PH 301 Dr. Cecilia Vogel Lecture 5

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

Velocity  u = dx/dt  u’ = dx’/dt’

What’s dt’/dt? SO…

Velocity Transformation  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

Using Velocity Transformation  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!

EXAMPLE A spaceship is approaching the planet Zorgon at a speed of 0.85 c. 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 0.95 c ? u = velocity of shuttle relative to the ship so… A is ________ and B is _____ Then C is _________

EXAMPLE A spaceship is approaching the planet Zorgon at a speed of 0.85 c. 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 0.95 c ? u’ = velocity of A relative to C so… u’ is velocity of ________ relative to _____ u’ = c

EXAMPLE A spaceship is approaching the planet Zorgon at a speed of 0.85 c. 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 0.95 c ? v = velocity of C relative to B, so… v is velocity of _______________ relative to _____ v = Ship relative to Zorgon is c, so

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

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

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

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

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

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

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

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

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.

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!

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

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

Spacetime Future  Given a particular event at x o, y o, z o, t o,  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> t o, AND  d<c  t  d = spatial distance between x, y, z and x o, y o, z o   t= t- t o

Spacetime Past  Spacetime Past of the event  the set of all spacetime points such that  t< t o, AND  d<c|  t|  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

Lightcone  An event’s lightcone  is the set of spacetime points such that  d=c|  t|  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

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

Elsewhere Elsewhere  The Elsewhere of the event  consists of all the other spacetime points (other than lightcone, past, future)  d>c|  t|  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

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

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

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