W1601 Lecture 23: W1601 Lecture 23: Special Relativity: December 8, 2015 Special Relativity: December 8, 2015

Today:Today: Special RelativitySpecial Relativity

Time DilationTime Dilation Length ContractionLength Contraction TransformationsTransformations Lorentz TransformationsLorentz Transformations Momentum and EnergyMomentum and Energy Doppler ShiftsDoppler Shifts

Transformations between observersTransformations between observers

Using the postulates of Special Relativity, we can start to work Using the postulates of Special Relativity, we can start to work out how to out how to transformtransform coordinates between different inertial coordinates between different inertial observers.observers.

What is a transformation? It’s a mathematical operation that What is a transformation? It’s a mathematical operation that takes us from one inertial observertakes us from one inertial observer’’s coordinate system into s coordinate system into anotheranother’’s.s.

The set of possible transformations between inertial reference The set of possible transformations between inertial reference frames are called the Lorentz Transformations. They form a frames are called the Lorentz Transformations. They form a group (in the mathematical sense of group (in the mathematical sense of ““group theorygroup theory””).).

The possible Lorentz Transformations: translations, rotations, and The possible Lorentz Transformations: translations, rotations, and boosts.boosts.

Translations (fixed displacements)Translations (fixed displacements)

In fixed translations, the two In fixed translations, the two observers have different observers have different origins, but donorigins, but don’’t move with t move with respect to each other.respect to each other.

In this case, the observersIn this case, the observers’’ clocks differ by a constant clocks differ by a constant bb00 and their positions differ and their positions differ by a constant vector by a constant vector bb::

0''

bttbxx O

O'

x y

z

x' y'

z'

b

x

'x

Rotations (fixed)Rotations (fixed) In fixed rotations, the two In fixed rotations, the two

observers have a common observers have a common origin and donorigin and don’’t move with t move with respect to each other.respect to each other.

In this case, the observersIn this case, the observers’’ coordinates are rotated with coordinates are rotated with respect to each other.respect to each other.

The spatial transformation can The spatial transformation can be accomplished with a be accomplished with a rotation matrix; measured rotation matrix; measured times are the same:times are the same:

'

'ttxRx �

OO'

xy

z

x'

y'

z'

One coordinate system rotatedwith respect to another.

BoostsBoosts In boosts, the two frame axes are In boosts, the two frame axes are

aligned, but the frames move at aligned, but the frames move at constant velocity with respect to constant velocity with respect to each other.each other.

The origins are chosen here to The origins are chosen here to coincide at time coincide at time tt in both in both frames.frames.

As you might expect, the fact that As you might expect, the fact that the observersthe observers’’ coordinates are coordinates are not fixed relative to each other not fixed relative to each other makes boosts more complex than makes boosts more complex than translations and rotations.translations and rotations.

It is in boosts that the constancy It is in boosts that the constancy of the speed of light plays a big of the speed of light plays a big role.role.

O

O'

x y

z

x' y'

z'

v

Observers’ axes are aligned, but one movesat constant velocity v with respect to the other.

Boosts: Galileo vs. LorentzBoosts: Galileo vs. Lorentz Suppose we have two observers Suppose we have two observers OO and and OO’’. . OO is at rest, and is at rest, and OO’’ moves along the moves along the xx

direction with constant velocity direction with constant velocity vv..

According to Galileo, the transformation between the coordinates of According to Galileo, the transformation between the coordinates of OO and and OO’’ is is pretty simple; but according to Lorentz and Einstein, we get complicated pretty simple; but according to Lorentz and Einstein, we get complicated expressions with many factors of expressions with many factors of cc involved: the so-called Lorentz transformations. involved: the so-called Lorentz transformations.

If an event occurs at position If an event occurs at position x,y,zx,y,z and time and time tt for observer for observer OO, what are the , what are the spacetime coordinates spacetime coordinates xx'',y,y'',z,z'' and and tt'' measured by measured by OO’’? Galileo and Lorentz say ? Galileo and Lorentz say the following:the following:

Note the Lorentz factor Note the Lorentz factor γγ in the Lorentz boosts. in the Lorentz boosts.

Lorentz (length) contractionLorentz (length) contraction Suppose a moving observer Suppose a moving observer OO’’ puts a rigid puts a rigid ““metermeter”” stick along the stick along the xx’’ axis: axis:

one end is at one end is at xx’’=0=0, and the other at , and the other at xx’’==LL’’..

Now an observer Now an observer OO at rest measures the length of the stick at time at rest measures the length of the stick at time tt=0=0, , when the origins of when the origins of OO and and OO’’ are aligned. are aligned.

