PH 103 Dr. Cecilia Vogel Lecture 14

Review Outline  Consequences of Einstein’s postulates  time dilation  simultaneity  Einstein’s relativity  1 st and 2 nd postulates of special relativity  equivalence principle of general relativity

Some Consequences Can be derived from constancy of speed of light:  Time interval between two events is different measured by different observers.  Order of events may be different measured by different observers.  Length of object or length of a trip is different measured by different observers.

Recall Classical Relativity  Suppose two observers time the pretzel you throw and catch. One observer on plane, one on Earth. Same pretzel. Go-stop. t=5 s Go stop. t =?

Recall Classical Relativity  At any point, let the velocity of the pretzel measured by the plane observer be v p.  The velocity measured by Earth observer is v plane + v p. Earth observer measures faster speed:  pretzel goes farther, faster, but same time (5s) Compared to this frame, in this frame, the pretzel goes... farther

Now Einstein’s Relativity  That worked for pretzels, what about light?  Person on super-plane shines light at mirror.  Suppose two observers time the light that shines and reflects. One observer on plane, one on Earth. Same light. Go-stop. t o =5  s Go stop. t=?

Now Einstein’s Relativity Compared to this frame, in this frame, light goes...  At any point, the velocity of the light measured by the plane observer is c  And the velocity measured by Earth observer is c, so Earth observer measures:  Light goes farther, same speed→ farther longer time! t>5  s

Time Dilation  Generalize result:   t and  t o are both the time between same events  measured by different observers  v is relative speed of the two observers  Notice that if v<<c, the two times are .  Hard part: which time is which?

Time Dilation  Units comment  If velocities are given as a fraction of c,  then the “c”units cancel.  Example:  If v= 0.8 c,  then (v/c) =(0.8c/c) = 0.8 (no units)

Proper time  What’s the difference between  t and  t o ?   t o is the “proper time”  It is always less than any other measured time.  Definition : proper time is time in frame in which the two events occur at same place  Events are  single instant in time, single point in space,  Events don’t move.  They don’t have reference frame. Occur in all frames  Time interval is time between two events  One event makes you start your timer  other even makes you stop you timer.

Proper time  What’s the difference between  t and  t o ?   t o is the “proper time”  Definition : proper time is time in frame in which events occur at same place  For example  If the two events happen to someone or something, their frame measures proper time  your heartbeat  If someone is at both events, (or same  x away) their frame measures proper time  person taking trip  particle’s lifetime  you and the clock on the wall

Time Trip  Nikos travels to a planet 12 light-years away at a speed of 0.6 c. Juan stays on Earth. Each measures the trip to take a different amount of time. Note:  A light-year is distance light travels in a yr  1 light-year = (c)(1yr) = 1c-yr  Consistent units: distance in light-years, speed as fraction of c, time in years  The values in example are relative to Earth  In that frame (in any one frame), the laws of physics hold, including d = vt Ex: 12c-yr/24yr = 0.5c

More Example  If Nikos makes a 12 light-year trip at 0.6c, Juan sees him moving at 0.6c for 20 years.  d = vt = (0.6c)(20 y) = 12c-yr.  Nikos sees himself moving at   Nikos sees the planet getting closer at 0.6c for 16 years.  d = vt = (0.6c)(16y) = 9.6 c-yr  Both are measuring the distance between Earth and planet, yet the distances are different!

Proper Length  What’s the difference between the two lengths?  One is the “proper length”  It’s always longer than any other measured length.  Def: proper length is length in frame in which object (or ends of trip) is at rest  For example  Object, or anyone at rest relative to it, measures object’s proper length.  Your own height  Length of ship you are riding on  Someone measures the proper length between two objects, if both are at rest relative to them  person on either planet, for a trip between planets

Just How Proper is it? If there is a proper time and a proper length, is there a proper reference frame?  NO!!!!!!!  Proper time of trip in example: Nikos  Proper length of trip in example: Juan  Proper time of astronaut’s heartbeat: Astronaut  Astronaut’s heartbeat looks SLOW to you.  Proper time of your heartbeat: You  Your heartbeat looks SLOW to astronaut.

Simultaneity  How do we know whether 2 events are simultaneous?  If signals coming from events arrive at the same time, the events might not have been simultaneous  how long did it take the signal to get here? demo

If t arrive, d, and v the same, conclude t event same

If t arrive, is smaller and d/v also smaller, can conclude t event same v eraser +v thrower V eraser -v thrower

If t arrive is earlier and d/v is same, must conclude t event is earlier! one observer says two events are simultaneous, other says they are not!! SIMULTANEITY IS NOT ABSOLUTE