into to modern physics › 2015 › 10 › ... · Modern Physics. Events and Frames of Reference An event is a physical happening that occurs at a certain place and time An observer

Modern Physics

Events and Frames of Reference

An event is a physical happening that occurs at a certain place and time

An observer uses a reference frame to describe an event

A frame of reference is the physical surroundings from which an observer observes and measures the world around him/her.

Types of reference frame - inertial and non-inertial

An event can be described differently by observers at 2 different frames of reference

Events and Frames of Reference

Classical Physics: a bit of history....

Aristotle

384-322 B.C.

Aristotle’s Physics...

Every sensible body is by its nature somewhere. (Physics,Book 3, 205a:10)Time is the numeration of continuous movement. (Physics, Book 4, 223b:1)


Aristotle’s Space and Time

There exists a Prime Mover, a privileged being in the state of Absolute RestThe position of everything else is measured with three numbers (x, y, z) with respect to the Prime Mover, who sits at (0,0,0). The time is measured by looking at the Prime Mover's clock

x

y

z

(x,y,z)


Galileo’s Challenge...

Galileo argued that there is no such thing as "Absolute Rest". In his view: The mechanical laws of physics are the same for every observer moving with a constant speed along a straight line (this is called "inertial observer" for short).

Galileo Galilei

1564 -1642


Galileo’s Space and Time

Every inertial observer could declare themselves "the Prime Mover", and measure the position of everything with respect to their own set of (x, y, z) The time is still measured by looking at the Prime Mover's clock!

x

y

z

(x,y,z)

x'

y'

(x',y',z')

z' v


K

z

x

y

K' v

x'

y'

z' A

y

vt y'

Galileo’s Transformation

(Newtonian - Galilean transformation)


Newtonian Mechanics...

Newton's laws of mechanics are in agreement with Galileo's relativity A body, not acted upon by any force, stays at rest or remains in uniform motion.To get an object to change its velocity, we need a force

Sir Isaac Newton

1642-1727

Newtonian Relativity

• We know how positions of an object transform when we go from one inertial frame of reference to another

velocity of an object in K is equal to its velocity in K',

plus the velocity of K’ with respect to K

= 0 as v = const

Accelerations are the same in both K and K’ frames!

So Newtonian forces will be the same in both frames

• What about accelerations?

• What about velocities?

“Newton’s Laws are the same in all inertial frames”

Newtonian Relativity

“Newton’s Laws are the same in all inertial frames”

39.1 The Principle of Galilean Relativity

To describe a physical event, we must establish a frame of reference. You should recallfrom Chapter 5 that an inertial frame of reference is one in which an object is observedto have no acceleration when no forces act on it. Furthermore, any system moving withconstant velocity with respect to an inertial frame must also be in an inertial frame.

There is no absolute inertial reference frame. This means that the results of anexperiment performed in a vehicle moving with uniform velocity will be identical tothe results of the same experiment performed in a stationary vehicle. The formalstatement of this result is called the principle of Galilean relativity:

1246 C H A P T E R 3 9 • Relativity

The laws of mechanics must be the same in all inertial frames of reference.

Let us consider an observation that illustrates the equivalence of the laws ofmechanics in different inertial frames. A pickup truck moves with a constant velocity,as shown in Figure 39.1a. If a passenger in the truck throws a ball straight up, and if aireffects are neglected, the passenger observes that the ball moves in a vertical path. Themotion of the ball appears to be precisely the same as if the ball were thrown by aperson at rest on the Earth. The law of universal gravitation and the equations ofmotion under constant acceleration are obeyed whether the truck is at rest or inuniform motion.

Both observers agree on the laws of physics—they each throw a ball straight up andit rises and falls back into their hand. What about the path of the ball thrown by theobserver in the truck? Do the observers agree on the path? The observer on theground sees the path of the ball as a parabola, as illustrated in Figure 39.1b, while, asmentioned earlier, the observer in the truck sees the ball move in a vertical path.Furthermore, according to the observer on the ground, the ball has a horizontalcomponent of velocity equal to the velocity of the truck. Although the two observersdisagree on certain aspects of the situation, they agree on the validity of Newton’s lawsand on such classical principles as conservation of energy and conservation of linearmomentum. This agreement implies that no mechanical experiment can detect anydifference between the two inertial frames. The only thing that can be detected is therelative motion of one frame with respect to the other.

(b)(a)

Figure 39.1 (a) The observer in the truck sees the ball move in a vertical path whenthrown upward. (b) The Earth observer sees the path of the ball as a parabola.

Principle of Galilean relativity

Quick Quiz 39.1 Which observer in Figure 39.1 sees the ball’s correct path?(a) the observer in the truck (b) the observer on the ground (c) both observers.

39.1 The Principle of Galilean Relativity

To describe a physical event, we must establish a frame of reference. You should recallfrom Chapter 5 that an inertial frame of reference is one in which an object is observedto have no acceleration when no forces act on it. Furthermore, any system moving withconstant velocity with respect to an inertial frame must also be in an inertial frame.

There is no absolute inertial reference frame. This means that the results of anexperiment performed in a vehicle moving with uniform velocity will be identical tothe results of the same experiment performed in a stationary vehicle. The formalstatement of this result is called the principle of Galilean relativity:


The laws of mechanics must be the same in all inertial frames of reference.

Let us consider an observation that illustrates the equivalence of the laws ofmechanics in different inertial frames. A pickup truck moves with a constant velocity,as shown in Figure 39.1a. If a passenger in the truck throws a ball straight up, and if aireffects are neglected, the passenger observes that the ball moves in a vertical path. Themotion of the ball appears to be precisely the same as if the ball were thrown by aperson at rest on the Earth. The law of universal gravitation and the equations ofmotion under constant acceleration are obeyed whether the truck is at rest or inuniform motion.

Both observers agree on the laws of physics—they each throw a ball straight up andit rises and falls back into their hand. What about the path of the ball thrown by theobserver in the truck? Do the observers agree on the path? The observer on theground sees the path of the ball as a parabola, as illustrated in Figure 39.1b, while, asmentioned earlier, the observer in the truck sees the ball move in a vertical path.Furthermore, according to the observer on the ground, the ball has a horizontalcomponent of velocity equal to the velocity of the truck. Although the two observersdisagree on certain aspects of the situation, they agree on the validity of Newton’s lawsand on such classical principles as conservation of energy and conservation of linearmomentum. This agreement implies that no mechanical experiment can detect anydifference between the two inertial frames. The only thing that can be detected is therelative motion of one frame with respect to the other.