Using the first boost equation Using the first boost equation xx’’==γγ((x-vtx-vt) ) at time at time tt=0,=0, it looks like the lengths it looks like the lengths are related by:are related by:

This is the Lorentz contraction: if an object has length This is the Lorentz contraction: if an object has length L’L’ when it is at rest, when it is at rest, then when it moves with speed then when it moves with speed vv in a direction parallel to its length, an in a direction parallel to its length, an observer at rest will measure its length as the observer at rest will measure its length as the shortershorter value value L’/L’/γγ..

Lorentz contractionLorentz contraction

The

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sics

Cla

ssro

om

Spaceship moving at 10% of speed of light

Spaceship moving at 99% of speed of light

Spaceship moving at 85% of speed of light

Spaceship moving at 99.99% of speed of light

Lorentz contractionLorentz contraction An example of Lorentz contraction in the case of collisions of two gold nuclei An example of Lorentz contraction in the case of collisions of two gold nuclei

at the RHIC collider at Brookhaven Lab on Long Island:at the RHIC collider at Brookhaven Lab on Long Island:

https://www.youtube.com/watch?v=Vyq_AYWctSo

These nuclei have a Lorentz factor of about 200.These nuclei have a Lorentz factor of about 200.

Velocity addition (briefly)Velocity addition (briefly) Finally, letFinally, let’’s briefly derive the rule for addition of relativistic velocities (we will s briefly derive the rule for addition of relativistic velocities (we will

need to use the boost equations…)need to use the boost equations…)

Suppose a particle is moving in the Suppose a particle is moving in the xx direction at speed direction at speed uu’’ with respect to with respect to observer observer OO’’. What is its speed . What is its speed uu with respect to with respect to OO??

Since the particle travels a distance Since the particle travels a distance ΔΔxx==γγ((ΔΔxx’’++vvΔΔtt’’)) – an – an ““inverseinverse”” boost – in boost – in time time ΔΔtt=γ=γ((ΔΔtt’’+(+(v/cv/c22))ΔΔxx’’)), the velocity in frame , the velocity in frame OO is: is:

where where vv is the is the relativerelative velocity of the two inertial frames. velocity of the two inertial frames.

Since Since u=u=ΔΔxx//ΔΔtt and and uu’’==ΔΔxx’’//ΔΔtt’’, we get the addition rule:, we get the addition rule:

)'/')(/(1)'/'(

')/('''

22 txcvvtx

xcvttvx

tx

ΔΔΔΔ

ΔΔ

ΔΔ

ΔΔ

vuucvuvuu

' ;)/'(1

'2 to compare

Now on to kinematics…Now on to kinematics… Finally, we can apply the algebra of 4-vectors to moving objects.Finally, we can apply the algebra of 4-vectors to moving objects.

We will talk about relativistic kinematics – this can be used to examine We will talk about relativistic kinematics – this can be used to examine object/particle collisions and also decays of unstable particles (recall our object/particle collisions and also decays of unstable particles (recall our discussion of cosmic ray muons).discussion of cosmic ray muons).

In the context of what we have discussed so far, we might start to think of In the context of what we have discussed so far, we might start to think of objects as moving objects as moving ““observersobservers””, and scientists as stationary observers. , and scientists as stationary observers.

The reference frame of the moving object is often called the The reference frame of the moving object is often called the ““object rest object rest frameframe””, while the frame in which the scientist sits at rest, studying the , while the frame in which the scientist sits at rest, studying the object, is called the object, is called the ““lab framelab frame””..

To begin, letTo begin, let’’s s definedefine (not derive) the notions of relativistic energy, (not derive) the notions of relativistic energy, momentum, and the mass-energy relation. These should reduce to momentum, and the mass-energy relation. These should reduce to classical expressions when velocities are very low (classical limit).classical expressions when velocities are very low (classical limit).

Relativistic momentumRelativistic momentum The relativistic momentum (a three-vector) of a particle is similar to The relativistic momentum (a three-vector) of a particle is similar to

the momentum youthe momentum you’’re familiar with, except for one of those factors re familiar with, except for one of those factors of of γγ::

The relativistic momentum agrees with the more familiar expression The relativistic momentum agrees with the more familiar expression in the so-called in the so-called ““classical regimeclassical regime”” where where vv is a small fraction of is a small fraction of cc. . In this case:In this case:

vmcvvmp

...

