Relativity & Thermodynamics
Our Understanding of Space and
Time
Credits:
www.phy.mtu.edu/~akantamn/Physics/lec3/Special_Relativity%20III.ppt
www.few.vu.nl/~ptn900/teaching_files/Relativity.ppt
www.serendip.brynmawr.edu/local/scisoc/time/7arrows.ppt
www.astro.ufl.edu/~vicki/AST3019/General_Relativity.ppt
The Relativity Principle
The Ptolemaic
Model
The Copernican
Model
Galileo Galilei
1564 - 1642
Problem: If the earth were
moving wouldn’t we feel it? – No
The Relativity Principle
v
A coordinate system moving at a
constant velocity is called an inertial
Galileo Galilei
1564 - 1642
reference frame.
The Galilean Relativity Principle:
All physical laws are the same in all inertial reference
frames.
The Relativity Principle
Other Examples:
Galileo Galilei
1564 - 1642
As long as you move at constant velocity you are in
an inertial reference frame.
Electromagnetism
A wave solution traveling at the
speed of light
c = 3.00 x 108 m/s
Maxwell: Light is an EM wave!
James Clerk
Maxwell
1831 - 1879
Problem: The equations don’t tell
what light is traveling with respect to
Einstein’s Approach to Physics
1. Gedanken (Thought) Experiments
E.g., if we could travel next to a light
wave, what would we see?
2. “The Einstein Principle”:
Albert Einstein
1879 - 1955
If two phenomena are
indistinguishable by experiments
then they are the same thing.
Einstein’s Approach to Physics
2. “The Einstein Principle”:
If two phenomena are
indistinguishable by experiments
then they are the same thing.
current
Albert Einstein
1879 - 1955
A magnet moving
towards a coil
current
A coil moving
towards a magnet
Both produce the same current
Implies that they are the same phenomenon
Einstein’s Approach to Physics
1. Gedanken (Thought) Experiments
E.g., if we could travel next to a light
wave, what would we see?
c
Albert Einstein
1879 - 1955
c
We would see an EM wave frozen in space next to us
Problem: EM equations don’t predict stationary waves
Electromagnetism
Another Problem: Every experiment measured the
speed of light to be c regardless of motion
The observer on the
ground should
measure the speed
of this wave as
c + 15 m/s
Conundrum: Both observers actually measure the
speed of this wave as c!
Special Relativity Postulates
Einstein: Start with 2 assumptions & deduce all else
1. The Relativity Postulate: The laws of physics are
the same in every inertial reference frame.
2. The Speed of Light Postulate: The speed of light
in vacuum, measured in any inertial reference
frame, always has the same value of c.
This is a literal interpretation
of the EM equations
Special Relativity Postulates
Looking through Einstein’s eyes:
Both observers
(by the postulates)
should measure
the speed of this
wave as c
Consequences:
• Time behaves very differently than expected
• Space behaves very differently than expected
Time Dilation
One consequence: Time Changes
Equipment needed: a light clock and a fast space ship.
Time Dilation
In Bob’s reference frame the time between A & B is Δt0
Ending Event B
Bob
Δt0
D
Beginning Event A
Sally
on earth
2D
t0 
c
Time Dilation
In Sally’s reference frame the time between A & B is Δt
Bob
Bob
Sally
on earth
A
Δt
B
Length of path for the light ray:
 v t 
2s  2 D  L  2 D  

 2 
2
2
2
2
and
2s
t 
c
Time Dilation
Length of path for the light ray:
 v t 
2s  2 D  L  2 D  

2


2
2
2
and
2
Solve for Δt: t 
t 
2D / c
1 v / c
2
t0  2 D / c
2
Time measured
by Bob
t0
1 v / c
2
2s
t 
c
2
Time Dilation
t 
Δt0 = the time between A
t0
1 v / c
2
& B measured by Bob
2
Δt = the time between A
& B measured by Sally
v = the speed of one
observer relative to the
other
If Δt0 = 1s, v = .999 c then: t 
1s
1  .999
Time Dilation = Moving clocks slow down
2
 500 s
Time Dilation
How do we define time?
The flow of time each observer experiences is measured
by their watch – we call this the proper time
• Sally’s watch always displays her proper time
• Bob’s watch always displays his proper time
• If they are moving relative to each other they will not
agree
Time Dilation
A Real Life Example: Lifetime of muons
Muon’s rest lifetime = 2.2x10-6 seconds
Many muons in the upper atmosphere (or in the
laboratory) travel at high speed.
If v = 0.999 c. What will be its average lifetime as
seen by an observer at rest?
t 
t0
1 v / c
2
2

