Developing a Theory of Gravity

Does the Sun go around the Earth or Earth around Sun?

Why does this happen?

Plato } Artistotle } Philosophy Ptolemy } & Models Copernicus } Galileo Kepler } } Experiment & Mathematical Description Newton } Physics

Einstein??

Timeline

How

do objects fall to the ground?

• Aristotle philosophies only [without looking for deeper explanations] • Galileo performs experiment – Observe – Deduce – Then philosophize

Scientific Methodology • experiment – free-fall (S= ½ at 2 ) fig. 5-4 • rule out philosophies through observations --> “scientific proof” Galileo performs experiments to proof his philosophical views -- he observes and deduces.

Experiment

Drop Objects What does the Fall depend on?

a) Constant Speed?

b) Dependence on shape?

c) Dependence on weight?

Test! Experiment! Look at Results! Then Philosophy!

Experimental Results

• Speed

increases

as bodies fall; they

accelerate

!

Mathematical Description

(based on Empirical Results) free-fall s= ½ at 2 • All bodies “fall” at the

same

– NO dependence on Weight – NO dependence on Shape rate

Intermission

Which of these cans rolls fastest?

Make a Prediction!

• Empty can • Can filled with liquid • Can filled with solid food

How

does Earth go around the Sun?

— Brahe: Observations — Kepler

analyzes

Brahe’s Observations: — Invent Laws empirically — describe orbits mathematically 

1.

Kepler’s 3 Laws: Orbits of planets are ellipses with the sun at one focus 2.

A line from a planet to the sun sweeps out equal areas in equal times 3.

A planet’s orbital period squared is proportional to its average distance from the Sun cubed.

P

2 

a

3

mathematical description -- no cause!

Why

does the Earth go around the Sun?

So far: Models

only

no

explanation • Galileo “Introduces Methodology” • Brahe "Observer" • Kepler "Analyst” • Newton “Provides Model” 

Why

Newton—Laws of Motion

lst law: Inertia

object at rest remains at rest object in motion remains in motion unless a force acts on it Absence of force

2 nd law: F=ma

Galileo—free fall  acceleration Acceleration is the result of a force acting on the mass

3 rd Law: Actio = Reactio

For every action there is an equal and opposite reaction Balance of forces

Why does the apple fall off the tree?

Aristotle motion of earthly things are determined by their natural tendencies to move towards their proper place in the cosmos, i.e. towards the center of the Earth. Newton A force acts on the apple This force makes the apple accelerate ==> Gravity acts on the body

…and one day the apple fell off the tree Aristotle – uses common sense The motions of earthly things are determined by their natural tendencies to move towards their proper place in the cosmos, i.e. towards the center of the Earth.

How

does the Apple fall? Galileo –

Experiment and Analysis

Observe and Experiment (e.g. Tower of Pizza) Deduce (& mathematical description; e.g. s= ½ g/t)

Why

does the Apple fall?

Newton --

Theory

A force acts on the apple This force makes the apple accelerate toward Earth ==> Gravity acts on the body

Philosophy Scientific Methodology and Scientific Thinking

Is Gravity a “Universal" Force?

— Why does the Apple fall off the Tree?

— Why does the Moon not fall towards the Earth?

Hypothesis: — Gravity acts on Moon — Gravity is counterbalanced by Centrifugal force (remember: Action = Re-actio) Is this correct? — How do you test this?

 (a) Cannon Ball Thought Experiment  (b) Mathematical Proof

Is Gravity a “Universal" Force?

— Why does the Apple fall off the Tree?

— Why does the Moon not fall towards the Earth?

Why doesn’t the Moon fall towards the Earth?

Why does the moon not travel in a straight line?

Recall Newton's first law --> A force must act on the moon What is this force?

Centrifugal force Effect of this force?

--> change in direction

The Resulting Path of the Moon… Gravitational pull (blue) Centrifugal force (red) Resulting Path (black)

Cannon Ball Thought Experiment • How fast would the cannon ball have to travel to go around the earth in a circle?

• If the cannon ball was at the moon's distance how fast would it have to travel then?

• What is the velocity of the Moon?

How would you test whether this hypothesis is correct?

Discussion

Speed of the Moon? For orbital motion have:

v



GM r

Does this agree to the observed speed?

v

 distance time 

circumfere nce

1 month  1 2 

r

month Yep!

The Universal Law of Gravity Gravity decreases with the inverse square law

F



Mass

1 

Mass

2 distance 2 This applies to all Objects, Apples, Cannon Balls, the Moon, and other Planets.

 UNIVERSAL LAW Newton provided an explanation of

WHY

the Moon goes around the Earth.

Quiz Questions

• Deriving Kepler's Third Law • Calculating Earth’s Mass (By measuring earth’s acceleration on its surface and distance from center of Earth)

Einstein Principle of Equivalence Acceleration pulls you down  

No

Gravity!! ONLY acceleration  Need New Theory of Gravity "General Relativity"

Test of General Relativity Eclipse in 1917

• Curvature strongest in vicinity of dense and massive objects • (black holes are theoretical playgrounds for

Einstein Ring

Difference between Newtonian Theory of Gravity and General Theory of Relativity • Newtonian: The Sun creates a gravitational field that exerts a force upon the Earth, which, in turn, causes it to orbit around the Sun rather than move in a straight line • General Relativity: The Mass-Energy Distribution of the Sun changes the geometry of space-time. The Earth is in free motion (no forces acting on it!) and travels on a geodesic of space-time. But because space-time is curved around the Sun, the Earth orbits the Sun.

From the Special Theory of Relativity to the General Theory of Relativity Newtonian Mechanics – 3 space coordinates -- no time coordinate  no relation between event 1 and event 1   need the Special Theory of Relativity need frames of reference -- need Lorentz Transformation However: Galaxy is accelerating due to -- other galaxies around it -- expansion of the Universe   Acceleration is due to forces Include “forces” into the Theory of Special Relativity    General Theory of Relativity

Task of General Relativity

Couple Geometry to the Mass distributions and motions How does matter affect the Geometry of Space-Time How do particles move in this Geometry (no forces!)

G ij

 8 

G c

2 

T ij

Geometry described by Robertson Walker Metric

dS

2 

c

2

dt

2 

R

2 (

t

) 

dr

2 1 

kr

2 

r

2 

d

 2 Stress Energy Tensor A constant  sin 2 

d

 2   Last two equations (only if coordinates are chosen cleverly)