Документ взят из кэша поисковой машины. Адрес оригинального документа : http://zebu.uoregon.edu/1999/ph123/lec03.html
Дата изменения: Thu Oct 7 22:32:21 1999
Дата индексирования: Tue Oct 2 03:35:27 2012
Кодировка:
Поисковые слова: п п п п п п п п п п п п п п п

Historical Cosmologies

Cosmology and the Origin of Life

Newtonian Gravity:

Gravitational Force is an attractive force which weakens with the inverse square of the distance between two mass points. Thus, double the distance and the gravitational force weakens by a factor of 4

F_g goes as 1/R²
We can derive Kepler's Third Law from Newton's equations as follows:

Step 1: Assume that M₁ is a small mass in a circular orbit about a much larger mass M₂ . We can write down the Force law on M₁ using Newton's formulations:

Step 2: Combining terms yields:
Step 3: In an orbit governed by a central force, the centripetal acceleration, a is given by:
Step 4: For a circular orbit, the circular velocity, V_c is the total distance traveled (the circumference of the circle) divided by the orbital period, P or V_c = 2p R/P which then yields:
in which the harmonic law of Kepler is now apparent as the R³/P² term.

The point of this derivation was to show that only the R^-2 force law can yield Kepler's Third Law. No other force law will work. If this gravitional force law is universal then Kepler's laws must also be universal.

The Distances to the Stars

From Herschel's catalog, the first accurate stellar parallax measurement of 0.32 seconds of arc for the star 61 Cygni was published by the F.W. Bessel in 1838.

This was followed in 1839 by Thomas Henderson's measurement of about 1 arc-second for Alpha Centauri and the 1840 measurement of Alpha Lyrae (Vega) at 0.26 arcseconds.

1 parsec is the distance a star would have to be at to have a parallactic angle of 1 arcsecond. This distance is equivalent to 3.26 light years.

By 1840, it became clear that even the closest stars were more than one million times farther away than our Sun.

The Universe was now a large place and the intrinsic energy output of the stars had to be huge in order that their light could reach us.

Now back to , who makes two mistakes:

Newton implicitly assumes space is flat

Newton explicitly assumes that communication time is instantaneous

Einstein Rescues Us:

Sets the speed of light to be independent of the motion of the source

provides absolute frame and finite communication time. If not, you could have this .

Thus, Einstein gives us causality - a very important thing

Suggests that mass causes curvature of space and which is what we call gravity. This means that time can act strangely in a strong gravitational field:
- An Orbit in Curved Space This is a prediction made by General Relativity and confirmed by direct observation of Mercury.
- The Ant Universe in Flat Space
- The Ant Universe in Curved Space

Next step is the observation of galaxy redshifts which will demand that the Universe is, in fact, expanding.

Okay so we live in curved spacetime and now you are telling me that the Universe is exanding:

Uniform Expansion of the Universe:

Hubble noticed a correlation between recessional velocity and distance. This is known as the Hubble law:

V = HD

where V is velocity (in km/s), D is distance (in megaparsecs), H is the Hubble constant (present day expansion rate of the Universe)

The line through the data is a "best fit" linear relationship which shows that there is a linear relationship between the the velocity at which a galaxy moves away from us and its distance. This linear relatinship is consistent with a model of uniform expansion for the Universe.

This simple relation implies something remarkable about the Universe.

At some earlier time, all the galaxies had to have been together in the same space at the same time the Universe was once really small. It is important to realize that the galaxies are stuck to the surface of the universe by gravity and its the surface that expands. The galaxies themselves are not moving but travel along with the surface as shown here .

Run time backwards and realize that all the galaxies used to be together.

So what do we know now:

Gravity: Curvature of SpaceTime: Mass causes this curvature; space is deformed around this mass; this deformation is gravity. The more mass there is, the more deformation (curvature) there is and gravitational field becomes stronger
Galaxy Redshifts indicate that galaxies are moving radially away from one another
V=HD demands that the expansion is uniform
Since the universe is a SURFACE (in spacetime) then it has no CENTER!
In the past the Universe was a smaller place but still had the same mass in it - in the way distant past, all of this mass was in the same place at the same time, except that it wasn't mass it was Energy
and you:

Photons have an energy related mass. Photons are therefore effected by gravity. Light is bent in a strong gravitational field as the surface of the universe goes from being flat (light travels in a straight line) to curved (light follow the curved trajectory). This principle is shown in this animation .

The distribution of mass on in the universe determines the detailed shape of the surface light is constrained to follow this surface and this allows the universe to be observed!

Ants with Flashlights in curved space time

The ants can only communicate by flashlight the photons in the flashlight must follow the curvature of the universe because of the rule and so the photons must follow the blue trajectory and not the red one

Returning now to the expansion of the Universe with the help of some handy demonstration ants:

Place them on the surface of a bomb and given them each a telescope. Explode the bomb. V is not related to D.
Place the ants on the surface of a balloon and inflated the balloon; let the ants randomly walk around the surface of the balloon V is not related to D
Get some glue. Glue the ants to the balloon and inflate it. The separation between ants increases and V is observed to be proportional to D. The velocity of the ants is completely determined by the expansion motion of the surface.
Now suppose the ants are only partially glued (one leg is glued down). The ants are thrashing around the glue spot and therfore have a random component to their motion. Hence the total motion of the ants is due both to this random component and the expansion component. For short separations, the random component could dominate over the expansion component.
Horizons and the Expansion Age of the Universe:
V = HD c = HD ==> D =c/H ==> This is our causal horizon - beyond this distance something would have to travel faster than the speed of light in order to communicate with us. All observers are surrounded by such a horizon.
Horizons are okay. Our assumption about homogeneity means that the stuff beyond the horizon is the same stuff we already know about. This assumption must be correct due to horizon overlaps and causality.

Back to the Ants glued to the balloon: