Archive for Albert Einstein

One Hundred Years of the Cosmological Constant

Posted in History, The Universe and Stuff with tags , , , , , , on February 8, 2017 by telescoper

It was exactly one hundred years ago today – on 8th February 1917 – that a paper was published in which Albert Einstein explored the cosmological consequences of his general theory of relativity, in the course of which he introduced the concept of the cosmological constant.

For the record the full reference to the paper is: Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie and it was published in the Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften. You can find the full text of the paper here. There’s also a nice recent discussion of it by Cormac O’Raifeartaigh  and others on the arXiv here.

Here is the first page:


It’s well worth looking at this paper – even if your German is as rudimentary as mine – because the argument Einstein constructs is rather different from what you might imagine (or at least that’s what I thought when I first read it). As you see, it begins with a discussion of a modification of Poisson’s equation for gravity.

As is well known, Einstein introduced the cosmological constant in order to construct a static model of the Universe. The 1917 paper pre-dates the work of Friedman (1923) and Lemaître (1927) that established much of the language and formalism used to describe cosmological models nowadays, so I thought it might be interesting just to recapitulate the idea using modern notation. Actually, in honour of the impending centenary I did this briefly in my lecture on Physics of the Early Universe yesterday.

To simplify matters I’ll just consider a “dust” model, in which pressure can be neglected. In this case, the essential equations governing a cosmological model satisfying the Cosmological Principle are:

\ddot{a} = -\frac{4\pi G \rho a }{3} +\frac{\Lambda a}{3}


\dot{a}^2= \frac{8\pi G \rho a^2}{3} +\frac{\Lambda a^2}{3} - kc^2.

In these equations a(t) is the cosmic scale factor (which measures the relative size of the Universe) and dots are derivatives with respect to cosmological proper time, t. The density of matter is \rho>0 and the cosmological constant is \Lambda. The quantity k is the curvature of the spatial sections of the model, i.e. the surfaces on which t is constant.

Now our task is to find a solution of these equations with a(t)= A, say, constant for all time, i.e. that \dot{a}=0 and \ddot{a}=0 for all time.

The first thing to notice is that if \Lambda=0 then this is impossible. One can solve the second equation to make the LHS zero at a particular time by matching the density term to the curvature term, but that only makes a universe that is instantaneously static. The second derivative is non-zero in this case so the system inevitably evolves away from the situation in which $\dot{a}=0$.

With the cosmological constant term included, it is a different story. First make \ddot{a}=0  in the first equation, which means that

\Lambda=4\pi G \rho.

Now we can make \dot{a}=0 in the second equation by setting

\Lambda a^2 = 4\pi G \rho a^2 = kc^2

This gives a static universe model, usually called the Einstein universe. Notice that the curvature must be positive, so this a universe of finite spatial extent but with infinite duration.

This idea formed the basis of Einstein’s own cosmological thinking until the early 1930s when observations began to make it clear that the universe was not static at all, but expanding. In that light it seems that adding the cosmological constant wasn’t really justified, and it is often said that Einstein regard its introduction as his “biggest blunder”.

I have two responses to that. One is that general relativity, when combined with the cosmological principle, but without the cosmological constant, requires the universe to be dynamical rather than static. If anything, therefore, you could argue that Einstein’s biggest blunder was to have failed to predict the expansion of the Universe!

The other response is that, far from it being an ad hoc modification of his theory, there are actually sound mathematical reasons for allowing the cosmological constant term. Although Einstein’s original motivation for considering this possibility may have been misguided, he was justified in introducing it. He was right if, perhaps, for the wrong reasons. Nowadays observational evidence suggests that the expansion of the universe may be accelerating. The first equation above tells you that this is only possible if \Lambda\neq 0.

Finally, I’ll just mention another thing in the light of the Einstein (1917) paper. It is clear that Einstein thought of the cosmological as a modification of the left hand side of the field equations of general relativity, i.e. the part that expresses the effect of gravity through the curvature of space-time. Nowadays we tend to think of it instead as a peculiar form of energy (called dark energy) that has negative pressure. This sits on the right hand side of the field equations instead of the left so is not so much a modification of the law of gravity as an exotic form of energy. You can see the details in an older post here.


Charles Ives & Albert Einstein: Parallel Lives

Posted in Music, The Universe and Stuff with tags , , , on October 20, 2015 by telescoper

I just noticed that today is the birthday of the great American modernist composer Charles Ives, who was born 141 years ago on this day. Some time ago I read The Life of Charles Ives by Stuart Feder, it’s a very interesting and informative biography of one of the strangest but most fascinating composers in the history of classical music so I thought I’d rehash an old piece I wrote about him to celebrate his birthday.

Charles Ives was by any standards a daring musical innovator. Some of his compositions involve atonal structures and some involve different parts of the orchestra playing in different time signatures. He also wrote strange and wonderful piano pieces, including some which involved re-tuning the piano to obtain scales involving quarter-tones. Among this maelstrom of modern ideas he also liked to add quotations from folk songs and old hymns which gives his work a paradoxically nostalgic tinge.

