P301 Lecture 4Postulates of Special Relativity

• I- The laws of physics are the same in all inertial reference frames of reference (this is the same statement that Galileo used)– Note that the extra bits in the book’s statement are not quite correct. The

laws of physics do not provide a preferred frame of reference, but the Cosmic Microwave Background does give us a way of detecting motion relative to the big bang (CoM), and is this sense it can provide a “preferred” local frame against which others could be compared).

• II- The speed of light is a law of physics (it is a constant: c=1/()1/2 =299,792,458 m/s; NOTE this is 0.983~1.0ft/ns)

– This is idea is contained within Maxwell’s equations, but it took Einstein to take the leap and say that it is a law of physics that should not depend on the reference frame (according to postulate I). This has far-reaching and VERY COUNTER INTUITIVE consequences. Understanding at least some of those consequences is our task for the next several lectures.

P301 Lecture 4Time dilation: Lorentz Contraction

• Any clock (see next slide) in the moving frame appears to advance more slowly than a clock in a stationary frame.

• Confusion arises in the treatments given in many texts, because you inevitably have to talk about events in different positions at different times in making a real comparison for time intervals and lengths.

• All observers must agree about coincident EVENTS (i.e. Things that happen at the SAME TIME AND PLACE), and on the INTERVAL (s) between those events:

s2 = (ct)2 – (r)2

• They need not agree on the time or spatial separation individually.

• In casual discussions of Lorentz contraction and time dilation, we tend to gloss over this feature, and that leads to much of the confusion.

• Consider what lines of constant ct’ (a “NOW” line) and constant x’ (a “HERE” line) appear on a graph of ct vs. x (a space-time diagram).

Muon Flux (quarknet result)

http://cosmic.lbl.gov/SnowMass/main.htmlExponent is 0.321/km

These data agree well with the expected decay for muons(t1/2=2.2sec) travelling at 0.96c IFF you account for time dilation.

P301 Lecture 4Tests of Lorentz Invariance

Excerpt from a Sept. 2004 Sci. Am. Article by IU Prof. V. A. Kostelecky, See also his website: http://www.physics.indiana.edu/~kostelec/faq.html

P301 Lecture 4Relative Simultaneity

How might I synchronize two clocks separated by a distance L?

Send a light flash from the point midway between the two, and start the clocks with the arrival of the light.

Does this really do the trick? What does it mean to say that the two clocks are truly synchronized if they are separated by a distance L?

L

P301 Lecture 4Relative Simultaneity

How might I synchronize two clocks separated by a distance L?

Send a light flash from the point midway between the two, and start the clocks with the arrival of the light.

Does this really do the trick? What does it mean to say that the two clocks are truly synchronized if they are separated by a distance L?I could use this same trick to link these events to a different reference frame (say) by having the laser beams bounce off mirrors and place marks on the other frame (say railway tracks with the mirrors and light source on the train). The two must agree that the two marks were made on the track and on what positions on the track they are made (say the track was ruled like a long meter stick).

L

P301 Lecture 4Relative Simultaneity

How might I synchronize two clocks separated by a distance L?

Send a light flash from the point midway between the two, and start the clocks with the arrival of the light.

Does this really do the trick? What does it mean to say that the two clocks are truly synchronized if they are separated by a distance L?

What do I conclude if I view this activity from a frame that is moving to the left at a high speed? Say that I have mirrors at the clocks, which deflect some of this light onto the moving object to see where and when they arrive, as viewed in that frame?

L

P301 Lecture 4Relative Simultaneity

Events that are simultaneous in the train’s frame are NOT simultaneousin the track frame! But we can’t say which one is standing still, so we cannotsay whether the two events did, in fact, take place at the same time*!!!

P301 Lecture 4Space-time diagrams

From Tipler. The path of a particleThrough this diagram (its “worldline”)Describes its location as a function of timeSuch diagrams play an important role in Feynman’s version of quantum mechanics,And they also help to visualize the distortionsIntroduced by the Lorentz transformation.

Light-cone

In this case events at O and A would have a “time-like” separation, and the world line connecting them is said to be time-like as well

Worldline

P301 Lecture 4Space-time diagrams

From Tipler. The path of a particleThrough this diagram (its “world line”)Describes its location as a function of timeSuch diagrams play an important role in Feynman’s version of quantum mechanics,And they also help to visualize the distortionsIntroduced by the Lorentz transformation.