(b)(a)

Figure 39.1 (a) The observer in the truck sees the ball move in a vertical path whenthrown upward. (b) The Earth observer sees the path of the ball as a parabola.

Principle of Galilean relativity

Quick Quiz 39.1 Which observer in Figure 39.1 sees the ball’s correct path?(a) the observer in the truck (b) the observer on the ground (c) both observers.

Then the clouds start to gather…

• For more than two centuries after its inception the Newtonian view of the world ruled supreme

• However, at the end of the 19th century problems started to appear

• The problematic issue can be reduced to these questions:

–What is light?

–How does it propagate?

–Is light a wave or a particle?

Here comes Maxwell...• Maxwell brought together the

knowledge of electricity and magnetism known in his day in a set of four elegant equations known as Maxwell's equations

• In the process, he introduced a new concept: electromagnetic waves, and found that they traveled at the speed of light

• Light is an electromagnetic phenomenon!

James C. Maxwell 1831-1879

Electromagnetic waves

electric field

magnetic field

Waves in general• The waves we are all familiar with require

something to propagate in

Sound waves are compressionsof air (water, etc.)

Spring compressions in a slinky

• What about light?

– The most natural assumption would be that it requires a medium, too!

• This mysterious medium for

light was called ETHER

• What would its properties be?

– We see light from distant starts,

so ether must permeate the

whole universe

– Must be very tenuous, or else

the friction would have stopped

the Earth long ago

Aether would be like a ghostly wind blowing through the Universe!

• Michelson and Morley attempted to detect ether by measuring the speed of light in two different directions: “upwind” and “downwind” with respect to ether.

Measuring the speed of light the Newtonian way

17

at rest with respect to the ether. The Galilean velocity transformation equation wasexpected to hold for observations of light made by an observer in any frame moving atspeed v relative to the absolute ether frame. That is, if light travels along the x axis andan observer moves with velocity v along the x axis, the observer will measure the light tohave speed c ! v, depending on the directions of travel of the observer and the light.

Because the existence of a preferred, absolute ether frame would show that lightwas similar to other classical waves and that Newtonian ideas of an absolute frame weretrue, considerable importance was attached to establishing the existence of the etherframe. Prior to the late 1800s, experiments involving light traveling in media moving atthe highest laboratory speeds attainable at that time were not capable of detectingdifferences as small as that between c and c ! v. Starting in about 1880, scientistsdecided to use the Earth as the moving frame in an attempt to improve their chancesof detecting these small changes in the speed of light.

As observers fixed on the Earth, we can take the view that we are stationaryand that the absolute ether frame containing the medium for light propagation movespast us with speed v. Determining the speed of light under these circumstances isjust like determining the speed of an aircraft traveling in a moving air current, orwind; consequently, we speak of an “ether wind” blowing through our apparatus fixedto the Earth.

A direct method for detecting an ether wind would use an apparatus fixed to theEarth to measure the ether wind’s influence on the speed of light. If v is the speed ofthe ether relative to the Earth, then light should have its maximum speed c " v whenpropagating downwind, as in Figure 39.3a. Likewise, the speed of light should have itsminimum value c # v when the light is propagating upwind, as in Figure 39.3b, and anintermediate value (c2 # v2)1/2 in the direction perpendicular to the ether wind, as inFigure 39.3c. If the Sun is assumed to be at rest in the ether, then the velocity of theether wind would be equal to the orbital velocity of the Earth around the Sun, whichhas a magnitude of approximately 3 $ 104 m/s. Because c % 3 $ 108 m/s, it isnecessary to detect a change in speed of about 1 part in 104 for measurements in theupwind or downwind directions. However, while such a change is experimentallymeasurable, all attempts to detect such changes and establish the existence of theether wind (and hence the absolute frame) proved futile! We explore the classicexperimental search for the ether in Section 39.2.

The principle of Galilean relativity refers only to the laws of mechanics. If it isassumed that the laws of electricity and magnetism are the same in all inertial frames, aparadox concerning the speed of light immediately arises. We can understand this byrecognizing that Maxwell’s equations seem to imply that the speed of light always hasthe fixed value 3.00 $ 108 m/s in all inertial frames, a result in direct contradiction towhat is expected based on the Galilean velocity transformation equation. According toGalilean relativity, the speed of light should not be the same in all inertial frames.

To resolve this contradiction in theories, we must conclude that either (1) the lawsof electricity and magnetism are not the same in all inertial frames or (2) the Galileanvelocity transformation equation is incorrect. If we assume the first alternative, then apreferred reference frame in which the speed of light has the value c must exist and themeasured speed must be greater or less than this value in any other reference frame, inaccordance with the Galilean velocity transformation equation. If we assume the secondalternative, then we are forced to abandon the notions of absolute time and absolutelength that form the basis of the Galilean space–time transformation equations.

39.2 The Michelson–Morley Experiment

The most famous experiment designed to detect small changes in the speed of light wasfirst performed in 1881 by Albert A. Michelson (see Section 37.7) and later repeatedunder various conditions by Michelson and Edward W. Morley (1838–1923). We state atthe outset that the outcome of the experiment contradicted the ether hypothesis.


c + v

(a) Downwind

(b) Upwind

(c) Across wind

vc

v

c – v

c

v

cc 2 – v 2!

Figure 39.3 If the velocity of theether wind relative to the Earthis v and the velocity of light relativeto the ether is c, then the speedof light relative to the Earth is(a) c " v in the downwinddirection, (b) c # v in the upwinddirection, and (c) (c2 # v2)1/2

in the direction perpendicular tothe wind.










c + v

(a) Downwind

(b) Upwind

(c) Across wind

vc

v

c – v

c

v

cc 2 – v 2!












c + v

(a) Downwind

(b) Upwind

(c) Across wind

vc

v

c – v

c

v

cc 2 – v 2!



Michelson-Morley Experiment• Michelson and Morley

used a very sensitive interferometer to detect the difference in the speed of light depending on the direction in which it travels.

• NO such dependence was found!

So NO aether?

Or an error in the measurements?

Another problem...

• Maxwell's equations introduce the speed of light, c = 3 x 108 m/s

– But they don't say with respect to what this velocity is to be measured!

• So what can we conclude?

–That light must move at speed c in all reference frames?

• But this contradicts Newtonian mechanics!

• If electromagnetism is governed by the same rules as Newtonian mechanics, the “addition of velocities” rule should also apply.

cBut what if uy’ = c and v = c?

