einstein’s theory of special relativity and the speed of light done by: yong yu wen 3o3(33)
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Introductory Video (Special relativity)Einstein's Relativity: The Famous
Equation E=mc2 http://www.youtube.com/watch?v=7h7tyQlpda4&fmt=18 (must watch)
Covers:◦ Time dilation◦ Length contraction◦ E=mc2◦ Space-time (partially)◦ Nearly everything I will be covering…
Special relativityTheory of the structure of space-timeBased on two postulates that are
contradictory in classical mechanics◦Laws of physics are the same for all observers
in any inertia frame of reference (frames of reference in uniform relative motion with respect to each other).
◦Speed of light in a vacuum is constant for all observers
Other consequences◦Time dilation and length contraction◦simultaneity
Lorentz transformation (background)Relationship between the
coordinates of a stationary timeframe (t, x, y, z) and the time frame moving at a constant speed (t’, x’, y’, z’)
Discovered by Hendrik Lorentz in 1890, by analyzing the behaviour of the electric and magnetic field of a charge moving with constant speed.
Hendrik Lorentz
Lorentz transformation (background)He found out certain
transformation of the space and time coordinates left the Maxwell’s equations unchanged.
That means, Maxwell’s equations have the same form in all inertial frames of reference.
Maxwell’s Equation
Lorentz factor
Known as γ (gamma), and given◦ is the velocity in terms of c
◦ where v is the velocity observed
Factor by which length contraction and time dilation occurs.
Difference from 1 is negligible for speeds much slower than c
◦ such as most everyday speeds
Increases at relativistic speeds and diverges to infinity as v approaches c.
222 1
1
vc
c
c
v
γ approaches infinity as v reaches the speed of light
Lorentz transformationProperties
◦Is a linear function of x and t.◦Does not change the y and z
coordinates◦Does not affect the speed of a light
wave◦Making a second transformation with
a speed v in the –x direction gives the original space-time coordinates
◦The transformation reduces to the Galilean transformation at small speeds (v << c).
Lorentz transformation
2
2
2
1
1
'
'
)('
)('
cv
y
zz
yy
vtxxc
xvtt
where
For small speeds (v << c), we have γ ≈ 1, and the Lorentz transformation reduces to the Galilean transformation.The form of the Lorentz transformation is a direct consequences of the fact that the speed of light is the same on all frames of referenceTo obtain the reverse transformation, all we need to do is to change the primed and unprimed variable and also the value of v
Information of the Speed of light It is the speed of electromagnetic radiation
in a vacuum.◦ Radio waves, visible light gamma rays etc.
Commonly known asExact value: 299,792,458 m/s Usually denoted by c, for "constant“Plays an important part is various branches
in physics, such as relativity.No non-hypothetical object can surpass the
speed of light.◦ Example: tachyons-hypothetical subatomic
particles.
18103 ms
Discovery
Rømer's observations of the occultation of Io from Earth
Before the 17th century, it was not known whether light was transmitted instantaneously or merely very quickly.
In the 17th century, Ole Rømer first demonstrated that it travelled at a finite speed by studying the apparent motion of Jupiter's moon Io.
In 1975 the speed of light was known to be 299,792,458 m/s.
In 1983, the metre was redefined as the distance travelled by light in vacuum in 1⁄299,792,458 of a second.
Alternate way of measuring the speed of light
http://www.youtube.com/watch?v=CbJjhZrT3EY
Interesting way using wavelengths, microwave oven and CHOCOLATE!!!
= 299 792 458 m / s
2mcE Mass of a body is a measure of its
energy content. Shows that energy can be converted
to matter and vice-versa, depending on a conversion factor, c.
Does not depend on any specific system of measurement units.◦ Speed of light is set equal to 1 in
natural units, and the formula becomes the identity E = m; hence the term mass–energy equivalence
Shows that energy always exhibits mass.
Energy applied on an object only partially increases its speed; it also increases its mass.
When an object reaches the speed of light, further energy will only increase its mass, and will not change its speed.
