new-relativity i.ppt

Phy 582- Modern Physics Modern Physics-Relativity I

Modern Physics-Relativity I

Special Theory of RelativityA Brief introduction




Classical PhysicsAt the end of the 19th century it looked as if Physics was pretty well wrapped up.

Newtonian mechanics and the law of Gravitation had explained how the planets moved and related that to how ordinary objects here on earth responded to forces.

Classical Physics (Cont)All this came to be known as classical physics.

Little did the physicist of 1900 realize what was in store during the next 100 years, when the ideas, theories, and results of modern physics were developed.

Twentieth Century Physics

Special Theory of Relativity

General Theory of Relativity

Quantum Theory

Special Theory of RelativityIntroduced a new way to view

Space

Time

Simultaneity

General Theory of Relativity

Re-interpreted gravitational theory in terms of space-time.

Quantum Theory

Introduced a new way to think about atomic processesReplaced absolute knowledge with probabilities

Helped clear up some problems that classical theories could not explain.

Galilean-Newtonian RelativityInertial Reference FrameInertial reference frames are those in which Newtons laws of motion are valid.

Relativity PrincipleThe basic laws of physics are the same in all inertial reference frames.

Both understood by Galileo and Newton

Inertial Reference FrameA reference frame is called an inertial frame if Newton laws are valid in that frame.Such a frame is established when a body, not subjected to net external forces, is observed to move in rectilinear motion at constant velocity.


Newtons laws of motion must be implemented with respect to (relative to) some reference frame.A reference frame is called an inertial frame if Newtons laws are valid in that frame.

Such a frame is established when a body, not subjected to net external forces, moves in rectilinear motion at constant velocity.

Newtonian Principle of RelativityIf Newtons laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system.

This is referred to as the Newtonian principle of relativity or Galilean invariance.

Basic Problems of Newtonian MechanicsNewtonian mechanics fails to describe properly the motion of objects whose speeds approach that of light.Newtonian mechanics is a limited theory.It places no upper limit on speed.It is contrary to modern experimental results.Newtonian mechanics becomes a specialized case of Einsteins special theory of relativity.When speeds are much less than the speed of light.

Galilean RelativityTo describe a physical event, a frame of reference must be established.There is no absolute inertial frame of reference.This means that the results of an experiment performed in a vehicle moving with uniform velocity will be identical to the results of the same experiment performed in a stationary vehicle.Reminders about inertial frames:Objects subjected to no forces will experience no acceleration.Any system moving at constant velocity with respect to an inertial frame must also be in an inertial frame.According to the principle of Galilean relativity, the laws of mechanics must be the same in all inertial frames of reference.

Galilean Relativity ExampleThe truck moves with a constant velocity with respect to the ground.The observer in the truck throws a ball straight up.It appears to move in a vertical path.The law of gravity and equations of motion under uniform acceleration are obeyed.

Galilean Relativity Example, cont.There is a stationary observer on the ground.Views the path of the ball thrown to be a parabolaThe ball has a velocity to the right equal to the velocity of the truck.

Galilean Relativity Example, conclusionThe two observers disagree on the shape of the balls path.Both agree that the motion obeys the law of gravity and Newtons laws of motion.Both agree on how long the ball was in the air.Conclusion: There is no preferred frame of reference for describing the laws of mechanics.

Views of an EventAn event is some physical phenomenon.Assume the event occurs and is observed by an observer at rest in an inertial reference frame.The events location and time can be specified by the coordinates (x, y, z, t).Consider two inertial frames, S and S.S moves with constant velocity along the common x and x axes.The velocity is measured relative to S.Assume the origins of S and S coincide at t = 0.

Galilean Space-Time Transformation EquationsAn observer in S describes the event with space-time coordinates (x, y, z, t).An observer in S describes the same event with space-time coordinates (x, y, z, t).The relationship among the coordinates are:x = x vty = yz = zt = t

Notes About Galilean Transformation EquationsThe time is the same in both inertial frames.Within the framework of classical mechanics, all clocks run at the same rate.The time at which an event occurs for an observer in S is the same as the time for the same event in S.This turns out to be incorrect when v is comparable to the speed of light.

