Lecture Physics A2: Einstein’s special relativity

Einstein’s Special Relativity

Pham Tan Thi, Ph.D. Department of Biomedical Engineering Faculty of Applied Sciences Ho Chi Minh University of Technology

• Statement for Special Relativity

• Reference Frame and Inertial Frame

• Newton Relativity or Galilean Invariance/Transformation

• A need of “Ether” (a Medium for propagating Light)

• Michelson-Morley Experiment

• Lorentz - FriztGerald Proposal

• The Problem of Simultaneity

• Lorentz Transformation

• Consequence of Lorentz Transformation

• Twin Paradox

Classic Picture for Relative Motion

Consider a Situation

Reference #1 Reference #2

Consider a Situation

Reference #1

Speed of light

0.5c

From the girl’s point of view on the platform, that light would not look like it is going faster than the speed of light. It would just look like it is moving at exactly the speed of light

Classical and Modern Physics

Modern Physics Small, Fast moving Object • Relativistic Mechanics • Quantum Mechanics

Classical Physics Large, Slow moving Object • Newtonian Mechanics • Electromagnetism and Waves • Thermodynamics

10% of c

• Below 10% of the speed of light, c, classical mechanics holds

(relativistic eﬀects are minimal)

• Above 10%, relativistic mechanics holds (more general theory)

SPECIAL THEORY OF RELATIVITY

Aims to answer some burning questions:

• Could Maxwell’s equations for electricity and magnetism

be reconciled with the laws of mechanics?

• Where was the ether?

History

Albert Einstein surprised the world in 1905 when

• He theorized that time and distance cannot be measured

absolutely

• They only have meaning when they are measured relative to

something

Einstein published his theory in two steps:

• Special theory of relativity (1905) ➔ How space and time are

interwoven

• General theory of relativity (1915) ➔ Eﬀects of gravity on space &

time

What is “relative” in relativity?

• Motion … all motions is relative • Measurements of motion (and space & time) make no sense unless we are told what they are being measured relative to

What is “absolute” in relativity?

• The laws of nature are the same for everyone • The sped of light, c, is the same for everyone

What is Relative?

• A plane ﬂies from Nairobi to Quito at 1,650 km/hr

• The Earth rotates at the equator at 1,650 km/hr

• An observer…

✦on the Earth’s surface sees the plane ﬂies westward overhead ✦at a far distance sees the plane stands still and the Earth rotate

underneath it

Origin of Special Theory of Relativity

Albert Einstein

(1879 - 1955)

• In 1905, Albert Einstein changed our

perception of the world forever.

• He published the paper on the

electrodynamics of a moving body

• In this, he presented what is now

Albert Einstein, Ann. Phys.

17, 891 (1905).

called the Special Theory of Relativity

Einstein’s Discussion

✴What was the background to this work? ✴What was the new idea that he proposed? ✴How was this experimentally conﬁrmed? ✴How does this inﬂuence our thinking today?

The Special Theory of Relativity

• The laws of Physics are known to be unchanged

(“invariant”) under rotations.

• A rotation mixes the space coordinates but does not

change the length of any object.

• So there should be a linear transformation.

The Special Theory of Relativity

• Special Relativity extends this invariance to certain

transformations of space and time together.

• Collect the space coordinates (x,y,z) as well as time t into a four

component vector.

• c is the speed of light. According to Relativity, it is the same in

every reference frame.

• Relativity states that all laws of physics are invariant under those

Mijxj

linear transformations:

→

j=1 X

which leave x2 + y2 + z2 - c2t2 unchanged

• This quantity is like a “length” in space-time, rather than just

space.

The Special Theory of Relativity

We will now examine the physical meaning of this statement, as well as how it came to be proposed by Einstein.

Electrodynamics

~E =

r ·

⇢ "0

Gauss’ Law • The crisis that motivated

Gauss’ Law for Magnetism

~B = 0

Einstein’s work was related to the laws of electricity and magnetism, or Electrodynamics

r · Faraday’s Law

~E = –

@ ~B @t

r ⇥

• These laws were known, thanks to Maxwell, and embodied in his famous equations.

