Notes on General Relativity and Cosmology
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What do we observer?
- The sky is dark (Olber's paradox: If the universe were infinite
then every line of sight would end on a star, eventually so the sky
should be bright, not dark).
- The universe is isotropic (looks the same in every direction, at least on a large scale > 200Mpc).
- The universe is homogeneous (looks the same at every location; we are not at a privileged location).
- There is a cosmic microwave background radiation which is very
uniform corresponding to a black body curve for an object at 2.725K +
or - 0.001K.
- Objects (stars, galaxies etc.) are moving faster the further away
from us they are in a nearly linear relation. We also can measure how
this rate changes over time since when we look out into space we are
looking backward in time (it isn't exactly linear- see below).
- There are no magnetic monopoles in the universe.
- The matter that we can detect in the universe is about 3/4 ths
hydrogen, 1/4 helium and a tiny fraction consisting of all the other
baryonic material. (i.e. normal every day stuff; protons neutrons, etc.).
- Galaxies and superclusters of galaxies do not fly apart (even
though it looks like they don't have enough mass to stay together, if
we only count matter that we can detect directly).
What is the current model (called the Benchmark Universe) of how the universe is constructed?
- The General theory of Relativity is the correct set of equations which describe the large scale structure of the universe.
- The universe started with a hot big bang.
- There was a period early in the universe where there was a sudden
expansion (called inflation) before settling down to a more steady
- The expansion rate of the universe is now accelerating and has been for about the past 4 billion years.
- Currently about 0.001% of the universe is made of photons.
- Currently about 4% + or - 1% of the universe is made of baryonic material (the stuff we can detect).
- Currently about 26% of the universe is made of dark matter (which
keeps galaxies and clusters from flying apart). This could be
neutrinos, black holes, WIMPS (weakly interacting massive particles) or
some other unknown mass.
- Currently about 70% of the universe is dark energy. This could be vacume energy or something else we don't yet know about.
How do the observations support the current model?
Evidence for Einstein's equations:
- Newton's equations told us: Mass tells gravity how to exert a force (F=GMm/r2)
and F=ma tells a mass how to move as a result of a force. He has no
explanation for why m is the same in both equations. Einstein's
equations tell us: Mass/energy tells space time how to curve and curved
space time tells mass/energy how to move.There is no difference between
inertia mass and gravitational mass as there is in Newton's theory.
- The three major tests we have made of Einstein's equations (the
shift in the perihelion of
Mercury, the red shift of light traveling away from a massive object,
the bending of light by the sun) turn out correct. Global Position
Satellite positioning and airplane navigation makes use of the theory
to give correct positions.
- Einstein's equations tell us there should be black holes and we
detect them (rather, we see evidence that they are there such as
- Einstein's equations tell us the universe should be either expanding or contracting; it is expanding.
the big bang has started, Einstein's equations give the correct amount
and time for galaxy and star formation (clumping due to gravitation).
Evidence for the hot big bang:
- The Doppler shift (red shift) of stars tells us how fast they are
moving. From this Hubble found that most stars and galaxies are moving
away from us and the ones further away are moving away faster. This
tells us the universe is expanding. (No, we are not at the center,
think about dots on an expanding balloon, they all move away from each
other and each dot will see the dots further away as moving faster.).
The rate of expansion has been approximately constant over time but not
quite (this small change is significant).
- An expanding universe solves Olber's paradox. The universe is not
infinite in time so light has not had enough time to reach us from
stars that are further than a certain distance (the horizon). Current
calculations of the horizon distance and the age of the universe are
consistent with each other.
- If we extrapolate backwards in time to a point when the matter
and energy density of the universe is very high (the first few minutes
of the universe) we can apply the equations of nuclear physics since
the universe at that time is a plasma (similar to what goes on in a
hydrogen bomb when it explodes). At first the energy density is so high
that quarks cannot come together to form protons and neutrons. At a
later time the energy density is low enough to form protons and
neutrons but not atoms because the photons present have enough energy
to keep all atoms completely ionized. At a still later time the energy
density drops to a point where photons decouple from matter and atoms
can form. All of this happens in the first three minutes of the
existence of the universe. When the equations of nuclear physics are
solved for this transition period we get a prediction that 3/4ths of
the matter in the universe should be hydrogen, 1/4 helium and small
amounts of isotopes of hydrogen (i.e. deuterium), lithium and
beryllium. These predictions are very precise and they give the exactly
quantities we see in the universe today, with the exception of heavier
elements which are created in stellar processes (fusion in stars and
nova). Nuclear physics (physics of the tiny) at the big bang determined
the makeup of the universe we see today.