Galilean Velocity Transformation EquationSuppose that a particle moves through a displacement dx along the x axis in a time dt.The corresponding displacement dx is

u is used for the particle velocity and v is used for the relative velocity between the two frames.

Galilean Transformation Equations Final NotesThe x and x axes coincide, but their origins are different.The y and y axes are parallel, but do not coincide.This is due to the displacement of the origin of S with respect to that of S.The same holds for z and z axes.These axes coincide only at one instant due to the time-varying displacement of the origin of S with respect to that of S.Time = 0 when the origins of the two coordinate system coincide.If the S frame is moving in the positive x direction relative to S, the v is positive. Otherwise, it is negative

Speed of LightGalilean relativity does not apply to electricity, magnetism, or optics.Maxwell showed the speed of light in free space is c = 3.00 x 108 m/s.Physicists in the late 1800s thought light waves moved through a medium called the ether.The speed of light would be c only in a special, absolute frame at rest with respect to the ether.

The Need for Ether The wave nature of light suggested that there existed a propagation medium called the luminiferous ether or just ether.

Ether had to have such a low density that the planets could move through it without loss of energy

It also had to have an elasticity to support the high velocity of light waves

Effect of Ether Wind on LightAssume v is the velocity of the ether wind relative to the earth.c is the speed of light relative to the ether.Various resultant velocities are shown.

Ether Wind, cont.The velocity of the ether wind is assumed to be the orbital velocity of the Earth.All attempts to detect and establish the existence of the ether wind proved futile.But Maxwells equations seem to imply that the speed of light always has a fixed value in all inertial frames.This is a contradiction to what is expected based on the Galilean velocity transformation equation.

An Absolute Reference System Ether was proposed as an absolute reference system in which the speed of light was this constant and from which other measurements could be made.The Michelson-Morley experiment was an attempt to show the existence of ether.

The Michelson-Morley Experiment

Albert Michelson (18521931) received the Nobel Prize for Physics (1907), and built an extremely precise device called an interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions.


Michelson-Morley experiment Used the Michelson interferometer with half-transparent glass Arm 2 is aligned along the direction of the Earths motion through space The effect should have been to show small, but measurable shifts in the fringe pattern


Michelson-Morley Expected ResultsThe speed of light measured in the Earth frame should be c - v as the light approaches mirror M2

The speed of light measured in the Earth frame should be c + v as the light is reflected from mirror M2

The experiment was repeated at different times of the year when the ether wind was expected to change direction and magnitude


Michelson-Morley ResultsNo fringe shift of the magnitude required was ever observed which means that speed of light (in vacuum) is the same in all inertial frames of referencesIn other words, speed of light is independent of ether speed and direction of motion of a frame of referenceThis contradicted our normal intuition and the Galileos principles of motion

What to do?Are Maxwells equations wrong?They correctly predicted so many observations that physicists were reluctant to give them up.

Ether is dragged along by the earth?Got the same results when the M&M experiment was carried out in balloons and on mountaintops.Each attempt to determine a way to find a preferred reference system seemed to be doomed to failure

There is a Way Out of This MessHenri Poincare finally concluded that such a complete conspiracy of nature must itself be regarded as a law of nature. i.e., the Principle of Relativity must be valid!!

This was the state of affairs in 1905 when Einstein presented his Theory of Relativity.

The Transition to Modern Relativity Although Newtons laws of motion had the same form under the Galilean transformation, Maxwells equations did not.In 1905, Albert Einstein proposed a fundamental connection between space and time and that Newtons laws are only an approximation.

Maxwells Equations

In Maxwells theory the speed of light, in terms of the permeability and permittivity of free space, was given by

Thus the velocity of light between moving systems must be a constant.

Enter Einstein - 1905

In 1905 Albert Einstein proposed that we accept the fact that the speed of light was the same in all reference systems(this was consistent with the M&M result) and was tantamount to doing away with the concept of the ether.

