Relativity

Chapter 1

Modern physics is the study of the two great revolutions in physics - relativity and quantum

mechanics.

Galilean Relativity All inertial reference frames are equivalent! Another way of stating this principle is that only relative motion can be detected.

Transformation Equations If you know what an observer in a particular reference frame observes then you can predict the observations made by an observer in any other reference frame. The equations that enable you to make these calculations are called Transformation Equations.

Invariance Since the labeling of your coordinate axis and its origin location is arbitrary, the equations of physics should have the same form regardless when you rotate or translate your axis set. Equations that have this property are said to be invariant to the transformation.

It was shown that Maxwell's Equations are not invariant under a Galilean Transformation so E&M and Mechanics are not consistent.

IV. Important Physics Problems of Late 19th CenturyModern Physics was developed as the solution to some extremely important problems in the late 19th century that stumped physicists. We will study these important problems and see how they have caused us to change our notions of time, space, and matter. Some of these important problems include a) the ether problem, b) stability of the atom, c) blackbody radiation, d) photoelectric effect, and e) atomic spectra.

A Brief Overview of Modern Physics

20th Century revolution:- 1900 Max Planck

Basic ideas leading to Quantum theory

- 1905 Einstein Special Theory of Relativity

21st Century Story is still incomplete

Basic Problems

Newtonian mechanics fails to describe properly the motion of objects whose speeds approach that of light

Newtonian mechanics is a limited theory– It places no upper limit on speed– It is contrary to modern experimental results– Newtonian mechanics becomes a specialized case of

Einstein’s special theory of relativity when speeds are much less than the speed of light

Galilean Relativity

To describe a physical event, a frame of reference must be established

There is no absolute inertial frame of reference– This means that the results of an experiment

performed in a vehicle moving with uniform velocity will be identical to the results of the same experiment performed in a stationary vehicle

Galilean Relativity Reminders about inertial frames– Objects subjected to no forces will experience no acceleration– Any system moving at constant velocity with respect to an

inertial frame must also be in an inertial frame According to the principle of Galilean relativity, the

laws of mechanics are the same in all inertial frames of reference

Galilean Relativity

The observer in the truck throws a ball straight up– It appears to move in a

vertical path– The law of gravity and

equations of motion under uniform acceleration are obeyed

Galilean Relativity

There is a stationary observer on the ground– Views the path of the ball thrown to be a parabola– The ball has a velocity to the right equal to the velocity of

the truck

Galilean Relativity – conclusion

The two observers disagree on the shape of the ball’s path

Both agree that the motion obeys the law of gravity and Newton’s laws of motion

Both agree on how long the ball was in the air Conclusion: There is no preferred frame of reference

for describing the laws of mechanics

The cornerstone of the theory of special relativity is the Principle of Relativity:

The Laws of Physics are the same in all inertial frames of reference.

We shall see that many surprising consequences follow from this innocuous looking statement.

Frames of Reference and Newton's Laws

Let us review Newton's mechanics in terms of frames of reference.

A "frame of reference" is just a set of coordinates - something you use to measure the things that matter in Newtonian mechanical problems - like positions and velocities, so we also need a clock.

A point in space is specified by its three coordinates (x,y,z) and an "event" like, say, a little explosion by a place and time – (x,y,z,t).

The "laws of physics" we shall consider are those of Newtonian mechanics, as expressed by Newton's laws of motion, with gravitational forces and also

contact forces from objects pushing against each other. _____________________________

For example, knowing the universal gravitational constant from experiment (and the masses involved), it is possible from Newton's second law,

force = mass x acceleration,

to predict future planetary motions with great accuracy.

Suppose we know from experiment that these laws of mechanics are true in one frame of reference. How do they look in another frame, moving with respect to the first frame? To figure out, we have to find how to get from position, velocity and acceleration in one frame to the corresponding quantities in the second frame.

Obviously, the two frames must have a constant relative velocity, otherwise the law of inertia

won't hold in both of them.

Let's choose the coordinates so that this velocity is along the x-axis of both of them.