K

z

x

y

K' v

x'

y'

z'

Houston, we've got a problem…

• Suppose that addition of velocities does work for light, too. Then imagine the following experiment:

v

I think the speed of light is v-c!

• If the car is moving with speed v, and light from the rear of the car is moving with speed c, we should measure speed of light = v - c.

– Then if we know c (and we do from other experiments), we should derive v.

• Numerous experiments tried to measure the speed of Earth based on this general idea -- with NO results whatsoever!!! Speed of light seemed always to be the same!

Maybe that’s fine?

What do we know so far?

• Newton's mechanics based on Galileo's relativity–All laws of mechanics are the same in different

inertial reference frames (frames moving with a constant speed along a straight line relative to one another)

• Maxwell's electrodynamics

–There is a fundamental constant of nature, the speed of light (c) that is always the same

• The fact that there is such a constant is inconsistent with Newton’s mechanics!

• Einstein was faced with the following choices:– Maxwell's equations are wrong. The

right ones would be consistent with Galileo's relativity

• That's unlikely. Maxwell's theory has been so well confirmed by numerous experiments!

– Galileo's relativity was wrong when applied to electromagnetic phenomena. There was a special reference frame for light.

• This was more likely, but it assumed light was like any other waves and required a medium for propagation. That medium was not found!

– There is a relativity principle for both mechanical and electromagnetic phenomena, but it's not Galileo's relativity.

Special Theory of Relativity• It required the genius and the

courage of Einstein to accept the third alternative. His special relativity is based on two postulates:

• All laws of nature are the same in all inertial frames

– This is really Galileo's relativity

• The speed of light is independent of the motion of its source

– This simple statement requires a truly radical re-thinking about the nature of space and time!

Albert Einstein 1879-1955

Consequences of Einstein’s Special Theory of Relativity

• Relativity of Simultaneity

• Relativity of Time

• Relativity of Length

• Mass - Energy Equivalence

25

Relativity of Simultaneity

26

coordinates and one time coordinate. Observers in different inertial frames willdescribe the same event with coordinates that have different values.

As we examine some of the consequences of relativity in the remainder of thissection, we restrict our discussion to the concepts of simultaneity, time intervals, andlengths, all three of which are quite different in relativistic mechanics from what theyare in Newtonian mechanics. For example, in relativistic mechanics the distancebetween two points and the time interval between two events depend on the frame ofreference in which they are measured. That is, in relativistic mechanics there is nosuch thing as an absolute length or absolute time interval. Furthermore, eventsat different locations that are observed to occur simultaneously in one frameare not necessarily observed to be simultaneous in another frame movinguniformly with respect to the first.

Simultaneity and the Relativity of Time

A basic premise of Newtonian mechanics is that a universal time scale exists that is thesame for all observers. In fact, Newton wrote that “Absolute, true, and mathematicaltime, of itself, and from its own nature, flows equably without relation to anythingexternal.” Thus, Newton and his followers simply took simultaneity for granted. In hisspecial theory of relativity, Einstein abandoned this assumption.

Einstein devised the following thought experiment to illustrate this point. A boxcarmoves with uniform velocity, and two lightning bolts strike its ends, as illustrated inFigure 39.5a, leaving marks on the boxcar and on the ground. The marks on theboxcar are labeled A! and B!, and those on the ground are labeled A and B. Anobserver O! moving with the boxcar is midway between A! and B!, and a groundobserver O is midway between A and B. The events recorded by the observers are thestriking of the boxcar by the two lightning bolts.

The light signals emitted from A and B at the instant at which the two bolts strikereach observer O at the same time, as indicated in Figure 39.5b. This observer realizesthat the signals have traveled at the same speed over equal distances, and so rightlyconcludes that the events at A and B occurred simultaneously. Now consider the sameevents as viewed by observer O!. By the time the signals have reached observer O,observer O! has moved as indicated in Figure 39.5b. Thus, the signal from B ! hasalready swept past O!, but the signal from A! has not yet reached O!. In other words, O!sees the signal from B! before seeing the signal from A!. According to Einstein, the twoobservers must find that light travels at the same speed. Therefore, observer O! concludesthat the lightning strikes the front of the boxcar before it strikes the back.

This thought experiment clearly demonstrates that the two events that appearto be simultaneous to observer O do not appear to be simultaneous to observer O!.


v

A' B'

OA B

(a)

v

A' B'

OA B

(b)

O'

O'

Figure 39.5 (a) Two lightning bolts strike the ends of a moving boxcar. (b) The eventsappear to be simultaneous to the stationary observer O, standing midway between Aand B. The events do not appear to be simultaneous to observer O!, who claims thatthe front of the car is struck before the rear. Note that in (b) the leftward-traveling lightsignal has already passed O! but the rightward-traveling signal has not yet reached O!.

! PITFALL PREVENTION 39.2 Who’s Right?You might wonder which observerin Fig. 39.5 is correct concerningthe two lightning strikes. Both arecorrect, because the principle ofrelativity states that there is nopreferred inertial frame of reference.Although the two observers reachdifferent conclusions, both arecorrect in their own referenceframe because the concept ofsimultaneity is not absolute. This,in fact, is the central point ofrelativity—any uniformly movingframe of reference can be used todescribe events and do physics.

coordinates and one time coordinate. Observers in different inertial frames willdescribe the same event with coordinates that have different values.

As we examine some of the consequences of relativity in the remainder of thissection, we restrict our discussion to the concepts of simultaneity, time intervals, andlengths, all three of which are quite different in relativistic mechanics from what theyare in Newtonian mechanics. For example, in relativistic mechanics the distancebetween two points and the time interval between two events depend on the frame ofreference in which they are measured. That is, in relativistic mechanics there is nosuch thing as an absolute length or absolute time interval. Furthermore, eventsat different locations that are observed to occur simultaneously in one frameare not necessarily observed to be simultaneous in another frame movinguniformly with respect to the first.

Simultaneity and the Relativity of Time

A basic premise of Newtonian mechanics is that a universal time scale exists that is thesame for all observers. In fact, Newton wrote that “Absolute, true, and mathematicaltime, of itself, and from its own nature, flows equably without relation to anythingexternal.” Thus, Newton and his followers simply took simultaneity for granted. In hisspecial theory of relativity, Einstein abandoned this assumption.