Classical Relativity1,000,000 ms-11,000,000 ms-1 How fast is Spaceship A approaching Spaceship B? Both Spaceships see the other approaching at 2,000,000 ms-1. This is Classical Relativity.

Einsteins Special Relativity1,000,000 ms-10 ms-1300,000,000 ms-1Both spacemen measure the speed of the approaching ray of light.How fast do they measure the speed of light to be?

Special RelativityStationary man300,000,000 ms-1Man travelling at 1,000,000 ms-1301,000,000 ms-1?Wrong!

The Speed of Light is the same for all observers

Einsteins Principle of Relativity(Postulates of the Special Theory of Relativity)Resolves the contradiction between Galilean relativity and the fact that the speed of light is the same for all observers.PostulatesFirst Postulate: The principle of relativity: The laws of physics must be the same in all inertial reference frames.Second Postulate: The constancy of the speed of light: the speed of light in a vacuum has the same value, c = 3.00 x 108 m/s, in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light.

The Constancy of the Speed of LightThis is required by the first postulate.Confirmed experimentally in many waysExplains the null result of the Michelson-Morley experimentRelative motion is unimportant when measuring the speed of light.We must alter our common-sense notions of space and time.

Consequences of Special RelativityRestricting the discussion to concepts of simultaneity, time intervals, and lengthThese are quite different in relativistic mechanics from what they are in Newtonian mechanics.In relativistic mechanicsThere is no such thing as absolute length.There is no such thing as absolute time.Events at different locations that are observed to occur simultaneously in one frame are not observed to be simultaneous in another frame moving uniformly past the first.

SimultaneityIn special relativity, Einstein abandoned the assumption of simultaneity.Thought experiment to show thisA boxcar moves with uniform velocity.Two lightning bolts strike the ends.The lightning bolts leave marks (A and B) on the car and (A and B) on the ground.Two observers are present: O in the boxcar and O on the ground

Simultaneity Thought Experiment Set-upObserver O is midway between the points of lightning strikes on the boxcar, A and B.

Observer O is moving with the boxcar midway between the points of lightning strikes on the ground, A and B.

Simultaneity Thought Experiment Results The light reaches observer O at the same time.He concludes the light has traveled at the same speed over equal distances.Observer O concludes the lightning bolts occurred simultaneously.

Simultaneity Thought Experiment Results, cont.By the time the light has reached observer O, observer O has moved.The signal from B has already swept past O, but the signal from A has not yet reached him.The two observers must find that light travels at the same speed.Observer O concludes the lightning struck the front of the boxcar before it struck the back (they were not simultaneous events).

Simultaneity Thought Experiment, SummaryTwo events that are simultaneous in one reference frame are in general not simultaneous in a second reference frame moving relative to the first.That is, simultaneity is not an absolute concept, but rather one that depends on the state of motion of the observer.In the thought experiment, both observers are correct, because there is no preferred inertial reference frame.

Simultaneity, Transit TimeIn this thought experiment, the disagreement depended upon the transit time of light to the observers and doesnt demonstrate the deeper meaning of relativity.In high-speed situations, the simultaneity is relative even when transit time is subtracted out.We will ignore transit time in all further discussions.

Time DilationA mirror is fixed to the ceiling of a vehicle.The vehicle is moving to the right with speed .An observer, O, at rest in the frame attached to the vehicle holds a flashlight a distance d below the mirror.The flashlight emits a pulse of light directed at the mirror (event 1) and the pulse arrives back after being reflected (event 2).

Time Dilation, Moving ObserverObserver O carries a clock.She uses it to measure the time between the events (tp).Model the pulse of light as a particle under constant speed.The observer sees the events to occur at the same place.tp = distance/speed = (2d)/c

Time Dilation, Stationary ObserverObserver O is in a second frame at rest with respect to the ground.He observes the mirror and O to move with speed v.By the time the light from the flashlight reaches the mirror, the mirror has moved to the right.The light must travel farther with respect to O than with respect to O.

Time Dilation, ObservationsBoth observers must measure the speed of the light to be c.The light travels farther for O.The time interval, t, for O is longer than the time interval for O, tp.

