ch-9

Chapter 9 Relativity v1

m1

F 21 F 12 m2 v2

F I G U R E 9.1

Two particles interact with each other. According to Newton’s third law, we must have : : F 12 F 21.

S

S′

y

y′

v P (event)

x′

vt x O

F I G U R E 9.2

x

O′

x′

An event occurs at a point P. The event is seen by two observers O and O in inertial frames S and S , where S moves with a velocity : v relative to S.

M1

Arm 1

Ether wind v M0 Arm 2 M2

Telescope

Figure 9.3 In the Michelson interferometer, the ether theory claims that the time of travel for a light beam traveling from the beam splitter to mirror M1 and back will be different from that for a light beam traveling from the beam splitter to mirror M2 and back. The interferometer is sufficiently sensitive to detect this time difference.

v

O'

A' A

O (a)

F I G U R E 9.4

v

B' B

O'

A' A

O

B' B

(b)

(a) Two lightning bolts strike the ends of a moving boxcar. (b) The events appear to be simultaneous to the observer at O, who is standing on the ground midway between A and B. The events do not appear to be simultaneous to the observer O riding on the boxcar, who claims that the front of the car is struck before the rear. Note that the leftwardtraveling light signal from B has already passed observer O , but the rightward-traveling light signal from A has not yet reached O .

v

v Mirror

y′

y′

d O′

O′ x′

O′

O

c ∆t 2 x′

v∆t (a)

O′

(b)

d

v∆t 2 (c)

Figure 9.5 A mirror is fixed to a moving vehicle, and two observers measure the time interval between two events: the leaving of a light pulse from a flashlight and the arrival of the reflected light pulse back at the flashlight. (a) Observer O , riding on the vehicle, sees the light pulse travel a total distance of 2d and measures a time interval between the events of tp . (b) Observer O is standing on the Earth and sees the mirror and O move to the right with speed v. Observer O measures the distance that the light pulse travels to be greater than 2d and measures a time interval between the events of t. (c) The right triangle for calculating the relationship between t and tp .

Muon is created ≈ 6.6 × 102 m Muon decays

(a) Muon is created

≈ 4.8 × 103 m

Muon decays (b)

F I G U R E 9.6

(a) Without relativistic considerations, muons created in the atmosphere and traveling downward with a speed of 0.99c would travel only about 6.6 102 m before decaying with an average lifetime of 2.2 s. Therefore, very few muons would reach the surface of the Earth. (b) With relativistic considerations, the muon’s lifetime is dilated according to an observer on Earth. As a result, according to this observer, the muon can travel about 4.8 103 m before decaying, which results in many of them arriving at the surface.

Speedo

F I G U R E 9.7

(a)

Goslo

Speedo

(b)

Goslo

(a) As Speedo leaves his twin brother, Goslo, on the Earth, both are the same age. (b) When Speedo returns from his journey to Planet X, he is younger than Goslo.

y′ Lp

O′

x′

(a)

y L

O

(b)

v

x

Figure 9.8 (a) According to an observer in a frame attached to the meter stick (that is, both the stick and the frame have the same velocity), the stick is measured to have its proper length Lp. (b) According to an observer in a frame in which the meter stick has a velocity : v relative to the frame, the stick is measured to be shorter than the proper length Lp by a factor (1 v 2/c 2)1/2.

y

y Lp (a)

F I G U R E 9.9

v L (b)

(Example 9.3) (a) When the spacecraft is at rest, its shape is measured as shown. (b) The spacecraft is measured to have this shape when it moves to the right with a speed v. Note that only its x dimension is contracted in this case.

S

S′

y′

y

vt O

x x O′

F I G U R E 9.10

v P (event) Q (event) x′

∆x ′ ∆x x′

Events occur at points P and Q and are observed by an observer at rest in the S frame and another in the S frame, which is moving to the right with a speed v.

y

y′

S (attached to the Earth)

S ′ (attached to A) –0.850c 0.750c A

O

F I G U R E 9.11

x

O′

B x′

(Example 9.4) Two spacecraft A and B move in opposite directions.

Police officer at rest in S

0.75c

0.90c

F I G U R E 9.12 z y x Emily

David

(Example 9.5) David moves to the east with a speed 0.75c relative to the police officer, and Emily travels south at a speed 0.90c relative to the officer.

K/mc 2

Relativistic case Nonrelativistic case

2.0 1.5 1.0 0.5

0.5c

F I G U R E 9.13

1.0c

1.5c

2.0c

u

A graph comparing relativistic and nonrelativistic kinetic energy of a particle. The energies are plotted as a function of speed u. In the relativistic case, u is always less than c.

vel = 0 ael = 0

ael = + g ˆj

vel = 0

ael = + g ˆj

g = – g ˆj

ael = 0

g = – g ˆj

(a)

(b)

(c)

(d)

(a) The observer is at rest in an elevator in a uniform gravitational field g g ˆj , directed downward. The observer drops his briefcase, which moves downward with acceleration g. (b) The observer is in a region where gravity is negligible, but the elevator moves upward with an acceleration : a el g ˆj . The observer releases his briefcase, which moves downward (according to the observer) with acceleration g relative to the floor of the elevator. According to Einstein, the frames of reference in parts (a) and (b) are equivalent in every way. No local experiment can distinguish any difference between the two frames. (c) In the accelerating frame, a ray of light would appear to bend downward due to the acceleration. (d) If parts (a) and (b) are truly equivalent, as Einstein proposed, part (c) suggests that a ray of light would bend downward in a gravitational field.

F I G U R E 9.14

:

Apparent direction to star

F I G U R E 9.15

Deflection of starlight passing near the Sun. Because of this effect, the Sun or other remote objects can act as a gravitational lens. In his general theory of relativity, Einstein calculated that starlight just Earth grazing the Sun’s surface should be deflected by an angle of 1.75 seconds of arc.

Deflected path of light from star

1.75" Sun

Light from star (actual direction)

(a)

(b)

Figure Q9.16

2.00 m 30.0° Direction of motion

Figure P9.17

S

S′ v = 0.800c u = 0.900c x

Figure P9.20

x′

(Courtesy of Garmin Ltd.)

Figure P9.45 This Global Positioning System (GPS) receiver incorporates relativistically corrected time calculations in its analysis of signals it receives from orbiting satellites, allowing the unit to determine its position on the Earth’s surface to within a few meters. If these corrections were not made, the location error would be about 1 km.