chapter 28 – special relativity

Chapter 28 – Special Relativity

What’s special about special relativity?

• In 1905, Einstein’s first paper on relativity dealt only with inertial reference frames (constant velocity).

• 10 years later, he published a more encompassing theory of relativity that considered accelerated motion and it’s connection to gravity. This was a discussion of “general” relativity.

• His earlier work was special in that it discussed the “special case” of inertial reference frames.

28.1 Events and Inertial Reference Frames

An event is a physical “happening” that occurs at a certain place and time. Liftoff!Reference Frame: Coordinate system in which observers may make measurements in time and space. Anyone at rest in the reference frame (as well as a video recorder, or other data acquisition device) can be considered an observer.

28.1 Events and Inertial Reference Frames

An inertial reference frame is one in which Newton’s law of inertia is valid. An inertial reference frame can be moving, but it cannot be accelerating. The plane moves at constant velocity. In spite of its centripetal acceleration, the Earth is treated as an inertial reference frame, because the effects of its rotation and orbit are relatively minor.

Reference Frames•Extend infinitely far in all directions. You can be thousands of miles away, yet still in the same reference frame

•Observers are at rest in their reference frames.

•A reference frame is not the same as a point of view. Therefore all observers at rest relative to each other share the same reference frame and will view time and length the same way.

Relative Velocity Simulation

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=140

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=140

3.4 Relative Velocity without relativity

One example of an inertial reference frame is on the ground. This observer is at rest in his reference frame.

The train traveling at a constant speed is another frame of reference. Anybody sitting down in the train is an observer at rest in their reference frame.

3.4 Relative Velocity

An observer at rest on the ground sees the train traveling at 9m/s relative to the ground: vTG = +9 m/s, where + indicates to the right (TG = Train relative to ground).

An observer at rest in the trains sees the man on the ground traveling at vGT = -9m/s, where – indicates to the left (GT = Ground relative to train).

Is one of these reference frames more “real” than the other?Considering that the Earth revolves around the Sun and the whole Universe is expanding, I think not.


A passenger inside the train starts to walk up to the bar car at the front of the moving train. Observers in their seats note that he is traveling at vPT = +2 m/s. The passenger is moving relative to the train’s frame of reference, therefore v PT

What is his velocity (v PG ) according to the ground-based observer? Recall vTG is +9.0 m/s

TGPTPG vvv

3.4 Relative Velocity (See Chapter 3)

A passenger inside the train starts to walk up to the bar car at the front of the moving train. Observers in their seats note that he is traveling at vPT = +2 m/s. The passenger is moving relative to the train’s frame of reference, therefore v PT

What is his velocity (v PG ) according to the ground-based observer? Recall vTG is +9.0 m/s

A. + 2m/s, B. +7 m/s C. +9 m/s D. +11m/s


Same situation, but now imagine the passenger moving to the rear of the train (that’s left). What is his speed vPG now?

A. + 2m/s, B. +7 m/s C. +9 m/s D. -7 m/s

An airplane is flying at a constant velocity, vPG = +200 m/s. In the diagram, the plane’s reference frame is labeled S’. Two sound waves travel towards the plane. Wave 1 travels left at vS1G = - 340 m/s, and Wave 2 travels right at vS2G = +340 m/s. What is the speed of each wave relative to the plane (vS1P) and (vS2P) ?

vPG = +200 m/s.

vS1G = - 340 m/s.

vS2G = + 340 m/s.

Find (vS1P) and (vS2P)

vSP = vSG + vGP

vPG = +200 m/s, and vGP = -200 m/s

vS1G = - 340 m/s

vS2G = + 340 m/s.

Find (vS1P) and (vS2P)

vS1P = vS1G + vGP =-340 m/s - 200 m/s = -540 m/s

vS2P = vS2G + vGP = +340 m/s - 200 m/s = +140 m/s

Ocean waves are approaching the beach at 10 m/s relative to the ground(vWG). A boat heading out to sea travels at 6 m/s relative to the ground (vBG ). How fast (speed only, not direction) are the waves moving in the boat’s reference frame (vWB )?

