Chap 1

 

Chapter 1

Relativity Relativity deals with measurement of events: where and when they happen, and by how much any two events are separated in space and time. In addition, relativity also deals with transforming such measurements between reference frames that move relative to each other, hence the name relativity .

The theory of relativity will take us into a world far beyond that of ordinary experience  –   –  the world of objects moving at speeds close to the speed of light.  Among other surprises, surprises, Einstein’s Einstein’s theory predicts that the rate at which a clock runs depends on how fast the clock is moving relative to the observer: the faster the motion, the slower the clock rate. This and other predictions of the theory have passed every experimental test, which led us to a deeper view of the nature of space and time.

 A reference frame  is a coordinate system relative to which physical measurements are taken. An inertial frame of reference  is one which moves with constant velocity (or one which is not accelerating).

The special theory of relativity  deals only with inertial reference frames in which Newton’s Laws are valid while the general theory of relativity deals with the more challenging situation in which the reference frames accelerate.

1.1 Special Theory of Relativity

The special theory of relativity was introduced by Albert Einstein in 1905 in an attempt to propose drastic revisions in the Newtonian concepts of space and time. This theory was based on two simple postulates:

Relativity Postulate: 1. The laws of physics are the same for observers in all inertial reference frames.The frames. –  –No No frame is preferred. 1

 

2. The Speed of light Postulate:  The speed of light in vacuum has the same value c  in  in all directions and in all inertial reference frames. Note that c  =  = 299,792,458 m/s = 3.0 x 10 8 m/s These propositions have the following implications: 1. Events that are simultaneous for one observer may not be simultaneous for another. 2. When two observers moving relative to each other measure a time interval or a length, they may not get the same results. 3. For the conservation principles of momentum and energy to be valid in all enertial systems, Newton’s second law and the equation for momentum and kinetic energy have to be revised. Relativity has important consequences in all areas of physics, including thermodynamics, electromagnetism, optics, atomic and nuclear physics, and high energy physics. Although many of the results derived in this section seem counter to our intuition, the theory is in good agreement with experimental observations.

1.2 Inertial Frames of Reference

S

y

S’

y’

 

 

 

x’  

x

P

  y

y’   

x

 

O

x’

O’    O’    Figure 1.1 Two reference frames S and S’. S’. The S-frame is defined by  x- and  yaxis while the S’ S’-frame is defined by x’ and y’  axes.  axes. 2

 

The position of particle P can be described by the coordinates x and y in the Sframe or by x’  by  x’  and   and y’  in   in the S’ S’-frame. S if for observers on Earth and S’ S ’ is for the moving spacecraft. S’ S’  moves relative to S with constant velocity u  along the common  x- x’   x’ -axis. The two origins O and O’ coincide at time t   = t’ = 0. Figure 1 shows that the position of particle P as described in S and S ’ are related by  by   x = x’ + ut  

 y = y’  

 = z’    z  =

(Galilean coordinate transformation) transformation )

(1.1)

This relation described by equation 1.1 is called the Galilean coordinate transformation. If particle P moves in the  x -direction, -direction, its velocity component as measured by and observer in S is given by v =dx/dt . Its velocity component in S’ S’ is v’ =dx’ /dt ’  ’.  It follows that = dx’   / dt   + u  dx / dt  dt   = dt ’’  +

or

 + u  v = v’  +

(Galilean velocity transformation) transformation)

(1.2)

Equation 1.2 is called the Galilean velocity transformation, where dx/dt is the   x’ w ith derivative of x with respect to time t, dx’/dt’ is the derivative of  x’ ith respect to t ’  ’,  and u  is the velocity of frame S’. 

1.3 Relativity of Simultaneity

Suppose that one observer (Jing) notes that two independent events (event red and event blue) occur at the same time. Suppose also that another observer (Dudz), who is moving at a constant velocity v  with   with respect to Jing, also records these same two events. Will they get the same result? The answer is no.

“Whether or not two events at different locations are simultaneous   depends on the state of motion of the observer.”  

 According to the principle  According principle of relativit relativity, y, no inertial frame of reference is more correct than any other in the formulation of physical laws. “Each observer is correct in his/her own frame of reference ” . In other words, simultaneity is not an absolute concept but a relative one depending on the motion of the observer. It follows that the time interval between two events may be different in different frames of reference. 3

 

1.4 Relativity of Time Intervals

If observers who move relative to each other measure the time interval (or temporal separation) between two events, they generally will find different results. Why? Because the spatial separation of the events can affect the time intervals measured by the observers.

