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PH 301

PH 301. Dr. Cecilia Vogel Lecture 5. Velocity transformation NOT simple addition Spacetime intervals, diagrams. Review. Lorentz transformations order of events twin paradox. Outline. u = dx/dt u’ = dx’/dt’. Velocity. What’s dt’/dt?. SO…. Note: Speed will never be bigger than c

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PH 301

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  1. PH 301 Dr. Cecilia Vogel Lecture 5

  2. Velocity transformation NOT simple addition Spacetime intervals, diagrams Review • Lorentz transformations • order of events • twin paradox Outline

  3. u = dx/dt u’ = dx’/dt’ Velocity

  4. What’s dt’/dt? SO…

  5. Note: Speed will never be bigger than c If u’ and v are <c, then u<c If |u’| or |v| =c, then u=c speed of light the same Pay attention to the sign of velocities Pay attention to order of frames Velocity Transformation

  6. Step 1: Let u = answer you seek. Step 2: u = velocity of A rel to B, so A and B are determined. Step 3: Identify frame C -- what’s left? Step 4: Determine u’ u’= velocity of A rel to C If you have C rel to A, use opposite sign Step 5: Determine v v = velocity of C rel to B If you have B rel to C, use opposite sign Step 6: Plug in the numbers to compute u. Step 7: Check that your answer makes sense! Using Velocity Transformation

  7. A spaceship is approaching the planet Zorgon at a speed of 0.85c. A diplomatic shuttle is sent ahead to arrive at the planet earlier. With what velocity should the shuttle be launched relative to the ship, in order to approach the planet at a rate of 0.95c? EXAMPLE u = velocity of shuttle relative to the ship so… A is ________ and B is _____ Then C is _________

  8. A spaceship is approaching the planet Zorgon at a speed of 0.85c. A diplomatic shuttle is sent ahead to arrive at the planet earlier. With what velocity should the shuttle be launched relative to the ship, in order to approach the planet at a rate of 0.95c? EXAMPLE u’ = velocity of A relative to C so… u’ is velocity of ________ relative to _____ u’ = +0.95 c

  9. A spaceship is approaching the planet Zorgon at a speed of 0.85c. A diplomatic shuttle is sent ahead to arrive at the planet earlier. With what velocity should the shuttle be launched relative to the ship, in order to approach the planet at a rate of 0.95c? EXAMPLE v = velocity of C relative to B, so… v is velocity of _______________ relative to _____ v = Ship relative to Zorgon is +0.85 c, so

  10. With what velocity should the shuttle be launched relative to the ship, in order to approach the planet at a rate of 0.95c? EXAMPLE u’ = +0.95 c, v = -0.85 c

  11. A proton is traveling at a speed of 0.75c relative to the lab. A neutron is to collide with it at a relative speed of 0.90c. With what velocity should the neutron go relative to the lab? EXAMPLE u = velocity of neutron relative to the lab so… A is ________ and B is _____ Then C is _________

  12. A proton is traveling at a speed of 0.75c relative to the lab. A neutron is to collide with it at a relative speed of 0.90c. With what velocity should the neutron go relative to the lab? EXAMPLE u’ = velocity of A relative to C so… u’ is velocity of ________ relative to _____ neutron proton u’ =

  13. A proton is traveling at a speed of 0.75c relative to the lab. A neutron is to collide with it at a relative speed of 0.90c. With what velocity should the neutron go relative to the lab? EXAMPLE v = velocity of C relative to B, so… v is velocity of _______________ relative to _____ proton lab v =

  14. With what velocity should the neutron go relative to the lab? EXAMPLE

  15. Space-time Considers time as a fourth dimension. An event is given by a 4-component vector: 3 space, 1 time I can’t draw in 4 dimensions Let’s consider 1 space & 1 time

  16. Space-time Diagrams An event is a point on the diagram. ct World-line of light • A world-line is the path of an object on the diagram. x • The steeper the slope, the slower it’s going. • Slope = 1, means speed c.

  17. Classical Invariant In classical relativity, everyone measures the same distance between two events in space: If • Then

  18. Invariant in Space-time In space-time, we let the 4-component vector be (x, y, z, ict) So that the space-time interval, is invariant.

  19. Spacetime • recall • special relativity (ch 2) • An event is something that occurs at a particular place and time – at a particular point in spacetime • Spacetime graph • Graphs an event as a point in 4-D spacetime • (x, y, z, t) • We will consider 1-D space, 1-D time • 2-D graphs are easier to draw!

  20. Worldline • Worldline of an object • Is the set of all spacetime points occupied by the object • Although t is ordinate, and x is abscissa, do not think of t(x) • Slope of worldline • d(ct)/dx = c/(dx/dt) • v/c = 1/slope • steeper  slower • vertical  stopped • |slope| = 1  |v|=c, worldline of light • generally, worldline |slope|>1

  21. Transform Worldlines • A Spacetime graph is drawn from a particular reference frame • In the spacetime graph drawn from a different reference frame • the slope of the worldline of a massive object is different • according to the velocity transformation equation • the slope of the worldline of light is not different • slope is still +1

  22. Spacetime Future • Given a particular event at xo, yo, zo, to, • the points on a spacetime graph are divided into 3 regions: its future, its past, and its elsewhere • Spacetime Future of the event • the set of all spacetime points such that • t> to, AND • d<cDt • d = spatial distance between x, y, z and xo, yo, zo • Dt= t- to

  23. Spacetime Past • Spacetime Past of the event • the set of all spacetime points such that • t< to, AND • d<c|Dt| • NOTE: • The event can be reached by a signal from its past • this event can be affected by events in the past • A signal from the event can reach points in the future • this event can affect events in its future

  24. Lightcone • An event’s lightcone • is the set of spacetime points such that • d=c|Dt| • It is the boundary of the future • and of the past • A signal from this event can only reach events in the lightcone • by traveling at the speed of light • This event can only be reached by a signal from an event in the lightcone • if the signal travels at speed of light

  25. Lightcone, Past, and Future lightcone |slope| =1 future past

  26. Elsewhere • The Elsewhere of the event • consists of all the other spacetime points (other than lightcone, past, future) • d>c|Dt| • The event cannot be reached by, nor can anything from the event reach, an event in its elsewhere • this event cannot affect nor be affected by events in its elsewhere

  27. Elsewhere is not… • This does NOT mean • that an event that is currently in our elsewhere can never affect us • that event may be in the past of future points on our worldline • It also does NOT mean • that an event that is currently in our elsewhere can never have been affected by us • that event may have been in the future of past points on our worldline

  28. Elsewhere Example • For example • if the sun had disappeared 4 minutes ago • that event is in our elsewhere right now • d= 8c-min, c|Dt| = 4 c-min, d>c|Dt| • BUT, four minutes from now, that event will be in our past, and we will be gravely affected! worldline of sunlight our worldline + 4 min sun disappearing - 4 min

  29. Transform lightcone • The lightcone of an event • is the same set of points in all reference frames • All observers agree on • which events are in the event’s future • and its past • and its elsewhere

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