Lecture 1. Relativistic kinematics

Outline:

Structure of the course “Modern Physics”Classical Mechanics, Galilean Principle of RelativitySpecial Relativity, Einstein’s Principle of RelativityLorentz TransformationsTime Dilation

Prof. Michael Gershenson, Office: Serin Physics 122WPhone 5-3180, E-mail: gersh@physics.rutgers.eduOffice hours: Mon. 2:30-3:30 PM and by appointment(e-mailed questions are encouraged)Textbook: A.Beiser, Concepts of Modern Physics, 6th Ed.Additional reading: T.A. Moore, Six Ideas that Shaped Physics, Unit Q(Particles behave like waves).

Some Comments

It’s a very fast-paced course (absolutely no time for “warm-up”). In particular, strictly stick to our schedule.

The most efficient learning approach: read the lecture notes and the textbook before the corresponding lecture (you will be able to formulate your questions in class and be more comfortable with PowerPoint format). I’ll be updating the lecture notes, but these changes (hopefully) will be minor.

Do not ignore office hours, a valuable learning resource.

When solving problems, don’t just look for a formula to plug in the numbers, think about underlying principles.

Exams: closed-textbook/lecture-notes, one sheet with formulae/notes is allowed.

Structure of the course “Modern Physics”

Classicalphysics

Relativisticmechanics,

El.-Mag.(1905)

Quantummechanics(1920’s-)

Relativisticquantum

mechanics(1927-)

v/c

h/s

s – the action=momentum×distance, units kg×m2/s

Maxwell’s Equations of electromagnetism (1873)

Newtonian Mechanics,ThermodynamicsStatistical Mechanics

Relativistic Mechanics (kinematics/dynamics)

Quantum (non-relativistic) Mechanics

Intro to Statistical Mechanics of bosons and fermions

Course“Modern Physics”

Newton’s Laws (1687)

Second LawObserved from an inertial reference frame, the acceleration of a particle is proportional to the net force on it and inversely proportional to its mass: a = F /m.Force is a vector quantity and the resultant force is found from all the forces present by vector addition.

Third LawWhenever a particle A exerts a force on another particle B, B simultaneously exerts a force on A with the same magnitude in the opposite direction. This law is often simplified into the sentence "Every action has an equal and opposite reaction."

First LawIt is possible to select a set of reference frames, called inertial reference frames, observed from which a particle moves without any change in velocity if no net force acts on it. This law is often simplified into the sentence "A particle will stay at rest or continue at a constant velocity unless acted upon by an external unbalanced force."

Galilean Principle of Relativity

Space and Time in Classical Mechanics:Space – uniform and isotropic.Time – uniform and absolute (the same in both IRFs and non-IRFs).

Reference frame – a set of 3 spatial coordinates (e.g. x,y,z) and a time coordinate t.

Inertial reference frames (IRFs) (experimental observation, the existence of IRFs is postulated by the 1st Newton’s Law): the frames where bodies removed from interaction with other bodies will maintain their state of rest or of uniform straight-line motion.

Significance of IRFs: Newton’s Laws have the same form in all these frames (the laws are invariant under the coordinate transformations that transform one IRF into another one).

Practical definition of IRF: a frame that moves with a constant velocity relative to very distant objects, e.g. distant stars.

(velocity – vector (!), speed – the length of this vector)

Galilean Principle of Relativity (1632) “The laws of classical mechanics are invariant in all inertial reference frames”

All this makes sense only when we consider inertial reference frames!

Galilean Transformations

tVrr

tt

−=′

=′ - the “absolute” time

- because the space is uniform and isotropic, the IRFs can move with respect to one another with constant velocity

The IRF transformations that preserve “invariance” of Newton’s Laws are known as Galilean Transformations (G.Tr.)

Let’s check that the 2nd Newton’s Law is invariant under Galilean Transformations:

1. differentiate with respect to the (absolute) time:

dr dr dr Vdt dt dt′ ′= = −

′ v v V′ = − - Galilean velocity addition rule

2. differentiate again :

dv dv a adt dt′

′= = - acceleration is the same in all IRFs

The force in N. mechanics can depend (only!) on the difference of two radius-vectors and velocities. Thus F F′ =

,x x′

z z′

y y′

r r′

V

Vt−

t tx x Vty yz z

′ =′ = −′ =′ =

(This is not the case in non-inertial RFs)

(at t=0 the origins coincide with one another)

N. Laws are invariant

under G. Tr.