“The time interval between two events depends on how far apart they

occur, in both space and time, that is, their spatial and temporal separations are entangled.”   If two successive events occur at the same place in an inertial reference frame, the time interval Δt  interval Δt 0   between them, measured on a single clock where they occur 0  between at the same point, is a proper time  between the events. Observers in frames moving relative to that frame will measure a larger value for this interval. For an observer moving with relative speed v , the measured time interval is

t  

t 0 

t 0



1  v / c    2

1   

 

2

 

t        t 0  

or

(Time dilation) dilation)

(1.3)

This effect (described by equation 1.3) is called time dilation. Here,  β   is the speed parameter  given  given by  β =v/c

(speed parameter )

(1.4)

1

  

1    2

 

(Lorentz factor )

(1.5)

   is called the Lorentz factor. 

Note that “there is only one frame of reference in which a clock is at rest and there are infinitely many in which it is moving.”  moving.”  

 An important important consequence consequence of time dilation is that moving clocks run slow as measured by an observer at rest.

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Example 1.1  You are on Earth as a spaceship flies past at a speed of 0.99c (about 2.97 x 108 m/s) relative to the Earth. A high-intensity signal light (perhaps a pulsed laser) on the ship blinks on and off; each pulse lasts 2.20 x 10 -6  s as measured on the spaceship. What do you measure as the duration of each light pulse?

Solution/Answer: Let S be the Earth’s frame of reference and let S’ be that of the spaceship. The time between laser pulses measured by an observer on the spaceship (in which the laser is at rest) is Δt 0 = 2.20 x 10-6 s. This is the proper time in S’ S’, referring to two events, the starting and stopping of the pulse, that occur at the same point relative to S’. S’. The corresponding interval Δt   that you measured on Earth (or the S frame) is given by

t  

t 0 1     2



6 2.20 x   10  s

 15.6 x10 6 s  

1  0.992

This result is a factor of seven greater than that of

Δt 0.

1.4.1 The Twin Paradox

There are twin astronauts named Almira and Hamila. Almira remained on Earth while Hamila takes off on a high-speed trip through the Galaxy. Because of time dilation, Hamila is younger than Almira when she returns to Earth.

Example 1.2  Two twins Roy and Joy are 25 years old when one of them (Joy) sets out on a journey through space at nearly constant speed. The twin in the spaceship measures time with an accurate watch. When she returns to Earth, she claims to be 31 years old, while the twin left on Earth (Roy) knows that she is 43 years old. What was the speed of Joy’s Jo y’s spaceship?  spaceship? 

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Solution/Answer: The spaceship clock as seen by the space-twin reads the trip time to be Δt 0 which is 6 years long. The Earth bound twin sees his sister age 6 years but his clock tell him that a time Δt   = 18 years has actually passed. Hence Δt = γΔt 0 becomes

2

6  18 1  v c

2

 

 

 v   1  0.111    c 

from which

  or v  0.943   c 2.83 x10 8 m  s  

1.5 Relativity of Length

 A length measured measured in the frame in which the body body is at rest (the rest frame of the body) is called a proper length; thus the length  L0  is a proper length in S’ and the length measured in any other frame moving relative to S’ is less than  L0. This can be expressed as

    L   L0   1

v 2  c

2



L0  

 

(1.6)

(Length contraction) contraction) 

This effect (described by equation 1.6) is called length contraction. When v   is very small compared to c,   approaches 1. Thus in the limit of small speeds, equation 1.6 reduces to  L = L0, a relation approaching Newtonian mechanics. This and the corresponding result for time dilation (equation 1.3) show that the Galilean coordinate transformation are sufficiently accurate for relative speeds smaller than c.

Note that length contraction only occurs when  L  is parallel to  L0. There is no length contraction when L   L is perpendicular to L   0.

Example 1.3  A crew member on the spaceship measures the spaceships length to be 400 m. If the spaceship’s speed is 0.99 c , what length do observers measure on Earth?

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Solution/Answer:  The 400-m length of the spaceship is the proper length   L0   because it is measured in the frame in which the spaceship is at rest. To get the length L measured by observers on Earth, we use equation 1.5 and get

2    L   L0 1  v 2  400 m 1  0.992  56   .4m   c

This value of  L = 56.4 m is smaller (contracted) than that of the proper length  L0   = = 400 m.

