What is Relativity?
Relating measurements in one reference frame to those in
a different reference frame moving relative to the first
1905 - Einstein’s first paper on relativity, dealt with inertial
reference frames (Special Relativity)
1915 - Einstein published theory that considered
accelerated motion and its connection to gravity (General
Relativity)
Special Relativity
GR describes black holes, curved spacetime, and the
evolution of the universe; very mathematical
SR deals with a “special case” case of motion - motion at a
constant velocity (acceleration is zero)
SR is restricted to inertial reference frames - relative
velocity is constant
Reference Frames
Inertial Reference Frame:
Reference Frames
Inertial Reference Frame:
A reference frame in which Newton’s first law is valid
Reference Frames
Which of these is an inertial reference frame (or a very
good approximation)?
a.
b.
c.
d.
e.
f.
Your bedroom
A car rolling down a steep hill
A train coasting along a level track
A rocket being launched
A roller coaster going over the top of a hill
A skydiver falling at terminal speed
Standard Reference Frames S and S’
Galilean Transformations of Position
If you know a position measured in one inertial reference
frame, you can calculate the position that would be
measured in any other inertial reference frame...
Suppose a firecracker explodes at time t. The
experimenters in reference frame S determine that the
explosion happened at position x. Similarly, the
experimenters in S’ (which moves at a velocity v) find that
the firecracker exploded at x’ in their reference frame. What
is the relationship between x and x’?
Galilean Transformations of Velocity
If you know the velocity of a particle in one inertial
reference frame, you can find the velocity that would be
measured in any other inertial reference frame...
Suppose the experimenters in both reference frames now
track the motion of an object by measuring its position at
many instants of time. The experimenters in S find that the
object’s velocity is u. During the same time interval Δt, the
experimenters in S’ measure the velocity to be u’.
Galilean Transformations of Velocity
Use u and u’ to represent the velocities of objects with
respect to reference frames S and S’.
Find the relationship between u and u’ by taking the time
derivatives of the position equations. (Recall: ux = dx/dt)
Example
An airplane is flying at speed 200 m/s with respect to the
ground. Sound wave 1 is approaching the plane from the
front, sound wave 2 is catching up from behind. Both waves
travel at 340 m/s relative to the ground. What is the speed
of each wave relative to the plane?
A simpler example...
Ocean waves are approaching the beach at 10 m/s. A boat
heading out to sea travels at 6 m/s. How fast are the waves
moving in the boat’s reference frame?
Einstein’s Principle of Relativity
All the laws of physics
are the same in all
inertial reference
frames.
Maxwell’s Contribution
● Maxwell’s equations are true in all inertial
reference frames
● Maxwell’s equations predict that
electromagnetic waves, including light, travel
at speed c = 3 x 108 m/s
● Therefore, light travels at speed c in all
inertial reference frames.
Implications
Implications
● Recent experiments use unstable
elementary particles, mesons, that
decay into high energy photons of
light.
● Every experiment designed to
compare the speed of light in different
reference frames has found that light
travels at speed c in every inertial
reference frame, regardless of how
the reference frames are moving with
respect to each other.
Example
Use a Galilean
transformation
to determine
the bicycle’s
velocity.
Example
Repeat your measurements but measure the
velocity of the light wave as it travels from the
tree to the lamppost.
Example
Repeat your measurements but measure the
velocity of the light wave as it travels from the
tree to the lamppost.
● Δx’ differs from Δx
● u’ differs from u
● BUT experimentally, u’ = u…
● What does this tell us about our assumptions
regarding the nature of time?
Events and Measurements
Event: a physical activity that takes place at a
definite point in space and a definite instant in
time.
Spacetime coordinates
(x, y, z, t)
Measurements
The (x, y, z) coordinates of
an event are determined
by the intersection of the
meter sticks closest to the
event.
The event’s time, t, is the
time displayed on the
clock nearest the event.
Stop and Think
A carpenter is working on a house two blocks away.
You notice a slight delay between seeing the
carpenter’s hammer hit the nail and and hearing the
blow. At what time does the event “hammer hits nail”
occur?
a. at the instant you hear the blow
b. at the instant you see the hammer hit
c. very slightly before you see the hammer hit
d. very slightly after you see the hammer hit
Synchronization of Clocks
Detection of light wave sent out from origin.
How long does it take for light to travel 300 m?
Finding the time of an event
Experimenter A in a reference frame S stands at the origin
looking in the positive x-direction. Experimenter B stands at
x = 900 m looking in the negative x-direction. A firecracker
explodes somewhere between them. Experimenter B sees
the light flash at t = 3.0 µs. Experimenter A sees the light
flash at t = 4.0 µs. What are the spacetime coordinates of
the explosion?
Finding the time of an event
Simultaneity
When two events occurring at different positions take place
at the same time.