Einstein devised the following thought experiment to illustrate this point. A boxcarmoves with uniform velocity, and two lightning bolts strike its ends, as illustrated inFigure 39.5a, leaving marks on the boxcar and on the ground. The marks on theboxcar are labeled A! and B!, and those on the ground are labeled A and B. Anobserver O! moving with the boxcar is midway between A! and B!, and a groundobserver O is midway between A and B. The events recorded by the observers are thestriking of the boxcar by the two lightning bolts.

The light signals emitted from A and B at the instant at which the two bolts strikereach observer O at the same time, as indicated in Figure 39.5b. This observer realizesthat the signals have traveled at the same speed over equal distances, and so rightlyconcludes that the events at A and B occurred simultaneously. Now consider the sameevents as viewed by observer O!. By the time the signals have reached observer O,observer O! has moved as indicated in Figure 39.5b. Thus, the signal from B ! hasalready swept past O!, but the signal from A! has not yet reached O!. In other words, O!sees the signal from B! before seeing the signal from A!. According to Einstein, the twoobservers must find that light travels at the same speed. Therefore, observer O! concludesthat the lightning strikes the front of the boxcar before it strikes the back.

This thought experiment clearly demonstrates that the two events that appearto be simultaneous to observer O do not appear to be simultaneous to observer O!.


v

A' B'

OA B

(a)

v

A' B'

OA B

(b)

O'

O'

Figure 39.5 (a) Two lightning bolts strike the ends of a moving boxcar. (b) The eventsappear to be simultaneous to the stationary observer O, standing midway between Aand B. The events do not appear to be simultaneous to observer O!, who claims thatthe front of the car is struck before the rear. Note that in (b) the leftward-traveling lightsignal has already passed O! but the rightward-traveling signal has not yet reached O!.

! PITFALL PREVENTION 39.2 Who’s Right?You might wonder which observerin Fig. 39.5 is correct concerningthe two lightning strikes. Both arecorrect, because the principle ofrelativity states that there is nopreferred inertial frame of reference.Although the two observers reachdifferent conclusions, both arecorrect in their own referenceframe because the concept ofsimultaneity is not absolute. This,in fact, is the central point ofrelativity—any uniformly movingframe of reference can be used todescribe events and do physics.

Two events that are simultaneous in one frame of reference are

in general not simultaneous in a

second 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

Relativity of Simultaneity

27

Time Dilation

28

S E C T I O N 3 9 . 4 • Consequences of the Special Theory of Relativity 1253

At the Active Figures linkat http://www.pse6.com, youcan observe the bouncing ofthe light pulse for variousspeeds of the train.

Einstein’s thought experiment demonstrates that two observers can disagree on thesimultaneity of two events. This disagreement, however, depends on the transittime of light to the observers and, therefore, does not demonstrate the deepermeaning of relativity. In relativistic analyses of high-speed situations, relativity showsthat simultaneity is relative even when the transit time is subtracted out. In fact, all ofthe relativistic effects that we will discuss from here on will assume that we are ignoringdifferences caused by the transit time of light to the observers.

Time Dilation

We can illustrate the fact that observers in different inertial frames can measuredifferent time intervals between a pair of events by considering a vehicle moving to theright with a speed v, such as the boxcar shown in Figure 39.6a. A mirror is fixed to theceiling of the vehicle, and observer O! at rest in the frame attached to the vehicle holdsa flashlight a distance d below the mirror. At some instant, the flashlight emits a pulse oflight directed toward the mirror (event 1), and at some later time after reflecting fromthe mirror, the pulse arrives back at the flashlight (event 2). Observer O! carries a clockand uses it to measure the time interval "tp between these two events. (The subscript pstands for proper , as we shall see in a moment.) Because the light pulse has a speed c, thetime interval required for the pulse to travel from O! to the mirror and back is

(39.5)

Now consider the same pair of events as viewed by observer O in a second frame, asshown in Figure 39.6b. According to this observer, the mirror and flashlight are moving tothe right with a speed v, and as a result the sequence of events appears entirely different.By the time the light from the flashlight reaches the mirror, the mirror has moved to theright a distance v "t/2, where "t is the time interval required for the light to travel fromO! to the mirror and back to O! as measured by O. In other words, O concludes that,because of the motion of the vehicle, if the light is to hit the mirror, it must leave the

"tp #distance traveled

speed#

2dc

two events that are simultaneous in one reference frame are in general notsimultaneous in a second frame moving relative to the first. That is, simultaneity isnot an absolute concept but rather one that depends on the state of motion ofthe observer.

v

O ! O ! O !

x !O

y !

v"t

(b)

d

v"t2

c"t2

(c)

vMirror

y !

x !

d

O !

(a)

Active Figure 39.6 (a) A mirror is fixed to a moving vehicle, and a light pulse is sentout by observer O! at rest in the vehicle. (b) Relative to a stationary observer O standingalongside the vehicle, the mirror and O! move with a speed v . Note that what observerO measures for the distance the pulse travels is greater than 2d . (c) The right trianglefor calculating the relationship between "t and "tp .

In other words,




Time Dilation


(39.5)



speed#

2dc


v

O ! O ! O !

x !O

y !

v"t

(b)

d

v"t2

c"t2

(c)

vMirror

y !

x !

d

O !

(a)


In other words,




Time Dilation


(39.5)



speed#

2dc


v

O ! O ! O !

x !O

y !

v"t

(b)

d

v"t2

c"t2

(c)

vMirror

y !

x !

d

O !

(a)


In other words,




Time Dilation


(39.5)



speed#

2dc


v

O ! O ! O !

x !O

y !

v"t

(b)

d

v"t2

c"t2

(c)

vMirror

y !

x !

d

O !

(a)


In other words,

flashlight at an angle with respect to the vertical direction. Comparing Figure 39.6a and b,we see that the light must travel farther in (b) than in (a). (Note that neither observer“knows” that he or she is moving. Each is at rest in his or her own inertial frame.)

According to the second postulate of the special theory of relativity, both observersmust measure c for the speed of light. Because the light travels farther according to O,it follows that the time interval !t measured by O is longer than the time interval !tpmeasured by O". To obtain a relationship between these two time intervals, it is conve-nient to use the right triangle shown in Figure 39.6c. The Pythagorean theorem gives

Solving for !t gives

(39.6)

Because !tp # 2d/c, we can express this result as

(39.7)

where

(39.8)

Because $ is always greater than unity, this result says that the time interval !tmeasured by an observer moving with respect to a clock is longer than the timeinterval !tp measured by an observer at rest with respect to the clock. This effectis known as time dilation.