211 2

2

(Taylor expansion)

Relativistic energyRelativistic energy The relativistic energy (excluding particle interactions) is quite a bit different The relativistic energy (excluding particle interactions) is quite a bit different

from the classical expression:from the classical expression:

When the particle velocity v is much smaller than c, we can expand the When the particle velocity v is much smaller than c, we can expand the denominator (Taylor expansion again) to get:denominator (Taylor expansion again) to get:

The second term here corresponds to the classical kinetic energy, while the The second term here corresponds to the classical kinetic energy, while the leading term is a constant. (This is not a contradiction in the classical limit, leading term is a constant. (This is not a contradiction in the classical limit, because in classical mechanics, we can offset particle energies by arbitrary because in classical mechanics, we can offset particle energies by arbitrary amounts.)amounts.)

The constant term, which survives even when The constant term, which survives even when vv, is called the , is called the rest energy rest energy of the particle; it is Einsteinof the particle; it is Einstein’’s famous equation:s famous equation:

...83

21...

83

21 2

422

4

4

2

22

cvmmvmc

cv

cvmcE

2rest mcE

Classical vs. relativistic collisionsClassical vs. relativistic collisions

In classical collisions, recall the usual conservation In classical collisions, recall the usual conservation laws:laws:

1)1) Mass is conserved; Mass is conserved; 2)2) Momentum is conserved;Momentum is conserved;3)3) Kinetic energy may or may not be conserved.Kinetic energy may or may not be conserved.

The types of collisions that occur classically include:The types of collisions that occur classically include:1)1) Sticky: kinetic energy decreasesSticky: kinetic energy decreases2)2) Explosive: kinetic energy increasesExplosive: kinetic energy increases3)3) Elastic: kinetic energy is conserved.Elastic: kinetic energy is conserved.

Classical vs. relativistic collisionsClassical vs. relativistic collisions

In relativistic collisions, the usual conservation laws In relativistic collisions, the usual conservation laws include:include:

1)1) Conservation of relativistic energy; Conservation of relativistic energy; 2)2) Conservation of relativistic momentum;Conservation of relativistic momentum;3)3) Kinetic energy may or may not be conserved.Kinetic energy may or may not be conserved.

Note that conservation of energy and momentum can be Note that conservation of energy and momentum can be encompassed into conservation of four-momentum.encompassed into conservation of four-momentum.

The types of relativistic collisions include:The types of relativistic collisions include:1)1) Sticky: kinetic energy decreases, rest energy and mass Sticky: kinetic energy decreases, rest energy and mass

increase.increase.2)2) Explosive: kinetic energy increases, rest energy and mass Explosive: kinetic energy increases, rest energy and mass

decrease.decrease.3)3) Elastic: kinetic energy, rest energy, and mass are conserved.Elastic: kinetic energy, rest energy, and mass are conserved.

Inelastic collisions: classical vs. relativisticInelastic collisions: classical vs. relativistic There is a difference in interpretation between classical and There is a difference in interpretation between classical and

relativistic inelastic collisions.relativistic inelastic collisions.

In the classical case, inelastic collisions mean that kinetic In the classical case, inelastic collisions mean that kinetic energy is converted into energy is converted into ““internal energyinternal energy”” in the system (e.g., in the system (e.g., heat).heat).

In special relativity, we say that the kinetic energy goes into In special relativity, we say that the kinetic energy goes into rest energy.rest energy.

Is there a contradiction? No, because the energy-mass Is there a contradiction? No, because the energy-mass relation relation E=mcE=mc22 tells us that all tells us that all ““internalinternal”” forms of energy are forms of energy are manifested in the rest energy of an object. (In other words, manifested in the rest energy of an object. (In other words, hot objects weigh more than cold objects. But this is not a hot objects weigh more than cold objects. But this is not a measurable effect even on the atomic scale!)measurable effect even on the atomic scale!)

Applications of mass-energy Applications of mass-energy equivalenceequivalence

SummarySummary Lorentz boosts to and from a moving reference frame:Lorentz boosts to and from a moving reference frame:

Relativistic momentum and energy:Relativistic momentum and energy:

Doppler Shift of LightDoppler Shift of Light Though the speed of light is constant in all reference frames, Though the speed of light is constant in all reference frames,

the the frequencyfrequency of light emitted by a source moving towards or of light emitted by a source moving towards or away from an observer will be shifted as measured by the away from an observer will be shifted as measured by the observer.observer.

If source is approaching observer (u > 0), frequency increases If source is approaching observer (u > 0), frequency increases light is light is blue-shiftedblue-shifted..

If source is receding from observer (u < 0), frequency If source is receding from observer (u < 0), frequency decreases decreases light is light is red-shiftedred-shifted..

For u << c, can make the approximation For u << c, can make the approximation ΔΔf/f f/f ≈ u/c≈ u/c