Option A - Relativity

Relativity

15 hours SL, 25 hours HL


A.1 – The beginnings of relativity (CORE)


Essential idea: Einstein’s study of electromagnetism revealed inconsistencies between the theory of Maxwell and Newton‘s mechanics. He recognized that both theories could not be reconciled and so choosing to trust Maxwell’s theory of electromagnetism he was forced to change long-cherished ideas about space and time in mechanics.
Nature of science: Paradigm shift: The fundamental fact that the speed of light is constant for all inertial observers has far-reaching consequences about our understanding of space and time. Ideas about space and time that went unchallenged for more than 2,000 years were shown to be false. The extension of the principle of relativity to accelerated frames of reference leads to the revolutionary idea of general relativity that the mass and energy that spacetime contains determines the geometry of spacetime.
Understandings:
• Reference frames
• Galilean relativity and Newton’s postulates concerning time and space
• Maxwell and the constancy of the speed of light
• Forces on a charge or current
Applications and skills:
• Using the Galilean transformation equations
• Determining whether a force on a charge or current is electric or magnetic in a given frame of reference
• Determining the nature of the fields observed by different observers
Guidance:
• Maxwell’s equations do not need to be described
• Qualitative treatment of electric and magnetic fields as measured by observers in relative motion. Examples will include a charge moving in a magnetic field or two charged particles moving with parallel velocities. Students will be asked to analyse these motions from the point of view of observers at rest with respect to the particles and observers at rest with respect to the magnetic field.
Data Booklet reference:
x’ = xvt
u’ = uv
• The x' represents the coordinate of an event or an object in the inertial reference frame (IRF) S' which is moving at velocity v relative to another IRF called S within which the event or object has a coordinate x. The t represents the elapsed time. The u' represents the velocity of the object in S' and u represents the velocity of the object in S. In this "classical" scheme, time t is absolute (it is the same in S and S' ). These two equations are known as the Galilean transformations.
Theory of knowledge:
• When scientists claim a new direction in thinking requires a paradigm shift in how we observe the universe, how do we ensure their claims are valid?
Aims:
Aim 3: this sub-topic is the cornerstone of developments that followed in relativity and modern physics


A.2 – Lorentz transformations (CORE)


Essential idea: Observers in relative uniform motion disagree on the numerical values of space and time coordinates for events, but agree with the numerical value of the speed of light in a vacuum. The Lorentz transformation equations relate the values in one reference frame to those in another. These equations replace the Galilean trans-formation equations that fail for speeds close to that of light.
Nature of science: Pure science: Einstein based his theory of relativity on two postulates and deduced the rest by mathematical analysis. The first postulate integrates all of the laws of physics including the laws of electromagnetism, not only Newton’s laws of mechanics.
Understandings:
• The two postulates of special relativity
• Clock synchronization
• The Lorentz transformations Warp drive proof-of-concept Relativity and Maxwell Dr. Smalley:Special relativity and E&M
• Velocity addition
• Invariant quantities (spacetime interval, proper time, proper length and rest mass)
• Time dilation
• Length contraction
• The muon decay experiment
Applications and skills:
• Using the Lorentz transformations to describe how different measurements of space and time by two observers can be converted into the measurements observed in either frame of reference
• Using the Lorentz transformation equations to determine the position and time coordinates of various events
• Using the Lorentz transformation equations to show that if two events are simultaneous for one observer but happen at different points in space, then the events are not simultaneous for an observer in a different reference frame
• Solving problems involving velocity addition
• Deriving the time dilation and length contraction equations using the Lorentz equations
• Solving problems involving time dilation and length contraction
• Solving problems involving the muon decay experiment
Guidance:
• Problems will be limited to one dimension
• Derivation of the Lorentz transformation equations will not be examined
• Muon decay experiments can be used as evidence for both time dilation and length contraction
Data Booklet reference:
g = ( 1 – v2 / c2 ) -1/2
x’ = g ( x – vt ); ∆x’ = g ( ∆x – vt )
t’ = g ( t – vx / c2 ); ∆t’ = g ( ∆t – vx / c2 )
u’ = ( uv ) / ( 1 – uv / c2 )
• ∆t = g t0
L = L0 / g
• ( ct’ )2 – ( x’ )2 = ( ct )2 – ( x )2
• The g represents the Lorentz factor which is used in many of the relativistic equations called the Lorentz transformations. The v represents the relative velocity between the inertial reference frames S and S' . The x and the x' represent the spatial coordinates of an event in the S and the S' IRFs, respectively. The t and the t' represent the time coordinates of an event in the S and the S' IRFs, respectively. The c is the speed of light. The t0 and the L0 represent the proper time and the proper length, respectively, while the ∆t and the L represent the observed time and the observed length by someone not measuring the proper entities. Proper time is that time measured by an observer for whom two events occur at the same spatial coordinates. Proper length is that length measured by an observer who is in the rest frame of the object whose length is being measured.
Utilization:
• Once a very esoteric part of physics, relativity ideas about space and time are needed in order to produce accurate global positioning systems (GPS)
Aims:
Aim 2: the Lorentz transformation formulae provide a consistent body of knowledge that can be used to compare the description of motion by one observer to the description of another observer in relative motion to the first
Aim 3: these formulae can be applied to a varied set of conditions and situations
Aim 9: the introduction of relativity pushed the limits of Galilean thoughts on space and motion