We can see that time dilation is not observed in our everyday lives by consideringthe factor $. This factor deviates significantly from a value of 1 only for very highspeeds, as shown in Figure 39.7 and Table 39.1. For example, for a speed of 0.1c, thevalue of $ is 1.005. Thus, there is a time dilation of only 0.5% at one-tenth the speed oflight. Speeds that we encounter on an everyday basis are far slower than this, so we donot see time dilation in normal situations.

The time interval !tp in Equations 39.5 and 39.7 is called the proper timeinterval. (In German, Einstein used the term Eigenzeit , which means “own-time.”) In

$ #1

!1 %v2

c 2

!t #!tp

!1 %v2

c 2

# $ !tp

!t #2d

!c 2 % v2

#2d

c !1 %v2

c 2

! c !t2 "2

# ! v !t2 "2

& d2


Time dilation

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

10

15

20

5

1

"

v(108 m/s )

Figure 39.7 Graph of $ versus v. As the speed approaches that of light, $ increasesrapidly.

v/c "

0.001 0 1.000 000 50.010 1.000 050.10 1.0050.20 1.0210.30 1.0480.40 1.0910.50 1.1550.60 1.2500.70 1.4000.80 1.6670.90 2.2940.92 2.5520.94 2.9310.96 3.5710.98 5.0250.99 7.0890.995 10.010.999 22.37

Approximate Values for "at Various Speeds

Table 39.1




(39.6)


(39.7)

where

(39.8)




$ #1

!1 %v2

c 2

!t #!tp

!1 %v2

c 2

# $ !tp

!t #2d

!c 2 % v2

#2d

c !1 %v2

c 2

! c !t2 "2

# ! v !t2 "2

& d2


Time dilation

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

10

15

20

5

1

"

v(108 m/s )


v/c "

0.001 0 1.000 000 50.010 1.000 050.10 1.0050.20 1.0210.30 1.0480.40 1.0910.50 1.1550.60 1.2500.70 1.4000.80 1.6670.90 2.2940.92 2.5520.94 2.9310.96 3.5710.98 5.0250.99 7.0890.995 10.010.999 22.37


Table 39.1

Time Dilation




(39.6)


(39.7)

where

(39.8)




$ #1

!1 %v2

c 2

!t #!tp

!1 %v2

c 2

# $ !tp

!t #2d

!c 2 % v2

#2d

c !1 %v2

c 2

! c !t2 "2

# ! v !t2 "2

& d2


Time dilation

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

10

15

20

5

1

"

v(108 m/s )


v/c "

0.001 0 1.000 000 50.010 1.000 050.10 1.0050.20 1.0210.30 1.0480.40 1.0910.50 1.1550.60 1.2500.70 1.4000.80 1.6670.90 2.2940.92 2.5520.94 2.9310.96 3.5710.98 5.0250.99 7.0890.995 10.010.999 22.37


Table 39.1




(39.6)


(39.7)

where

(39.8)




$ #1

!1 %v2

c 2

!t #!tp

!1 %v2

c 2

# $ !tp

!t #2d

!c 2 % v2

#2d

c !1 %v2

c 2

! c !t2 "2

# ! v !t2 "2

& d2


Time dilation

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

10

15

20

5

1

"

v(108 m/s )


v/c "

0.001 0 1.000 000 50.010 1.000 050.10 1.0050.20 1.0210.30 1.0480.40 1.0910.50 1.1550.60 1.2500.70 1.4000.80 1.6670.90 2.2940.92 2.5520.94 2.9310.96 3.5710.98 5.0250.99 7.0890.995 10.010.999 22.37


Table 39.1

Because ! is always greater than unity, this result says that the time interval "t measured by an observer moving with respect to a clock is longer than the time interval "tp measured by an observer at rest with respect to the clock.

proper time interval !tp is the time interval between two events measured by an observer who sees the events occur at the same point in space

• Time in a moving system slows down comparing to a stationary system!

• all physical processes, including chemical and biological ones, are measured to slow down when those processes occur in a frame moving with respect to the observer.

• For example, the heartbeat of an astronaut moving through space would keep time with a clock inside the spacecraft. Both the astronaut’s clock and heartbeat would be measured to slow down according to an observer on Earth comparing time intervals with his own clock (although the astronaut would have no sensation of life slowing down in the spacecraft).

No time dilation

8 m

300 m

With time dilation

!+

!+

What does this mean?

Muons


high in the atmosphere where they are produced. However, experiments show that alarge number of muons do reach the surface. The phenomenon of time dilationexplains this effect. As measured by an observer on Earth, the muons have a dilatedlifetime equal to ! "tp. For example, for v # 0.99c, ! ! 7.1 and ! "tp ! 16 $s. Hence,the average distance traveled by the muons in this time as measured by an observeron Earth is approximately (0.99)(3.0 % 108 m/s)(16 % 10&6 s) ! 4.8 % 103 m, as indi-cated in Figure 39.8b.

In 1976, at the laboratory of the European Council for Nuclear Research (CERN)in Geneva, muons injected into a large storage ring reached speeds of approximately0.999 4c. Electrons produced by the decaying muons were detected by counters aroundthe ring, enabling scientists to measure the decay rate and hence the muon lifetime.The lifetime of the moving muons was measured to be approximately 30 times as longas that of the stationary muon (Fig. 39.9), in agreement with the prediction of relativityto within two parts in a thousand.

Muonat rest

Muon movingat 0.999 4c

50 100 150

0.5

1.0

Frac

tion

of m

uons

rem

aini

ng

t( s)µ

Figure 39.9 Decay curves for muonsat rest and for muons traveling at aspeed of 0.999 4c.

Example 39.1 What Is the Period of the Pendulum?

The period of a pendulum is measured to be 3.00 s in thereference frame of the pendulum. What is the period whenmeasured by an observer moving at a speed of 0.950c rela-tive to the pendulum?

Solution To conceptualize this problem, let us changeframes of reference. Instead of the observer moving at 0.950c,we can take the equivalent point of view that the observer is atrest and the pendulum is moving at 0.950c past the stationaryobserver. Hence, the pendulum is an example of a clockmoving at high speed with respect to an observer and we cancategorize this problem as one involving time dilation.