~B = µ0 ~J + µ0"0

@ ~E @t

r ⇥

Ampere’s Law

Electrodynamics

These equations depend on the speed of light, c.

• In what frame is this speed to be measured?

• It was thought that light propagates via a medium called “ether”,

much as sound waves propagate via air or water.

• In that case, the speed of light should change when we move with

respect to the ether - just as for sound in air.

• So c would be the speed of light as measured while one is at rest

relative to the ether.

Reference Frames

Two or more objects which do not move relative to each other share the same reference frame.

• they experience time and measure distance & mass in the same

way

Objects moving relative to the other are in diﬀerence reference frames

• like the plane and ground • they experience time and measure distance and mass in diﬀerent

ways

Reference Frames

A reference frame in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of object in it.

Inertial frames

• in which no accelerations are observed

Non-Inertial frames • that is accelerating with respect to

in the absence of external forces

an inertial reference frame

• bodies have acceleration in the

absence of applied forces

• that is not accelerating • Newton’s laws hold in all inertial

reference frames

Inertial Reference Frame

• A reference frame is called an inertial frame if Newton

laws are valid in that frame.

• Such a frame is established when a body, not subjected to

net external forces, is observed to move in rectilinear motion at constant velocity.

NEWTONIAN PRINCIPLE OF RELATIVITY

• If Newton’s law are valid in one reference frame, then they are

also valid in another reference frame moving at a uniform velocity.

• This is referred to as the Newtonian Principle of Relativity or

Galilean Invariance

Inertial Frame K and K’

~v • K is at rest and K’ is moving with velocity

• Axes are parallel

• K and K’ are said to be inertial coordinate systems

The Galilean Transformation

For a point P

In system K: P = (x, y, z, t)

In system K’: P = (x’, y’, z’, t’)

P

Conditions of the Galilean Transformation

• Parallel axes

• K’ has a constant relative velocity in the x-direction with

respect to K

~vt

x0 = x y0 = y z0 = z t0 = t

• Time (t) for all observers is a Fundamental invariant, i.e.

the same for all inertial observers.

• Inverse relation:

x = x0 + ~vt

y0 = y

z0 = z

t0 = t

Drawbacks of the Galilean Transformation

• Velocity component

v0x =

= vx

= vy

v0y =

= vz

v0z =

dx0 dt0 dy0 dt0 dz0 dt0

DRAWBACKS:

Violets both of the postulates of the special theory of relativity:

(i) Same equations of physics in K and K’, but the

equations of electricity and magnetism are entirely diﬀerent

(ii) c’ = c - v

The Transition to Modern Relativity

• Although Newton’s laws of motion had the same form under the

Galilean transformation, Maxwell’s did not.

• In 1905, Albert Einstein proposed a fundamental connection between space and time; and that Newton’s law are only an approximation

The Need for Ether

The wave nature of light suggested that there existed a propagation medium called the luminiferous or just ether.

• Ether had to have such a low density that the planets could move

through it without loss of energy

• It also had to have an elasticity to support the high velocity of

light waves

Maxwell’s Equations

In Maxwell’s theory, the speed of light, in terms of permeability and permittivity of free space, was given by

v = c =

1 pµo"o

Thus the velocity of light between moving systems must be a constant.

An Absolute Reference System

• Ether was proposed as an absolute reference system in which the

speed of light was the constant of 3 x 108 m/s and from which other measurements could be made.

• The Michelson-Morley experiment was an attempt to show the

existence of ether.

The Michelson-Morley Experiment

• Albert Michelson built an extremely

precise device called an interferometer to measure the minute phase diﬀerence between two light waves traveling in mutually orthogonal directions.

• Experiments were performed to

Albert Abraham Michelson

(1852 - 1931)

compare the speed of light when moving along or against the ether.