An experimenter in reference frame S stands at the origin
looking in the positive x-direction. At t = 3.0 µs she sees
firecracker 1 explode at x = 600 m. A short time later, at t =
5.0 µs, she sees firecracker 2 explode at x = 1200 m. Are
the two explosions simultaneous? If not, which firecracker
exploded first?
Stop and Think
A tree and pole are 3000 m apart. Each is suddenly hit by a
bolt of lightning. Mark, who is standing at rest midway
between the two, sees the two lightning bolts at the same
instant in time. Nancy is at rest under the tree. Define event
1 to be “lightning strikes tree” and event 2 to be “lightning
strikes pole.” For Nancy, does event 1 occur before, after,
or at the same time as event 2?
A “Thought Experiment”...
A long railroad car is traveling to the right with a velocity v.
A firecracker is attached to each end of the car, just about
the ground. Each firecracker will make a burn mark on the
ground when where they explode. Ryan is standing on the
ground; Peggy is standing in the exact center of the car
with a light detector.
The Event in Ryan’s Frame
The Event in Peggy’s Frame
The real sequence
of events in
Peggy’s reference
frame
Relativity of Simultaneity
Two events occurring simultaneously in
reference frame S are not simultaneous in any
reference frame S’ moving relative to S.
Stop and Think
A tree and a pole are 3000 m apart. Each is hit by a bolt of
lightning. Mark, who is standing at rest midway between the
two, sees the two lightning bolts at the same instant of time.
Nancy is flying her rocket at v = 0.5c in the direction from
the tree toward the pole. The lightning hits the tree just as
she passes by it. Define event 1 to be “lightning strikes
tree” and event 2 to be “lightning strikes pole.” For Nancy,
does event 1 occur before, after, or at the same time as
event 2?
Time Dilation
Time is no longer an absolute quantity: it is not
the same for two reference frames moving
relative to each other.
● Time interval between two events
● Whether two events are simultaneous
Depends on the observer’s reference frame.
Time Dilation - A light clock
The light source emits a very
short pulse of light that travels
to the mirror and reflects back
to a light detector next to the
source. The clock advances
one “tick” each time the
detector receives a light pulse
and the light source
immediately emits the next
light pulse.
Time Dilation - A light clock
Two experimenters measure the interval between two
clicks of the light clock. The clock is at rest in reference
frame S’. Reference frame S’ moves to the right with
velocity v relative to reference frame S.
Time Dilation - A light clock
Event 1: the emission of a light pulse
Event 2: the detection of that light pulse
In frame S, Δt = t2 - t1
In frame S’, Δt’ = t’2 - t’1
Time Dilation - A light clock
In terms of h and c, what is the time interval, Δt’, in the
clock’s rest frame, S’?
Compare this to the time interval, Δt, in reference frame S.
(Use the classical analysis approach in which the speed of
light does depend on the motion of the reference frame
relative to the light source.)
Time Dilation - A light clock
Time Dilation - A light clock
Time Dilation - A light clock
A classical analysis finds that the clock ticks at exactly the
same rate in both frame S and frame S’.
Show that the time intervals are not the same according to
the principle of relativity.
Time Dilation - A light clock
Time Dilation
Δτ is the time interval between two events that occur at the
same position and called proper time.
Clocks moving relative to an observer are measured by that
observer to run more slowly.
(Or, the time interval between two ticks is the shortest in
the reference frame in which the clock is at rest.)
Example - from the Sun to Saturn
Saturn is 1.43 x 1012 m from the sun. A rocket
travels along a line from the sun to Saturn at a
constant speed 0.9c relative to the solar
system. How long does the journey take as
measured by an experimenter on Earth? As
measured by an astronaut on the rocket?
Stop and Think
Molly flies her rocket past Nick at constant velocity v.
Molly and Nick both measure the time it takes the
rocket, from nose to tail, to pass Nick. Which of the
following is true?
a. Both Molly and Nick measure the same amount of
time.
b. Molly measures a shorter time interval than Nick.
c. Nick measures a shorter time interval than Molly.
The Twin Paradox
George and Helen are twins. On their 25th
birthday, Helen departs on a starship voyage to a
distant star. Her starship accelerates almost
instantly to a speed of 0.95c and that she travels
9.5 light years (9.5 ly) from Earth. Upon arriving,
she discovers that the planets circling the star are
inhabited by fierce aliens, so she immediately turns
around and heads home at 0.95c.
The Twin Paradox
A light year is the distance that light travels in
one year.
According to George:
● how old will he be when his sister returns?
● how old will Helen be when she returns to Earth?
According to Helen:
● how old will she be when she returns to Earth?
● how old will George be when she returns to Earth?
Reconciling the Twin Paradox
It is logically impossible for each to be younger
than the other at the time they are reunited.
● Are the situations truly symmetrical?
