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Einstein’s Universe
Tuesday, February 5
Einstein – Newton smackdown!
Two different ways of thinking
about gravity and space.
The Way of Newton:
Space is static (not expanding
or contracting) and flat.
(“Flat” means that all Euclid’s
laws of geometry hold true.)
“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?
“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.
 M g mg 
Fg  G 2 
r


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)
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.
a  F / mi
If a gravitational force is applied to an
object with gravitational mass mg and
inertial mass mi, its acceleration is
 M g  m g 

a  G 2 
r
m

 i 
Objects falling side-by-side have the
same acceleration (the same mg/mi).
Truly astonishing and
fundamental fact of physics:
m g = mi
for every known object!
This equality is known as the
“equivalence principle”.
The equivalence principle
(which Newton leaves unexplained)
led Einstein to devise his theory of
General Relativity.
Let’s do a “thought
experiment”, of the kind
beloved by Einstein.
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.
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.
Einstein’s insight:
Gravity affects the paths of photons,
even though they have no mass!
Mass and energy are
interchangeable: E = mc2
Newton
Einstein
Mass & energy are
very different things.
Mass & energy are
interchangeable:
E = mc2
Space & time are
very different things.
Space & time are
interchangeable:
part of 4-dimensional
space-time.
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!
A third way of thinking about a bear:
3) No forces are acting on the bear; it’s
merely following the shortest distance
between two points in space-time.
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.
Objects with lots of mass
(& energy) curve space
(& distort time) in their vicinity.
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.
The cosmological principle:
On large scales (bigger than stars,
galaxies, clusters of galaxies) the
universe is homogeneous and isotropic.
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; I’ll show 2-d analogs.)
(1) This 2-d space is
flat (or Euclidean):
(2) This 2-d space is
positively curved:
(3) This 2-d space is
negatively curved:
Measuring curvature is easy…
in principle.
Flat: angles of
triangle add to 180°
Positive: angles
add to >180°
Negative: angles
add to <180°
Curvature is hard to detect on scales
smaller than the radius of curvature.
Flat = good
approximation
Flat = bad
approximation
Parallax (flat space)
August
p
d
a
p
February
d = Sun-star distance
a=Sun-Earth distance (1 A.U.)
p = a/d
Parallax (positive curvature)
August
p
p
February
p < a/d
Parallax (negative curvature)
August
p
p
February
p > a/d
As d→infinity, p→a/R
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.
We need Bigger triangles to
measure the curvature accurately!
d

=L/d (flat)
>L/d (positive)
<L/d (negative)
L
In a negatively curved universe,
The most distant objects we can
see are the hot & cold spots on the
Cosmic Microwave Background.
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.
But we’ve measured the size of the hot
& cold spots, and they are 1° across!
Consistent with flat (Euclidean) space.
What Would Einstein Say?
Density > critical density:
Positive curvature
Density < critical:
Negative curvature
Since our universe is close to flat,
the density must be close to the
critical density.
Thursday:
Midterm Exam
Bring your calculator and your
favorite writing utensil.