo went on the journey Speedo thinks he was at rest and Goslo and the earth raed away from him on a 6.5 year journey and then headed bak toward him for another 6.5 years The paradox whih twin is the traveler and whih is really older? The Twin Paradox The Resolution Relativity applies to referene frames moving at uniform speeds The trip in this thought experiment is not symmetrial sine Speedo must experiene a series of aelerations during the journey Therefore, Goslo an apply the time dilation formula with a proper time of 4 years This gives a time for Speedo of 13 years and this agrees with the earlier result There is no true paradox sine Speedo is not in an inertial frame Length Contration The measured distane between two points depends on the frame of referene of the observer The proper length, L p, of an objet is the length of the objet measured by someone at rest relative to the objet The length of an objet measured in a referene frame that is moving with respet to the objet is always less than the proper length This effet is known as length ontration LP v L LP 1 Length ontration takes plae only along the diretion of motion Four-Dimensional Spae-Time Spae and time are even more intriately onneted. Spae has three dimensions, and time is a fourth. When viewed from different referene frames, the spae and time oordinates an mix. Example 3: A starship is measured to be 15 m long while it is at rest with respet to an observer. If the starship now flies past the observer at a speed of 0.99, what length will the observer measure for the starship? L Lp 1 v / (15 m) 1 (0.99 ) / 17.6m 10
Example 4: An observer on Earth sees a spaeship at an altitude of 4,350 km moving downward toward Earth with a speed of 0.970. (a) What is the distane from the spaeship to Earth as measured by the spaeship s aptain? L L 1 v / 4,350km 1 (0.970 ) / p 3 1.06x10 km (b) After firing his engines, the aptain measures his ship s altitude as 67 km, while the observer on Earth measures it to be 65 km. What is the speed of the spaeship at this instant? L L 1 v / p L L Lp(1 v / ) 1 v / L p v L L km km 1 ( / p) 1 (67 / 65 ) 0.904 Relativisti Definitions To properly desribe the motion of partiles within speial relativity, Newton s laws of motion and the definitions of momentum and energy need to be generalized These generalized definitions redue to the lassial ones when the speed is muh less than Relativisti Momentum To aount for onservation of momentum in all inertial frames, the definition must be modified mv p mv 1 v v is the speed of the partile, m is its mass as measured by an observer at rest with respet to the mass When v <<, the denominator approahes 1 and so p approahes mv Relativisti Addition of Veloities Galilean relative veloities annot be applied to objets moving near the speed of light Einstein s modifiation is vad vdb vab vadvdb 1 The denominator is a orretion based on length ontration and time dilation 11
Relativisti Corretions Remember, relativisti orretions are needed beause no material objets an travel faster than the speed of light A basi result of speial relativity is that nothing an equal or exeed the speed of light. This would require infinite momentum not possible for anything with mass. Example 5: Suppose that Kirk s spaeraft is traveling at 0.600 in the positive x-diretion, as measured by a nearby observer, while Sotty is traveling in his own vehile diretly toward Kirk in the negative x-diretion at -0.800 relative the nearby observer. What s the veloity of Kirk relative to Sotty? vks vso v K = Kirk S = Sotty KS vksvso 1 O = Observer vks 0.800 0.800vKS 0.600 1 0.600 vks 0.800 vks ( 0.800 ) 1 0.600 0.480v v 0.800 v KS 0.946 KS KS E = m ; Mass and Energy At relativisti speeds, not only is the formula for momentum modified; that for energy is as well. The total energy an be written: Where the partile is at rest, Combining the relations for energy and momentum gives the relativisti relation between them: 1
