1. Stephen Hawking Math 305 Tom Kreitinger 11/18/2009

2. Stephen Hawking Commercial Overview:
Part I – The shoulders of giants: Newton, Einstein, Schwarzschild, Friedmann, Lemaitre and Oppenheimer.
Part II – Two of Stephen Hawking's major contributions:
Big Bang Theory
Black Hole Radiation

3. Isaac Newton
”If I have seen further it is by standing on the shoulders of giants.”
World famous ”Principia” (1687) - Laws of motion and gravitation
Applies to all motion in the solar system
Yes, actually, it is rocket science!

4. Isaac Newton's Law of Universal Gravity
F(force) = G [ (m1 * m2) / R² ]
Example:
G = 6.67e-11 Newtons-metres² / kilograms²
m1 = gravitational mass 1 - A person 60 kg
m2 = gravitational mass 2 - Earth 5.98e24 kg
R² = radius - Earth 6.37e6 metres
F(force) = Weight = 590 Newtons = 132 lb

5. Isaac Newton's Law of Universal Gravity
F = G [ (m1 * m2) / R² ]
Gravitational attraction is weak - The attraction between two sumo wreslers is 10,000 times less than the pull necessary to pick up a square of toilet tissue

6. Isaac Newton's Second Law of Motion
F(force) = m * a (acceleration) or
a (acceleration) = F(force) / m
Newton's Law of Gravitation:
F(force) = G [ (m1 * m2) / R² ]
Is there any difference between ”inertial mass” and ”gravitational mass”?

7. Albert Einstein 1879-1955 Spooky Action at a Distance
According to Newton if the Sun disappeared in an instant, so would its gravitational field at the Earth, millions of miles away; yet light from the Sun, with its finite speed, would continue for another 8 minutes - this troubled Einstein.

8. Albert Einstein Not a force but space itself ?
Maybe it is not a force at all. Since the motion of a freely falling object does not depend on the object’s mass or composition (as Galileo discovered) gravitation might be due to certain properties of the medium it's falling in -- that is, space itself.

9. Albert Einstein's Happiest Thought - 1907
An observer falling freely from the roof of a house will not feel his own weight. If that person drops other bodies while falling, then these remain in a state of rest relative to him.
Gravitational mass is the same as inertial mass, gravity equals acceleration

10. Albert Einstein's Relativity - Field Equations
Produces a set of 20 extremely difficult simultaneous equations with 10 unknown quantities.
Matter tells space how to curve and then space tells matter how to move.
This becomes "the mountain” for others to visit.

11. Karl Schwarzschild 1873-1916 Provides exact solution
Visits the mountain...
Provides an exact solution to Einstein's Field Equations
System of ten equations connecting twenty quantities, resulting in hundreds of terms
A math super hero

12. Karl Schwarzschild
The math showed how the space curvature around a mass varies with distance
Schwarzschild Radius - the critical point at which curvature is so strong that matter can no longer escape
R = 2GM / c
G- gravitational constant
c- speed of light

13. What about Stephen Hawking?
Several solutions to the field equations were fundamental to Stephen Hawking's work
It is currently 1915 and Hawking will not even be born for another 27 years!
On the right we see (or don't see?) a black hole.

14. Alexander Friedmann
Einstein introduced an artificial term (lambda) in his Field Equations, essentially to ”stop the expansion of the Universe”; at the time astronomers were telling him the Universe was static.
Friedmann dropped the lambda constant and got an expanding universe.

15. Alexander Friedmann
By 1929 Edwin Hubble, using a 100-inch telescope, had confirmed that the universe is, in fact, expanding.
Einstein later called his ``cosmological constant” (lambda) the biggest mistake of his life.

16. Abbe Lemaitre 1894-1966 Precursor to Big Bang
First to use Friedmann- type solutions to formulate a model for the beginning of the Universe which he called the Primordial Atom or Cosmic Egg.

17. Abbe Lemaitre - 1931 Precursor to Big Bang
He thought that the universe at one time existed in a very small space, before the expansion took place several billion years ago, and that it is still expanding today
Gave a seminar on his model Universe to Hubble and Einstein

18. Singularity – In Mathematics
A point at which a function cannot be defined, diverges to infinitely large values
y=1/x has x=0 as a point of singularity
Small change in x from to results in y changing from +1 million to -1 million

19. Singularity – In Relativity
In general relativity singularity is a region of space-time in which the curvature becomes so strong that the laws of general relativity break down and the laws of quantum gravity take over.