A.3 – Spacetime diagrams (CORE)


Essential idea: Spacetime diagrams are a very clear and illustrative way to show graphically how different observers in relative motion to each other have measurements that differ from each other.
Nature of science: Visualization of models: The visualization of the description of events in terms of spacetime diagrams is an enormous advance in understanding the concept of spacetime.
Understandings:
• Spacetime diagrams Spacetime tutorial / simulation (there is a set of exercises that go with this tutorial below in the Problem Set section.)
• Worldlines
• The twin paradox
Applications and skills:
• Representing events on a spacetime diagram as points
• Representing the positions of a moving particle on a spacetime diagram by a curve (the worldline)
• Representing more than one inertial reference frame on the same spacetime diagram
• Determining the angle between a worldline for specific speed and the time axis on a spacetime diagram
• Solving problems on simultaneity and kinematics using spacetime diagrams
• Representing time dilation and length contraction on spacetime diagrams
• Describing the twin paradox
• Resolving of the twin paradox through spacetime diagrams
Guidance:
• Examination questions will refer to spacetime diagrams; these are also known as Minkowski diagrams
• Quantitative questions involving spacetime diagrams will be limited to constant velocity
• Spacetime diagrams can have t or ct on the vertical axis
• Examination questions may use units in which c = 1
Data Booklet reference:
q = tan -1( v / c )
Theory of knowledge:
• Can paradoxes be solved by reason alone, or do they require the utilization of other ways of knowing?
Aims:
Aim 4: spacetime diagrams allow one to analyse problems in relativity more reliably


A.4 – Relativistic mechanics (AHL)


Essential idea: The relativity of space and time requires new definitions for energy and momentum in order to preserve the conserved nature of these laws.
Nature of science: Paradigm shift: Einstein realized that the law of conservation of momentum could not be maintained as a law of physics. He therefore deduced that in order for momentum to be conserved under all conditions, the definition of momentum had to change and along with it the definitions of other mechanics quantities such as kinetic energy and total energy of a particle. This was a major paradigm shift.
Understandings:
• Total energy and rest energy
• Relativistic momentum
• Particle acceleration Particle discovery timeline
• Electric charge as an invariant quantity
• Photons Photon packet in slow motion
• MeV c-2 as the unit of mass and MeV c-1 as the unit of momentum
Applications and skills:
• Describing the laws of conservation of momentum and conservation of energy within special relativity
• Determining the potential difference necessary to accelerate a particle to a given speed or energy
• Solving problems involving relativistic energy and momentum conservation in collisions and particle decays
Guidance:
• Applications will involve relativistic decays such as calculating the wave-lengths of photons in the decay of a moving pion
• The symbol m0 refers to the invariant rest mass of a particle
• The concept of a relativistic mass that varies with speed will not be used
• Problems will be limited to one dimension
Data Booklet reference:
E = g m0c2
E0 = m0c2
EK = ( g - 1 )m0c2
p = g m0v
E2 = p2c2 + m02c4
qV = ∆EK
• The g represents the Lorentz factor which is defined in subtopic A2 above. The E represents the relativistic energy of a particle, the m0 represents the rest mass or invariant mass of the particle, and the speed of light c = 3.00x108 ms-1. The E0 represents the rest energy of a particle. The EK represents the relativistic kinetic energy of the particle. The v represents the velocity of the particle relative to the observer. The p represents the relativistic momentum of the particle. The q represents the charge of a particle accelerated through a potential difference V.
Theory of knowledge:
• In what ways do laws in the natural sciences differ from laws in economics?
Utilization:
• The laws of relativistic mechanics are routinely used in order to manage the operation of nuclear power plants, particle accelerators and particle detectors
Aims:
Aim 4: relativistic mechanics synthesizes knowledge on the behaviour of matter at speeds close to the speed of light
Aim 9: the theory of relativity imposes one severe limitation: nothing can exceed the speed of light


