Einstein’s theory of Special Relativity and the speed of lightDone by: Yong Yu Wen 3o3(33)

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

SPEED OF LIGHT

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

PROOF THAT C IS THE UPPER

LIMIT OF SPEED

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

Summary (speed of light)

Speed

•299,792,458 m/s

•Speed of light in a vacuum is same for everyone

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