General Relativity and
The Expanding Universe
Michael Marshall & Michael Szopiak
“Matter tells space how to curve. Space tells matter how to
move.” – American physicist John Wheeler
Special Theory of Relativity
General Theory of Relativity
2 Postulates (Einstein, 1905):
Equivalence Principle (Einstein,
1907):
1) The form of the laws of physics
is the same in all inertial
reference frames
In a small region of space-time
(locally), it is not possible to
distinguish operationally between
a frame ‘at rest’ in a uniform
gravitational field and a frame
uniformly accelerated through
empty space
2) The speed of light has the same
constant value for all inertial
observers
Consequences of the Equivalence Principle
1) Inertial Mass = Gravitational Mass
2) Light is affected (‘bent’) by gravitational fields
Inertial Reference Frames
Performed locally, no experiment inside the free-falling elevator could
distinguish it from an inertial frame.
Since gravitational and
accelerated frames are
equivalent, we cannot
determine an absolute inertial
frame.
Mach’s Influence on General Relativity
It was supposed that Mach’s concept (of an inertial frame determined
with a reference to the fixed stars) was incorporated into General
Relativity – but these showed otherwise.
De Sitter’s (Empty, Expanding)
Gödel’s (Absolute Rotation)
Experimental Tests
• Einstein’s theory of general relativity and the equivalence
principle are tested with three well known experiments
– Bending of light rays by a gravitational field
– Gravitational red shift of Light
– Advance of the perihelion of Mercury
Bending of Light Rays by a
Gravitational Field
• Mass equivalence principle
– “The form of each physical law is the same in all locally
inertial frames.” (Harris 45)
• Therefore, if light bends in an accelerating frame, any
experiment should conclude that light bends in a gravitationally
equivalent frame
Light Bends in an Accelerating Frame
Bending of Light Rays
• The mass equivalent principle provides another reason for
thinking light would bend in a gravitational field
• E = mc2 can be rearranged to m = E/c2
• Therefore, light (which carries energy) has an effective ‘mass’ in
a gravitational field
The Experiment
• Travel to Russia and South America
• Observe solar eclipse
• Calculate apparent position of star at point 2 and the actual
position of the star at point 1
Einstein’s Prediction
• In a 1911 paper, he postulated the bending of the starlight to be
about 0.83” (seconds of arc) for the given example
• Einstein recognizes theoretical problem
• Revised prediction of 1.7” before any experiment was
performed
• Actual data from 1919 confirms Einstein’s prediction of 1.7”
An Elegant Theory
“Now I am fully satisfied, and I no longer doubt the correctness
of the whole system, whether the observation of the eclipse
succeeds or not ... The sense of the thing is too convincing.”
(Einstein, 1914)
“When I was giving expression to my joy that the results
coincided with his calculations, he said quite unmoved, ‘But I
knew that the theory was correct’, and when I asked, what if
there had been no confirmation of his prediction, he countered,
‘Then I would have been sorry for the dear Lord – the theory is
correct.’” (Einstein’s student, 1919)
Interesting Questions
• What if the expedition had gotten the initial results of 1.7” before
Einstein adjusted his numbers?
• How do you think this would have affected the scientific
community’s view of general relativity?
• How does the concept of falsifiability connect to this experiment if
the expedition calculated 1.7” before Einstein adjusted his
prediction?
• Even if Einstein were to correct his error, would his theory have the
same strength?
– Should it have the same strength?
Gravitational Redshift
The Doppler Effect (Revisited)
Gravitational Redshift
The Doppler Effect (Revisited)
1 ( vc )
red 

v
1 ( c )
1   2
1
1
2
2
 f source
  f source1  vc  1 vc 
1 
1
f observed
v
c
v
c
 f obs  f source1  12 vc 1  12 vc 
 
f obs  f source1  vc   f source 1  c 2
gl
 gl
This is equivalent to :
 2
 c
for v 
gl
c
Gravitational Redshift
The Doppler Effect (Revisited)
Real-World Example: Global Positioning System (GPS)
f 
1
t
 