To analyze the problem, note that the proper timeinterval, measured in the rest frame of the pendulum, is"tp # 3.00 s. Because a clock moving with respect to anobserver is measured to run more slowly than a stationaryclock by a factor !, Equation 39.7 gives

To finalize this problem, we see that indeed a movingpendulum is measured to take longer to complete a period

9.60 s# (3.20)(3.00 s) #

"t # ! "tp #1

!1 &(0.950c)2

c2

"tp #1

!1 & 0.902 "tp

than a pendulum at rest does. The period increases by afactor of ! # 3.20. We see that this is consistent with Table39.1, where this value lies between those for ! for v/c # 0.94and v/c # 0.96.

What If? What if we increase the speed of the observer by5.00%? Does the dilated time interval increase by 5.00%?

Answer Based on the highly nonlinear behavior of ! as afunction of v in Figure 39.7, we would guess that theincrease in "t would be different from 5.00%. Increasing vby 5.00% gives us

vnew # (1.050 0)(0.950c) # 0.997 5c

(Because ! varies so rapidly with v when v is this large, we willkeep one additional significant figure until the final answer.)If we perform the time dilation calculation again, we find that

Thus, the 5.00% increase in speed has caused over a 300%increase in the dilated time!

# (14.15)(3.00 s) # 42.5 s

"t # ! "tp #1

!1 &(0.997 5c)2

c 2

"tp #1

!1 & 0.995 0 "tp

general, the proper time interval is the time interval between two eventsmeasured by an observer who sees the events occur at the same point in space.

If a clock is moving with respect to you, the time interval between ticks of themoving clock is observed to be longer than the time interval between ticks of anidentical clock in your reference frame. Thus, it is often said that a moving clock ismeasured to run more slowly than a clock in your reference frame by a factor !. Thisis true for mechanical clocks as well as for the light clock just described. We cangeneralize this result by stating that all physical processes, including chemical andbiological ones, are measured to slow down when those processes occur in a framemoving with respect to the observer. For example, the heartbeat of an astronaut moving through space would keep time with a clock inside the spacecraft. Both the astronaut’s clock and heartbeat would be measured to slow down according to an observer on Earth comparing time intervals with his own clock (although the astronautwould have no sensation of life slowing down in the spacecraft).


Quick Quiz 39.3 Suppose the observer O" on the train in Figure 39.6 aims herflashlight at the far wall of the boxcar and turns it on and off, sending a pulse of lighttoward the far wall. Both O" and O measure the time interval between when the pulseleaves the flashlight and it hits the far wall. Which observer measures the proper timeinterval between these two events? (a) O" (b) O (c) both observers (d) neither observer.

Quick Quiz 39.4 A crew watches a movie that is two hours long in a space-craft that is moving at high speed through space. Will an Earthbound observer, who iswatching the movie through a powerful telescope, measure the duration of the movieto be (a) longer than, (b) shorter than, or (c) equal to two hours?

! PITFALL PREVENTION 39.3 The Proper Time

IntervalIt is very important in relativisticcalculations to correctly identifythe observer who measures theproper time interval. The propertime interval between two eventsis always the time interval mea-sured by an observer for whomthe two events take place at thesame position.

Strange as it may seem, time dilation is a verifiable phenomenon. An experimentreported by Hafele and Keating provided direct evidence of time dilation.4 Timeintervals measured with four cesium atomic clocks in jet flight were compared withtime intervals measured by Earth-based reference atomic clocks. In order to comparethese results with theory, many factors had to be considered, including periods ofspeeding up and slowing down relative to the Earth, variations in direction of travel,and the fact that the gravitational field experienced by the flying clocks was weakerthan that experienced by the Earth-based clock. The results were in good agreementwith the predictions of the special theory of relativity and can be explained in terms ofthe relative motion between the Earth and the jet aircraft. In their paper, Hafele andKeating stated that “Relative to the atomic time scale of the U.S. Naval Observatory, theflying clocks lost 59 # 10 ns during the eastward trip and gained 273 # 7 ns duringthe westward trip. . . . These results provide an unambiguous empirical resolution ofthe famous clock paradox with macroscopic clocks.”

Another interesting example of time dilation involves the observation of muons,unstable elementary particles that have a charge equal to that of the electron and amass 207 times that of the electron. (We will study the muon and other particles inChapter 46.) Muons can be produced by the collision of cosmic radiation with atomshigh in the atmosphere. Slow-moving muons in the laboratory have a lifetime which ismeasured to be the proper time interval $tp % 2.2 &s. If we assume that the speed ofatmospheric muons is close to the speed of light, we find that these particles can travela distance of approximately (3.0 ' 108 m/s)(2.2 ' 10(6 s) ! 6.6 ' 102 m before theydecay (Fig. 39.8a). Hence, they are unlikely to reach the surface of the Earth from

4 J. C. Hafele and R. E. Keating, “Around the World Atomic Clocks: Relativistic Time GainsObserved,” Science, 177:168, 1972.

(a)

! 4.8 " 103 m

(b)

! 6.6 " 102 m

Muon is created

Muon decays

Muon is created

Muon decays

Figure 39.8 (a) Without relativis-tic considerations, muons createdin the atmosphere and travelingdownward with a speed of 0.99ctravel only about 6.6 ' 102 mbefore decaying with an averagelifetime of 2.2 &s. Thus, very fewmuons reach the surface of theEarth. (b) With relativisticconsiderations, the muon’s lifetimeis dilated according to an observeron Earth. As a result, according tothis observer, the muon can travelabout 4.8 ' 103 m before decaying.This results in many of themarriving at the surface.











(a)

! 4.8 " 103 m

(b)

! 6.6 " 102 m

Muon is created

Muon decays

Muon is created

Muon decays












(a)

! 4.8 " 103 m

(b)

! 6.6 " 102 m

Muon is created

Muon decays

Muon is created

Muon decays


The twin "paradox"

• On their 20th birthday, Jane gets her space ship driver's license and takes off from Earth at 0.95c. Her twin brother Joe stays home.

• Upon his return, Jane is shocked to discover that Joe has aged 42 years and is now 62 years old. Jane, on the other hand, has aged only 13 years.

• The "paradox" lies in the fact that from Jane's point of view, it was Joe who traveled. Shouldn’t he be younger, then?

x

ct

Length ContractionFor example, suppose that a meter stick moves past a stationary Earth observer with

speed v, as in Figure 39.11. The length of the stick as measured by an observer in a frameattached to the stick is the proper length Lp shown in Figure 39.11a. The length of thestick L measured by the Earth observer is shorter than Lp by the factor (1 ! v2/c2)1/2.Note that length contraction takes place only along the direction of motion.