1.6 The Lorentz Transformation

The Lorentz coordinate transformation is a relativistic generalization of the Galilean coordinate. It also relates the spacetime coordinates of a single event as seen by observers in two inertial frames, S and S’, where S’ is moving relative to S with velocity u in the positive x-x’ x-x’ direction. The four coordinates are related by  x'      x  ut   

 y '    y  

 z '    z  

t '      t   ux c 2   

(1.7) 

If we wish to go the other way, we get  x      x'  ut '  

The Lorentz factor   

t       t ' ux   ' c 2  

and

1 2

(1.8)

 

1  u2 c

For one-dimensional one-dimensional motion, a particle’s velocity v  in S and velocity v’   in S’ are related by v' 

vu 1  uv c 2

  and

v

v'u 1  uv' c 2

  (1.9)

(relativistic velocity transformation) transformation )

Equation 1.9 is called relativistic velocity transformation. In this equation, it is assumed that t   t '  0 when the origin of S and S’ coincide. Note that the spatial value  x and the temporal value t   are bound together as shown in equation 1.7.

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This entanglement entanglement of space and time was a prime message of Einstein’s theory, a message that was long rejected by many of his contemporaries.

Example 1.4  A  spaceship moving away from the Earth with speed 0.90 c  fires  fires a robot space probe in the same direction as its motion, with peed 0.70 c relative to the spaceship. What is the probe’s speed relative to the Earth?  

Solution/Answer:  Let the Earth’s frame of reference be S, and that of the spaceship be S’. Then u=0.90c and v’ =0.70 =0.70c. The nonrelativistic Galilean velocity addition formula would give a speed relative to the Earth of 1.60 c; but this value is greater than the speed of light and must be incorrect. The correct relativistic result from equation 1.8 is v ' u

v

1  uv ' c 2



0.70c  0.90c      0  .98c   1  0.90c 20.70c  c

Example 1.5  A scoutship from the Earth tries to catch up with the missile-firing spaceship of example 1.4 by traveling at 0.95 c  relative to the Earth. What is its speed relative to the spaceship?

Solution/Answer:  Again Again we let the Earth’s frame of reference be S  and that of the spaceship be S’. Again we have u=0.90c, but now v=0.95c. According to nonrelativistic velocity addition, the scoutship’s velocity relative to the spaceship

would be 0.05c. We get the correct result from equation 1.8 as

v' 

vu 1  uv c 2



0.95c  0.90c      0  .34c   0.90c 0.95c 

1

c2

Here, the relativistically correct value of the relative velocity is nearly even times as large as the incorrect Newtonian value.

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1.7 The Doppler Effect for Electromagnetic Waves

The Doppler Effect is the frequency shift from a source due to the relative motion of source and observer. For sound waves traveling in air, the Doppler effect depends on two velocities: the velocity of the source and the velocity of the detector with respect to the air. Air is the medium which transmits the waves.

That is not the situation with light waves, for they and other electromagnetic waves require no medium, being able to travel even through vacuum. The Doppler effect for light waves depend only on one velocity, the relative velocity between source and detector a measured from the reference frame of either.

1.7.1 Relativistic Doppler Effect

If a source emitting light waves of frequency f 0   moves directly away from a detector with relative radial speed v  (and  (and speed parameter  β   = = v / c), the frequency f  measured  measured by the detector is

1    

 f     f  0

1    

 

(1.10)

(source and detector separating ) 

If the source moves directly toward the detector, the signs in front of both  β   symbols in equation 1.10 are reversed. 1      f     f   1       0

(1.11)

(source and detector moving closer ) 

1.7.2 Astronomical Doppler Effect

In astronomical observations of stars, galaxies, and other sources of light, we can determine how fast the sources are moving, either directly away from us or directly toward us, by measuring the Doppler shift of the light that reaches us.

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For astronomical observations, the Doppler Effect is measured in wavelengths  λ. For speeds much less than c , equation 1.10 leads to

v

   

c 

(1.12)

where Δ λ  is the Doppler shift in wavelength (the magnitude of the change in wavelength) due to the motion. Here  f =c/ λ  λ  and  f 0  = c/ λ  λ0. Here,  λ0  is the proper wavelength.

1.7.3 Transverse Doppler Effect

If the relative motion of the light source is perpendicular to a line joining the source and detector, the Doppler frequency formula is

 f      f  0   1     2  

(1.13)

(Transverse Doppler Effect ) 

Equation 1.13 is called transverse Doppler Effect which is due to time dilation. If we rewrite equation 1.10 in terms the period T of oscillation of the emitted light wave instead of the frequency, we have

Since

T = 1/ f   f  

then

T  

T 0     T 0   2 1    

(1.14)

 f 0  is the proper period of the source. If we compare equation 1.3 in which T0  = 1/ f  with that of equation 1.13 this equation is simply the time dilation formula, since a period is a time interval.

For low speeds (  β