Notice we also throw in a clock with each frame.

Now what are the coordinates of the event (x,y,z,t) in S'? It's easy to see t' = t - we synchronized the clocks when O‘passed O. Also, evidently, y' = y and z' = z, from the figure.We can also see that x = x' +vt. Thus (x,y,z,t) in Scorresponds to (x',y',z', t' ) in S', where

That's how positions transform - these are known as the Galilean transformations.

tt

zz

yy

vtxx

What about velocities ? The velocity in S' in the x' direction

This is just the addition of velocities formula

vuvdt

dxvtx

dt

d

dt

xd

td

xdu xx

)(

vuu xx

How does acceleration transform?

dt

duvu

dt

d

dt

ud

td

ud xx

xx

)(

the acceleration is the same in both frames. This again is obvious - the acceleration is the rate of change of velocity, and the velocities of the same particle measured in the two frames differ by a constant factor - the relative velocity of the two frames.

xx aa

Since v is constant we have

If we now look at the motion under gravitational forces, for example,

we get the same law on going to another inertial frame because every term in the above equation stays the same.

rr

mGmam ˆ

221

1

Note that acceleration is the rate of change of momentum - this is the same in both frames. So, in a collision, if total momentum is conserved in one frame (the sum of individual rates of change of momentum is zero) the same is true in all inertial frames.

Maxwell’s Equations of Electromagnetismin Vacuum

0q

AdE 0 AdB

dt

ddE B

dt

dIdB E 000

Gauss’ Law for ElectrostaticsThe total electric flux through any closed surface equals

the net charge inside that surface divided by ε0

Gauss’ Law for MagnetismThe net magnetic flux through a closed

surface is zero

Faraday’s Law of InductionThe line integral of the electric field around any closed path (the emf), equals the rate of change of magnetic flux through surface area bounded

by that path

Ampere’s LawThe line integral of the magnetic field around any closed path is the sum of μ0 times the net current

through that path and ε0μ0 times the rate of change of electric flux through any surface bounded

by that path

4

The Equations of Electromagnetism

dt

ddE B

dt

ddB E

00

3

.. if you change a magnetic field you induce an electricfield.........

.. if you change a magnetic field you induce an electricfield.........

Faraday’s Law

Ampere’s Law

.. if you change an electric field you induce a magneticfield.........

.. if you change an electric field you induce a magneticfield.........

B dl ddtE 0 0

E dl ddtB

Electromagnetic Waves

Faraday’s law: dB/dt electric fieldMaxwell’s modification of Ampere’s law dE/dt magnetic field

These two equations can be solved simultaneously.

The result is:E(x, t) = EP sin (kx-t)

B(x, t) = BP sin (kx-t) z

j

Plane Electromagnetic Waves

x

Ey

Bz

E(x, t) = EP sin (kx-t)

B(x, t) = BP sin (kx-t) z

j

c

Plane Electromagnetic Waves

x

Ey

Bz

E(x, t) = EP sin (kx-t)

B(x, t) = BP sin (kx-t) z

j

cNotes: Waves are in Phase, but fields oriented at 900. k=2π/λ. Speed of wave is c=ω/k (= fλ)

smc /103/1 800

At all times

E=cB

It was recognized that the Maxwell equations did not obey the principles of Newtonian relativity. i.e. the equations were not invariant when transformed between the inertial reference frames using the Galilean transformation.

Lets consider an example of infinitely long wire with a uniform negative charge density λ per unit length and a point charge q located a distance y1 above the wire.

The observer in S and S’ see identical electric field at distance y1=y1’ from an infinity long wire carrying uniform charge λ per unit length. Observers in both S and S’ measure a force on charge q due to the line of charge.

1

2

y

kE

1

2

y

kqF

However, the S’ observer measured and additional force due to the magnetic field at y1’ arising from the motion

of the wire in the -x’ direction. Thus, the electromagnetic force does not have the same form in different inertial systems, implying that Maxwell’s equations are not invariant under a Galilean transformation.