The proper length and the proper time interval are defined differently. The properlength is measured by an observer for whom the end points of the length remain fixed inspace. The proper time interval is measured by someone for whom the two events takeplace at the same position in space. As an example of this point, let us return to thedecaying muons moving at speeds close to the speed of light. An observer in the muon’sreference frame would measure the proper lifetime, while an Earth-based observer wouldmeasure the proper length (the distance from creation to decay in Figure 39.8). In themuon’s reference frame, there is no time dilation but the distance of travel to the surfaceis observed to be shorter when measured in this frame. Likewise, in the Earth observer’sreference frame, there is time dilation, but the distance of travel is measured to be theproper length. Thus, when calculations on the muon are performed in both frames, theoutcome of the experiment in one frame is the same as the outcome in the other frame—more muons reach the surface than would be predicted without relativistic effects.


Lp

y!

O !(a)

x !

Ly

O(b)

x

v

Active Figure 39.11 (a) A meterstick measured by an observer in aframe attached to the stick (that is,both have the same velocity) has itsproper length Lp. (b) The stickmeasured by an observer in a framein which the stick has a velocity vrelative to the frame is measured tobe shorter than its proper lengthLp by a factor (1 ! v2/c2)1/2.

At the Active Figures linkat http://www.pse6.com, youcan view the meter stick fromthe points of view of twoobservers to compare themeasured length of the stick.

Quick Quiz 39.6 You are packing for a trip to another star. During thejourney, you will be traveling at 0.99c. You are trying to decide whether you should buysmaller sizes of your clothing, because you will be thinner on your trip, due to lengthcontraction. Also, you are considering saving money by reserving a smaller cabin to sleepin, because you will be shorter when you lie down. Should you (a) buy smaller sizes ofclothing, (b) reserve a smaller cabin, (c) do neither of these, or (d) do both ofthese?

Quick Quiz 39.7 You are observing a spacecraft moving away from you. Youmeasure it to be shorter than when it was at rest on the ground next to you. You alsosee a clock through the spacecraft window, and you observe that the passage of time onthe clock is measured to be slower than that of the watch on your wrist. Compared towhen the spacecraft was on the ground, what do you measure if the spacecraft turnsaround and comes toward you at the same speed? (a) The spacecraft is measured to belonger and the clock runs faster. (b) The spacecraft is measured to be longer and theclock runs slower. (c) The spacecraft is measured to be shorter and the clock runsfaster. (d) The spacecraft is measured to be shorter and the clock runs slower.

Space–Time Graphs

It is sometimes helpful to make a space–time graph, in which ct is the ordinate andposition x is the abscissa. The twin paradox is displayed in such a graph in Figure 39.12

World-line of Speedo

World-line of light beamWorld-lineof Goslo

ct

x

Figure 39.12 The twin paradox on aspace–time graph. The twin who stays onthe Earth has a world-line along the ct axis.The path of the traveling twin throughspace–time is represented by a world-linethat changes direction.

For example, suppose that a meter stick moves past a stationary Earth observer withspeed v, as in Figure 39.11. The length of the stick as measured by an observer in a frameattached to the stick is the proper length Lp shown in Figure 39.11a. The length of thestick L measured by the Earth observer is shorter than Lp by the factor (1 ! v2/c2)1/2.Note that length contraction takes place only along the direction of motion.

The proper length and the proper time interval are defined differently. The properlength is measured by an observer for whom the end points of the length remain fixed inspace. The proper time interval is measured by someone for whom the two events takeplace at the same position in space. As an example of this point, let us return to thedecaying muons moving at speeds close to the speed of light. An observer in the muon’sreference frame would measure the proper lifetime, while an Earth-based observer wouldmeasure the proper length (the distance from creation to decay in Figure 39.8). In themuon’s reference frame, there is no time dilation but the distance of travel to the surfaceis observed to be shorter when measured in this frame. Likewise, in the Earth observer’sreference frame, there is time dilation, but the distance of travel is measured to be theproper length. Thus, when calculations on the muon are performed in both frames, theoutcome of the experiment in one frame is the same as the outcome in the other frame—more muons reach the surface than would be predicted without relativistic effects.


Lp

y!

O !(a)

x !

Ly

O(b)

x

v

Active Figure 39.11 (a) A meterstick measured by an observer in aframe attached to the stick (that is,both have the same velocity) has itsproper length Lp. (b) The stickmeasured by an observer in a framein which the stick has a velocity vrelative to the frame is measured tobe shorter than its proper lengthLp by a factor (1 ! v2/c2)1/2.

At the Active Figures linkat http://www.pse6.com, youcan view the meter stick fromthe points of view of twoobservers to compare themeasured length of the stick.

Quick Quiz 39.6 You are packing for a trip to another star. During thejourney, you will be traveling at 0.99c. You are trying to decide whether you should buysmaller sizes of your clothing, because you will be thinner on your trip, due to lengthcontraction. Also, you are considering saving money by reserving a smaller cabin to sleepin, because you will be shorter when you lie down. Should you (a) buy smaller sizes ofclothing, (b) reserve a smaller cabin, (c) do neither of these, or (d) do both ofthese?

Quick Quiz 39.7 You are observing a spacecraft moving away from you. Youmeasure it to be shorter than when it was at rest on the ground next to you. You alsosee a clock through the spacecraft window, and you observe that the passage of time onthe clock is measured to be slower than that of the watch on your wrist. Compared towhen the spacecraft was on the ground, what do you measure if the spacecraft turnsaround and comes toward you at the same speed? (a) The spacecraft is measured to belonger and the clock runs faster. (b) The spacecraft is measured to be longer and theclock runs slower. (c) The spacecraft is measured to be shorter and the clock runsfaster. (d) The spacecraft is measured to be shorter and the clock runs slower.

Space–Time Graphs

It is sometimes helpful to make a space–time graph, in which ct is the ordinate andposition x is the abscissa. The twin paradox is displayed in such a graph in Figure 39.12

World-line of Speedo

World-line of light beamWorld-lineof Goslo

ct

x

Figure 39.12 The twin paradox on aspace–time graph. The twin who stays onthe Earth has a world-line along the ct axis.The path of the traveling twin throughspace–time is represented by a world-linethat changes direction.


contradiction due to the apparent symmetry of the observations. Which twin hasdeveloped signs of excess aging?

