Archive for Hubble’s Law

Voting Matters

Posted in Maynooth, Politics, The Universe and Stuff with tags , , , , , on October 4, 2018 by telescoper

At last I have this afternoon free of teaching and other commitments, and having fortified myself with lunch in Pugin Hall, I’m preparing to make an attempt on the summit of the Open Journal of Astrophysics now that all the outstanding administrative obstacles have been cleared. Before shutting myself away to do up the loose ends, however, I thought I’d do a quick post about a couple of electoral matters.

The first relates to this, which arrived at my Maynooth residence the other day:

This document reminded me that there is a referendum in Ireland on the same day as the Presidential election I mentioned at the weekend. The contents of the booklet can be found here. In brief,

At present, the Constitution says that publishing or saying something blasphemous is an offence punishable under law. Blasphemy is currently a criminal offence. The referendum will decide if the Constitution should continue to say that publishing or saying something blasphemous is a criminal offence. If the referendum is passed, the Oireachtas will be able to change the law so that blasphemy is no longer a criminal offence.

Having read the booklet thoroughly and thereby having understood all the issues, and the implications of the vote,  I have decided that I will vote in favour of making blasphemy compulsory.

The other matter being put to a vote is something I just found out about today when I got an email from the International Astronomical Union concerning an electronic vote on Resolution B4, that the Hubble Law be renamed the Hubble-Lemaître law. For background and historical references, see here. I don’t really have strong opinions on this resolution, nor do I see how it could be enforced if it is passed but, for the record, I voted in favour because I’m a fan of Georges Lemaître




Hubble versus Slipher

Posted in History, The Universe and Stuff with tags , , , , , on September 15, 2012 by telescoper

Since I’m here at a conference celebrating the scientific achievements of Vesto M. Slipher, I thought I’d take the opportunity to make a few remarks about Slipher’s work and legacy.

I often use this picture in popular talks to illustrate the correlation between distance (x-axis) and apparent recession velocity (y-axis) that has become universally known as Hubble’s Law. This is an early version of such a plot published by Edwin Hubble in 1929.

In public talks I rarely have time to go into the details of this, but it is worth saying that only the results on the x-axis were Hubble’s own measurements. Hubble only contributed half of the above plot, i.e. the distance measurements, and these turned out to be wrong by a factor of about 10 owing to an incorrect identification of the stars used as standard candles. All the recession velocities on the y-axis – obtained by looking at the displacement of lines in the target galaxy’s spectrum – were in fact obtained by Vesto Slipher at the Lowell Observatory here in Flagstaff, Arizona. Hubble used these data from Slipher with permission, but gave no credit to Slipher in the references to his 1929 work. A later, and more convincing, version of this plot published in 1931 by Hubble and Humason, was accompanied by a generous acknowledgement to Slipher’s contribution. However, by then, Hubble’s name was firmly associated with the plot and Slipher’s contribution was largely forgotten for many years subsequently.

This episode isn’t at all atypical of Hubble’s behaviour. He was an extremely ambitious man who was an expert in the art of promoting himself and the Mount Wilson Observatory where he worked. Slipher was a very different type of man: quiet, self-effacing, and very much a team player, dedicated to scientific accuracy rather than his own reputation.

It’s worth saying further that the key observation that led to the understanding that the Universe is expanding is the fact that most of the spectra obtained by Slipher, over the years subsequent to his first measurement of the spectrum of the Andromeda Nebula (M31) celebrated by this conference, showed a redshift indicating velocity away from the observer. Even without distance measurements this leads directly an interpretation in terms of cosmic expansion. Ironically, the first spectrum he obtained, M31 shows a blue shift, as do a few others plotted with negative velocities in the above diagram, but the more distant sources exclusively show a redshift.

As a scientist should be, Slipher was very careful about the interpretation of this result. The more distant objects are fainter and thus more difficult to observe. Could it arise from some systematic artifact? Or could there be an unknown physical effect that produces a redshift dependent on the size of the source? These questions could only be answered when accurate distances to the nebulae were established, so Hubble’s contribution was by no means negligible. It’s completely untrue, however, to say that Hubble discovered the expansion of the Universe, so there’s yet another example of Stigler’s Law of Eponymy whenever anyone talks about the Hubble expansion.