Light-cone

Line of “now”

Line of “now’ ”

P301 Lecture 4Space-time diagrams

From Tipler: note that the x’ axis is defined by the condition ct’=0, which is equivalent to ct=x according to the Lorentz transformation. You read info off the primed axes by looking along lines parallel to those axes; note that both frames agree on the speed of light 1 space unit (say a “foot”) in one ct unit (also a “foot”, or 1 ns of time)

P301 Lecture 4Space-time diagrams

From Tipler: note that the x’ axis is defined by the condition ct’=0, which is equivalent to ct=x according to the Lorentz transformation. You read info off the primed axes by looking along lines parallel to those axes; note that both frames agree on the speed of light 1 space unit (say a “foot”) in one ct unit (also a “foot”, or 1 ns of time)

Time dilation shows up here

P301 Lecture 4Space-time diagrams

From Tipler: note that the x’ axis is defined by the condition ct’=0, which is equivalent to ct=x according to the Lorentz transformation. You read info off the primed axes by looking along lines parallel to those axes; note that both frames agree on the speed of light 1 space unit (say a “foot”) in one ct unit (also a “foot”, or 1 ns of time)

NOTE: things that are simultaneous in one frame are not in the other!

P301 Lecture 4Space-time diagrams

For more practice with these sorts of diagrams, consider the website:http://www.cco.caltech.edu/~phys1/java/phys1/Einstein/Einstein.html This is useful for seeing some of the results, and it is used a lot in Feynman’s approach to QM, and in QFT in general.

P301 Lecture 5Proper time and length

• Recall that the interval is and invariant and is defined through:

s2 = (ct)2 – (r)2

• Different reference frames need not agree on the time or spatial separation individually.

• We define the PROPER TIME interval () between two events that take place at the same position in a reference frame as the time interval measured by a clock that is at rest in that same frame. (i.e. s/c).

• We define the PROPER LENGTH as the distance between two points measured at rest.

P301 Lecture 4Space-like and Time-like intervals

For some moving frame x’ I can line up any event below the light cone along the x’ axis by choosing the speed properly, so it appears to have happened at the same time as the event at O.

(space-like separation).

O

P301 Lecture 4Space-like and Time-like intervals

Similarly I can line up any event above the light cone along the ct’ axis by choosing the speed properly, so it appears to have happened at the same place (i.e. same x’ coordinate) as the event at O.

(time-like separation).

O

P301 Lecture 5Momentum and Energy

Consider a particular two-D elastic collision in two different reference frames:•Center of mass frame•The frame of particle B•What is the x-component of the final velocities in the second case?Is momentum conserved in each?

P301 Lecture 5Momentum and Energy

Consider a particular two-D elastic collision in two different reference frames:•Center of mass frame•The frame of particle B•What is the x-component of the final velocities in the second case?Is momentum conserved in each?In the second case it is NOT conserved for large enough v, if you assume that p=mov

P301 Lecture 5Relativistic Energy and momentum

• The Lorentz-invariant momentum is defined as:

p=mov• The Lorentz-invariant energy is defined as:

E=moc2

• NOTE: kinetic energy is then defined as the difference between E and the rest energy moc2. There is opportunity for confusion when E is only a little greater than the rest energy (some people get lazy and forget about the rest energy and use E when they really mean kinetic energy).

• Where mo is the mass of the system measured at rest.

• In some books (but not Thornton and Rex), a relativistic mass is defined as m = mo (and therefore depends on the object’s speed). T&R use m to refer to the rest mass only. I will endeavour to always use mo when referring to rest mass, and to always write mo rather than just m, but be fore-warned of possible confusion when looking at other texts.

P301 Lecture 5JITT

How much energy is released in the mutual annihilation of a positron and electron? (at rest)

– Most had little trouble with this (assuming they survived my screw-up on due date), but a few warnings:

P301 Lecture 5JITT

How much energy is released in the mutual annihilation of a positron and electron? (at rest)

– Most had little trouble with this (assuming they survived my screw-up on due date), but a few warnings:

– 8.2E-14 + 8.2E-14 = 1.6E-13 (what is wrong with this?)

P301 Lecture 5JITT

How much energy is released in the mutual annihilation of a positron and electron? (at rest)

– Most had little trouble with this (assuming they survived my screw-up on due date), but a few warnings:

– 8.2E-14 + 8.2E-14 = 1.6E-13 (what is wrong with this?)– E = 7.2e10 Joules (probably the same problem as above, but

his time the student was not lucky. UNITS!!!)– K=mc^2(gamma-1) (the Q asked for energy not K, in this case

the final state has two photons and the initial state has gamma=1, so K=0. Be careful with K and E.)

