In a book by Alexander Moszkowski, there is an Einstein quote about a hypothetical size of the universe (100 million Light years). Moszkowski claims that Einstein had deduced that from the gravitational constant and the mass distribution of the Milky Way, as known around 1920. My question:
What was the data and the best estimate of the mass density of the Milky Way, back then in 1920?
This is a part answer to your question, based on some recent research of the photochemical behaviour modelled and observed for Titan's tholin haze and modelling of Titan's stratosphere.
The process appears to begin, according to the paper Ice condensation layers in Titan’s Stratosphere (Barth, 2012) (Abstract only - paywalled), with
Photochemical destruction of methane along with the destruction of nitrogen molecules from energetic electrons in Titan’s upper atmosphere result in the production of a number of hydrocarbon and nitrile compounds which may be capable of condensing at the colder temperatures of Titan’s lower stratosphere.
then, according to the paper Laboratory experiments of Titan tholin formed in cold plasma at various
pressures: implications for nitrogen-containing polycyclic aromatic
compounds in Titan haze (Imanaka et al. 2004), in particular in reference to Titan's stratosphere,
. In the stratosphere (100–
300 km), further chemical reactions are induced by the catalytic
$CH_4$ dissociation by such molecules as $C_2H_2$ and
$C_4H_2$ absorbing the long UV (> 155 nm) irradiation
The significance of these UV absorbing molecules is explained in the article Photochemical activity of Titan’s low-altitude condensed haze (Gudipati et al. 2013) (Abstract only - paywalled), they state that tholin haze could form on condensed aerosols in Titan's atmosphere, demonstrating that, at least part of Titan's atmosphere is photochemically active. Through modelling, they found that
Detected in Titan’s atmosphere, dicyanoacetylene ($C_4N_2$) is used in our laboratory simulations as a model system for other larger unsaturated condensing compounds. We show that $C_4N_2$ ices undergo condensed-phase photopolymerization (tholin formation) at wavelengths as long as 355 nm pertinent to solar radiation reaching a large portion of Titan’s atmosphere, almost close to the surface.
and evidence of these ices is suggested in the article Titan’s aerosol and stratospheric ice opacities between 18 and 500 μm: Vertical and spectral characteristics from Cassini CIRS (Anderson and Samuelson, 2011) stating that the ices and aerosols
appear to be located over a narrow altitude range in the stratosphere centered at ∼90 km. Although most abundant at high northern latitudes, these nitrile ice clouds extend down through low latitudes and into mid southern latitudes, at least as far as 58°S.
Having botched the first attempt at answering this question and not wanting to delete the evidence, let me try again here.
The "tetrad postulate" is independent from metricity and from the condition that the connection be torsion-free. It is simply the equivalence (via the vielbein) of two connections on two different bundles. Here are the details. $M$ is a smooth $n$-dimensional manifold.
First of all we have an affine connection $nabla$ on $TM$ with connection coefficients $Gamma^rho_{munu}$ relative to a coordinate basis -- that is,
$$nabla_{partial_mu} partial_nu = Gamma_{munu}^rho partial_rho,$$
with $partial_mu$ an abbreviation for $partial/partial x^mu$ where $x^mu$ is a local chart on $M$.
Then we have a connection on an associated vector bundle to the frame bundle $P_{mathrm{GL}}(M)$. The frame bundle is a principal $mathrm{GL}(n)$-bundle and given any representation $rho: mathrm{GL}(n) to mathrm{GL}(V)$ of $mathrm{GL}(n)$ we can define a vector bundle
$$P_{mathrm{GL}}(M) times_rho V.$$
Take $V$ to be the defining $n$-dimensional representation and call the resulting bundle $E$. Relative to a local frame $e_a$ for $E$, a connection $hatnabla$ defines connection one-form $omega$ by
$$hatnabla_{partial_mu} e_a = omega_{mu~a}^b e_b.$$
Now the vielbein defines a bundle isomorphism $TM buildrelcongoverlongrightarrow E$ and all the "tetrad postulate" says is that the two connections $nabla$ and $hatnabla$ correspond. In fact, the "tetrad postulate" is just the statement that the vielbein is a parallel section of the bundle $T^*M otimes E$ relative to the tensor product connection.
This works for any affine connection $nabla$ on any smooth manifold $M$. No metric is involved.
A special case of this construction is when $(M,g)$ is a riemannian manifold and $nabla$ is the Levi-Civita connection (i.e., the unique torsion-free, metric connection on $TM$).
You can without loss of generality restrict to orthonormal frames, which defines a principal $mathrm{O}(n)$ (or $mathrm{O}(p,q)$ depending on signature) bundle. The representation $V$ restricts to an irreducible rep of the orthogonal group, possessing an invariant bilinear form $eta$. This relates $g$ and $eta$ as in your question.
To expand a little more, yes the Earth would hang in the same spot in the sky, moving around in a small circle as the moon rotated around it over the course of each of its 28 day orbits. It would have phases, full Earth when the moon is between it and the sun, new Earth when the Earth is between the moon and the sun, and wax and wane between these two points. Each time the Earth was full, a different part of it would be visible from the moon, depending on the season and how it happened to line up when full Earth occurred.