One of the great things about coming to this meeting was the chance to meet Alan Slipher, grandson of Vesto Slipher. He and other members of his family refer to Vesto as “VM”, by the way, which I hadn’t realised before. VM lived a long life, dying in 1969 just short of his 94th birthday, so Alan knew him well until age 17 or so. He spoke most warmly and movingly after yesterday’s conference dinner about his memories of his grandfather, who he clearly looked up to. His words confirmed the impression I’d already formed, that Slipher was an extremely cautious and serious scientist as well as a kindly and humble man.

The contrasting personalities of Slipher and Hubble are further illustrated by correspondence between the two that is archived at the Lowell Observatory. Slipher comes across as kindly and cooperative, Hubble as pompous and self-regarding. I know which of the two I admire the best, both and scientist and human being.

A Piece on a Paradox

Posted in The Universe and Stuff with tags , , , , , , , , , on March 7, 2012 by telescoper

Not long ago I posted a short piece about the history of cosmology which got some interesting comments, so I thought I’d try again with a little article I wrote a while ago on the subject of Olbers’ Paradox. This is discussed in almost every astronomy or cosmology textbook, but the resolution isn’t always made as clear as it might be. The wikipedia page on this topic is unusually poor by the standards of wikipedia, and appears to have suffered a severe attack of the fractals.

I’d be interested in any comments on the following attempt.

One of the most basic astronomical observations one can make, without even requiring a telescope, is that the night sky is dark. This fact is so familiar to us that we don’t imagine that it is difficult to explain, or that anything important can be deduced from it. But quite the reverse is true. The observed darkness of the sky at night was regarded for centuries by many outstanding intellects as a paradox that defied explanation: the so-called Olbers’ Paradox.

The starting point from which this paradox is developed is the assumption that the Universe is static, infinite, homogeneous, and Euclidean. Prior to twentieth century developments in observation (Hubble’s Law) and theory  (Cosmological Models based on General Relativity), all these assumptions would have appeared quite reasonable to most scientists. In such a Universe, the intensity of light received by an observer from a source falls off as the inverse square of the distance between the two. Consequently, more distant stars or galaxies appear fainter than nearby ones. A star infinitely far away would appear infinitely faint, which suggests that Olbers’ Paradox is avoided by the fact that distant stars (or galaxies) are simply too faint to be seen. But one has to be more careful than this.

Imagine, for simplicity, that all stars shine with the same brightness. Now divide the Universe into a series of narrow concentric spherical shells, in the manner of an onion. The light from each source within a shell of radius r  falls off as r^{-2}, but the number of sources increases in the same manner. Each shell therefore produces the same amount of light at the observer, regardless of the value of r.  Adding up the total light received from all the shells, therefore, produces an infinite answer.

In mathematical form, this is

I = \int_{0}^{\infty} I(r) n dV =  \int_{0}^{\infty} \frac{L}{4\pi r^2} 4\pi r^{2} n dr \rightarrow \infty

where L is the luminosity of a source, n is the number density of sources and I(r) is the intensity of radiation received from a source at distance r.

In fact the answer is not going to be infinite in practice because nearby stars will block out some of the light from stars behind them. But in any case the sky should be as bright as the surface of a star like the Sun, as each line of sight will eventually end on a star. This is emphatically not what is observed.

It might help to think of this in another way, by imagining yourself in a very large forest. You may be able to see some way through the gaps in the nearby trees, but if the forest is infinite every possible line of sight will end with a tree.

As is the case with many other famous names, this puzzle was not actually first discussed by Olbers. His discussion was published relatively recently, in 1826. In fact, Thomas Digges struggled with this problem as early as 1576. At that time, however, the mathematical technique of adding up the light from an infinite set of narrow shells, which relies on the differential calculus, was not known. Digges therefore simply concluded that distant sources must just be too faint to be seen and did not worry about the problem of the number of sources. Johannes Kepler was also interested in this problem, and in 1610 he suggested that the Universe must be finite in spatial extent. Edmund Halley (of cometary fame) also addressed the  issue about a century later, in 1720, but did not make significant progress. The first discussion which would nowadays be regarded as a  correct formulation of the problem was published in 1744, by Loys de Chéseaux. Unfortunately, his resolution was not correct either: he imagined that intervening space somehow absorbed the energy carried by light on its path from source to observer. Olbers himself came to a similar conclusion in the piece that forever associated his name with this cosmological conundrum.

Later students of this puzzle included Lord Kelvin, who speculated that the extra light may be absorbed by dust. This is no solution to the problem either because, while dust may initially simply absorb optical light, it would soon heat up and re-radiate the energy at infra-red wavelengths. There would still be a problem with the total amount of electromagnetic radiation reaching an observer. To be fair to Kelvin, however, at the time of his writing it was not known that heat and light were both forms of the same kind of energy and it was not obvious that they could be transformed into each other in this way.

To show how widely Olbers’ paradox was known in the nineteenth Century, it is worth also mentioning that Friedrich Engels, Manchester factory owner and co-author with Karl Marx of the Communist Manifesto also considered it in his book The Dialectics of Nature. In this discussion he singles out Kelvin for particular criticism, mainly for the reason that Kelvin was a member of the aristocracy.

In fact, probably the first inklings of a correct resolution of the Olbers’ Paradox were contained not in a dry scientific paper, but in a prose poem entitled Eureka published in 1848 by Edgar Allan Poe. Poe’s astonishingly prescient argument is based on the realization that light travels with a finite speed. This in itself was not a new idea, as it was certainly known to Newton almost two centuries earlier. But Poe did understand its relevance to Olbers’ Paradox.  Light just arriving from distant sources must have set out a very long time ago; in order to receive light from them now, therefore, they had to be burning in the distant past. If the Universe has only lasted for a finite time then one can’t add shells out to infinite distances, but only as far as the distance given by the speed of light multiplied by the age of the Universe. In the days before scientific cosmology, many believed that the Universe had to be very young: the biblical account of the creation made it only a few thousand years old, so the problem was definitely avoided.

Of course, we are now familiar with the ideas that the Universe is expanding (and that light is consequently redshifted), that it may not be infinite, and that space may not be Euclidean. All these factors have to be taken into account when one calculates the brightness of the sky in different cosmological models. But the fundamental reason why the paradox is not a paradox does boil down to the finite lifetime, not necessarily of the Universe, but of the individual structures that can produce light. According to the theory Special Relativity, mass and energy are equivalent. If the density of matter is finite, so therefore is the amount of energy it can produce by nuclear reactions. Any object that burns matter to produce light can therefore only burn for a finite time before it fizzles out.

Imagine that the Universe really is infinite. For all the light from all the sources to arrive at an observer at the same time (i.e now) they would have to have been switched on at different times – those furthest away sending their light towards us long before those nearby had switched on. To make this work we would have to be in the centre of a carefully orchestrated series of luminous shells switching on an off in sequence in such a way that their light all reached us at the same time. This would not only put us  in a very special place in the Universe but also require the whole complicated scheme to be contrived to make our past light cone behave in this peculiar way.

With the advent of the Big Bang theory, cosmologists got used to the idea that all of matter was created at a finite time in the past anyway, so  Olber’s Paradox receives a decisive knockout blow, but it was already on the ropes long before the Big Bang came on the scene.

As a final remark, it is worth mentioning that although Olbers’ Paradox no longer stands as a paradox, the ideas behind it still form the basis of important cosmological tests. The brightness of the night sky may no longer be feared infinite, but there is still expected to be a measurable glow of background light produced by distant sources too faint to be seen individually. In principle,  in a given cosmological model and for given assumptions about how structure formation proceeded, one can calculate the integrated flux of light from all the sources that can be observed at the present time, taking into account the effects of redshift, spatial geometry and the formation history of sources. Once this is done, one can compare predicted light levels with observational limits on the background glow in certain wavebands which are now quite strict .

