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Title: Cosmology

Syllabus ----------------------------------------
--------------------------- Special theory of
relativity. General theory of relativity. Friedman
n universe models, Hubble constant, cosmological
red shift Elementary particles and
interactions. Modern cosmology, phases and
processes near after the Big Bang. Evolution of
the Universe with dark matter and dark
energy. -----------------------------------------
The study of the Universe at large, its origins
and evolution. It is distinct from cosmogony,
which is the study of the origin and evolution of
objects within the Universe, such as
galaxies. Although the roots of cosmology can be
traced back to ancient myths and legends, and the
Greek study of the way the planets move, modern
cosmology is essentially a mathematical
description of the behaviour of spacetime on the
largest scale, using equations derived from
Albert Einsteins general theory of relativity.
So the date on which modern cosmology was born
can be set very precisely, as 1917, the year
Einstein first applied those equations to
describe the Universe at large. Although some
theorists have developed alternative theories of
gravity and spacetime, which have led to
cosmological models different from those derived
from Einsteins theory, these alternative
cosmologies have now been ruled out by
observations. Within the framework of Einsteins
theory, there have been two great cosmological
hypotheses, the Big Bang and the Steady
State. Both these models conform to the
observational evidence that the Universe is
expanding as spacetime stretches, but in the big
bang model this is seen as evidence that the
Universe was born out of singularity a finite
time ago, while in the Steady State model it was
required that new matter should be created
continuously to fill the gap between the galaxies
as they moved apart, so that the overall
appearance of the Universe stayed the same.
The simple Steady State hypothesis has been found
to be incorrect, because there is now clear
observational evidence that the Universe changes
as time passes. This leaves a variety of possible
Big Bang models, which can be considered as
possible descriptions of the real Universe. To
within the limits of observational accuracy, our
Universe is indistinguishable from flat model
which obeys the laws of Euclidean geometry. This
is the simplest possible Universe allowed by
Einsteins equations.
Theory of the relativity
Newtons mechanics
Electromagnetic field theory
Special theoryof relativity
General theory of relativity
Relation between classical mechanics,
electromagnetic field theory and theory of
Special theory of relativity
Special theory of relativity Description of the
relationship and interactions between moving
object, developed by Albert Einstein early in the
20th century. The theory was first published in
1905, in a mathematical form based on equations
its implications can be more clearly visualized,
however, using a geometrical description of
events taking place In a fourdimensional
spacetime, first applied in this context by
Herman Minkowski in 1908. The special theory
gets its name because it applies only to be the
special case of objects moving at constant speeds
in straight lines that is, at constant
velocities. It does not deal with accelerated
motions, including the acceleration caused by
gravity. The later extensions of Einsteins
theory to deal with gravity and other
accelerations (the general theory of relativity)
developed the geometrical model of spacetime
further. The key features of the special theory
are that, from the point of view of an observer
who is regarded as stationary (in his or her own
inertial frame), time recorded on a moving clock
will run slow, a moving object will shrink in the
direction of motion, and the moving object will
gain mass. The speed of light is the same for any
observer in any inertial frame, no matter how he
or she is moving relative to the source of the
light, and it is impossible to accelerate an
object from below the speed of light up to the
speed of light. All of these predictions have
been tested and verified many times in
experiment. It is the special theory that says
that mass and energy can be interchanged in line
with Einsteins equation E0 m0c2 this too has
been confirmed by experiments.
General theory of relativity
General theory of relativity Theory of gravity
developed by Albert Einstein in the early part of
the 20th century, and presented to the Prussian
Academy of Sciences in 1915. Because gravity is
the dominant force in the universe at large
(thanks to its very long range), the theory is
also a theory of cosmology, and underpins all
modern models of how the Universe got to be the
way it is. Einsteins special theory of
relativity, published in 1905, deals with the
dynamical relationships between objects moving at
constant speeds in straight lines. It does not
deal with accelerations, or with gravity, which
is why it is called the special (meaning
restricted) theory. Einstein always intended to
generalize his theory to deal with accelerations
and gravity, but it took him ten years (not all
the time devoted exclusively to the general
theory) to find a satisfactory mathematical
description of the dynamics of the Universe and
everything in it. Indeed, the whole point about
Einsteinss theory is that it gives us a physical
picture of how gravity works Isaac Newton
discovered the inverse square law of gravity, but
explicitly said that he offered no explanation of
why gravity should follow an inverse square law.
