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Newton, Einstein, and Gravity
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• Chapter 5

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Guidepost
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• If only Renaissance astronomers had understood
  gravity, they wouldnt have had so much trouble
  describing the motion of the planets, but that
  insight didnt appear until three decades after
  the trial of Galileo. Isaac Newton, starting from
  the work of Galileo, devised a way to explain
  motion and gravity, and that allowed astronomers
  to understand orbital motion and tides. Then, in
  the early 20th century, Albert Einstein found an
  even better way to describe motion and gravity.
• This chapter is about gravity, the master of the
  universe. Here you will find answers to five
  essential questions
• What happens when an object falls?
• How did Newton discover gravity?
• How does gravity explain orbital motion?
• How does gravity explain the tides?
• How did Einstein better describe motion and
  gravity?

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Guidepost (continued)
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• Gravity rules. The moon orbiting Earth, matter
  falling into black holes, and the overall
  structure of the universe are dominated by
  gravity. As you study gravity, you will see
  science in action and find answers to three
  important questions
• How do we know? What are the differences among a
  hypothesis, a theory, and a law?
• How do we know? Why is the principle of cause
  and effect so important to scientists?
• How do we know? How are a theorys predictions
  useful in science?
• The rest of this book will tell the story of
  matter and gravity. The universe is a swirling
  waltz of matter dancing to the music of gravity,
  and you are along for the ride.

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Outline
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I. Galileo and Newton A. Galileo and Motion B.
Newton and the Laws of Motion C. Mutual
Gravitation II. Orbital Motion and Tides A.
Orbits B. Orbital Velocity C. Calculating
Escape Velocity D. Kepler's Laws Re-examined E.
Newton's Version of Kepler's Third Law F. Tides
and Tidal Forces G. Astronomy After Newton III.
Einstein and Relativity A. Special
Relativity B. The General Theory of
Relativity C. Confirmation of the Curvature of
Space-Time
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A New Era of Science
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Mathematics as a tool for understanding physics
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Isaac Newton (1643 - 1727)
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• Building on the results of Galileo and Kepler

• Adding physics interpretations to the
  mathematical descriptions of astronomy by
  Copernicus, Galileo and Kepler

Major achievements

1. Invented Calculus as a necessary tool to solve
   mathematical problems related to motion

1. Discovered the three laws of motion

1. Discovered the universal law of mutual gravitation

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Velocity and Acceleration
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Acceleration (a) is the change of a bodys
velocity (v) with time (t)
a
a Dv/Dt
Velocity and acceleration are directed quantities
(vectors)!
v
Different cases of acceleration

1. Acceleration in the conventional sense (i.e.
   increasing speed)

1. Deceleration (i.e. decreasing speed)

1. Change of the direction of motion (e.g., in
   circular motion)

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Acceleration of Gravity
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Acceleration of gravity is independent of the
mass (weight) of the falling object!
Iron ball
Wood ball
Difficult to verify on Earth because of air
resistance but astronauts could verify it easily
on the moon
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Newtons Laws of Motion (1)
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1. A body continues at rest or in uniform motion in
   a straight line unless acted upon by some net
   force.

An astronaut floating in space will continue to
float forever in a straight line unless some
external force is accelerating him/her.
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Newtons Laws of Motion (2)
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1. The acceleration a of a body is inversely
   proportional to its mass m, directly proportional
   to the net force F, and in the same direction as
   the net force.

a F/m ? F m a
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Newtons Laws of Motion (3)
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1. To every action, there is an equal and opposite
   reaction.

The same force that is accelerating the boy
forward, is accelerating the skateboard backward.
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The Universal Law of Gravity
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• Any two bodies are attracting each other through
  gravitation, with a force proportional to the
  product of their masses and inversely
  proportional to the square of their distance

Mm
F - G
r2
(G is the Universal constant of gravity.)
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Understanding Orbital Motion
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The universal law of gravity allows us to
understand orbital motion of planets and moons
Example

• Earth and moon attract each other through
  gravitation.

Dv

• Since Earth is much more massive than the moon,
  the moons effect on Earth is small.

v
v

• Earths gravitational force constantly
  accelerates the moon towards Earth.