Time Dilation, Time ComparisonsThe time interval t is longer than the time interval tp

is always greater than unity

Time Dilation, SummaryThe time interval t between two events measured by an observer moving with respect to a clock is longer than the time interval tp between the same two events measured by an observer at rest with respect to the clock.This effect is known as time dilation.

g Factor Time dilation is not observed in our everyday lives.For slow speeds, the factor of g is so small that no time dilation occurs.As the speed approaches the speed of light, g increases rapidly.

g Factor Table

Identifying Proper TimeThe time interval tp is called the proper time interval.The proper time interval is the time interval between events as measured by an observer who sees the events occur at the same point in space.You must be able to correctly identify the observer who measures the proper time interval.

Time DilationTo understand time dilation the idea of proper time must be understood:

The term proper time, is the time difference between two events occurring at the same position in a system as measured by a clock at that position.

Same location

Not Proper Time

Beginning and ending of the event occur at different positions Time Dilation

Time Dilation Generalization If a clock is moving with respect to you, the time interval between ticks of the moving clock is observed to be longer that the time interval between ticks of an identical clock in your reference frame.All physical processes are measured to slow down when these processes occur in a frame moving with respect to the observer.These processes can be chemical and biological as well as physical.

Example Ship A passes ship B with a relative velocity of 0.8c (eighty percent of the velocity of light). A woman aboard Ship B takes 4 s to walk the length of her ship. What time is recorded by the man in Ship A?Find relative time DtDt = 6.67 s

Time Dilation Verification Time dilation is a very real phenomenon that has been verified by various experiments.These experiments include:Airplane flightsMuon decayTwin Paradox

Airplanes and Time DilationIn 1972 an experiment was reported that provided direct evidence of time dilation.Time intervals measured with four cesium clocks in jet flight were compared to time intervals measured by Earth-based reference clocks.The results were in good agreement with the predictions of the special theory of relativity.

Time Dilation Verification Muon DecaysMuons are unstable particles that have the same charge as an electron, but a mass 207 times more than an electron.Muons have a half-life of tp = 2.2 s when measured in a reference frame at rest with respect to them (fig. a).Relative to an observer on the Earth, muons should have a lifetime of tp (fig. b)A CERN experiment measured lifetimes in agreement with the predictions of relativity.

Muon Decay3

Muon DecayModern Physics-Relativity I

The Twin Paradox The SituationA thought experiment involving a set of twins, Speedo and Goslo.Speedo travels to Planet X, 20 light years from the Earth.His ship travels at 0.95c.After reaching Planet X, he immediately returns to the Earth at the same speed.When Speedo returns, he has aged 13 years, but Goslo has aged 42 years.

The Twins PerspectivesGoslos perspective is that he was at rest while Speedo went on the journey.Speedo thinks he was at rest and Goslo and the Earth raced away from him and then headed back toward him. The paradox which twin actually ages more slowly?

The Twin Paradox The ResolutionRelativity applies to reference frames moving at uniform speeds.The trip in this thought experiment is not symmetrical since Speedo must experience a series of accelerations during the journey.Therefore, Goslo can apply the time dilation formula with a proper time of 42 years.This gives a time for Speedo of 13 years and this agrees with the earlier result.There is no true paradox since Speedo is not in an inertial frame.

Length ContractionThe measured distance between two points depends on the frame of reference of the observer.The proper length, Lp, of an object is the length of the object measured by someone at rest relative to the object.The length of an object measured in a reference frame that is moving with respect to the object is always less than the proper length.This effect is known as length contraction.

More About Proper LengthVery important to correctly identify the observer who measures proper length.The proper length is always the length measured by the observer at rest with respect to the points. Often the proper time interval and the proper length are not measured by the same observer.

Length ContractionSince time is affected by relative motion, length will also be different:Lo is proper length L is relative length Moving objects are foreshortened due to relativity.

Example : A meter stick moves at 0.9c relative to an observer. What is the relative length as seen by the observer?Length recorded by observer:L = 43.6 cmIf the ground observer held a meter stick, the same contraction would be seen from the ship.

