|Feb1-13, 05:56 AM||#18|
Meaning of a "Flat," "Open," and "Closed" Universe
The spatial geometry refers to the sum of angles = 180° of a triangle at a certain instant of time. Or assuming the expansion is "frozen". The triangle shall be formed by an object of a known size (from one edge to the other) and the distance between this object and our worldline. Both distances are true, meaning measured with rulers.
At t = t1 the angular size of said object is β1. Together with it't true size the sum of the angles is 180°, the geometry of the space is euclidean.
Then the universe expands (whereby the object participates) for x billion years and stops to be "frozen" again.
At t = t2 "watching" the frozen state nothing remarkable has changed, except that the triangle is much larger. But the sum of angles is still 180° and β2 = β1. The reason is that in contrast to the space-time curvature the spatial curvature doesn't change over time. If I am right, the space-time curvature is constant only in case the expansion is linear over time.
At t = t2 we see - while the universe expands - the light of the object, which was emitted at t1 x billion years back in time and measure it's angular size βexp.
My question is how is βexp related to β1 and to β2 resp.?
It seems the only given quantities are the true size of the object at t = t1, it's angular size βexp at t = t2 and and it's redshift, which is related to the look-back time (t2 - t1). Perhaps I missed something.
How do we calculate the spatial geometry from this (whereby I suspect that the measured angle βexp does not coincide with β1, β2 resp.)?
I have the notion that an angle measured by light rays says because of the dynamics involved something about the space-time curvature, but not about the spatial curvature. I might be wrong, but am much interested to learn why.
Any help appreciated.
|Feb1-13, 10:00 AM||#19|
Instead what we do is take many different observations, which are usually some form of correlating redshift with distance, and using the fact that the observed distance depends upon the spatial curvature. With enough such observations at different redshifts, we can obtain constraints on all of the parameters that make up these models (in this case, the parameters are primarily the dark energy density, the total matter density (normal + dark), and the expansion rate, with curvature being determined by the relationship between the total energy density and the expansion rate).
The CMB is a bit different, because the overall densities of normal and dark matter also impact the nature of the sound waves in the early universe that set up the pattern of hot and cold spots that we see.
Max Tegmark has a selection of movies that show how the various cosmological parameters impact the pattern of hot and cold spots:
(The plot shown here is a plot of the "power spectrum" of the CMB, which is the average amplitude of oscillations of a particular size on the sky. Long-wavelength oscillations are on the left, short-wavelength oscillations are on the right. The biggest peak is close to one degree on the sky, which means that most of the variation from place to place on the CMB occurs at one degree scales, but there are variations on all scales and the detailed pattern has a lot of power to constrain cosmological variables.)
|Feb3-13, 10:54 AM||#20|
Anyhow, thank you for your patience.
|Feb4-13, 08:18 AM||#21|
dchartier: As you can tell from the discussion above, one question leads to another to another......
Here is an visual illustration of shapes of the universe from NASA and some introductory discussion you may find helpful:
And another nice introductory source is here:
|Feb4-13, 08:33 AM||#22|
Minkowski geometry adds TIME as a fourth dimension to three of space creating SPACETIME....but still flat, no gravity....
This is where you often see light-cones like those pictured in the above link. You might want to make a mental note that the light-cone traces out what turns out to be a type of 'horizon'...similar to those encountered in GR as that of a black hole for example.
Horizons in cosmology are REALLY fun, but not so easy to understand at first.
In general relativity, things like pressure, energy and mass CURVE SPACETIME. At slow speeds, say the earth rotating around the sun, or the moon about the earth, even our space probes, most of the 'spacetime curvature' is the curvature of time. And that's VERY close to Newtonian descriptions and is why most spaceflights use Newtonian calculations.
For a real world application of GR, check Wikipedia about the GPS satellite system to get a feel for how TIME corrections are necessary due to relativistic effects: satellite time and earth surface time tick at different rates and those differences must be corrected for accurate positioning!
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