20. Robert Oppenheimer 1904-1939 Gravitational Collapse – 1939
The third solution of Einstein's equations & important to Hawking
Stars may eventually burn out and begin to collapse under gravitational contraction.
Einstein refused to accept that relativity could describe collapsed stars (e.g. Black holes)‏

21. Robert Oppenheimer 1904-1939 Gravitational Collapse – 1939
Massive star exhausts nuclear fuel
Star implodes to point of no return, as defined by the Schwarzschild Radius (R=2GM/c)‏
Then cuts itself off from the rest of the universe (i.e. singularity)‏.

22. Hitler's troops invade Poland triggering second World War
The threat of Nazi rule postponed work on cosmology. In 1942 physicist began to focus on other projects (e.g. The Manhattan project)‏
Einstein had opposed all the radical cosmological predictions of his own equations as developed by Schwarzschid, Friedmann and Oppenheimer.
It would be 20 years before this work was resumed and the consequences of these solutions appreciated.

23. Stephen Hawking 1942-
In 1962 Hawking found it difficult to tie his shoelace: he had contracted ALS - Lou Gehrig's Disease.
Doctors gave him only 2 years to live.
Yet he came to Cambridge University from Oxford to study cosmology, and
married Jane Wilde at Cambridge in July 1965.

24. Stephen Hawking
Hawking called his doctoral thesis ”Properties of Expanding Universes”
The steady state model proposed that matter is continuously being created as the universe expands in utter disagreement with the big bang notion of an infinitely dense initial state
In the 60's steady state was more accepted than big bang

25. Stephen Hawking Big Bang
Roger Penrose determined that a singularity exists when a giant star collapses to form a black hole.
Hawking adapted Penrose’s results to the expanding universe, as a giant collapsing star in reverse.
Hawking showed that if general relativity is correct there must have been a singularity in the past which was the beginning of time.

26. Stephen Hawking Big Bang
Today everyone assumes the Universe started with a Big Bang (background micro).
This is Hawkings major world wide contribution to Big Bang cosmology.
Thus by 1970, five years after receving his PhD. Hawkings was an internationally known cosmologist

27. Stephen Hawking A new way of thinking
By the early 1970s Hawking's hands were largely paralyzed: he could not draw pictures or write down equations.
His research has to be done in his head. Because the loss of his faculties was so gradual, he has trained his mind to think in new types of intuitive mental pictures and equations which replaced written work – an astounding adaptation.

28. Stephen Hawking Black hole - Eureka Moment
Hawking has used his mental pictures to gain insight into the study of the surface areas of black holes.
He was getting ready for bed one night and he saw in a flash that the surface area of a black hole can never decrease.
He concluded that the combined area of two black holes must always be greater than or equal to the sum of the two areas.

29. Stephen Hawking Black Holes
Hawking explores the boundary between the black hole and the vacuum of inter-stellar space, and envisions how the intense gravity at the surface might affect any particles that might appear there, whether real or virtual.

30. Uncertainty Principle & Virtual Particles
The uncertainty principle states that there are limits on how accurately we can observe certain physical quantities (e.g. position, energy and time).
The limit is not related to our instruments, but an inherent characteristic of the Universe, which does not reveal any quantity with absolute precision.

31. Black Holes Emit Radiation
Hawking has in effect combined quantum mechanics and general relativity in a single calculation to show that Black Holes have temperature, entropy, and produce radiation just like any other thermodynamic body.

32. Hawking Recognition
He has received Britain's highest academic honour: Fellow of The Royal Society.
He received the Pope Pius XI Medal – Why? Because the Big Bang model appeals to the church as a creation event! We’ve come a long way since Galileo....

33. Hawking Summary
Using general relativity (and standing on the shoulders of other giants) Hawking showed that the classical concept of time must have begun with a singularity at the Big Bang, and thus the Universe existed at one time in a hot dense state.
Hawking discovered that Black Holes radiate like thermodynamic bodies (now called Hawking Radiation) and possess a temperature and an entropy.
He has written books that make these complex subjects easier to understand.

34. The End
“We are just an advanced breed of monkeys on a minor planet of a very average star. But we can understand the Universe. That makes us something very special.”
- Stephen Hawking

35. Sources
A Brief History of Time & The Universe in a Nutshell – Stephen Hawking ISBN
Introducing Stephen Hawking by J.P McEvoy and Oscar Zarate ISBN