All the formulas presented here beome the usual Newtonian kinemati formulas when the speeds are muh smaller than the speed of light. There is no rule for when the speed is high enough that relativisti formulas must be used it depends on the desired auray of the alulation. Example 6: When two moles of hydrogen and one mole of oxygen reat to form two moles of water, the energy released is 484 kj. How muh does the mass derease in this reation? E ( 484x10 J) 1 m 5.38x10 kg 8 (3.00x10 m/ s) The initial mass of the system is 0.00 kg + 0.016 kg = 0.018 kg. Thus the hange in mass is relatively very tiny and an normally be negleted. Relativisti Addition of Veloities Relativisti veloities annot simply add; the speed of light is an absolute limit. The relativisti formula is: Example 7: alulate the speed of roket. 0.60 0.60 1.0 u 0.88 (0.60 )(0.60 ) 1 1.36 Relativisti Energy The definition of kineti energy requires modifiation in relativisti mehanis KE = m m The term m is alled the rest energy of the objet and is independent of its speed The term m is the total energy, E, of the objet and depends on its speed and its rest energy Relativisti Energy Consequenes A partile has energy by virtue of its mass alone A stationary partile with zero kineti energy has an energy proportional to its inertial mass The mass of a partile may be ompletely onvertible to energy and pure energy may be onverted to partiles 13
Energy and Relativisti Momentum It is useful to have an expression relating total energy, E, to the relativisti momentum, p E = p + (m ) When the partile is at rest, p = 0 and E = m Massless partiles (m = 0) have E = p This is also used to express masses in energy units Mass of an eletron = 9.11 x 10-31 kg = 0.511 Me Conversion: 1 u = 931.494 MeV/ Pair Prodution An eletron and a positron are produed and the photon disappears A positron is the antipartile of the eletron, same mass but opposite harge Energy, momentum, and harge must be onserved during the proess The minimum energy required is m e = 1.0 MeV In pair annihilation, an eletron-positron pair produes two photons The inverse of pair prodution It is impossible to reate a single photon Momentum must be onserved Mass Inertial vs. Gravitational Mass has a gravitational attration ' for other masses mm g g Fg G r Mass has an inertial property that resists aeleration F i = m i a The value of G was hosen to make the values of m g and m i equal 14
Einstein s Reasoning Conerning Mass That m g and m i were diretly proportional was evidene for a basi onnetion between them No mehanial experiment ould distinguish between the two He extended the idea to no experiment of any type ould distinguish the two masses Postulates of General Relativity All laws of nature must have the same form for observers in any frame of referene, whether aelerated or not In the viinity of any given point, a gravitational field is equivalent to an aelerated frame of referene without a gravitational field This is the priniple of equivalene Impliations of General Relativity Gravitational mass and inertial mass are not just proportional, but ompletely equivalent A lok in the presene of gravity runs more slowly than one where gravity is negligible The frequenies of radiation emitted by atoms in a strong gravitational field are shifted to lower frequenies This has been deteted in the spetral lines emitted by atoms in massive stars A gravitational field may be transformed away at any point if we hoose an appropriate aelerated frame of referene a freely falling frame Einstein speified a ertain quantity, the urvature of spaetime, that desribes the gravitational effet at every point Curvature of Spaetime There is no suh thing as a gravitational fore Aording to Einstein Instead, the presene of a mass auses a urvature of spaetime in the viinity of the mass This urvature ditates the path that all freely moving objets must follow General Relativity Summary Mass one tells spaetime how to urve; urved spaetime tells mass two how to move John Wheeler s summary, 1979 The equation of general relativity is roughly a proportion: Average urvature of spaetime a energy density The atual equation an be solved for the metri whih an be used to measure lengths and ompute trajetories 15
General Relativity predits that a light ray passing near the Sun should be defleted by the urved spaetime reated by the Sun s mass The predition was onfirmed by astronomers during a total solar elipse Explanation of Merury s orbit Explained the disrepany between observation and Newton s theory Time delay of radar bouned off Venus Gradual lengthening of the period of binary pulsars due to emission of gravitational radiation Blak Holes If the onentration of mass beomes great enough, a blak hole is believed to be formed In a blak hole, the urvature of spae-time is so great that, within a ertain distane from its enter, all light and matter beome trapped The radius is alled the Shwarzshild radius Also alled the event horizon It would be about 3 km for a star the size of our Sun At the enter of the blak hole is a singularity It is a point of infinite density and urvature where spaetime omes to an end 16