A.5 – General relativity (AHL)


Essential idea: General relativity is applied to bring together fundamental concepts of mass, space and time in order to describe the fate of the universe.
Nature of science: Creative and critical thinking: Einstein’s great achievement, the general theory of relativity, is based on intuition, creative thinking and imagination, namely to connect the geometry of spacetime (through its curvature) to the mass and energy content of the spacetime. For years it was thought that nothing could escape a black hole and this is true but only for classical black holes. When quantum theory is taken into account a black hole radiates like a black body. This unexpected result revealed other equally unexpected connections between black holes and thermodynamics.
Understandings:
• The equivalence principle Einstein online
• The bending of light Deflection of light by delay
• Gravitational redshift and the Pound–Rebka–Snider experiment
• Schwarzschild black holes
• Event horizons
• Time dilation near a black hole Epic rap battle between Einstein and Hawking.
• Applications of general relativity to the universe as a whole Interactive relativity site LIGO background
Applications and skills:
• Using the equivalence principle to deduce and explain light bending near massive objects
• Using the equivalence principle to deduce and explain gravitational time dilation
• Calculating gravitational frequency shifts
• Describing an experiment in which gravitational redshift is observed and measured
• Calculating the Schwarzschild radius of a black hole
• Applying the formula for gravitational time dilation near the event horizon of a black hole
Guidance:
• Students should recognize the equivalence principle in terms of accelerating reference frames and freely falling frames
Data Booklet reference:
• ∆f / f =gh / c2
Rs = 2GM / c2
• ∆t = ∆t0 / sqrt( 1 – Rs / r )
• The f represents change in frequency f of light caused by the gravitational red shift in a gravitationa field having a strength g over a change in height h. The Schwartzchild Radius Rs is that radius a mass M must be in order to have an escape velocity of c (i.e.: light cannot escape from it and the mass is called a "black hole"). The Universal gravitational constant G = 6.67x10-11 N m2 kg -2. The t represents the gravitational time dilation at a distance r from the center of a black hole having Rs. The t0 represents the proper time.
Theory of knowledge:
• Although Einstein self-described the cosmological constant as his “greatest blunder”, the 2011 Nobel Prize was won by scientists who had proved it to be valid through their studies on dark energy. What other examples are there of initially doubted claims being proven correct later in history?
Utilization:
• For the global positioning system to be so accurate, general relativity must be taken into account in calculating the details of the satellite’s orbit
• The development of the general theory of relativity has been used to explain the very large-scale behaviour of the universe as a whole with far-reaching implications about the future development and fate of the universe
Aims:
Aim 2: the general theory of relativity is a great synthesis of ideas that are required to describe the large-scale structure of the universe
Aim 9: it must be appreciated that the magnificent Newtonian structure had serious limitations when it came to the description of very detailed aspects of planetary motion



OPTION A PROBLEM SET

This is the complete problem set for OPTION A - the same one I hand out. If you lose yours, you can download this one to replace it.


Because of the fascination we have with spacetime geometry, here is a set of exercises covering Minkowski space, compliments of IB instructor Jonathan Crymes from Norcross High School. There is a link within the document to a simulation that is very visual.



OPTION A FORMATIVE ASSESSMENTS

These are the Formative Assessments (practice) that you will do in order to prepare yourself for the Summative Assessments (evidence of proficiency). You can expect to receive a mark of at least Proficient on the Summative Assessment if you understand everything on these Formative Assessments.





OPTION A PROJECTS

Project marks are meant to replace summative assessment marks. Projects are your last opportunity to demonstrate your proficiency in meeting the standards of the assessment criteria.



EXTENSION NOTES FOR ENRICHMENT

The following extension notes cover relativity in more detail. There is a smattering of calculus in some of the notes. For each document there is an associated problem set. I have not yet developed the answer keys for the problems. These documents were originally designed by me during a summer program at the University of Wisconsin - Milwaukee, as independent study material for AP Physics (calculus treatment) students. I would recommend that you try these in the order they are displayed, if you try them at all.

SPECIAL RELATIVITY




















GENERAL RELATIVITY









BOOKS
Spooky Action At A Distance - George Musser

This book is a fun and mind-expanding read on alternatives to Einstein's universe of spacetime. Einstein himself considered the possibility of action at a distance (rather than field). He viewed the particles of the universe as responding to local interactions - basing his theory of relativity on this fundamental belief, but had a feeling that they could also be subjected to nonlocal interactions. You may have heard of this nonlocal interaction being called "quantum entanglement." The book delves into entanglement, and its possible consequences on spacetime.