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Chapter 37p

RelativityRelativity

PowerPoint® Lectures forUniversity Physics, Twelfth Edition

– Hugh D. Young and Roger A. Freedman

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

g g g

Lectures by James Pazun

37. Relativity1. Maxwell’s equations (and especially the wave equation) are valid in all inertial reference frames (observers in

t t l ti ti ) ti th t th d f li ht i i th i ll i ti l fconstant relative motion u) suggesting that the speed of light in vacuum c is the same in all inertial reference frames(IRF’s).

2. This special constancy in the speed of light leads to time dilation/length contraction (a clock appears to run slower, and a rod appears to be shorter, according to an observer in constant motion u relative to the clock, and with the rod’s length l oriented parallel to u). 0ll =0t tγΔ = Δ

2 2

1γ =

3. “Two” postulates: (1) The laws of physics are the same in all IRF’s. (2) The speed of light in vacuum c is the same in all IRF’s.

4. Lorentz transformation (position, velocity, …): A linear transformation, relating quantities in one reference frame to those of any IRF, consistent with (or derived from) the postulates of relativity. For simplicity, assume that u points

γ0γ2 21 /u c

γ−

along the +ve x-axis. . .

5. The Doppler effect: This is an observed phenomenon that may be described by the Lorentz transformation for period T, as time interval, and the wavelength λ, as displacement, of a sinusoidal wave. The emitted frequency f_0 and received frequency f are related by

2' ( ), ' , ' , ' ( / )x x ut y y z z t t xu cγ γ= − = = = − 2 2 2' , ' , ' 1 / (1 / ) (1 / )

yx zx y z

x x x

vv u vv v vuv c uv c uv cγ γ−

= = =− − −

(moving closer)c uf f+= (moving apart)c uf f−

=ece ed eque cy a e e a ed by

6. Special relativistic mechanics (Newton’s 2nd law and the work-energy relation): The first postulate requires that Newton’s 2nd law of motion be revised. . .

net net net or ( ) , , (proper time) d p d dE dtF mv F v F dd dt dt

γ γ ττ γ= = = ⋅ =

0 (moving closer)f fc u

=− 0 (moving apart)f f

c u=

+

2 2 , E mc mc Ke p mvγ γ= = + = 2 2 2 2( ) ( )E mc pc= +

7. General Relativity: The laws of physics need to be valid not just for inertial observers but for all observers (ie. In all local coordinate systems). Now, just as one could not experimentally distinguish between IRF’s, locally it is impossible to experimentally distinguish between a force field ( eg. gravitational field) in an IRF and a non-IRF (curved space-time coordinate system). Therefore, gravity may be seen as a result of the curving of space-time by matter.


Goals for Chapter 37

• To consider the invariance of physical laws simultaneity• To consider the invariance of physical laws, simultaneity

• To study time dilation

• To study length contraction

• To see how the Lorentz transformation can show how• To see how the Lorentz transformation can show how different frames of reference explain vexing observations

T id h th D l ff t li t l t• To consider how the Doppler effect applies not only to sound but also to EM waves

• To explain how relativistic motion changes momentum

• To see where Newtonian Mechanics fit into the Special and


General Theories of Relativity

Introduction• Imagine a gedanken experiment g g p

where two identical twins in their early 30s meet at the launch pad of a new spacecraft. John-boy puts

hi h l d h k h don his helmet and shakes hands with Billy-bob. John-boy flies away in a ship that travels at 0 999c for a one year journey to a0.999c for a one-year journey to a distant planet. When he comes back, a year older, he meets Billy-bob for lunch and sees a man across the table who’s approaching retirement. WHAT?

• Experiments at Brookhaven can• Experiments at Brookhaven can accelerate particles to near the speed of light. Unfortunately, none of them are wearing


none of them are wearing timepieces.

Einstein’s First Postulate• The laws of physics applyThe laws of physics apply

in the same fashion everywhere. Newton’s L k ttLaws work no matter what your point of view.

A t i i il• A magnet moving in a coil of wire will induce a current, but … who’s to say if the coil is moving over the magnet or if the magnet is moving throughmagnet is moving through the coil.


Einstein’s Second Postulate• The speed of light is always the same It matters not how fast youThe speed of light is always the same. It matters not how fast you

are going or in which direction you travel, the speed of light is always the same.


Simultaneity—watching lightning• Perception of one reference frame from another makes it hard to• Perception of one reference frame from another makes it hard to

correctly interpret perceptions of physical events.


Relativity of time intervals• A “light clock” can be used to gain a graphic understanding of theA light clock can be used to gain a graphic understanding of the

gamma term that modifies the size of any quantity modified by relativistic effects.

• Refer to Figure 37.6 below.


The gamma factor’s exponential impact• As velocity approaches c the value of gamma grows quicklyAs velocity approaches c, the value of gamma grows quickly.

• Refer to Figure 37.8 below.

• Read Problem-Solving Strategy 37.1.g gy

• Refer to Example 37.1—effect of travel at 0.99c.

• Refer to Example 37.2—effect of travel at aircraft velocities.

• Refer to Example 37.3—travel at 0.6c.


Lengths parallel or perpendicular to motion• Motions perpendicular to the relativistic velocities are notMotions perpendicular to the relativistic velocities are not

contracted. Motions parallel are directly involved.

• Figure 37.12 (below) illustrates the effect.g ( )

• Read Problem-Solving Strategy 37.2.


Contraction examples• Follow Example 37.4, illustrated by Figure 37.13 below.Follow Example 37.4, illustrated by Figure 37.13 below.

• Follow Example 37.5—length contraction.

F ll C t l E l 37 6• Follow Conceptual Example 37.6.


Appearance of objects in relativistic motion• How does an object moving at velocities appear to an observer?• How does an object moving at velocities appear to an observer?

• Refer to Figure 37.14.


The Lorentz transformation• The transformation allows us to determine adjustments for eachThe transformation allows us to determine adjustments for each

dimension (x, y, or z) that’s involved in the problem.

• Refer to Figure 37.15 below at left.g

• Read Problem-Solving Strategy 37.3.

• Follow Example 37 7• Follow Example 37.7.

• Follow Example 37.8, illustrated by Figure 37.16, below at right.


The Doppler effect applied to EM waves• The Doppler effect can cause enough change in the velocity ofThe Doppler effect can cause enough change in the velocity of

light to change the wavelength perceived by an observer at a distance. See Figure 37.17 along the bottom of the slide.


Relativistic momentum• As velocities near the speed of light the gamma factor would causeAs velocities near the speed of light, the gamma factor would cause

momentum/mass/kinetic energy needed to cause the motion toapproach infinity. It’s simply impossible. This is explained graphically in Figure 37.20 in the lower left corner for momentum and Figure 37.21 in the g glower right corner for kinetic energy.

• Follow Example 37.10.


Relativistic examples• Consider Example 37 11 motion of an electron• Consider Example 37.11—motion of an electron.

• Consider Example 37.12—relativistic collision, illustrated by Figure 37 23 belowFigure 37.23 below.


Newtonian mechanics and relativity• How would an astronaut perceive travel at (or near) the speed ofHow would an astronaut perceive travel at (or near) the speed of

light? See Figure 37.24.


Newtonian mechanics and relativity II• Figure 37.25 speculates on the effect of motion near c on a 2-D mapFigure 37.25 speculates on the effect of motion near c on a 2 D map

of outer space.


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