Importance of Vectors in Classical Mechanics

That makes vectors so useful in classical mechanics: if one can formulate a law that looks like “vector” = “vector” , this automatically means that this law is

invariant under G.Tr. !

All laws of classical physics must have the following forms to be invariant under G.Tr.:

In other words, if one side of an equation is a scalar (vector), the other side must also be a scalar (vector) to satisfy Galilean Principle of Relativity.

In particular, G.Tr. do not affect the length of a vector: the length of a vector is invariant under G.Tr.

t tx x Vty yz z

′ =′ = −′ =′ =

( ) ( ) ( )

Galilean transformations do not affect the relations between vectors.

2 2 22 1 2 1 2 1length x x y y z z= − + − + −

( ) ( ) ( )2 2 22 1 2 1 2 1length x x y y z z length′ ′ ′ ′ ′ ′ ′= − + − + − =

Invariant: Let’s formulate some physical law in the form A=B. If a coordinate transformation affects neither A nor B, we say that this law is invariant under the transformation.

“scalar A” = “scalar B” “vector A” = “vector B”

Maxwell’s Equations: challenge to Galilean PR

Some odd things:At first glance, there is a built-in asymmetry: a charge in motion produces a magnetic field, whereas a charge at rest does not. Also, it follows from M.Eqs that the speed of light is the same in all IRFs, at odds with Galilean velocity addition.

This asymmetry brought into being an idea of a unique stationary RF (ether), with respect to which all velocities are to be measured, and where M. Eqs can be written in their usual form. However, the famous Michelson-Morley experiment (1887) did not detect any motion of the Earth with respect to the ether.

In 1873, Maxwell formulated Equations of Electromagnetism. On one hand, Maxwell’s Equations describe very well all observed e.-m. phenomena, on the other hand, they are not invariant under G.Tr.!

What are the options? At least one of the following statements must be wrong: (a) the principle of relativity applies to both mechanical and e.-m. phenomena; (b) M. Eqs are correct; (c) G. Tr. are correct.

0

0 0 0

/

0

E

B

BEt

EB Jt

ρ ε

μ ε μ

∇⋅ =

∇⋅ =

∂∇× = −

∂∂

∇× = +∂

0 0 2

1c

ε μ =

Einstein’s Principle of Relativity

One of the consequence of Einstein’s Principle of Relativity (being applied to Maxwell’s Equations): the speed of light in vacuum is the same in all IRFs and doesn't depend on the motion of the source of light or an observer (in line with the experimental evidence that the ether does not exist).

Thus, Maxwell’s Equations agree with Einstein’s Principle of Relativity. Conclusion: Galilean Transformations must go. The idea of universal and absolute time is wrong ! One has to come up with correct transformations that “work” for both mechanical and e.-m. phenomena (any speed up to ~c). Consequently, the laws of mechanics have to be modified to be covariant under new (correct) transformations.

Einstein's Principle of Relativity (the first postulate of the Special Theory of Relativity): “The laws of physics are the same (covariant) in all IRFs”.

Covariance is less restrictive than invariance: Let A=B. If, under RF transformation, both A and B are transformed into A’ and B’, but still A’=B’, than the law is covariant.

Einstein (1905) assumed that (a) and (b) are correct, and put forward the following

The second postulate: “The speed of light in vacuum is the same for all inertial observers, regardless of the motion of the source”.

However, this applies to all (not necessarily e.-m.) phenomena. Thus,

Lorentz TransformationsThe class of transformations that maintain the covariance of Maxwell’s equations was already derived by Lorentz by that time (1904) (though Lorentz suggested that the ether wind physically compresses all matter in just the right way to conceal the variations of c in Michelson-Morley experiment, he still believed in absolute time).