The situation in our current problem is actually not symmetrical. To resolve thisapparent paradox, recall that the special theory of relativity describes observationsmade in inertial frames of reference moving relative to each other. Speedo,the space traveler, must experience a series of accelerations during his journeybecause he must fire his rocket engines to slow down and start moving backtoward Earth. As a result, his speed is not always uniform, and consequently heis not in an inertial frame. Therefore, there is no paradox—only Goslo, who isalways in a single inertial frame, can make correct predictions based on specialrelativity. During each passing year noted by Goslo, slightly less than 4 monthselapses for Speedo.

Only Goslo, who is in a single inertial frame, can apply the simple time-dilationformula to Speedo’s trip. Thus, Goslo finds that instead of aging 42 yr, Speedo agesonly (1 ! v2/c2)1/2(42 yr) " 13 yr. Thus, according to Goslo, Speedo spends 6.5 yrtraveling to Planet X and 6.5 yr returning, for a total travel time of 13 yr, in agreementwith our earlier statement.

Quick Quiz 39.5 Suppose astronauts are paid according to the amountof time they spend traveling in space. After a long voyage traveling at a speedapproaching c, would a crew rather be paid according to (a) an Earth-based clock,(b) their spacecraft’s clock, or (c) either clock?

Length Contraction

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

Consider a spacecraft traveling with a speed v from one star to another. Thereare two observers: one on the Earth and the other in the spacecraft. The observerat rest on the Earth (and also assumed to be at rest with respect to the two stars)measures the distance between the stars to be the proper length Lp. According tothis observer, the time interval required for the spacecraft to complete the voyage is#t " Lp/v. The passages of the two stars by the spacecraft occur at the sameposition for the space traveler. Thus, the space traveler measures the proper timeinterval #tp. Because of time dilation, the proper time interval is related tothe Earth-measured time interval by #tp " #t/$. Because the space travelerreaches the second star in the time #tp, he or she concludes that the distance Lbetween the stars is

Because the proper length is Lp " v #t, we see that

(39.9)

where is a factor less than unity. If an object has a proper length Lpwhen it is measured by an observer at rest with respect to the object, then whenit moves with speed v in a direction parallel to its length, its length L ismeasured to be shorter according to .L " Lp !1 ! v

2/c 2 " Lp / !

!1 ! v 2/c

2

L "Lp

$" Lp !1 !

v 2

c 2

L " v #tp " v #t$

! PITFALL PREVENTION 39.4 The Proper LengthAs with the proper time interval,it is very important in relativisticcalculations to correctly identifythe observer who measures theproper length. The properlength between two points inspace is always the length mea-sured by an observer at rest withrespect to the points. Often theproper time interval and theproper length are not measuredby the same observer.

Length contraction




(39.6)


(39.7)

where

(39.8)




$ #1

!1 %v2

c 2

!t #!tp

!1 %v2

c 2

# $ !tp

!t #2d

!c 2 % v2

#2d

c !1 %v2

c 2

! c !t2 "2

# ! v !t2 "2

& d2


Time dilation

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

10

15

20

5

1

"

v(108 m/s )


v/c "

0.001 0 1.000 000 50.010 1.000 050.10 1.0050.20 1.0210.30 1.0480.40 1.0910.50 1.1550.60 1.2500.70 1.4000.80 1.6670.90 2.2940.92 2.5520.94 2.9310.96 3.5710.98 5.0250.99 7.0890.995 10.010.999 22.37


Table 39.1

• An observer moving along an object will find it shorter than it would be if the observer was standing still!

• Length contraction takes place only along the direction of motion

• So a space ship moving with 9/10 the speed of light along a lattice will find that the lattice is shorter than it was when the ship was at rest!

L

L'

What does this mean?

Traveling faster that light: a catch!

• Notice, however, that special relativity only precludes things from traveling faster than light in vacuum.

• In media (e.g., water or quartz) particles can travel faster than light can in that medium.

• This results in the so-called Cherenkov radiation, which is a very beautiful phenomenon widely used by physicists

BaBar experiment's DIRC:Detector of InternallyReflected Cherenkov Radiation

How does kinematics cope with relativity?

• It’s all very well to say that nothing can move faster than light, but Newtonian mechanics says that:

• So if we apply more and more force to an object, we can increase its speed more and more, and nothing tells us that it can’t move faster than light!

• This means that Newton’s second law must be modified in relativity. It becomes:

Mass m is no longer constant!

• It can be shown from first principles (conservation of energy and momentum) and relativity postulates that mass becomes dependent on velocity at large speeds:

• If velocity v is very small comparing to c, then this formula becomes

• Such considerations led Einstein to say that the mass of an object is equal to the total energy content divided by c2

m0 = rest mass

kinetic energy

faster means heavier!

The world’s most famous equation

• The equivalence of energy and mass has been confirmed by numerous experiments

m0 m0

An electron and an anti-electron (positron) of mass m0

collide and annihilate, and two photons, each with energy = m0c

2, come out!

Fermilab’s accelerators

Relativity and anti-matter

• Given the relativistic equations for energy, mass, and momentum, we can obtain the following relation:

• Note that this means that E has two solutions, one with plus and one with minus sign.

• But what does negative energy means? How can anything have negative energy?

• It was this kind of problem that eventually lead people to the idea of anti-matter.

Experimental verifications of special relativity

• Special relativity has been around for almost 100 years, and has brilliantly passed numerous experimental tests– Special relativity is a "good" theory in the sense that it

makes definite predictions that experimentalists are able to verify.

• Things like time dilation, length contraction, equivalence of mass and energy are no longer exotic words -- they are simple tools that particle physicists use in their calculations every day.

• One should remember that special relativity was not something that Einstein just came up with out of the blue -- it was based on existing experimental results.

Is there anything left of Newton’s laws, then?

• Einstein himself felt obliged to apologize to Newton for replacing Newton’s system with his own. He wrote in his Autobiographical notes:

• However, special relativity does not make Newton’s mechanics obsolete. In our slow-moving (comparing to the speed of light) world, Newton’s mechanics is a perfect approximation to work with.

Newton, forgive me. You found the only way which, in your age, was just about possible for a man of highest thought and creative power.

Conclusions• Special relativity revolutionized our understanding

of space and time

– There is no "space" and "time" by themselves -- there is only four-dimensional space-time!

• It describes the motion of particles close to the speed of light

– No massive particles can ever exceed the speed of light

– Massless particles move at the speed of light

• Special relativity has been extremely well-tested by experiment.

• At everyday speeds, Newton's mechanics is a good approximation to work with.

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into to modern physics › 2015 › 10 › ... · Modern Physics. Events and Frames of Reference An event is a physical happening that occurs at a certain place and time An observer