Einsteins Universe

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Einsteins Universe

• Tuesday, February 5

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Einstein Newton smackdown!
Two different ways of thinking about gravity and
space.
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The Way of Newton
Space is static (not expanding or contracting)
and flat.
(Flat means that all Euclids laws of geometry
hold true.)
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Objects have a natural tendency to move on
straight lines at constant speed.
However, we see planets moving on curved orbits
at varying speed.
How do you explain that, Mr. Newton?
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There is a force acting on the planets the
force called GRAVITY.
The gravitational force depends on a property
that we may call the gravitational mass, mg.
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Fg gravitational force
Mg mass of one object
mg mass of other object
r distance between centers of
objects G universal constant of gravitation
(G 6.7 10-11 Newton meter2 / kg2)
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Newton gave another law that gives the
acceleration in response to any force (not just
gravity)!
The acceleration depends on a property that we
may call the inertial mass, mi.
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If a gravitational force is applied to an object
with gravitational mass mg and inertial mass mi,
its acceleration is
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Objects falling side-by-side have the same
acceleration (the same mg/mi).
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Truly astonishing and fundamental fact of physics
mg mi
for every known object!
This equality is known as the equivalence
principle.
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The equivalence principle (which
Newton leaves unexplained) led Einstein to
devise his theory of General Relativity.
Lets do a thought experiment, of the kind
beloved by Einstein.
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Two ways of thinking about a bear
1) Bear has constant velocity, box is accelerated
upward. 2) Box
has constant velocity, bear is accelerated
downward by gravity.
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Two ways of thinking about light
1) Light has constant velocity, box is
accelerated upward.
2) Box has constant velocity, light is
accelerated downward by gravity.
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Einsteins insight
Gravity affects the paths of photons, even though
they have no mass!
Mass and energy are interchangeable E mc2
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Einstein
Newton
Mass energy are very different things.
Mass energy are interchangeable E mc2
Space time are interchangeable part of
4-dimensional space-time.
Space time are very different things.
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Light takes the shortest distance between two
points.
In flat space, the shortest distance between two
points is a straight line.
In the presence of gravity, light
follows a curved line.
In the presence of gravity, space is not flat,
but curved!
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A third way of thinking about a bear
3) No forces are acting on the bear its merely
following the shortest distance between two
points in space-time.
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The Way of Newton
Mass tells gravity how much force to exert force
tells mass how to move.
The Way of Einstein
Mass-energy tells space-time how to curve curved
space-time tells mass-energy how to move.
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Objects with lots of mass (
energy) curve space ( distort
time) in their vicinity.
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The Big Question
How is the universe curved on
large scales (bigger than stars, bigger than
galaxies, bigger than clusters of galaxies)?
That depends on how mass energy are distributed
on large scales.
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The cosmological principle
On large scales (bigger than stars, galaxies,
clusters of galaxies) the universe is homogeneous
and isotropic.
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There are three ways in which space can have
homogeneous, isotropic curvature on large scales.
(Apologetic aside describing the curvature of
3-dimensional space is difficult Ill show 2-d
analogs.)
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(1) This 2-d space is flat (or Euclidean)
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(2) This 2-d space is positively curved
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(3) This 2-d space is negatively curved
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Measuring curvature is easyin principle.
Flat angles of triangle add to 180
Positive angles add to 180
Negative angles add to 27
Curvature is hard to detect on scales smaller
than the radius of curvature.
Flat good approximation
Flat bad approximation
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Parallax (flat space)
August
p
p
February
p a/d
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Parallax (positive curvature)
August
p
p
February
p 30
Parallax (negative curvature)
August
p
p
p a/d As d?infinity, p?a/R
February
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Bright idea

• The smallest parallax you measure puts a lower
  limit on the radius of curvature of negatively
  curved space.

Hipparcos measured p as small as 0.001
arcsec radius of curvature is at least 1000
parsecs.
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We need Bigger triangles to measure the curvature
accurately!
d
?
L
?L/d (flat) ?L/d (positive) ? 33
In a negatively curved universe,
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The most distant objects we can see are the hot
cold spots on the Cosmic Microwave Background.
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Universe became transparent when it was 350,000
years old.
The largest hot cold spots are about 700,000
light years across.
Hot cold spots should be 1 across if
the universe is flat.
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But weve measured the size of the hot cold
spots, and they are 1 across!
Consistent with flat (Euclidean) space.
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What Would Einstein Say?
Density critical density Positive curvature
Density Since our universe is close to flat, the density
must be close to the critical density.
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Thursday
Midterm Exam
Bring your calculator and your favorite writing
utensil.