A. 4 m/sB. 6 m/sC.16 m/s D.10 m/s

28.2 The Postulates of Special Relativity

THE POSTULATES OF SPECIAL RELATIVITY

1. The Relativity Postulate. The laws of physics are the samein every inertial reference frame. All inertial reference frames are

equally valid (and all are just as valid as the earth’s reference frame.

2. The Speed of Light Postulate. The speed of light in a vacuum,measured in any inertial reference frame, always has the same valueof c, no matter how fast the source of light and the observer are moving relative to one another.

Number 2 seems to contradict our idea of inertial reference frames. How could this be true. Why should a light wave be different from a sound wave?

Einstein’s Principle of RelativityEinstein’s Principle of Relativity

• Maxwell’s equations are considered to be laws of physics

• Maxwell’s equations are true in all inertial reference frames.

• Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c =

3.00 × 108 m/s. • Therefore, light travels at speed c in all

inertial reference frames.Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other.

28.2 The Postulates of Special Relativity

THE POSTULATES OF SPECIAL RELATIVITY

1. The Relativity Postulate. The laws of physics are the samein every inertial reference frame. All inertial reference frames are

equally valid (and all are just as valid as the earth’s reference frame.

2. The Speed of Light Postulate. The speed of light in a vacuum,measured in any inertial reference frame, always has the same valueof c, no matter how fast the source of light and the observer are moving relative to one another.

The only way #2 could be true is if Δt is NOT the same in all inertial reference frames. This was the great genius of Einstein’s Theory of Relativity. Time and length are not always what they seem to be!

28.3 The Relativity of Time: Time Dilation

TIME DILATION

•If you are in the same reference frame as the light clock, • Δt0 = 2D/c where D is the distance between mirror and receiver, and c is the speed of light.


An observer on the earth measures the light pulse traveling a greater distance between ticks (2s > 2D). She still measures the speed of light in the spaceship as c (postulate #2).Therefore, her clock reads a Δt = 2s/c instead of 2D/c and Δt > Δt0 The observer in the earth’s frame of reference measures an expanded, or dilated time.


The time interval between two events that occur at the same position is called the proper time interval (Δt0)In general, the proper time interval between events is the time interval measured by an observer who is at rest relative to the events.In the light clock example, the proper time interval was measured by the astronaut, because from his reference frame, both events (light leaving source, light hitting detector) happened at the same position.


PROPER TIME INTERVAL

The time interval between two events that occur at the same position is called the proper time intervalIn general, the proper time interval between events is the time interval measured by an observer who is at rest relative to the events.In the light clock example, the proper time interval was measured by the astronaut, because from his reference frame, both events (light leaving source, light hitting detector) happened at the same position. leaving but not by the earth-bound observer.

Proper time interval = ot

221 cv

tt o

Dilated time interval = t


221 cv

tt o

This is the time interval the earth-based observer would read. Since v< c, the denominator is less than 1, so Δt (Earth-based observer time interval) is be greater than Δt0 (astronaut observer time interval). It can be shown that:

From the Sun to SaturnFrom the Sun to Saturn

Who measures the proper time interval?


This implies it is the astronaut who measures the proper time interval, Δt0 .


221 cv

tt o

to find the proper time interval,Δt0

220 1 cvtt

Δt0 = 2310 s


220 1 cvtt

to find the proper time interval, Δt0

Δt0 = 2310s

28.4 The Relativity of Length: Length Contraction

The shortening of the distance between two points is oneexample of a phenomenon known as length contraction.

