Albert Einstein

The Special and General Theory of Relativity and his Thought Experiments

By Leiwen Wu

A Little About Albert Einstein

• Born: 14 March 1879 in Ulm, Württemberg, Germany

• Died: 18 April 1955 in Princeton, New Jersey, USA

• Einstein contributed more than any other scientist to the modern vision of physical reality. His special and general theories of relativity are still regarded as the most satisfactory model of the large-scale universe that we have.

Brief Overview of the Genius• 1879: Einstein born Ulm, Germany. • 1885 - 1925: Michelson and Morley began a series of

puzzling experiments which made the Newtonian Universe impossible.

• 1900: Max Planck shocked the physics community with the concept of quantization

• 1905: The miracle year in physics: Einstein published papers on Brownian motion as well as the seminal papers on his theory of relativity. He developed the Special Theory of Relativity in which he described how space and time are relative or related to each other.

• 1915: Einstein extended his discussion of relativity to include gravity and thereby explained the problem of Mercury. He developed the general theory of relativity which dealt with gravity and acceleration and a 4 dimensional space in which everything is related to each other.

• 1919: Eddington confirms Einstein's prediction concerning deflection of starlight.

• 1915 - 1925: Einstein was a co-leader in the birth and development of quantum mechanics

• 1925 - 1935: Einstein and Bohr engaged in a fascinating series of "debates" over the interpretations of physics especially the notion of determinism (God does not play dice)

• 1930 - 1955: Einstein searches for a unified theory of the universe

• 1933 - Hubble and Humanson discover the recessional nature of galaxies - Einstein's theories of the universe take shape.

• 1955: Einstein dies, Princeton, N.J.

Einstein’s Personal Life

• Einstein marries Mileva in 1903

• Mileva divorced Einstein in 1914

• Einstein married his cousin Elsa in 1919

Newton, Einstein, and Gravity

Newton’s Laws of Motion

I. A body continues at rest or in uniform motion in a straight line unless acted on by some net force.  

III. To every action, there is an equal and opposite reaction.

II. The acceleration of a body is inversely proportional to its mass, directly proportional to the net force, and in the same direction as the net force.

F = m a

Newton’s Law of Gravitation:

where G is the “gravitational constant,” M is the mass of the larger body, m is the mass of the smaller body, r is the separation between them. 

G M m

r 2F =

Gravity and Orbital Motion

The gravitational attraction between the Earth and the Moon causes the Moon to orbit around the Earth rather than moving in a straight line.

If a rocket is fired up from Earth, gravity will slow it down so that it either:

• falls back to Earth

• enters a closed orbit around the Earth

• escapes from Earth

Newton's Laws: Gravity and Motion

Cannon ball applet: http://zebu.uoregon.edu/~js/ast122/lectures/lec03.html

vesc =2 G M

r

Newton and Kepler

Newton showed that objects moving along closed orbits under the influence of gravity follow elliptical paths. 

Recall: Kepler’s First Law

Newton and Kepler

Newton also showed that objects in these orbits conserve angular momentum.

Recall: Kepler’s Second Law

QuickTime™ and a YUV420 codec decompressor are needed to see this picture.

An object orbiting in a circle around mass M has speed

The orbital period of this object is the circumference of its orbit divided by its speed :

so

Recall: Kepler's Third Law

Newton and Kepler

Quantum Theory of Light

• Before Einstein, people thought that space, the empty space in our universe, composed of things called ether.

• Einstein thought that light came in tiny packets, or particles called photons.

• It was the most shocking idea about our universe:• We live in a quantum universe built out of tiny, discrete

chunks of energy and matter.• Einstein would later try to combine the theory of relativity

and quantum mechanics in his unified theory which would explain our whole universe. Unfortunately, he died before he was able to complete it.

What is relativity?

The making of the special Theory of Relativity

• The Special theory of Relativity disproves Newton’s long held idea that space and time are absolute

• It creates a whole new way of thinking about our universe

• It creates a four dimensional universe where everything is related

• Special relativity is much simpler mathematically than general relativity, but harder to visualize and imagine. Einstein was 26 when he devised

the special theory of relativity

The Special theory of Relativity Explained

• The special theory of relativity states that time and space(distance) is relative or depends on each other

• Everything is related to each other and is not absolute.