The Michelson-Morley Experiment

2L/c

v2/c2

Using traditional mechanics, it follows that the transit times are:

There should be an observed discrepancy:

Michelson Interferometer

The original experiment compared the back-and-forth travel time of light, parallel and perpendicular to the supposed ether:

Michelson Interferometer

1. AC is parallel to the motion of the Earth inducing an “ether wind”

2. Light from source S is split by mirror C and D in mutually perpendicular directions

3. After reﬂection the beams recombine at A with slightly out of phase due to the “ether wind” as viewed by telescope E.

Interferogram of Michelson Interferometer

Typical interferometer fringe pattern expected when the system is rotated by 90°

The Analysis

2cl1

t1 =

Assuming the Galilean Transformation

v2 =

2l1 c

1 v2/c2

◆

✓

Time t1 from A to C and back: l1 c + v

t2 = t1 =

t = t2

l1 v2/c2 !

1 l1 l2 v2/c2 1 v2/c2 v2/c2 ! 1 1

2l2 2l2 l2 2 2 = = t1 = t = t2 pc2 c v2 c c v2/c2 1 So that the change in time is: p

p p p

t1 =

t = t2

l2 v2/c2

2 c

l1 v2/c2 !

Time t2 from A to D and back:

The Analysis

t01 =

t0 = t02

l2 v2/c2

2 c

l1 v2/c2 !

Upon rotating the apparatus, the optical path lengths l1 and l2 are interchanged producing a diﬀerent change in time (note the change in denominators)

l1 + l2

t =

v2/c2

l1 + l2 1

2 c

v2/c2 !

Thus a time diﬀerence between rotations is given by

v2(l1 + l2)/c3

⇡

and upon a binomial expansion, assuming v/c << 1, the equation reduces to

Result

Using the Earth’s orbital speed as:

v = 3 x 104 m/s

together with

l1 = l2 = 1.2 m

so that the time diﬀerence becomes

∆t’ - ∆t ≈ v2 (l1 + l2)/c3 = 8 x 10-17 s

Although a very small number, it was within the experimental range of measurement for light waves.

Michelson’s Conclusion

• Michelson noted that he should be able to detect a phase shift of light due to the time diﬀerence between path lengths but found none.

• He concluded that the hypothesis of the stationary ether must be

incorrect (no medium for propagating).

• After several repeats and reﬁnements with assistance from

Edward and Morley (1893-1923), again a null result.

• Thus ether does not seem to exist!

Possible Explanations

Many explanations were proposed but the most popular was the ether drag hypothesis

• This hypothesis suggested that the Earth somehow “dragged” the ether along as it rotates on its axis and revolves about the sun.

• This was contradicted by stellar aberration wherein telescopes had to be titled to observe starlight due to the Earth’s motion. If ether was dragged along, this tilting would not exist.

Lorentz-FritzGerald Contraction

v2/c2

Another hypothesis proposed independently by both H. A. Lorentz and G. F. FitzGerald suggests that the length l1 in the direction of the motion was contracted by a factor of

… thus making the path lengths equal account for the zero phase shift.

This hypothesize, however, was an ad hoc assumption that could not be experimentally tested.

Einstein’s Two Postulates

Albert Einstein was only two years old when Michelson reported his ﬁrst null measurement for the existence of the ether.

At the age of 16th, Einstein began thinking about the form of Maxwell’s equations in moving inertial systems.

In 1905, at the age of 26th, he published his startling proposal about the principle of relativity, in which he believed to be fundamental.

Einstein’s Two Postulates

With the belief that Maxwell’s equations must be valid in all inertial frames, Einstein proposes the followings postulates:

1. The principle of relativity: The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.

2. The constancy of the speed of light: Observers in all inertial systems measure the same value for the speed of light in a vacuum.

Re-evaluation of Time

In Newtonian physics, we previous assumed at t = t’

• Thus with “synchronized” clocks, events in K and K’ can be

considered simultaneously

BUT Einstein realized that each system must have its own observers with their own clocks and meter sticks

• Thus events considered simultaneous in K may not be in K’

The Problem of Simultaneity

Frank at rest is equidistant from events A and B:

Frank “sees” both ﬂashbulbs go oﬀ simultaneously.