L.Tr. for a 1D motion along the x axis:

,x x′

z z′

y y′

r r′

V

Vt−

2

2

2

2

2

1

1

Vt xctVc

x VtxVc

y yz z

−′ =

−

−′ =

−

′ =′ =

or, if we introduce

2 2

2

1 111 V

c

γβ

= =−−

( )

t t xc

x x cty yz z

βγ

γ β

⎛ ⎞′ = −⎜ ⎟⎝ ⎠

′ = −

′ =′ =

Minkowski: “Space of itself, and time of itself will sink into mere shadows, and only a kind of union between them will survive”.

( )( )

ct ct x

x x cty yz z

γ β

γ β

′ = −

′ = −

′ =′ =

Note the symmetry btw the transformations of space and time coordinates.

Linearity of L.Tr. reflects the fact that the space is uniform and isotropic, the time – uniform.

For small V<<c (β<<1, γ~1) – L.Tr. are reduced to G.Tr.

Vc

β =

ct ctx x ct x Vty yz z

β′ =′ = − = −′ =′ =

Example

( )

( )9

1.25 0.75 0.511

1.25 0.75 0.5

(thus 1.67 10 )

x x ct x ct my y mz z mct ct x ct x m

t s

γ β

γ β−

A flash of light occurs at x = 1m, y = 1m, z = 1m, and ct = 1m (so t = 3.3×10-9s). Locate this event in the primed RF, which moves at V/c = 0.6 to the right. (At t = 0, the origins of these RFs coincide).

V/c ≡ β =0.6, γ =1.25, so βγ =0.75

′ = − = − =

′ = =′ = =′ = − = − =

′ = ×

Lorentz Tr.Galilean Tr.

9

0.6 0.411

1(thus 3.33 10 )

x x Vt x ct my y mz z mct ct m

t s−

′ = − = − =′ = =′ = =′ = =

′ = ×

Lorentz transformations (similar to G.Tr.) show how to calculate the coordinates (t’,x’,y’,z’) of some physical event in one IRF if one knows the coordinates of the same event in another IRF.

,x x′

z z′

y y′

V

Vt−

The relativity of simultaneityOne of the striking consequences of Einstein’s postulates in the relativity of simultaneity.

Two events that are simultaneous in one IRF are not, in general, simultaneous in another.

K

K – the rest reference frame of the car. By reckoning of an observer in this IRF, light from the bulb in the middle of the car reaches the car ends [events (a) and (b)] simultaneously.

K’ – an observer on the ground. By this observer’s reckoning these two events are notsimultaneous: for as the light travels from the bulb, the train itself moves forward, and thus, event (a) happens before event (b).

K’

(a) (b) (a) (b)

Second postulate: the speed of light is the same in all IRFs!

Time dilation

“Light-clock”:

The time interval measured in the moving system K’ is greater than the time interval measured in system K where these two events occur at the same place (the proper time is the minimum time interval).

mirror

y photon

mirror

y y′ = photon

2tV Δ

2tc Δ

Second postulate: the speed of light is the same in all IRFs!

02ytc

Δ = ( )22 / 22

y V tt

c+ Δ

Δ =

2 20

2 22

t tc Vt

c

Δ Δ⎛ ⎞ ⎛ ⎞+ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠Δ =

( ) ( ) ( )2 2 20c t c t V tΔ = Δ + Δ

The proper time (interval), Δt0 : the time interval between two events occurring at the same position as measured by a clock at rest (with respect to these two events).

We want to know the time interval between the same two events occurring at the same position as measured by (synchronized) clocks in a moving reference frame:

K K’

002

21

tt tVc

γΔΔ = = Δ

−

Time dilation (cont’d)

K0

sparklerlit K0

sparklergoes out

Proper time interval:

K

1 1,t x

V K

1x2 2,t x

( ) ( )0 0 0 02 1 2 12

2 1 2 21 /

Vt t x xct tV c

− − −− =

−but

0

2 21 /tt

V cΔ

Δ =−

0 00 2 1

0 02 1 0

t t t

x x

Δ = −

− =0 01 1,t x 0 0

2 1,t x

0 02 1x x=

2

21

21tt t ββ

⎛ ⎞Δ′Δ = ≈ Δ +⎜ ⎟− ⎝ ⎠

To observe this effect, the relative speed of the reference frames should be large. For the fastest spacecraft, the speed is ~10-4c, and the effect is of an order of 10-8:

20

11

tt βΔ

=Δ −

Of course, the same results stems directly from L.Tr.:/V cβ =

1

1

0 02

2

21

Vt xctVc

−=

−

20

2

21

Vt xctVc

+=

−

Hint: the math is simplified if we write L.Tr. for the “proper” time interval in the right-hand side.

