Special Relativityand

General Relativity

Special andGeneral Relativity

Einstein’s Physics

Objectives

• Be familiar with the Michelson-Morley experiment.

• Understand what the results of the experiment mean in terms of the “ether” and the speed of light.

Michelson-Morley Experiment• James Clerk Maxwell (1860):

light is e/m waves traveling at c.• Waves require a medium, so

light must travel through an “ether.”

• Michelson and Morley (1880s): looked for the ether using an interferometer.

Concept of the Interferometer

• Two boats will travel 24 m forward and back at 4 m/s. The river current is 2 m/s eastward.

• North-South blue route: (24 m / 4 m/s) x 2 = 12 s.

• East-West red route: (24 m / 6 m/s) + (24 m / 2 m/s) = 16 s.

• Blue boat wins!• But, if the river flows

northward, the red boat would win.

Michelson-Morley Experiment• As the earth moves

through the ether, the “wind” will act like the river current, affecting the motion of the light waves.

• Rotating the experiment will cause interference fringes to change, proving the existence of the ether.

Michelson-Morley Experiment• When they conducted their

experiment, no fringes were observed to change.

• No ether exists!• A secondary outcome of the

experiment was that c is always 3.00 x 108 m/s.

• Lorenz proposed that the ether wind affected the distance between the mirrors by a factor of

)/(1 22 cv

Einstein’s Question• Light propagates through

space by changing electric and magnetic fields.

• As a student, Albert Einstein wondered what would happen if you could travel along with a light wave? Would the changing fields occur? Would the light propagate?

• Einstein devoted his life to understanding light.

Hmm...

Objectives

• Know the two postulates of Einstein’s theory of relativity.

• Understand how the constancy of the speed of light affects our concept of time.

• Understand and apply the concept of space-time.

Einstein’s Postulates of Relativity1. All the laws of nature are the same in all uniformly

moving frames of reference. You cannot detect absolute uniform motion (no ether for reference).

2. The speed of light equals c and is independent of the speed of the source or the observer. C is absolute.

The evidence for #2:

pion

detectormeasuresenergy pion

movingat 0.99c

detectormeasuresSAME energy

Simultaneity• Einstein imagined lightning hitting two poles. • A stationary observer midway between the poles sees

the light hit the two poles simultaneously.• A moving observer midway between the poles sees the

light hit the pole that he is moving toward first, and the other pole afterwards.

• The two observers cannot agree on the order of events:• Time is relative! Only the speed of light is absolute!

Space-Time• speed = distance / time. • Applied to light, c = d / t.

If c is absolute, and time is relative, then distance (space) must be relative too.

• Einstein reasoned that the concepts of space and time are woven together into what he called space-time.

Think about it: any event takes place at a specific time and a specific place(in 4 dimensions)

Traveling in Space-Timetime

space (distance)

We travel mostly through time, but not through much space.

As an object approaches c, it travels mostly through space, but through little time.

A fast-moving spacecrafttravels through more spaceand thus through less time.

slope = t/d, and 1/v = t/d. As velocity goes up, slope goes down

Objectives

• Understand the concept of time dilation.• Be able to calculate time dilation.• Be familiar with evidence for time dilation.• Understand the implications of time dilation.

Time Dilation

to = proper time

t = dilated time (or td)

Time and distance measured by observerinside the spaceship.

Time and distance measured by observeroutside the spaceship.

Imagine two scientists measuring a light-pulse inside a moving spaceship. One is inside the spaceship, the other is outside the spaceship…

Time Dilation

v · t

c · tc · to

t is dilated time, clock in motion with respect to events to is “proper time”, clock at rest with respect to events

2

2

2

2

22

22

22

22

222

22222

222222

222

222

1

)1(

)1(

)(

)(

)()()(

)()()(

cv

tt

cv

tt

tcvt

tcvct

tcvct

tctvtc

ctvtct

vtctct

o

o

o

o

o

o

o

o

Calculating Time DilationProxima Centauri is the closest star to our solar system. If a spacecraft were sent to Proxima Centauri traveling at 75% of the speed of light (0.75 c), the trip would take 3.72 years according to the clocks onboard the ship. How long would the trip take according to people on Earth?

Time Dilation: The Evidence• In 1971, two atomic clocks were

placed on commercial jets and two “reference” atomic clocks were placed in a building. The clocks were synchronized.

• The jets traveled around the world twice (once east, once west)

• The clocks that traveled through more space (in jets) recorded less time than the stationary clocks, as predicted by Einstein.

The Twin “Paradox”

• One twin travels at relativistic speeds away from the earth, turns around, and returns at relativistic speeds.

• She will be younger than her twin brother!

• The twin brother experiences the dilated time.

Twin Paradox: The Evidence• 1976 at CERN• Muons normally decay in 2.2 ms

(to) A muon should only be able to make 15 revolutions around the accelerator in this time.

• When traveling at 0.9994 c, a muon will make 432 revolutions and decay in 63.5 ms (td), outlasting a twin stationary muon by a factor of 29.

Length Contraction

• length contraction: moving objects appear to contract along the direction of motion.

• Looking at a clock and meter-stick inside the spaceship, you would see less time pass for a beam of light to travel one meter; since c = d/t, distance must be less. 2

2

oc cv1LL

Lo = proper length

LC = contracted length

Length Contraction Calculation

All distances are contracted when you travel at relativistic speeds. Thus, Pluto, which is 39 AU away, would be “closer” if you traveled at 0.95 c. What is the contracted distance?

Relativistic MomentumNewton p = mv

Einsteinp = mv

particle acceleratordata supports Einstein

true only at non- relativistic speeds

What is the momentum of a proton (1.67 x 10-27 kg) traveling at 0.999c (2.997x108 m/s) according to Newton? What about to Einstein?

measured value =Einstein’s value

Relativistic Dynamics• Why can’t v > c?• As v → c, p → ∞

• Impulse-momentum theorem• F·t = m·v = p• If p → ∞, either F → ∞ or t → ∞• It either takes an infinite force or a

finite force applied for an infinite period of time to reach the speed of light!

Mom

entu

m (p

)

Speed (v) c

Einsteinp = mv

Newtonp = mv

The answer to Einstein’s question: it is not possibleto ride a light beam, so thereis no paradox.

Eo = mc2

• rest energy: the energy an object possesses due to its mass

• mass ≈ “frozen energy”• objects gain/lose mass when they

absorb/emit energy• The sun converts 4 billion kg/s into

energy through the process of nuclear fusion (4 H → He + energy)

• E = mc2 = (4 x 109 kg)(3 x 108 m/s)2

= 3.6 x 1026 J each second!= 360 heptillion W light bulb

General Relativity

Equivalence Principle• Einstein’s “happiest

thought” was that you don’t feel the force of gravity when you fall.

• But artificial gravity exists in an accelerating spacecraft.

• Gravity and acceleration are “equivalent.”

• An experiment done on earth or done when accelerating at g in a spacecraft will yield the same results! (general relativity).

Light and the Equivalence Principle• A scientist in an accelerating

spacecraft observes a horizontal beam of light to curve downward.

• According to the equivalence principle, gravity should curve light in a similar manner.

Sun

acceleration

gravity Astronomical observations after WWI showed that the sun did indeed bend starlight, supporting Einstein.

Curved Space

Newton’s laws could notfully explain the orbital motionof Mercury; however, Einsteinused his general theory to properly calculate the orbit.

If mass bends light, and light moves in a straight line, then mass must warp or curve space.

Warped Space and Orbital Motion

Newton (Law #1) said that an object will move in straight line unless acted on by unbalanced force. Einstein suggested that the object moves in a “straight line” through curved space!