Length contraction: L0 is the proper length, the length between 2 points measured by an observer at rest with respect to them.Note that the observer who experiences the proper time interval, is not the one who measures the proper length

2

2

1c

vLL o


Example 4 The Contraction of a Spacecraft

An astronaut, using a meter stick that is at rest relative to a cylindricalspacecraft, measures the length and diameter to be 82 m and 21 m respectively. The spacecraft moves with a constant speed of 0.95crelative to the earth. What are the dimensions of the spacecraft,as measured by an observer on earth.


m 2695.01m 821 2

2

2

ccc

vLL o

For this problem, the earthbound observer would determine the distance to Alpha Centaur to be the proper length, L0, as shown in the picture, and the observer in the spaceship would see the contracted length. However, it is the astronaut that sees the proper length of the spaceship she is traveling in, while experimenters will measure it at the contracted length.

The distance from the sun to The distance from the sun to SaturnSaturn


2

2

1c

vLL o


2

2

1c

vLL o

L = 0.62 x 1012 m

Beth and Charles are at rest relative to each other. Anjay runs past at velocity v while holding a long pole parallel to his motion. Anjay, Beth, and Charles each measure the length of the pole at the instant Anjay passes Beth. Rank in order, from largest to smallest, the three lengths LA, LB, and LC.

A. LA = LB = LC

B. LB = LC > LA

C. LA > LB = LC

D. LA > LB > LC

E. LB > LC > LA

Conservation of momentum• The Newtonian

momentum of an object is defined as the product of its mass and velocity (mv).

• Conservation of momentum of a system of objects before and after they interact, is a law of physics that is valid in all inertial reference frames.

28.5 Relativistic Momentum

When the speed of an object is close to c, the effects of relativity must be taken into account.The calculation for relativistic momentum is:

The graph shows the effects of relativity are not significant until the objects moves at some fraction of cThe relativistic momentum is always larger than its non-relativistic counterpart.

221 cv

mvp

28.5 Relativistic Momentum

Often it is convenient to ask for the ratio of relativistic to non-relativistic momenta for an object. This number can be thought of as the “relativistic factor” and can be calculated as:

This number is always larger than one!

221

1

cvmv

p

A collision of an electron with a target in a particle accelerator produces a muon that moves forward with a speed of 0.95c relative to the laboratory. The muon’s mass is 1.90 x 10-28 kg.

Determine the “relativity factor”, the factor by which the relativistic momentum is greater than the classical momentum.

http://www.wiley.com/college/cutnell/0470223553/image_gallery/ch28/pages/fig_28_09.htm

A collision of an electron with a target in a particle accelerator produces a muon that moves forward with a speed of 0.95c relative to the laboratory. The muon’s mass is 1.90 x 10-28 kg. The relativistic factor is:

= 3.20

The relativistic momentum is 3.20 times that of the non-relativistic momentum.

221

1

cvmv

p

http://www.wiley.com/college/cutnell/0470223553/image_gallery/ch28/pages/fig_28_09.htm

28.6 The Equivalence of Mass and Energy

In a short addendum to his original paper, Einstein showed that the total energy, and not only the kinetic energy, of an object was dependent on its speed and mass:

If the object is at rest relative to its reference frame, the denominator becomes 1 and we get the the most famous equation in the world:

The ratio of E/E0 is the relativistic factor:

22

2

1 cv

mcE

2mcEo

Relativistic Kinetic Energy

• It can be shown, by use of a binomial expansion of the square root term that when v<<c, KE = ½ mv2.

oEE KE

1

1

122

2

cvmcKE

Kinetic energy and total Kinetic energy and total energyenergy


Note the difference between the values for resting energy and kinetic energy for an object moving at non-relativistic speeds.


For the electron, m = 9.11 x 10-31 kg, start by calculating the relativistic factor E/E0:

220 1

1

cvE

E

EXAMPLE 37.12 Kinetic EXAMPLE 37.12 Kinetic energy and total energyenergy and total energy


Example 8 The Sun is Losing Mass

The sun radiates electromagnetic energy at a rate of 3.92x1026W.What is the change in the sun’s mass during each second that it is radiating energy? What fraction of the sun’s mass is lost duringa human lifetime of 75 years.


kg1036.4

sm103.00

s 0.1sJ1092.3 928

26

2

c

Em o

1230

79

sun

100.5kg1099.1

s1016.3skg1036.4

m

m

chapter 28 – special relativity

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