Classical Doppler Effect

The frequency shifts up if the source and observer are getting closer to each other, and down if they are receding from each other. Think of old “murder on the train movies”, or the sound of an Indy car as it goes by.

But for sound this is explicitly different for a moving source (right fig.) or moving observer (left fig.), because vs is fixed w.r.t. the medium (usually taken to be stationary). For light, this cannot be the way to view the problem!!

Doppler Effect for light

In this case, we have to remember that there are several different times involved: TS (the proper period of the source clock); TO (the period of the source clock as observed by the receiver); TR (the time interval between the arrival of pulses at the receiver)

In the time TO, the receiver sees the source move a distance vTO, and therefore we obtain the relationship:

cTO + vTR = cTR

See also: http://en.wikipedia.org/wiki/Relativistic_Doppler_effect

Source

pulses

Source Frame

cTs

cTo

vTR

cTR

cTR

Receiver Frame

t=0

t=TR

t=2TR

t=0, TS, 2TS

P301 Lecture 5Our motion through the Universe

http://apod.nasa.gov/apod/ap010128.html

Dipole asymmetry in the Cosmic Microwave Background caused by Earth’s motion w.r.t. that background radiation (this speed is some 600 km/s)

P301 Lecture 6“Headlight effect”

This effect is important in medium-energy physics experiments (lots of the “action” is in the forward direction), and for “synchrotron radiation” sources (very bright sources of x-rays obtained from highly relativistic electrons or positrons). See also the spreadsheet “P301_angle_transf.xls” on the web site.

P301 Lecture 6“Headlight effect”

http://www.google.com/imgres?imgurl=http://www.physik.uni-kiel.de/kekm-bilder/articles/Keule.jpg&imgrefurl=http://forums.myspace.com/t/4704111.aspx%3Ffuseaction%3Dforums.viewthread&usg=__InY_kykyRqgFy8rnHvsZUEeBsr4=&h=434&w=740&sz=155&hl=en&start=0&zoom=1&tbnid=RvcNUuMdTbzkBM:&tbnh=111&tbnw=190&prev=/images%3Fq%3Dsynchrotron%2Bradiation%26um%3D1%26hl%3Den%26client%3Dfirefox-a%26hs%3DmJy%26sa%3DX%26rls%3Dorg.mozilla:en-US:official%26biw%3D1263%26bih%3D762%26tbs%3Disch:1&um=1&itbs=1&iact=rc&dur=565&ei=ef6ETL65BcH68Ab3qIBt&oei=ef6ETL65BcH68Ab3qIBt&esq=1&page=1&ndsp=24&ved=1t:429,r:2,s:0&tx=96&ty=67

P301 Lecture 6Synchrotron Radiation facilities

The Advanced Photons Source at Argonne National Lab (Chicago) has 40 or more beam lines devoted to materials research, biochemistry, …

http://www.aps.anl.gov/Beamlines/Beamlines_Map/index.html

P301 Lecture 6“Examples from the Text”

•2-51 A spacecraft travelling out of the solar system at 0.92c sends back information at a rate of 400 Hz. At what rate do we receive the information?•2-78 Calculate the energy needed to accelerate a 10,000kg spacecraft to 0.3c for intergalactic exploration. Note: the current annual GLOBAL energy consumption is roughly 1021 J (DVB: suppose you took a year to reach this speed, how many nuclear reactors would be needed to provide the required power. If you had an antimatter drive, how much fuel would you need?).

P301 Lecture 6“Michelson Interferometer”

http://hyperphysics.phy-astr.gsu.edu/HBASE/PHYOPT/michel.html

•This is a classic piece of apparatus (in a more sophisticated form it is now being used to search for gravity waves, see LIGO).•The version used by M&M used many more bounces to increase its sensitivity (see text)•Basic idea is that a change in length (or transit time) of either arm will shift the position of the observed interference fringes.•LIGO looks for change in length on the order of 10-18 m out of arms that are 4x103 m long•Also the device at the heart of an FTIR spectrometer

P301 Lecture 6“LIGO”

http://www.jb.man.ac.uk/research/pulsar/images/Ligo_hanford.jpg

•This is one of two large Michelson interferometers built in the US (one in Richland Washington state, the other in Livingston Louisiana).