Foreshortening of ObjectsNote that it is the length in the direction of relative motion that contracts and not the dimensions perpendicular to the motion.If meter stick is 2 cm wide, each will say the other is only 0.87 cm wide, but they will agree on the length.Assume each holds a meter stick, in example.

LengthContractionA fast-moving plane at different speeds.v = 10% c

Why Length ContractionSuppose that 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.

By time dilation, only 500s/3.2=156s pass on the ship. The crew observes the Earth coming at them at 0.95cThis means that the sun-earth distance according to the crew must be reduced by 3.2!Ship covers 150,000,000 km in 500 sAs seen by earthbound observerEarth covers 47,000,000 km in 156 sAs seen by crew member observer

"When you sit with a nice girl for two hours, it seems like two minutes. When you sit on a hot stove for two minutes, it seems like two hours, that's relativity. -Albert Einstein

We have seen:Galilean Space-Time Transformation EquationsAn observer in S describes the event with space-time coordinates (x, y, z, t).An observer in S describes the same event with space-time coordinates (x, y, z, t).The relationship among the coordinates are:x = x vty = yz = zt = t

The Lorentz TransformationsThe special set of linear transformations that: preserve the constancy of the speed of light (c) between inertial observers; and, account for the problem of simultaneity between these observers known as the Lorentz transformation equations

The Lorentz TransformationsThe Lorentz coordinate transformations, is a set of formulas that relates the space and time coordinates of two inertial observers moving with a relative speed .

Seen two consequences of the Lorentz transformation in the time dilation and length contraction formulas.

The lorentz velocity transformation is the set of formulas that relate the velocities components ux, uy, uz of an object moving in frame S to the velocity components ux, uy, uz of the same object measured in frame S, which is moving with a speed relative to S.

The Lorentz transformation formulas provide a formal, concise, and almost mechanical method of solution of relativity problem.

Lorentz Transformation Equations

Lorentz Transformation Equations A more symmetric form:

Properties of

Recall = v/c < 1 for all observers.

equals 1 only when v = 0.

Graph: (note v c)

DerivationUse the fixed system K and the moving system KAt t = 0 the origins and axes of both systems are coincident with system K moving to the right along the x axis. A flashbulb goes off at the origins when t = 0. According to postulate 2, the speed of light will be c in both systems and the wavefronts observed in both systems must be spherical.

K K

DerivationSpherical wavefronts in K:

Spherical wavefronts in K:

Note: these are not preserved in the classical transformations with

Let x = (x vt) so that x = (x + vt)We want a linear equation (1 solution!!)

By Einsteins first postulate :

The wavefront along the x,x- axis must satisfy: x = ct and x = ct Thus ct = (ct vt) and ct = (ct + vt)

Solving the first one above for t and substituting into the second... Derivation

Derivation

from which we derive:

Gives the following result:

Finding a Transformation for tRecalling x = (x vt) substitute into x = (x + vt) and solving for t we obtain:

with:

t may be written in terms of (= v/c):

The complete Lorentz TransformationIf v

Addition of VelocitiesTaking differentials of the Lorentz transformation, relative velocities may be calculated (dv=0 because we are in inertial systems):

Addition of VelocitiesTaking differentials of the Lorentz transformation [here between the rest frame (K) and the space ship frame (K)], we can compute the shuttle velocity in the rest frame (ux = dx/dt):Suppose a shuttle takes off quickly from a space ship already traveling very fast (both in the x direction). Imagine that the space ships speed is v, and the shuttles speed relative to the space ship is u. What will the shuttles velocity (u) be in the rest frame?