● Do both observers spend the entire time in
an inertial reference frame?
● Who is actually younger?
Length Contraction
Consider the rocket that traveled from the sun
to Saturn in the previous example.
What is the length of the spatial interval in both
the S and S’ reference frames?
Length Contraction
Length Contraction
The distance between two objects in reference
frame S’ is not the same as the distance
between the same two objects in reference
frame S.
Where l is the proper length.
The distance from the sun to Saturn
A rocket traveled along a line from the sun to
Saturn at a constant speed of 0.9c relative to
the solar system. The Saturn-to-sun distance
was given as 1.43 x 1012 m. What is the
distance between the sun and Saturn in the
rocket’s reference frame?
Another Paradox?
Vignesh and David are in their physics lab
room. They each select a meter stick, lay the
two side by side, and agree that the meter
sticks are exactly the same length. They then
go outside and run past each other, in opposite
directions, at a relative speed v = 0.9c.
● Determine the length of each meter stick as
they move past one another.
Another Paradox? No!
Relativity allows us to compare the same
events as they’re measured in two different
reference frames.
The events by which Vignesh measures the
length (in Vignesh’s frame) of David’s meter
stick are not the same events as those by
which David measures the length (in David’s
frame) of Vignesh’s meter stick.
Another Paradox? No!
If this weren’t the case, then we could tell which
reference frame was “really” moving and which
was “really” at rest.
The principle of relativity doesn’t allow us to
make that distinction.
Each is moving relative to the other, so each
should make the same measurement for the
length of the other’s meter stick.
Length Contraction at v << c
Using the length contraction equation, the
length of a 1.00 m arrow (as measured at rest)
is calculated to be 1.00 m when it moves at 300
m/s relative to an observer.
Isn’t length contraction supposed to make the
measured length less than 1.00 m?
Binomial Expansion
Most calculators do not have the precision for
this calculation. We can use binomial
expansion to get around this limitation.
(1 - x)1/2 ≈ 1 - ½ x
What is the amount of length contraction for the
arrow?
The Spacetime Interval
A firecracker explodes at the origin of an inertial reference
frame. Then, 2.0 μs later, a second firecracker explodes
300 m away. Astronauts in a passing rocket measure the
distance between the explosions to be 200 m. According to
the astronauts, how much time elapses between the two
explosions?
The Spacetime Interval
Show that
The Spacetime Interval
A firecracker explodes at the origin of an inertial reference
frame. Then, 2.0 μs later, a second firecracker explodes
300 m away. Astronauts in a passing rocket measure the
distance between the explosions to be 200 m. According to
the astronauts, how much time elapses between the two
explosions?
Everything is relative?
Time intervals & space intervals
Not spacetime intervals - the spacetime interval
s between two events is not relative. It is
agreed upon by experiments in inertial
reference frames.
Stop and Think
Beth and Charles are at rest relative to each
other. Anjay runs past at velocity v while
holding a long pole parallel to his motion.
Anjay, Beth, and Charles each measure the
length of the pole at the instant Anjay passes
Beth. Rank in order, from largest to smallest,
the three lengths LA, LB, LC.
Lorentz Transformations
In classical relativity, t’ = t and the Galilean
transformation lets us calculate the position of
an event in frame S’.
Is there a similar transformation that lets us
calculate an event’s spacetime coordinates in
frame S’?
Lorentz Transformations
Transformation must:
1. agree with Galilean transformations at v << c
2. transform both space and time coordinates
3. ensure that c is the same in all reference
frames
Lorentz Transformations
where ɣ is a dimensionless
function of velocity that
goes to 1 as velocity goes
to 0.
Event 1 - light is emitted
from the origin of both
reference frames
Event 2 - light strikes a
light detector
Lorentz Transformations
Event 1: x = x’ = 0 and t = t’ = 0
Event 2: (x, t) and (x’, t’)
What is the position of event 2 in each reference frame?
Substitute these expressions into the transformation
equations and solve for ɣ.
Lorentz Transformations
Derive the Lorentz transformations for t and t’
Hint: require x = x and transform a position from
S to S’ and then back to S
Lorentz Transformations
Peggy and Ryan revisited
Peggy is standing in the center of a long, flat railroad car
that has firecrackers tied to both ends. The car moves past
Ryan, who is standing on the ground, with velocity v = 0.8c.
Flashes from the exploding firecrackers reach him
simultaneously 1.0 μs after the instant that Peggy passes
him, and he later finds burn marks on the track 300 m to
either side of where he had been standing.
Peggy and Ryan revisited
a. According to Ryan, what is the distance between the
two explosions, and at what times do the explosions
occur relative to the time that Peggy passes him?
b. According to Peggy, what is the distance between the
two explosions, and at what times do the explosions
occur relative to the time that Ryan passes her?
Lorentz Transformations
p. 945-946
#13-23