Proof that c is the upper limit of speedEnergy of an object with rest mass of m and
speed v is given by When v=0, γ=1, resulting in the famousSince shown above that γ approaches infinity
as v reaches c, the speed of light, it would take infinite amount of energy to accelerate an object with mass to the speed of light.
When energy is Travelling faster than the speed of light
would violate causality and would travel back in time. (will be explained later after I have covered space-time)
2mc2mcE
Proof that c is the upper limit of speed (Lorentz transformation)
Consider a particle in frame S moving with a velocity of dx/dt in the x direction.
In a frame S’, defined to be moving with a velocity v in the x direction relative to the frame S, the particle has the speed dx’/dt’.
The determination of dx’/dt’ in terms of dx/dt is obtained by differentiation of the coordinates )1(
)(
'
'
22 dtdx
cv
vdtdx
cvdx
dt
vdtdx
dt
dx
When the speed is small compared to c, v/c ≈0, we get
vdt
dx
dt
dx
'
'
Which is the Galilean addition rule
When the particle is a photon. Then dx/dt = c, and the transformed photon speed is
cc
cvvc
dt
dx
21'
'
The speed of the photon is unchanged, which is the second postulate of special relativity.
Proof that c is the upper limit of speed (Lorentz transformation)We can see that when v=c the amount
of energy and the mass of the particle is infinite, as , so only massless particles such as the photon can move at the speed if light.
From here we can see that v cannot exceed c, for if it does, the equation will lead to where n is positive, which is imaginary.
0
n
in
Example of “superluminal” travel
http://www.youtube.com/watch?v=FCOqZfDElIQ
In water, speed of light is reduced by 75%. In a nuclear reactor, charged particles in
water travels faster than the reduced speed of light.
Because these particles contain an electric charge, they emit energy (Cherenkov radiation). Any particles they bump into become radioactive, giving the water a characteristic blue glow.
The water has a bluish glow because of the radiation produced by charged particles moving faster than the speed of light in water.
The word “superluminal” in the title is in inverted comas, because slowing the speed of light to beat it is cheating.
So, at the moment, nothing can travel faster than 299,792,458 m/s
http://onwardstate.com/wp-content/uploads/2009/04/nuclear2.jpg
Galaxies moving faster than the light
http://www.youtube.com/watch?v=fxNbXjBbzEo part 1
http://www.youtube.com/watch?v=MoTNGmlOO2g&annotation_id=annotation_127900&feature=iv part 2 (more relevant)
In models of the expanding universe, the farther galaxies are from each other, the faster they drift apart.
This receding is not due to motion through space, but rather to the expansion of space itself. For example, galaxies far away from Earth appear to be moving away from the Earth with a speed proportional to their distances.
Beyond a boundary called the Hubble sphere, this apparent recessional velocity becomes greater than the speed of light.
http://startswithabang.com/wp-content/uploads/2008/04/cosmic.jpg
Length contraction, time dilation and simultaneity http://www.youtube.com/watch?v=88WTEQwvJ9g
Length contraction
physical phenomenon of a decrease in length detected by an observer in objects that travel at any non-zero velocity relative to that observer.
negligible at everyday speeds
Definition:
•Where L is the length of the object at rest.•L’ is the length observed by an observer in relative motion with respect to the object.•v is the relative velocity between the observer and the moving object.•c is the speed if light.
22 /1)(
' cLL
L
Length contraction
Consider a stick moving with speed v in the frame S
The length of the stick L may be determined by measuring the time for the stick to pass a stationary clock,
Moving Stick
Clock at rest
v
tvL
Length contractionThe length measurement L’ in a
frame in which the stick is at rest, is
'' tvL
Stick at rest
Moving Clockv
Length contraction
The 2 time intervals are related by the time dilation rule: The time interval is longer by a factor of γ in the frame where the clock is moving,
'
''
,
'
L
t
tLL
Therefore
tt
The length of the stick is longest in the frame where the stick is at rest (L’>L). In a frame where the stick is measured to be shorter by a factor of γ. Length contraction applies to any two points in space where any two points can be connected with an imaginary stick.