- The energy of the photons when they decouple from matter in the
first three minutes continue to travel through the universe but they
are very red shifted because of the gravity of the 'stuff' in the
universe. These photons are what we see as the cosmic microwave
background. Saying the photons are red shifted as they travel through
the universe is exactly equivalent to saying the universe cools over
time and it's black body radiation shifts to lower frequency. The
current temperature of the cosmic microwave background is exactly what
is predicted from the temperature at the time of the photon decoupling
and subsequent cooling since then.
- Einstein's equations predict either an expanding or contracting
universe, not a static one. All tests of the equations have been
successful so an expanding universe is consistent with Einstein's
Evidence for inflation:
- A sudden expansion of the early universe solves the flatness
problem: The universe appears flat today (i.e. not much acceleration-
the rate of expansion is fairly constant). In order to be this flat it
had to be pretty flat to start with. Imagine an ant crawling on a
balloon. How could we make the balloon appear more flat to the ant?
Make it bigger. A sudden inflation of the early universe does this for
- A sudden expansion of the early universe solves the horizon
problem: How did the universe become isotropic? For the microwave
background to be that smooth, the universe had to be well mixed yet we
know that some parts of the universe have not had time to communicate
(mix) with other parts. So how did the universe become isotropic? If
the universe was smaller than we thought at the beginning the various
parts could mix before losing contact with other parts in a sudden
- A sudden expansion solves the monopole problem: We don't see any
monopoles although theory says they should have existed in large
numbers at the beginning. A sudden expansion means the density of
monopoles is now very low, so low we do not expect to find any.
- The very small variations in the current cosmic microwave
tell us the amount of 'clumpyness' of the early universe if we
extrapolate backwards in time. This clumpyness fits nicely with the
idea that the universe at the very beginning was smooth but quantum
fluctuations (due to Heisenberg's uncertainty principle) caused it to
be just a
little lumpy. The lumps got stretched out into the current observed
microwave variations by inflation. This same lumpiness also made star
and galaxy formation possible (a perfectly smooth universe would not
form stars). So due to inflation, the quantum fluctuations of the early
universe led to galaxy formation and variations in the microwave
background which are observed.
Evidence for 'dark matter':
- Without a halo of dark matter (something which has
there is not enough gravity to keep galaxies and super clusters of
galaxies together. Based only on the material we can see, galaxies
should fly apart but they don't. Gravity gives us a specific value(26%
of the universe) for how much dark matter there must be to keep things
Evidence the universe is accelerating and there is 'dark energy':
- We can get the rate of expansion of the universe from red shift
data and also the rate of change of expansion (the acceleration). The
acceleration is determined by how much 'stuff' there is in the universe
(much like if you were to shoot a rocket straight upward; a bigger
earth would pull it down, a smaller earth would not because it would
cause less acceleration). The current acceleration is very small (i.e.
the expansion rate is fairly constant- but see the next bullet).
This means two things: the universe has very little curvature (is flat)
and there ought to be a lot more 'stuff' (70% more!) than what we
detect as either baryons or 'dark matter' to account for the small
acceleration we do measure.
- In 1998 two separate groups using different techniques measured
the red shift for 'standard candles' (astronomical objects with known
light emitting properties) for galaxies very far away. These were Type
Ia supernova and are the furthest away of anything we can measure as a
standard candle. Because of the expansion of the universe we expect
objects which are further way to be moving faster (and therefore have a
higher red shift). These groups expected to measure a slowing of the
expansion (a deceleration) and get a better value for the missing
'stuff' mentioned in the last bullet. But these Type Ia supernova are
moving a little TOO fast for their distance. This means the universe
has to be accelerating, not decelerating.
- In order to get an expanding universe from Einstein's equations
we have to add a term corresponding to negative energy (Einstein
originally tried to put his term in to get a static universe but then
took it out when he found out the universe was expanding). Positive
mass/energy tends to slow (decelerate) the expansion (this was in fact
what the astronomers were trying to measure when the found the
acceleration). So not only is 70% of the universe stuff that we can't
detect and have no theory for, it has to have negative energy!
- Fluctuations in the cosmic microwave background are also very
consistent with a flat universe where 70% of the 'stuff' is missing.
Notice this is two totally different sets of measurements (supernova
and microwaves) giving the same result (70% of the universe is missing).
- In quantum mechanics we will learn that because of the Heisenberg
uncertainty principle, what we think of as a vacume can have
mass/energy fluctuating into existence and then out of existence for
short times. On a small scale this makes very little difference
(although it has been measured- see the Casmir effect). On the scale of
the universe, however, this is a lot of 'stuff'. Unfortunately the
current calculations give a number that is way TOO BIG (more than the
missing 70% by a factor of millions!!!). Keep in mind, however, that
General Relativity and quantum mechanics are not compatible. So it is
possible that at some point in the future a quantum version of General
Relativity will correctly predict the amount of missing stuff.
Physics at IUS: http://physics.ius.edu/
Contact Dr. K. Forinash,