• Newton thought that time is the same every where.

• Special relativity disproves

100 years old

Imagine two twin brothers.

One in a space ship and one on the launch pad

Now the spaceship travels at 99.9% of light for 100 years earth time

Time to both are very different.

1 year old

Postulates of the Special Theory of Relativity

1. Observers cannot detect absolute uniform motion, only motion relative to other objects

– or –

The laws of physics are the same for all observers.

2. The speed of light is the same for all observers, independent of their motion relative to the source of the light.

If c were not absolute, you’d see car A reach the collisionpoint before car B! You would see a different event!

Consequences of anabsolute speed of light

for all observers:

time dilationtime dilation

The Equivalence Principle

In his General Theory of Relativity, Einstein explained the force of attraction between massive objects in this way:  

“Mass tells space-time how to curve, and the curvature of space-time tells masses how to accelerate.”

Einstein’s View of Gravitation

Orbits in Curved Space-Time QuickTime™ and a Cinepak decompressor are needed to see this p icture.

Gravity à la Einstein

Einstein’s general theory of relativity predicted that light paths should be affected by massive objects.

Einstein’s predictions were confirmed when the positions of stars near the sun were observed to be shifted during a 1919 solar eclipse.

Gravity à la Einstein

The Making of the General Theory of Relativity

• Einstein was 36 when he developed the theory of relativity.

• Einstein came up with this when he imagined a man falling of the roof.

The General Theory of Relativity Explained

• Einstein Discovered in his General Theory of Relativity that gravity and acceleration are the same phenomenon.

Imagine an elevator and a person standing in it.

What would happen to the person if the elevator free-falls?

The person would be floating in the elevator while it is free-falling.

Now Imagine that person in a space ship far away from any gravitational force. He would be floating in the ship.

If the ship the person is in accelerates at the right amount of speed, the person would feel the same as if gravity was pulling on him.

General Relativity Conclusion

• Einstein concluded that 4 dimensional

Principle of Equivalence: Einstein 1907

Box stationary in gravity field

Box falling freely

g

g

g

Box acceleratesin empty space

Box moves through space at constant velocity

Equivalence Principle

• Special relativity: all uniformly moving frames are equivalent, i.e., no acceleration

• Equivalence principle: Gravitational field = acceleration

freely falling frames in GR = uniformly moving frames in SR.

Aberration of Light

Moral: direction of light beam is relative

Gravitational deflection of Light

Now assume boxes are accelerating

Light path is curved

Light ray curved in accelerating frame

Principle of Equivalence (acceleration = gravity)

Gravity attracts light!

Paradox: How can gravity attract light if light has no mass?

MASS-ENERGY EQUIVALENCE

Gravity extracts energy from escaping matter

Gravity extracts energy from escaping light

Gravitational redshift, time dilation

Other points of view same result:– accelerating frames of reference - apply special relativity– spacetime is curved

2mcE

E=mc• This equation is the most

important single result of relativity theory

• It’s the idea that mass and energy are equivalent.

• Energy = mass times the speed of light squared

• E=m, the c squared is just to express how much energy can be made from one unit of mass.

2

Curved Spacetime

• Remember: Gravity warps time

slow

fast

BUT: in spacetime, time and space are not separable

=> Both space and time are curved (warped)

This is a bit hard to vizualize (spacetime already 4D…)

Tides

• Problem:

• Gravity decreases with distance => stretch…

r1

r2

moon

• Tides = gravity changes from place to place

freely falling

not freely falling

not freely falling

?

???

Tides

CURVATURE OF SPACETIME

• How to tell difference between accelerating frame and gravity?– tidal forces curvature

• Eliminates Newton’s “action at a distance”

• Freely moving bodies follow “shortest path”– not necessarily a straight line

GENERAL RELATIVITY: EINSTEIN 1915

• Matter + energy determine curvature of spacetime

• Curvature of spacetime determines motion of matter + energy

Light Rays and Gravity II

• In SR: light rays travel on straight lines

=> in freely falling frame, light travels on straight

lines

• BUT: to stationary observer light travels on curved paths

=> Maybe gravity has something to do with…

curvature of space ?