The Problem of Simultaneity

Mary, moving to the right with a speed v, observes events A and B in diﬀerent order:

Mary “sees” event B, then A

Lorentz Transformations

The special set of linear transformations that:

1. preserve the constancy of the speed of light, c, between inertial observers

and

2. account for the problem of simultaneity between these observers

x0 =

vt v2/c2

y0 = y

z0 = z

t0 =

(vx/c2) v2/c2

known as the Lorentz transformation equations

Lorentz Transformation Equations

ct)

x0 = (x

y0 = y

z0 = z

1 v2/c2

x/c)

t0 = (t

A more symmetric form: v c

PROPERTIES OF γ:

Recall β = v/c < 1 for all observers

1) γ equals 1 only when v = 0.

2) Graph of γ (note v ≠ c)

Derivation

• Use the ﬁxed system K and the moving system K’

• At t = 0, the origins and axes of both systems are coincident with

system K’ moving to the right along the x axis.

• A ﬂashbulb goes oﬀ at the origins when t = 0.

• According to postulate 2, the speed of light will be c in both systems

and the wavefronts observed in both systems must be spherical.

Derivation

• Use the ﬁxed system K and the moving system K’

• At t = 0, the origins and axes of both systems are coincident with

system K’ moving to the right along the x axis.

• A ﬂashbulb goes oﬀ at the origins when t = 0.

• According to postulate 2, the speed of light will be c in both systems

and the wavefronts observed in both systems must be spherical.

Derivation

Spherical wavefronts in K:

x2 + y2 + z2 = c2t2

Spherical wavefront in K’:

x’2 + y’2 + z’2 = c2t’2

Note these are not preserved in the classical transformations with

x’ = x - vt

y’ = y

z’ = z

t’ = t

Derivation

1. Let x’ = γ(x - vt) so that x = γ’(x’ - vt’)

2. By Einstein’s ﬁrst postulate: γ = γ’

3. The wavefront along the x, x’-axes must satisfy:

x = ct and x’ = ct’

4. Thus ct’ = γ(ct - vt) and ct = γ’(ct’ + vt’)

5. Solving the ﬁrst one above for t’ and substituting into the second gives the following result:

1 +

t0 = 2t0

v c

⇣

⌘ ⇣

⌘

from which we derive:

2 =

1 v2/c2

Finding a Transformation for t’

Recalling x’ = γ(x - vt) substitute into x = γ(x’ + vt) and solving for t’ we obtain:

2) + t

t0 =

x v

which may be written in terms of β (= v/c)

t0 =

(vx/c2) 2 1

The Complete Lorentz Transformation

For the moving frame:

For the stationary frame:

x0 =

x =

vt 2

x0 + vt0 2

y0 = y

y = y0 p

z0 = z

z = z0

t0 + (vx0/c2)

t =

t0 =

(vx/c2) 2 1

Remarks

• If v << c, i.e. β ≈ 0 and γ ≈ 1, we see these equations reduce to the

familiar Galilean transformation

• Space and time are now not separated.

• For non-imaginary transformations, the frame velocity cannot

exceed c.

m0 =

v2/c2

Consequence of Lorentz Transformation: Time Dilation

Time Dilation:

• Clocks in K’ run slow with respect to stationary clocks in K.

Time Dilation

To understand time dilation, the idea of proper time must be understood:

• The term proper time, To, is the time diﬀerence between two events

occurring at the same position in a system as measured by a clock at that position.

Total length is 10 m

Time Dilation

To understand time dilation, the idea of proper time must be understood:

• The term proper time, To, is the time diﬀerence between two events

occurring at the same position in a system as measured by a clock at that position.

Time Dilation

To understand time dilation, the idea of proper time must be understood:

• The term proper time, To, is the time diﬀerence between two events

occurring at the same position in a system as measured by a clock at that position.

Consequence of Lorentz Transformation: Length Contraction

Length Contraction:

• If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving.

• Lengths in K’ are contracted with respect to the same lengths

stationary in K.

Consequence of Lorentz Transformation: Length Contraction

Length Contraction:

• If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving.

• Lengths in K’ are contracted with respect to the same lengths

stationary in K.

Consequence of Lorentz Transformation: Length Contraction

Length Contraction:

• If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving.

• Lengths in K’ are contracted with respect to the same lengths

stationary in K.

Consequence of Lorentz Transformation: Length Contraction

Length Contraction:

• If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving.

• Lengths in K’ are contracted with respect to the same lengths

stationary in K.

