Relativity & ThermodynamicsOur Understanding of Space and

Time

Credits:www.phy.mtu.edu/~akantamn/Physics/lec3/Special_Relativity%20III.pptwww.few.vu.nl/~ptn900/teaching_files/Relativity.pptwww.serendip.brynmawr.edu/local/scisoc/time/7arrows.ppt www.astro.ufl.edu/~vicki/AST3019/General_Relativity.ppt

The Relativity Principle

Galileo Galilei 1564 - 1642

Problem: If the earth were moving wouldn’t we feel it? – No

The Copernican Model

The Ptolemaic Model

The Relativity Principle

A coordinate system moving at a constant velocity is called an inertial reference frame.

v

The Galilean Relativity Principle: All physical laws are the same in all inertial reference frames.

Galileo Galilei 1564 - 1642

The Relativity PrincipleOther Examples:

As long as you move at constant velocity you are in an inertial reference frame.

Galileo Galilei 1564 - 1642

Electromagnetism

James Clerk Maxwell 1831 - 1879

A wave solution traveling at the speed of light

c = 3.00 x 108 m/s

Maxwell: Light is an EM wave!

Problem: The equations don’t tell what light is traveling with respect to

Einstein’s Approach to Physics

Albert Einstein 1879 - 1955

1. Gedanken (Thought) Experiments

E.g., if we could travel next to a light wave, what would we see?

2. “The Einstein Principle”:

If two phenomena are indistinguishable by experiments then they are the same thing.

Einstein’s Approach to Physics2. “The Einstein Principle”:

If two phenomena are indistinguishable by experiments then they are the same thing.

A magnet moving A coil moving towards a magnet

Both produce the same currentImplies that they are the same phenomenon

towards a coil

Albert Einstein 1879 - 1955

current current

Einstein’s Approach to Physics1. Gedanken (Thought) Experiments

E.g., if we could travel next to a light wave, what would we see?

c

c

We would see an EM wave frozen in space next to usProblem: EM equations don’t predict stationary waves

Albert Einstein 1879 - 1955

ElectromagnetismAnother 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

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.

Einstein: Start with 2 assumptions & deduce all else

This is a literal interpretation of the EM equations

Special Relativity PostulatesLooking 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 DilationOne consequence: Time Changes

Equipment needed: a light clock and a fast space ship.

Time DilationIn Bob’s reference frame the time between A & B is Δt0

Sallyon earth

Bob

Beginning Event A

Ending Event B

cDt 2

0

D

Δt0

Bob

Time DilationIn Sally’s reference frame the time between A & B is Δt

Bob

A BSallyon earth

22 2 22 2 2

2v ts D L D

Length of path for the light ray:

cst 2

and

Δt

Time Dilation2

2 2 22 2 22v ts D L D

Length of path for the light ray:

cst 2

and

Solve for Δt:22 /1

/2

cv

cDt

cDt /20

Time measured by Bob

220

/1 cv

tt

Time Dilation

220

/1 cv

tt

Δt0 = the time between A & B measured by Bob

Δt = the time between A & B measured by Sally

v = the speed of one observer relative to the other

Time Dilation = Moving clocks slow down

If Δt0 = 1s, v = .999 c then: s 500999.1s 1

2

t

Time Dilation

• Bob’s watch always displays his proper time

• Sally’s watch always displays her proper time

How do we define time?

The flow of time each observer experiences is measured by their watch – we call this the proper time

• If they are moving relative to each other they will not agree

Time DilationA 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?

s 101.1999.1

s 102.2/1

3

2

6

220

cv

tt

Length Contraction

Bob’s reference frame:The distance measured by the spacecraft is shorter

Sally’s reference frame:Sally

Bob

0

0

LLvt t

The relative speed v is the same for both observers:

220

/1 cv

tt

220 /1 cvLL

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.

• It follows this path because the spacetime surface on which it must stay is curved.

• To fully represent the trajectory, need all 4 space-time dimensions curving into a 5th dimension(!)

• Hard to visualize, but still possible to measure

t t

x x

No gravity gravity

Principle of EquivalenceA 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 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

• Third, there is gravity and the elevator is also in free-fall

• Fs - mg = -mg or Fs = 0• “Weightless”

Let upward forces be positive, thus gravity is -g

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 RelativityOrbiting 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 ArrowEntropy 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 lawBut 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 canarrow of time arise?

Prigogine’s Solution:

Second Law is fundamental.Can’t derive.

Chaotic systems manifest irreversibleBehavior.

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?