GR: Einstein, 1915

• Einstein: mass/energy squeeze/stretch spacetime away

from being “flat”

• Moving objects follow curvature (e.g., satellites, photons)

• The equivalence principle guarantees:

spacetime is “locally” flat

• The more mass/energy there is in a given volume, the

more spacetime is distorted in and around that volume.

GR: Einstein, 1915

• Einstein’s “field equations” correct “action at a

distance” problem:

Gravity information propagates at the speed of light

=> gravitational waves

r?

• Imagine being an ant… living in 2D

• You would understand:

left, right, forward, backward,

but NOT up/down…

• How do you know your world is curved?

Curvature in 2D…

• In a curved space, Euclidean geometry does not apply:

- circumference 2 R

- triangles 180°

- parallel lines don’t stay parallel

<2R

R R

2R

=180

Curvature in 2D…

Curvature in 2D…

Curvature in 2D…

Geodesics

• To do geometry, we need a way to measure distances

=> use ant (let’s call the ant “metric”), count steps it

has to take on its way from P1 to P2 (in spacetime, the

ant-walk is a bit funny looking, but never mind

that)

• Geodesic: shortest line between P1 and P2

(the fewest possible ant steps)

P1 P2

ant

• To the ant, the geodesic is a straight line,

i.e., the ant never has to turn

• In SR and in freely falling frames, objects

move in straight lines (uniform motion)

• In GR, freely falling objects

(freely falling: under the influence of gravity only,

no rocket engines and such; objects: apples, photons,

etc.)

move on geodesics in spacetime.

Geodesics

Experimental Evidence for GR

• If mass is small / at large distances, curvature is weak

=> Newton’s laws are good approximation

• But: Detailed observations confirm GR

1) Orbital deviations for Mercury (perihelion precession)

Newton: Einstein:

2) Deflection of light

Experimental Evidence for GR

Experimental Evidence for GR

• What happens as the star shrinks / its mass increases? How much can spacetime be distorted by a very massive object?

• Remember: in a Newtonian black hole, the escape speed simply exceeds the speed of light

=> Can gravity warp spacetime to the point where even light cannot escape its grip?

That, then, would be a black hole.

Black Holes

Black Holes

• Time flows more slowly near a massive object,

space is “stretched” out (circumference < 2R)

• Critical: the ratio of circumference/mass of the object.

If this ratio is small, GR effects are large (i.e., more mass within same region or same mass within smaller region)

Black Holes

???

???

1) massive 2) small

• GR predicts: If mass is contained in a circumference smaller than a certain size

space time within and around that mass concentration qualitatively changes. A far away observer would locate this critical surface at a radius

• Gravitational time dilation becomes infinite as one approaches the critical surface.

kilometersMMc

GMR SunS )/(3

22

2

22

c

GMnceCircumfere

gravitational constant

speed of light

criticalcircumference

mass

Schwarzschild radius

The Schwarzschild Radius

• To a stationary oberserver far away, time flow at the critical surface (at RS) is slowed down infinitely.

• Light emitted close to the critical surface is severely red-shifted (the frequency is lower) and at the critical surface, the redshift is infinite.

From inside this region

no informationcan escape red-shifted

red-shifted intooblivion

Black Holes

• Inside the critical surface, spacetime is so warped that objects cannot move outward at all, not even light.

=> Events inside the critical surface can never affect the region outside the critical surface, since no

information about them can escape gravity.

=> We call this surface the event horizon

because it shields the outside completely from any events on the inside.

Black Holes

• Critical distinction to the Newtonian black hole:

Nothing ever leaves the horizon of a GR black hole.

• Lots of questions…

What happens to matter falling in?

What happens at the center?

Can we observe black holes anyway?

And much, much more…

Newton Einstein

Black Holes

Conclusion: Unified Theory• The unified theory is Einstein’s attempt to combine

quantum mechanics and his two Theories of Relativity• Einstein wanted this to be the most perfect idea of the

universe• Einstein never finished and died.• Einstein moved to Princeton, NJ to escape the Nazis• Einstein was Jewish, but did not practice the religion

until the end. He believed that the universe was so complex that only God could have created it.