The general theory of relativity also says that
gravity obeys an inverse square law (except in
extremely strong gravitational fields), but it
tells us why this should be so. That is why
Einsteins theory is better than Newtons even
thought it includes Newtons theory within
itself, and gives the same answers as Newtons
theory everywhere except where the gravitational
field is very strong.
Practices, demonstrations
Practices in special theory of relativity -
Lorentzs transformation, Lorentzs matrix,
relativistic factor, Practices in general
theory of relativity - description of the curved
spacetime, metric tensor, Schwarzschilds
some demonstrations http//demonstrations.wolfram
usCurvedSpacetime/ http//demonstrations.wolfram.c
General theory of relativity
  • Observed effects of general theory of relativity
  • advance of perihelion
  • of Mercury
  • of binary pulsars
  • deflection of light
  • by Sun, observed first in 1919 by Arthur
  • gravitational lensing, from stars (low
    probability) and galaxies often observed
  • gravitational time dilatation
  • gravitational redshift direct observed by
    Pound Rebka experiment
  • atomic clock shift by flight experiments
  • GPS in fact permanent operating general
    relativity experiment
  • gravitational radiation
  • indirect observations in binary pulsars

General theory of relativity
Advance of perihelion of Mercury The orbit of
Mercury around the Sun does not trace out the
same path every time, but shifts slightly from
one orbit to the next. Each orbit is an ellipse,
with the Sun at one focus of the ellipse. In each
orbit, at the closest approach of Mercury to the
Sun (perihelion), the ellipse shifts sideways by
a tiny amount. This advance of the perihelion was
predicted by Albert Einsteins general theory of
relativity but cannot be explained by Isaac
Newtons theory of gravity. The perihelion
precession of Mercury is 5600 arc seconds per
century. Newtonian mechanics, taking into account
all the effects from the other planets, predicts
a precession of 5557 seconds of arc per century.
In the early 20th century, Albert Einsteins
General Theory of Relativity provided the
explanation for the observed precession. The
effect is very small the Mercurian relativistic
perihelion advance excess is just 42.98
arcseconds per century, similar, but much
smaller, effects operate for other planets 8.62
arcseconds per century for Venus, 3.84 for Earth,
1.35 for Mars (http//
In 1859, the French mathematician and astronomer
Urbain Le Verrier reported that the slow
precession of Mercurys orbit around the Sun
could not be completely explained by Newtonian
mechanics and perturbations by the known planets.
He suggested, among possible explanations, that
another planet (or perhaps instead a series of
smaller 'corpuscules') might exist in an orbit
even closer to the Sun than that of Mercury, to
account for this perturbation. (Other
explanations considered included a slight
oblateness of the Sun.) The success of the search
for Neptune based on its perturbations of the
orbit of Uranus led astronomers to place faith in
this possible explanation, and the hypothetical
planet was even named Vulcan. However, no such
planet was ever found. (http//
General theory of relativity
Deflection of light One of first key tests of
the general theory of relativity which predicted
that light from a distant star passing close by
the Sun would be bent by a certain amount. The
only way to observe this is during an eclipse,
when the bright light of the Sun itself is
blocked by the Moon, and stars can be seen on the
sky around the eclipsed Sun. Albert Einstein
published his paper predictihg this effect in
1916, abd British astronomer Arthur Eddington
organiyed an expedition to observe the eclipse
from Principe, off the west coast of Africa,
while a second team observed it from Brazil.