Is Space Expanding?

Posted in The Universe and Stuff with tags , , , , , , , , , on August 19, 2011 by telescoper

I think I’ve just got time for a quick post this lunchtime, so I’ll pick up on a topic that rose from a series of interchanges on Twitter this morning. As is the case with any interesting exchange of views, this conversation ended up quite some distance from its starting point, and I won’t have time to go all the way back to the beginning, but it was all to do with the “expansion of space“, a phrase one finds all over the place in books articles and web pages about cosmology at both popular and advanced levels.

What kicked the discussion off was an off-the-cuff humorous remark about the rate at which the Moon is receding from the Earth according to Hubble’s Law; the answer to which is “very slowly indeed”. Hubble’s law is v=H_0 d where v is the apparent recession velocity and d the distance, so for very small distance the speed of expansion is tiny. Strictly speaking, however, the velocity isn’t really observable – what we measure is the redshift, which we then interpret as being due to a velocity.

I chipped in with a comment to the effect that Hubble’s law didn’t apply to the Earth-Moon system (or to the whole Solar System, or for that matter to the Milky Way Galaxy or to the Local Group either) as these are held together by local gravitational effects and do not participate in the cosmic expansion.

To that came the rejoinder that surely these structures are expanding, just very slowly because they are small and that effect is counteracted by motions associated with local structures which “fight against” the “underlying expansion” of space.

But this also makes me uncomfortable, hence this post. It’s not that I think this is necessarily a misconception. The “expansion of space” can be a useful thing to discuss in a pedagogical context. However, as someone once said, teaching physics involves ever-decreasing circles of deception, and the more you think about the language of expanding space the less comfortable you should feel about it, and the more careful you should be in using it as anything other than a metaphor. I’d say it probably belongs to the category of things that Wolfgang Pauli would have described as “not even wrong”, in the sense that it’s more meaningless than incorrect.

Let me briefly try to explain why. In cosmology we assume that the Universe is homogeneous and isotropic and consequently that the space-time is described by the Friedmann-Lemaître-Robertson-Walker metric, which can be written

ds^{2} = c^{2} dt^{2}-a^{2}(t) d\sigma^{2}

in which d\sigma^2 describes the (fixed) geometry of a three-dimensional homogeneous space; this spatial part does not depend on time. The imposition of spatial homogeneity selects a preferred time coordinate t, defined such that observers can synchronize watches according to the local density of matter – points in space-time at which the matter density is the same are defined to be at the same time.

The presence of the scale factor a(t) in front of the spatial 3-metric allows the overall 4-metric to change with time, but only in such a way that preserves the spatial geometry, in other words the spatial sections can have different scales at different times, but always have the same shape. It’s a consequence of Einstein’s equations of General Relativity that a Universe described by the FLRW metric must evolve with time (at least in the absence of a cosmological constant). In an expanding universe a(t) increases with t and this increase naturally accounts for Hubble’s law, with  H(t)=\dot{a}/a but only if you define velocities and distances in the particular way suggested by the coordinates used.

So how do we interpret this?

Well, there are (at least) two different interpretations depending on your choice of coordinates.  One way to do it is to pick spatial coordinates such that the positions of galaxies change with time; in this choice the redshift of galaxy observed from another is due to their relative motion. Another way to do it is to use coordinates in which the galaxy positions are  fixed; these are called comoving coordinates.  In general relativity we can switch between one view and the other and the observable effect (i.e. the redshift) is the same in either.

Most cosmologists use comoving coordinates (because it’s generally a lot easier that way), and it’s this second interpretation that encourages one to think not about things moving but about space itself expanding. The danger with that is that it sometimes leads one to endow “space” (whatever that means) with physical attributes that it doesn’t really possess. This is most often seen in the analogy of galaxies being the raisins in a pudding, with “space” being the dough that expands as the pudding cooks taking the raisins away from each other. This analogy conveys some idea of the effect of homogeneous expansion, but isn’t really right. Raisins and dough are both made of, you know, stuff. Space isn’t.

In support of my criticism I quote:

 Many semi-popular accounts of cosmology contain statements to the effect that “space itself is swelling up” in causing the galaxies to separate. This seems to imply that all objects are being stretched by some mysterious force: are we to infer that humans who survived for a Hubble time [the age of the universe] would find themselves to be roughly four metres tall? Certainly not….In the common elementary demonstration of the expansion by means of inflating a balloon, galaxies should be represented by glued-on coins, not ink drawings (which will spuriously expand with the universe).

(John Peacock, Cosmological Physics, p. 87-8). A lengthier discussion of this point, which echoes some of the points I make below, can be found here.

To get back to the original point of the question let me add another quote:

A real galaxy is held together by its own gravity and is not free to expand with the universe. Similarly, if [we talk about] the Solar System, Earth, [an] atom, or almost anything, the result would be misleading because most systems are held together by various forces in some sort of equilibrium and cannot partake in cosmic expansion. If we [talk about] clusters of galaxies…most clusters are bound together and cannot expand. Superclusters are vast sprawling systems of numerous clusters that are weakly bound and can expand almost freely with the universe.

(Edward Harrison, Cosmology, p. 278).

I’d put this a different way. The “Hubble expansion” describes the motion of test particles in a the coordinate system I described above, i.e one  which applies to a perfectly homogeneous and isotropic universe. This metric simply doesn’t apply on the scale of the solar system, our own galaxy and even up to the scale of groups or clusters of galaxies. The Andromeda Galaxy (M31),  for example, is not receding from the Milky Way at all – it has a blueshift.  I’d argue that the space-time geometry in such systems is simply nothing like the FLRW form, so one can’t expect to make physical sense trying to to interpret particle motions within them in terms of the usual cosmological coordinate system. Losing the symmetry of the FLRW case  makes the choice of appropriate coordinates much more challenging.

There is cosmic inhomogeneity on even larger scales, of course, but in such cases the “peculiar velocities” generated by the lumpiness can be treated as a (linear) correction to the pure Hubble flow associated with the background cosmology.  In my view, however, in highly concentrated objects that decomposition into an “underlying expansion” and a “local effect” isn’t useful. I’d prefer simply to say that there is no Hubble flow in such objects. To take this to an extreme, what about a black hole? Do you think there’s a Hubble flow inside one of those, struggling to blow it up?

In fact the mathematical task of embedding inhomogeneous structures in an asymptotically FLRW background is not at all straightforward to do exactly, but it is worth mentioning that, by virtue of Birkhoff’s theorem,  the interior of an exactly spherical cavity (i.e. void)  must be described by the (flat) Minkowski metric. In this case the external cosmic expansion has absolutely no effect on the motion of particles in the interior.

I’ll end with this quote from the Fount of All Wisdom, Ned Wright,in response to the question Why doesn’t the Solar System expand if the whole Universe is expanding?

This question is best answered in the coordinate system where the galaxies change their positions. The galaxies are receding from us because they started out receding from us, and the force of gravity just causes an acceleration that causes them to slow down, or speed up in the case of an accelerating expansion. Planets are going around the Sun in fixed size orbits because they are bound to the Sun. Everything is just moving under the influence of Newton’s laws (with very slight modifications due to relativity). [Illustration] For the technically minded, Cooperstock et al. computes that the influence of the cosmological expansion on the Earth’s orbit around the Sun amounts to a growth by only one part in a septillion over the age of the Solar System.

The paper cited in this passage is well worth reading because it demonstrates the importance of the point I was trying to make above about using an appropriate coordinate system:

In the non–spherical case, it is generally recognized that the expansion of the universe does not have observable effects on local physics, but few discussions of this problem in the literature have gone beyond qualitative statements. A serious problem is that these studies were carried out in coordinate systems that are not easily comparable with the frames used for astronomical observations and thus obscure the physical meaning of the computations.

Now I’ve waffled on far too long so  I’ll just finally  recommend this paper entitled Expanding Space: The Root of All Evil and get back to work…