Galilean Tr.:t t′ =

0t tΔ = Δ

Time dilation (cont’d)The effect of time dilation is “symmetric”: by the earth’s observer reckoning, all processes in a rocket are slowed down, and vice versa, by the rocket’s observer reckoning, all processes on the Earth are slowed down.

There is no contradiction however: the observers measure different things!

time dilation on Earth

time dilation on the rocket

ground clock A ground clock B

rocket clock rocket clock A rocket clock B

ground clock

The “ground” observer compares two ground clocks with one rocket clock.

The “rocket” observer compares two rocket clocks with one ground clock.

Clocks that are properly synchronized in one system will not be synchronized when observed from another system.

Twin Paradox

The symmetry will be broken if we consider the flight back and forth to the Earth. The RF of the rocket going back and forth is a non-inertial RF! Non-inertial RFs are considered in the General Theory of Relativity (Einstein, 1916).

IVI=const

Interestingly, the result of the GTR and that of a (wrong!) application of the STR coincide (e.g., one can consider a circular orbit with a constant speed (but not velocity, there is a constant acceleration that rotates the vector of velocity). As a result, a twin traveling on a rocket on arrival will be younger than his twin on Earth (“twin paradox”).

ProblemThe lifetime of the unstable particle in its rest frame is 2×10-8s. The particles are generated in the center of a sphere of a radius of 8 m. Calculate the minimum speed (in the lab frame) that the particle should have in order to reach the sphere’s surface before it decays.

R=8m

02 21 /

v tR v tv cΔ

= Δ =−

Requirementfor the min. speed

( )( )

2202

2 2 22 20

1 / /

v t RR vv c R c t

Δ= =

− + Δ

Step 1: make a nice drawing, label all quantities.

Step 2: identify reference frames you are working with.

β0 quantities – the particle’s rest RF (Δt0 is the proper life time)

β quantities – the lab RF (Δ t is the paricle’s life time in the lab RF)

02 21 /tt

v cΔ

Δ =−

Step 3: write the equation(s) and solve it.

( ) ( )8

22 8

8 2.4 10 / 0.88 / 3 / 2 10

mv m s cm cm s s−

= = × =+ ×

Step 4: plug in numbers.

Example: decay of cosmic-ray muons

82.994 10 / 0.998 0.998v m s c β= × = =N0– the number of muons generated at high altitude

N – the number of muons measured in

the sea-level lab

60 2.2 10t s−Δ = ×In the muon’s rest frame

By ignoring relativistic effects (wrong!), we get the decay length:

( )

6 80

0 0

2.2 10 3 10 / 66020,000exp exp 30

660

L t c s m s m

N N N

−= Δ × = × × × =

⎛ ⎞= − = −⎜ ⎟⎝ ⎠

In fact, the decay length is much greater, the muons can be detected at the sea level!

~20 km

Because of the time dilation, in the RF of the lab observer the muon’slifetime is: 6 835 10 3 10 / 10.5L s m s km−= × × × =

Muons are created at high altitudes due to collisions of fast cosmic-ray particles (mostly protons) with atoms in the Earth atmosphere. (Most cosmic rays are generated in our galaxy, primarily in supernova explosions).

Muon – an electrically charged unstable elementary particle with a rest energy ~ 207 times greater than the rest energy of an electron. The muon has an average half-life of 2.2 ×10-6 s.

602

35 101

tt sβ

−ΔΔ = ≈ ×

−

altit

ude

( )0 020,000exp exp 210,500

N N N⎛ ⎞= − ≈ −⎜ ⎟⎝ ⎠

Homework Assignment

Problems: 4,5,9,10,11,12,14,18,20,21

HW #1, due 09/09/2010

Please make sure that all required reference frames are clearly identified!