1

20

2 y

qv

Speed of the Light

It was postulated in the nineteenth century that electromagnetic waves, like other waves, propagated in a suitable material media, called the ether.

In according with this postulate the ether filed the entire universe including the interior of the matter.

It had the inconsistent properties of being extremely rigid (in order to support the stress of the high electromagnetic wave speed), while offering no observable resistance to motion of the planet, which was fully accounted for by Newton’s law of gravitation.

Speed of the LightThe implication of this postulate is that a light

wave, moving with velocity c with respect to the ether, would travel at velocity c’=c +v with respect to a frame of reference moving through the ether at v.

This would require that Maxwell’s equations have a different form in the moving frame so as to predict the speed of light to be c’, instead of

00

1

c

Conflict Between Mechanics and E&M

A. MechanicsGalilean relativity states that it is impossible for an observer to experimentally distinguish between uniform motion in a straight line and absolute rest. Thus, all states of uniform motion are equal.

Conflict Between Mechanics and E&M

B. E&MInitially-

The initial interpretation of the speed of light in Maxwell's theory was this c was the speed of light seen by observers in absolute rest with respect to the ether.

In other reference frames, the speed of light would be different from c and could be obtained by the Galilean transformation.

Problem-

It would now be possible for an observer to distinguish between different states of uniform motion by measuring the speed of light or doing other electricity, magnetism, and optics experiments.

Possible Solutions

1. Maxwell's theory of electricity and magnetism was flawed. It was approximately 20 years old while Newton's mechanics was approximately 200 years old.

2. Galilean relativity was incorrect. You can detect absolute motion!

3. Something else was wrong with mechanics (I.e Galilean transformation).

Experimental Results

Most physicists felt that Maxwell's equations were probably in error.

Numerous experiments were performed to detect the motion of the earth through the ether wind.

The most famous of these experiments was the Michelson-Morley experiment. Because of the tremendous precision of their interferometer, it was impossible for Michelson and Morley to miss detecting the effect of the earth's motion through the ether unless mechanics was flawed!

The Michelson-Morley experiment is a race between light beams. The incoming light beam is split into two beams by a half-silvered mirror. The beams follow perpendicular paths reflecting off full mirrors before recombining back at the half mirror. Time differences are seen in the interference pattern on the screen.

Theory

We will simplify the calculations by assuming that L1 = L2 = L.

The time required to complete path 1 (horizontal path) is given by

where we have used the Galilean transformation and velocity = distance/time.

uc

L

uc

LT1

22221

uc

2Lcucuc

uc

LT

21

c

u1

1

c

2LT

V. Binomial Approximation

The Binomial Expansion is a powerful method for approximating small effects in physics and engineering problems. It is extremely useful in both special relativity and electromagnetism problems even when you have a calculator.

The expansion of the nth power of (1+x) is given by

The Binomial approximation states that when x << 1

...x

2

1-nnxn1nx1 2

xn1x1 n

Since u<<c, we can use the binomial approximation:

2

1 c

u1

c

2LT

We can determine the time required to complete path 2 (vertical path) using the distance diagram below:

Using the Pythagorean theorem, we have:

2

222

2

2

TuL

2

Tc

u(T2/2) u(T2/2)

4

TuL

4

Tc 22

22

22

2

222

22 L4Tuc

2

2

2

22

22

2

cu

1c

4L

uc

4LT

22

c

u1c

L2T

Again, using the binomial approximation:

2

2 c

u

2

11

c

L2T

Thus, the time difference for the two paths is approximately

We can now calculate the phase shift in terms of wavelengths as follows:

22

21 c

u

c

L

c

u

2

12T-TΔT

c

L

cλf cT

λ cTλ

Thus, the phase shift in terms of a fraction of a wavelength is given by

2

c

uLTcλ

2

c

u

λ

L

λ

λ

Using a sodium light, = 590 nm, and a interferometer with L = 11 m, we have

This was a very large shift (20%) and couldn't have been overlooked.

0.210m10x5.90

m11

λ

λΔ 24

7

Result - No shift was ever observed regardless of when the experiment was performed or how the interferometer was orientated!