 t
1  ,
 t
1  
f obs  f source 1  c 2
gl
t lower
t lower

gl
higher
c
higher
1
c2
2
where g  GM
r2
GM
rea rth
 rsaGM
tellite

Advance of the Perihelion of Mercury
• Newtonian mechanics fails to
properly predict orbital paths for
high gravitational fields
– Mercury provides the perfect
example
• (Exaggerated picture)
Perihelion Precession
• Over one century, Mercury’s orbit appears to advance 5600” of
arc per century
• 5026” of arc are due to wobbling of Earth on axis
• 531” of arc are due to perturbation from planets
• Newtonian mechanics cannot explain about 43” of the
precession of Mercury
• General relativity is able to account for the missing 43”
General Relativity’s Success
• Three experiments validate general relativity
• The elegance with which general relativity explains each of these
difficult phenomena gives it strength as a theory
Review of the Classical Universe
• Newton believed the universe to be
– Euclidean (have zero curvature)
– Infinite
– Finite amount of matter
• Then changed position to uniform distribution of matter
throughout an infinite space in order to prevent collapse
of the universe (“Fixed Stars”)
– The stability of the universe was in question
– Newton postulated that our universe is stable because of the
action of God
Olbers’ Paradox
• Heinrich Olbers, an eighteenth century German astronomer,
asks the question:
– Why is it dark at night?
• This question seeks to understand what it would mean if the
universe is actually infinite in both mass and size.
• Olbers suggests that the sky should appear as bright as the
noonday sun.
Olbers’ Paradox
• The sky is not as bright as the noonday sun!
• Thus, we have a contradiction when we say we have an infinite,
static, homogeneous universe and a sky that is not always bright.
• This paradox lacked a solution for over 100 years, and led many
to research a finite universe.
A Review of Curvature
•
•
•
•
Metric
2 dimensions are easier to conceptualize
Then you can generalize to higher dimensions
Geodesics are straight lines in a space
Einstein’s Static Universe
Friedmann’s Solutions
To Einstein’s Field Equations
Possible Universes:
Solutions To Einstein’s Field Equations
Lemaître’s Universe:
A Big Bang
 Finite past – expanded from a hot, dense beginning
 Positive curvature
 Positive cosmological constant Λ > ΛE
Vesto Slipher
• From 1912 to 1925, Slipher
measured spectral lines of light from
distant galaxies
• Red shift!
• Unique spectral lines
• Great majority of galaxies are
receding
• Did not appear to be random
Edwin Hubble
• American (1889-1953)
• Had much more accurate ways of
telling distance
• By finding independent
determinations of distances to
distant galaxies, he determined
that the recessional velocity of a
galaxy is directly proportional to
its distance from the Earth
Hubble Constant
•
•
•
•
Doppler-shift
Hubble’s discovery
Hubble’s Law
Age of the Universe
– 13.7 Billion Years
Hubble’s Law
• Space is expanding
• The universe is not static
• Each observer perceives himself
as the center of the universe
• Universe’s expansion is slowing
down, which implies the universe
used to be more compact
– Supports notion of Big Bang
Olbers’ Paradox (Revisited)
• Problem of red shifting
– The light from distant
stars would not be in the
visible spectrum
• Problem of Hubble radius
– If recessional velocity is
equal to the speed of
light, then there are parts
of the universe we can
never know about
• Problem of the beginning
– Olbers’ paradox relies on
the notion that light has
had an infinite amount of
time to reach the Earth
– Since Hubble’s Law
supports the Big Bang,
then the universe has not
existed forever as we
know it
http://www.youtube.com/watch?v=gxJ4M7tyLRE
Dark Matter
• Interestingly, it appears that there is not enough mass for the
universe to be expanding like it is. This fact has led to the
postulation of dark matter.
– Dark matter is matter that we do not ordinarily encounter,
but its postulated affects can be observed through
gravitational lensing
A Modern Model of the Universe
 Our Galaxy: Disc-shaped; 100,000 ly wide in diameter
 Cosmic Background Radiation: T = 2.7 K
 Black Holes: Strong gravitational field (when a massive body
undergoes gravitational collapse) is dense enough so the escape
velocity is greater than the speed of light
 The Multiverse: Parallel universes?
Works Cited
Barrow, John D. The Book of Universes: Exploring the Limits of the Cosmos. New York: W. W. Norton & Company
Inc., 2011.
Harris, Randy. Modern Physics. 2nd ed. San Francisco: Pearson/Addison Wesley, 2008. Print.
Kehoe, Robert. “Ch. 4 Relativity.” Phsyics.smu.edu. Southern Methodist Univeristy, May 2006. Web. 04 Apr. 2013.
http://www.physics.smu.edu/kehoe/1301S06/Ch4Relativity.pdf.>.
Nave, Carl Rod. "HyperPhysics." HyperPhysics. Georgia State University, Aug. 2000. Web. 08 Apr. 2013.
<http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html>.
Minute Physics YouTube Videos:
Olber’s Paradox: http://www.youtube.com/watch?v=gxJ4M7tyLRE
Multiverse: http://www.youtube.com/watch?v=Ywn2Lz5zmYg
Pictures (from Internet):
http://www8.garmin.com/graphics/24satellite.jpg
http://marketplayground.com/wp-content/uploads/2010/12/garmin-nuvi-1340-sat-nav-garmin-nuvi-1340-gps.jpg
http://www.astro.virginia.edu/class/whittle/astr553/Topic01/t1_slipher.html
http://scienceworld.wolfram.com/biography/Hubble.html