Photographs of the positions of the stars near
the Sun on the sky were then compared with
photographs of the same part of the sky taken 6
months earlier, when the Earth was on the other
side of the Sun in its orbit and those stars were
visible at night. The comparison showed that the
stars photographed during the eclipse seemed to
have been shifted sideways slighthly by the
deflection of light by exactly the amount that
Einstein had predicted. In principle, this
deflection occurs whenever light passes near a
massive object, although usually the effect is
too small to be measurable. It is caused by the
curvature of spacetime associated with the mass.
A more extreme version of the same effect causes
the gravitational lens phenomenon, and in the
ultimate extreme light is trapped completely
within a black hole.
General theory of relativity
Gravitational time dilatation, gravitational red
shift Slowing down of clocks caused by a
gravitational field, as predicted by the general
theory of relativity.
(linear approximation)
With this effect must be for example calculated
also by GPS navigation system ?1 10,23000000000
MHz ?2 10,22999999543 MHz (also included
effect of special theory of relativity
transversal Doppler shift).
General theory of relativity
  • Binary pulsars
  • A binary pulsar exists when two neutron stars,
    one of which is a pulsar, are in orbit around one
    another, forming a binary star system. The term
    is also used to refer to a pulsar in orbit about
    any other star for example, a white dwarf. More
    than twenty binary pulsars are now known, but
    astronomers reserve the term the binary pulsar
    for the first one to be discovered, which is also
    known by its catalogue number, as PSR 191316.
    This pulsar has provided the most accurate test
    yet of Albert Einsteins general theory of
    relativity, and is the most accurate clock yet
  • The binary pulsar was discowered in 1974 by
    Russell Hulse and Joseph Taylor, of the
    University of Massachusetts, working in the
    Arecibo radio telescope in Puerto Rico.
  • Measured effects
  • advance of the perihelion
  • time dilatation (STR effect)
  • losing energy as a result of gravitational
  • Fro this results Hulse and Taylor won the 1993
    Nobel Prize in Physics.
  • For other information see http//

Equivalence principle
Equivalence principle That the effects of
acceleration are indistinquishable from the
effects of a uniform gravitational field. This
equivalence results from the equivalence between
gravitational mass and inertial mass. It led
Albert Einstein to the development of general
theory of relativity, when he realized that a
person falling from a roof would not feel the
effects of gravity the acceleration of their
fall would exactly cancel out the feeling of
Acceleration is equivalent to a uniform
gravitational field
In modern language, the equivalence is best
described in terms of a spaceship being
accelerated through space by constant firing of
its rocket motors. When the motors are not
firing, everything inside the spaceship floats
about in free fall, just as weightless as the
person falling from a roof. In principle, the
acceleration of the rocket could be adjusted so
that everything inside felt a force exactly as
strong as the force of gravity on Earth (or any
other strength you chose), pushing things to the
back of the vehicle as it moved forward through
space. Any scientific experiments carried out in
this accelerating frame of reference would give
exactly the same results as if the spaceship were
standing on its launch pad on Earth, and not
accelerating at all.
Schwarzschild metric
Eukleidian metric in kartesian and polar
Schwarzschild metric (space out of spherical
symetrical spread matter)
Minkowski metric in polar coordinates (space
without matter or in special theory of relativity)
Where is the
You can see that the Minkowski metric is limit of
the Scwarzschild metric for for r gtgt rg.
Friedman metric
Friedman metric (space with matter regularry
spreded with constant density)
You can see that the Minkowski metric is limit of
the Scwarzschild metric for for r gtgt rg.
The final test
will take place the last week of semester, on
Tuesday 17.5.2001 at 1615 after the last
lecture, during the time for practise in the room
T2C2-81. Later you can only arrange the
alternative date and time of the final test with
me individually For other information see
ysics_final_test.html 12 questions, each have 4
possibilities, maximum gain is 48 points,
Grading scale in per cent in per cent in points in points
Grading scale from to from to
Excellent (A) 90 100 43 48
Very good (B) 80 90 38 42
Good (C) 70 80 33 37
Satisfactory (D) 60 70 28 32
Sufficient (E) 50 60 24 27
Fail (F) 0 50 0 23