The Earth would roll in the sky according to the seasons on Earth and the precession of the moon's orbit. The moon tilts only 1.5 degrees from the ecliptic and so in essence has no seasons. The orbit of the moon is tilted 5 degrees from the ecliptic, so when it is farthest to the south of the plane of the solar system, more of the south of our planet is visible, and vice-versa when it is farthest north. But the Earth is tilted 23 degrees from the ecliptic, so this is what would mostly determine what part of it was visible from the moon. Here is a good summary of the relationship of the Earth and Moon. The distance between the two bodies is not to scale, but the relative size of the one to the other is to scale.
In December, Antarctica would be visible - as it was during the famous Blue Marble photo from Apollo 17, taken on Dec. 7, 1972.
(This version of the photo is supposed to be as it was in the original photograph. They took it on the way to the moon, so the orientation of the camera was the only thing that determined which way was 'up'. If you were on the moon at far southern latitudes, this is in fact how it would look from your point of view.)
In July, the Arctic would be visible. Apollo 11 took this photo on July 16, 1969, when they were half-way to the moon.
Here actually you don't see much of the Arctic. The moon's orbit had carried it a bit south of the Earth at that point. Heading to it meant going south a bit, so when they looked back at the Earth, they saw less of the far north.
Just to be clear, something you would NEVER see is this, which has been widely misunderstood as a photo of the whole planet from space. Actually it is a super-wide-angle photo taken from low earth orbit - the horizon line is for the part of the world the satellite could see, it is not one hemisphere of the Earth.
If you think about it for a second you realize this can't possibly be true. The United States is gigantic, if it was really this size Canada would occupy the whole arctic and roll over onto the other side of the world, displacing most of Russia. Argentina and Chile would take the place of Antarctica.
Miranda's book on Riemann surfaces ignores the analytical details of proving that compact Riemann surfaces admit nonconstant meromorphic functions, preferring instead to work out the algebraic consequences of (a stronger version of) that assumption. Shafarevich's book on algebraic geometry has this to say:
A harmonic function on a Riemann surface can be conceived as a description of a stationary state of some physical system: a distribution of temperatures, for instance, in case the Riemann surface is a homogeneous heat conductor. Klein (following Riemann) had a very concrete picture in his mind:"This is easily done by covering the Riemann surface with tin foil... Suppose the poles of a galvanic battery of a given voltage are placed at the points $A_1$ and $A_2$. A current arises whose potential $u$ is single-valued, continuous, and satisfies the equation $Delta u = 0$ across the entire surface, except for the points $A_1$ and $A_2$, which are discontinuity points of the function."
Does anyone know of a good reference on Riemann surfaces where a complete proof along these physical lines (Shafarevich mentions the theory of elliptic PDEs) is written down? How hard is it to make this appealing physical picture rigorous? (The proof given in Weyl seems too computational and a little old-fashioned. Presumably there are now slick conceptual approaches.)
It is great that you have an interest in astronomy.
Unfortunately, the UK will be on the wrong side of the Earth to see the lunar eclipse on April 15. If you you want to experiment with various places on earth that will see it, and what they will see, get a planetarium application like Stellarium or SkySafari or TheSkyX.
A lunar eclipse is always easy to spot - it is where the full moon is. Also, because it is the result of the Earth getting in the way of the Sun shining on the Moon, it is always after the sun sets and after the full moon rises in the East. That is, if you're going to be in a place to see the whole sequence, not just a part of it.
Just so you know, the tetrad is not one but a series of lunar eclipses over 2 years, this is just the first lunar eclipse in the series. The last tetrad was 2003-2004, so you were around, but probably not paying as much attention at that time :-)
Unfortunately, Cambridge UK will not see any of the first 3 eclipses in this tetrad. However the 28 Sept 2015 eclipse will be visible from that location.
Incidentally, one of the better places to see this lunar eclipse will be Easter Island (Rapa Nui) which will see the whole eclipse, and with the totality occuring high in the sky, near the meridian. Also, not much light pollution.
There is one rule that is generally true for all deep sky objects (nebulae, stars, galaxies,...): Aperture matters!
For solar system objects, aperture is not that important.
The second most important thing is: What size are the objects you want to look at: Small objects need long focal lengths and high magnifications, large objects need short aperture for low magnifications.
With 400mm you could watch objects like:
With 900mm you could watch objects like
Note that 60 and 70mm aperture are still very small for telescopes! The aperture influences two things:
The magnification is created by the eyepiece. E.g. when you have a 400mm focal length telescope and use a 10mm eyepiece, you get 400mm/10mm = 40x magnification.
Note: the shorter the eyepiece focal length, the more difficult it is to build. Good 5mm eyepieces can cost 100 USD and up. I personally started with a 750mm Newtonian with 150mm aperture and 25mm and 10mm eyepieces. That's a good allrounder, even though planets will appear rather small with the 10mm eyepiece. But you can later invest more money in good eyepieces, which you can re-use on better telescopes which you may buy later on.
Edit: One more thing -- the telescope mount is equally important as are the eyepieces and the telescope itself. A mount that fits the telescope easily is as expensive as the optical tube assembly itself. Hence many beginners start out with a Dobson telescope, which uses a very, very simple yet sturdy mount.