So thatdefining velocities as: ux = dx/dt, uy = dy/dt, ux = dx/dt, etc. it is easily shown that:

With similar relations for uy and uz:

The Lorentz Velocity TransformationsIn addition to the previous relations, the Lorentz velocity transformations for ux, uy , and uz can be obtained by switching primed and unprimed and changing v to v:

Relativistic velocity addition

Example 1 20.60c with respect to Earth0.50c with respect to rocket 1Calculate the speed of rocket 2 with respect to earth.


vRg = velocity of Romulans relative to galaxyvtR = velocity of torpedo relative to RomulansvEg = velocity of Enterprise relative to galaxyvRg = 1/2cvEg = 3/4cvtR = 1/3cRomulansEnterprisetorpedoExample: Lorentz velocity transformationCapt. Kirk decides to escape from a hostile Romulan ship at 3/4c, but the Romulans follow at 1/2c, firing a matter torpedo, whose speed relative to the Romulan ship is 1/3c. Question: does the Enterprise survive?


We need to compute the torpedo's velocity relative to the galaxy and compare that with the Enterprise's velocity relative to the galaxy. Using the Galilean transformation, we simply add the torpedos velocity to that of the Romulan ship:Galileos addition of velocities


We need to compute the torpedo's velocity relative to the galaxy and compare that with the Enterprise's velocity relative to the galaxy. Using the Galilean transformation, we simply add the torpedos velocity to that of the Romulan ship:Galileos addition of velocities


vRg = velocity of Romulans relative to galaxyvtR = velocity of torpedo relative to RomulansvEg = velocity of Enterprise relative to galaxyvRg = 1/2cvEg = 3/4cvtR = 1/3cRomulansEnterprisetorpedo

Einsteins addition of velocitiesDue to the high speeds involved, we really must relativistically add the Romulan ships and torpedos velocities:The Enterprise survives to seek out new worlds and go where no one has gone before


A space ship moving away from the earth with speed 0.90 c fires a robot space probe in the same direction as its motion, with speed 0.70 c relative to the space ship.What is the probes velocity relative to the earth?

A scoutship tries to catch up with the space ship by traveling at 0.95 c relative to the earth. What is the speed of the scoutship relative to the space ship?0.982 c0.345c

***********************VT = V1 + V2Mention the Ether*Moving with respect to the Ether*Mention the Michelson Morley experimentAlbert Michelson & Edward MorleyLead Einstein to propose his Theory of Special Relativity****************************v*http://scholar.uwinnipeg.ca/courses/38/4500.6-001/Cosmology/SpecialRelativity.htm**http://images.google.com/imgres?imgurl=http://www.abcteach.com/SolarSystem/images/spaces1.gif&imgrefurl=http://www.abcteach.com/SolarSystem/spaceshipshape.htm&h=533&w=592&sz=16&tbnid=q_LtYly4jHCVWM:&tbnh=119&tbnw=133&hl=en&start=2&prev=/images%3Fq%3Dspace%2Bship%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DGGLD,GGLD:2004-43,GGLD:en

http://images.google.com/imgres?imgurl=http://www.janetdavis.net/flashonline/images/imagess3.gif&imgrefurl=http://www.janetdavis.net/flashonline/1_spaceshipRacecar.htm&h=214&w=398&sz=4&tbnid=XFy_ZNWgG44uFM:&tbnh=64&tbnw=120&hl=en&start=202&prev=/images%3Fq%3Dspace%2Bship%26start%3D200%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DGGLD,GGLD:2004-43,GGLD:en%26sa%3DN

http://images.google.com/imgres?imgurl=http://www.schools.pinellas.k12.fl.us/gallery/cartoon/spaceship.gif&imgrefurl=http://www.schools.pinellas.k12.fl.us/gallery/cartoon/Page0005.html&h=91&w=231&sz=8&tbnid=k9--bJqaz4B7qM:&tbnh=40&tbnw=103&hl=en&start=21&prev=/images%3Fq%3Dcartoon%2Bspace%2Bship%26start%3D20%26svnum%3D10%26hl%3Den%26lr%3D%26rls%3DGGLD,GGLD:2004-43,GGLD:en%26sa%3DN*http://www.physics.unc.edu/~rowan/PHYS25-05/p25units/unit29/WCHAP29-5.html

Note: this plot exaggerates the difference between the two laws; also, its not possible to relabel the u axis to fix it. But its not off by much (at most a few per cent).****

new-relativity i.ppt

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