6
2.2 10 s
1  .999
2
3
 1.110 s
Length Contraction
The distance measured by the spacecraft is shorter
Sally’s reference frame:
Bob’s reference frame:
Sally
Bob
The relative speed v is the
same for both observers:
L0
L
v

t0 t
t 
t0
1 v2 / c2
L  L0 1  v 2 / c 2
Twirling Pole Paradox
You hold a really long pole. You hold one end firmly and
twirl the pole so that the free end goes around in a big
circle. Can the free end go faster than c?
1. No. Every physical object must travel less than c.
2. Yes, for a long enough pole twirled fast enough, the free
end must go faster than c.
3. Where's the barn? I heard this paradox had a barn.
Twirling Pole Paradox
1. No. Every physical object must travel less than c.
The pole end is real and cannot move at v > c. The problem
is that such a pole cannot be perfectly rigid.
Information cannot move ALONG the pole at v > c so that
the end of the pole cannot know that the inner parts of the
pole are twirling. A perfectly rigid pole would break. A
very elastic pole would twist into a spiral pattern with the
free end constrained to move at v < c.
Ladder Paradox
Also called the "Barn and the Pole" paradox.
You hold a long ladder and run toward a short garage.
If you run fast, can you trap the ladder in the garage?
or
?
Ladder Paradox
Yes, you can trap the ladder in the garage.
The information that the front end of the ladder has hit the
back end of the garage can only move along the ladder at v
< c. As this information moves, the back end of the ladder
can pass into the garage and the garage door can be
closed. We then get to see if the ladder is stronger than the
door.
Twin Paradox
One twin stays home.
One twin rockets away and then comes back.
Special relativity implies time dilation for moving objects,
but each moved only as seen by the other. Which twin is
older?
1. The twin who stayed home is older.
2. The twin who rocketed away and came back is older.
3. Symmetry demands they are the same age.
Twin Paradox
2. The twin who rocketed away & came back is younger.
The symmetry is broken because the leaving twin had to
accelerate to come back, whereas the staying twin
experienced no acceleration.
Is this a way to travel into the future? Yes. Time travel this
way is permissible. There is no way to use the twin paradox
to travel BACK in time.
General Relativity
• GR is Einstein’s theory of gravitation that builds
on the geometric concept of space-time
introduced in SR.
• Is there a more fundamental explanation of
gravity than Newton’s law?
• GR makes specific predictions of deviations from
Newtonian gravity.
Curved space-time
• Gravitational fields alter the rules of geometry in
space-time producing “curved” space
• For example the geometry of a simple triangle
on the surface of sphere is different than on a
flat plane (Euclidean)
• On small regions of a sphere, the geometry is
close to Euclidean
How does gravity curve space-time?
•With no gravity, a ball thrown upward continues
upward and the worldline is a straight line.
•With gravity, the ball’s worldline is curved.
No gravity
gravity
t
t
x
x
Principle of Equivalence
 A uniform gravitational field in some direction is
indistinguishable from a uniform acceleration in
the opposite direction
 Keep in mind that an accelerating frame
introduces pseudo-forces in the direction
opposite to the true acceleration of the frame
(e.g. inside a car when brakes are applied)
Elevator experiment
•First, elevator is supported and not
moving, but gravity is present. Equate
forces on the person to ma (=0 since a=0)
•Fs - mg = 0
so
Let upward
forces be
positive,
thus gravity
is -g
Fs = mg
•Fs gives the weight of the person.
•Second, no gravity, but an upward
acceleration a. The only force on the
person is Fs and so
•Fs = ma or Fs = mg if “a” value is the
same as “g”
•Person in elevator cannot tell the
difference between gravitational field and
accelerating frame
See also http://www.pbs.org/wgbh/nova/einstein/relativity/
Einstein was bothered by what he saw as a dichotomy in the
concept of "mass." On one hand, by Newton's second law
(F=ma), "mass" is treated as a measure of an object’s
resistance to changes in movement. This is called inertial mass.
On the other hand, by Newton's Law of Universal Gravitation,
an object's mass measures its response to gravitational
attraction. This is called gravitational mass. As we will see,
Einstein resolved this dichotomy by putting gravity and
acceleration on an equal footing.
The principle of equivalence is
really a statement that inertial
and gravitational masses are
the same for any object.
This also explains why all objects have the same
acceleration in a gravitational field (e.g. a feather
and bowling ball fall with the same acceleration in
the absence of air friction).
Tests of General Relativity
 Orbiting bodies - GR predicts slightly different
paths than Newtonian gravitation
 Gravitational Lensing
 Gravitational Red shifting
 Gravitational Waves
Arrows of Time
1. Memory Arrow
2. Cosmological Arrow
3. Entropy Arrow
1. Memory Arrow
Memory only
works backwards
(We don't remember
the future)
2. Cosmological Arrow
We live in an expanding universe, not a
contracting one.
Would time change direction if universe
started contracting?
3. Entropy Arrow
Entropy always increases
(second law of thermodynamics)
Entropy
Entropy can be thought of as waste heat
generated in any realistic process
“Disorder” in a system
Second Law:
Entropy of a system always
increases under realistic
(“irreversible”) process
Only law of nature that
exhibits direction of time!
Question:
Are arrows independent?
Hawking: Memory and entropy arrows
linked
(Requires energy to read one bit,
increases entropy by certain amount)
“Thermodynamic system”:
System with lots of particles
(Gas)
Obeys second law
But individual particles obey
Newtonian physics
Central Paradox:
Gas is a thermodynamic
system;
obeys second law.
But if gas particles individually
obey time-symmetric
Newtonian physics, how can
arrow of time arise?
Prigogine’s Solution:
Second Law is fundamental.
Can’t derive.
Chaotic systems manifest irreversible
Behavior.
Entropy increase tied up with chaos
Core Questions
•Do time and space exist independently of the
mind?
•Do they exist independently of one another?
•What accounts for time's apparently
unidirectional flow ?
•Do times other than the present moment
exist?
•What do these say about divine action in the
world?