Time Dilationhttp://www.youtube.com/watch?v
=HHRK6ojWdtU&fmt=18
Time dilationConsider a time interval Δt
measured at a fixed position x0 in a stationary frame,
If we use the Lorentz trans formation to calculate the time interval in a frame moving with speed v, we arrive at the result
12 ttt
tttc
vxt
c
vxt
ttt
)(
)()(
'''
12
20
120
2
12
d
A
B
d
A
B
t=t0 t=t0 +Δt
vΔt’ v
c22 vc
a) Clock at rest
b) Moving Clock
d
A
B
d
A
B
The clock is at rest. The time taken for the light to travel from point A to point B is . The time mentioned in the frame where the clock is at rest is called the proper time.
cdt /
The clock is moving with the speed v. Because the speed of light is constant, the time for the light to travel from point A to point B is 22
'vc
dt
------------------------------------------------------------------------------------------------------------------------
Figure 1.1
Time dilationThe time interval is longer in the moving
frame. This result is known as time dilation. In figure a)*, the clock is a apparatus that
detects the speed of light. The length of time (Δt) that it takes light to travel from point A to B is
In figure b)*, since the speed of light is constant, the horizontal component of velocity is v and the net speed is c, that the vertical component of velocity is (c2-v2)1/2
Therefore,
cdt /
c
d
cv
c
d
vc
dt
2
222
1
'
*previous slide
SimultaneityEvents that occur simultaneously
in one frame of reference does not necessary occur at the same time in another frame.
Example, Figure 1.2
vD2D1
Speed=c
Speed=c
-L Lx
D2D1
Speed=c
Speed=c
-L’ L’x’
vv
v
v=c
a) Frame SDetectors D1 and D2 are hit at the same
time
b) Frame S’Detectors D2 is hit before D1
Figure 1.2
Figure 1.2a) In frame S, a particle is produced with a
speed v, and when it is at x=0, the particle decays into 2 photons. Detectors D1 and D2 are located at x= -L and x=L. Since the speed of each photon is c, the photons arrive at the 2 detectors at the same time.
b) The same event is analysed when the particle is at rest (frame S’). Detector D2 is moving towards the photon and D1 is moving away from the photon that it hits. The speed of each photon is c, thus D2 gets hit be for D1. Events that are simultaneous in S are not simultaneous in S’
SimultaneityIn 1.2, a pion (π0) is produced with
speed v. The pion decays and the photon hits D1 and D2 at L/c each. Therefore, in frame S, the 2 photons strike the detectors simultaneously.
Simultaneity In frame S’, to determine the space-time
coordinates in the moving frame, we have to use the Lorentz transformation, which gives and c
vxctct 1
11 '
c
Lv
c
xxvc
vxct
c
vxct
ctcttc
2)(
)()(
'''
12
11
22
12
The time difference is
Where we have used the fact that (t2-t1) = 0. Therefore,
2
2'
c
Lvyt
c
vxctct 2
22 '
=>
The photons do not strike the detectors at the same time as t2’ is longer than t1’, so D2 is struck before D1. This is because D2 is moving towards the pion while D1 is moving away.
Violating causalityImagine a gun that can shoot faster
than light bullets. There will be a break down of cause and effect as the target dies before the person shoots.
Also, some observers would see the bullet hit the target before they saw the shooter fire the gun. Since one of the guiding principles of relativity is that all physical laws are the same to all observers, this violation of causality would be a big problem."
Bibliography Rohlf, J.W. (1994). Modern physics from a to z.
Canada: Johm Wiley & Sons inc. Crummett, W.P, & Western, A.B. (1994).
University physics models and applications. Dubuque, Iowa: Wm. C. Brown Communications inc..
http://en.wikipedia.org/wiki/Special_relativityhttp://
www2.slac.stanford.edu/vvc/theory/relativity.html
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Special_relativity.html
http://en.wikipedia.org/wiki/Lighthttp://en.wikipedia.org/wiki/Speed_of_light
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