Consequence of Lorentz Transformation: Length Contraction

Length Contraction:

• If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving.

• Lengths in K’ are contracted with respect to the same lengths

stationary in K.

Consequence of Lorentz Transformation: Length Contraction

Length Contraction:

• If something is moving relative to you, its length in the direction that it is moving will seem to shorter that it would if it was not moving.

• Lengths in K’ are contracted with respect to the same lengths

stationary in K.

Relativistic Energy

Due to the new idea of relativistic mass, we must now re-deﬁne the concepts of work and energy.

We modify Newton’s second law to include our new deﬁnition of linear momentum, and force becomes:

m~v

~F =

(m~v) =

d~p dt

d dt

v2/c2 !

For simplicity, let the particle start from rest under the inﬂuence of the force and calculate the kinetic energy K after the work is done.

W = K =

(m~v)

~vdt

K = m

(~v)

d dt

K = m

vd(v)

Relativistic kinetic energy:

K = mc2

mc2 = mc2

1 v2/c2

Relativistic Energy

Classic Kinetic energy

For speed v << c, we expand in a binomial series as:

1/2

Classic Kinetic energy

K = mc2

mc2

✓

= mc2

mc2

✓ K = mc2 +

◆ v2 c2 + ... 1 + ◆ Total Energy and Rest Energy mv2 mc2 =

mv2

u2 v2 1 2 1 Total Energy and Rest Energy 2

1 2

Total energy and Rest energy:

the rest energy

the rest energy mc2

The total energy is denoted by E and is given by

= K + mc2

mc2 =

The total energy is denoted by E and is given by 1

v2/c2

the rest energy:

Eo = mc2

the total energy:

mc2

E = mc2 =

= K + Eo

v2/c2

Relativistic Energy

Even when a particle has no velocity and therefore no kinetic energy, it still has energy by virtue of its mass.

The laws of conservation of energy and conservation of mass must be combined into one law: Law of conservation of mass-energy

Relativistic energy and momentum: Massless particles:

• For a particle having no mass:

E = pc

E2 = p2c2 + E2 o E2 = p2c2 + m2c4

• For a particle having no mass:

v = c

Twins Paradox TWIN PARADOX

a thought experiment

A longer life, but it will not seem longer

The Set-up Twins Mary and Frank at age 30 decide on two career paths: Mary decides to become an astronaut and to leave on a trip 8 lightyears (ly) from the Earth at a great speed and to return; Frank decides to reside on the Earth.

The Problem Upon Mary’s return, Frank reasons that her clocks measuring her age must run slow. As such, she will return younger. However, Mary claims that it is Frank who is moving and consequently his clocks must run slow. The Paradox Who is younger upon Mary’s return?

50 Yr

70 Yr

20 Yr

Twins Paradox However, this scenario can be resolved within the standard framework of special relativity.

The clear implication is that the travelling twin would indeed be younger, but the scenario is complicated by the fact that the travelling twin must be accelerated up to travelling speed, turned around, and decelerated again upon return to Earth.

Accelerations are outside the realm of special relativity and require general relativity.

The Resolution Frank’s clock is in an inertial system during the entire trip; however, Mary’s clock is not. As long as Mary is traveling at constant speed away from Frank, both of them can argue that the other twin is aging less rapidly

When Mary slows down to turn around, she leaves her original inertial system and eventually returns in a completely different inertial system.

Mary’s claim is no longer valid, because she does not remain in the same inertial system. There is also no doubt as to who is in the inertial system. Frank feels no acceleration during Mary’s entire trip, but Mary does.

General Theory of Relativity

What about non-inertial reference frames?

Einstein waited 10 years to publish his General Theory of Relativity

General Theory of Relativity

Describes the relationship between gravity and the geometrical structure of space and time

PRINCIPLE

General relativity is the extension of special relativity. It includes the eﬀects of accelerating objects and their mass on space-time.

It is based on two concepts: (i) the principle of equivalence, which is an extension of Einstein’s ﬁrst postulate of special relativity and (ii) the curvature of space-time due to gravity.

Lecture Physics A2: Einstein’s special relativity - PhD. Pham Tan Thi