Moon
F

• This acceleration is constantly changing the
  moons direction of motion, holding it on its
  almost circular orbit.

Earth
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Orbital Motion (2)
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In order to stay on a closed orbit, an object has
to be within a certain range of velocities
Too slow gt Object falls back down to Earth
Too fast gt Object escapes Earths gravity
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Geosynchronous Orbits
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Keplers Third Law Explained by Newton
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Balancing the force (called centripetal force)
necessary to keep an object in circular motion
with the gravitational force ? expression
equivalent to Keplers third law
Py2 aAU3
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The Tides
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Caused by the difference of the Moons
gravitational attraction on the water on Earth
Forces are balanced at the center of the Earth
Excess gravity pulls water towards the moon on
the near side
Excess centrifugal force pushes water away from
the moon on the far side
? 2 tidal maxima
? 12-hour cycle
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Spring and Neap Tides
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• The Sun is also producing tidal effects, about
  half as strong as the Moon.
• Near Full and New Moon, those two effects add up
  to cause spring tides.
• Near first and third quarter, the two effects
  work at a right angle, causing neap tides.

Spring tides
Neap tides
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Acceleration of the Moons Orbital Motion
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Earths tidal bulges are slightly tilted in the
direction of Earths rotation.
Gravitational force pulls the moon slightly
forward along its orbit.
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Einstein and Relativity
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Einstein (1879 1955) noticed that Newtons laws
of motion are only correct in the limit of low
velocities, much less than the speed of light.
? Theory of Special Relativity
Also, revised understanding of gravity
? Theory of General Relativity
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Two Postulates Leading to Special Relativity (1)
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1. Observers can never detect their uniform motion,
   except relative to other objects.

This is equivalent to
The laws of physics are the same for all
observers, no matter what their motion, as long
as they are not accelerated.
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Two Postulates Leading to Special Relativity (2)
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1. The velocity of light, c, is constant and will be
   the same for all observers, independent of their
   motion relative to the light source.

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Basics of Special Relativity
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The two postulates of special relativity have
some amazing consequences. Consider thought
experiment
Motion of stationaryobserver
Assume a light source moving with velocity v
relative to a stationary observer
v
v
v
c Dt
c Dt
Light source
c Dt
v Dt
Seen by an observer moving along with the light
source
Seen by the stationary observer
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Basics of Special Relativity (2)
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Now, recall that the speed of light, c, is the
same for all observers.
? The times Dt and Dt must be different!
Then, the Pythagorean Theorem gives
(cDt)2 (cDt)2 (vDt)2
or Dt (Dt)/g where g 1/(1 v/c2)1/2 is
the Lorentz factor.
c Dt
c Dt
v Dt
This effect is called time dilation.
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Other Effects of Special Relativity
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• Length contraction Length scales on a rapidly
  moving object appear shortened

• Relativistic aberration Distortion of angles

• The energy of a body at rest is not 0. Instead,
  we find
• E0 m c2

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General Relativity
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A new description of gravity
Postulate Equivalence Principle Observers can
not distinguish locally between inertial forces
due to acceleration and uniform gravitational
forces due to the presence of massive bodies.
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Another Thought Experiment
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Imagine a light source on board a rapidly
accelerated space ship
Time
Time
a
Light source
a
a
a
g
As seen by a stationary observer
As seen by an observer on board the space ship
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Thought Experiment (2)
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For the accelerated observer, the light ray
appears to bend downward!
Now, we cant distinguish between this inertial
effect and the effect of gravitational forces
Thus, a gravitational force equivalent to the
inertial force must also be able to bend light!
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Thought Experiment (Conclusion)
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This bending of light by the gravitation of
massive bodies has indeed been observed
During total solar eclipses The positions of
stars apparently close to the sun are shifted
away from the position of the sun.
? New description of gravity as curvature of
space-time!
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Another manifestation of bending of light
Gravitational lenses
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A massive galaxy cluster is bending and focusing
the light from a background object.
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Other Effects of General Relativity
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• Perihelion advance (in particular, of Mercury)

• Gravitational red shift Light from sources near
  massive bodies seems shifted towards longer
  wavelengths (red).