His pieces are often extremely diffficult to play (so I’m told) and sometimes not that easy to listen to, but while he’s often perplexing he can also be exhilarating and very moving. Other composers might play off two musical ideas against each other, but Ives would smash them together and to hell with the dissonance. I think the wholeheartedness of his eccentricity is wonderful, but I know that some people think he was just a nut.. You’ll have to make your own mind up on that.

My favourite quote of his can be found scrawled on a hand-written score which he sent to his copyist:

Please don’t try to make things nice! All the wrong notes are right. Just copy as I have – I want it that way.

But the point of adding this post to my blog was that in the course of reading the biography, it struck me that there is a strange parallel between the life of this controversial and not-too-well known composer and that of Albert Einstein who is certainly better known, especially to people reading what purports to be a physics blog.

For one thing their lifespans coincide pretty closely. Charles Ives was born in 1874 and died in 1954; Albert Einstein lived from 1879 to 1955. Of course the former was born in America and the latter in Germany. One inhabited the world of music and the other science; Ives, in fact, made his living in the insurance business and only composed in his spare time while Einstein spent most of his career in academia, after a brief period working in a patent office. Not everything Ives wrote was published professionally and he also rewrote things extensively, so it is difficult to establish exact dates for things, especially for a non-expert like me. In any case I don’t want to push things too far and try to argue that some spooky zeitgeist acted at a distance to summon the ideas from each of them in his own sphere. I just think it is curious to observe how similar their world lines were, at least in some respects.

We all know that Einstein’s “year of miracles” was 1905, during which he published classic papers on special relativity, Brownian motion and the photoelectric effect. What was arguably Ives’ greatest composition, The Unanswered Question, was completed in 1906 (although it was revised later). This piece is subtitled “A Cosmic Landscape” and it’s a sort of meditation on the philosophical problem of existence: the muted strings (which are often positioned offstage in concert performances) symbolize silence while the solo trumpet evokes the individual struggling to find meaning within the void. Here’s a fine recording of this work, featuring the New York Philharmonic conducted by Leonard Bernstein:

The Unanswered Question is probably Ives’ greatest masterpiece, but it wasn’t the only work he composed in 1906. A companion piece called Central Park in the Dark also dates from that year and they are sometimes performed together as a kind of diptych which offers interesting contrasts. While the former is static and rather abstract, the latter is dynamic and programmatic (in that it includes realistic evocations of night-time sounds).

Einstein’s next great triumph was his General Theory of Relativity in 1915, an extension of the special theory to include gravity and accelerated motion, which which came only after years of hard work learning the required difficult mathematics. Ives too was hard at work for the next decade which resulted in other high points, although they didn’t make him a household name like Einstein. The Fourth Symphony is an extraordinary work which even the best orchestras find extremely difficult to perform. Even better in my view is Three Places in New England (completed in 1914) , which contains my own favourite bit of Ives. The last movement, The Housatonic at Stockbridge is very typical of his unique approach, with a beautifully paraphrased hymn tune floating over the top of complex meandering string figures until the piece ends in a tumultuous crescendo.

After this period, both Einstein and Ives carried on working in their respective domains, and even with similar preoccupations. Einstein was in search of a unified field theory that could unite gravity with the other forces of nature, although the approach led him away from the mainstream of conventional physics research and his later years he became an increasingly marginal figure.

By about 1920 Ives had written five full symphonies (four numbered ones and one called the Holidays Symphony) but his ambition beyond these was perhaps just as grandiose as Einstein’s: to create a so-called “Universe Symphony” which he described (in typically bewildering fashion) as

A striving to present – to contemplate in tones rather than in music as such, that is – not exactly within the general term or meaning as it is so understood – to paint the creation, the mysterious beginnings of all things, known through God to man, to trace with tonal imprints the vastness, the spiritual eternities, from the great unknown to the great unknown.

I guess such an ambitious project – to create an entirely new language of “tones” that could give expression to timeless eternity, a kind of musical theory of everything – was doomed to failure. Although Ives was an experienced symphonic composer he couldn’t find a way to realise his vision. Only fragments of the Universe Symphony remain (although various attempts have been made by others to complete it).

In fact, the end of Ives’ creative career was much more sudden and final than Einstein who, although he never again reached the heights he had scaled in 1915 – who could? – remained a productive and respected scientist until his death. Ives had a somewhat melancholic disposition and from time to time suffered from depression. By 1918 he already felt that his creative flame was faltering, but by 1926 the spark was extinguished completely. His wife, appropriately named Harmony, remembered the precise day when this happened at their townhouse in New York:

He came downstairs one day with tears in his eyes, and said he couldn’t seem to compose anymore – nothing went well, nothing sounded right.

Although Charles Ives lived almost another thirty years he never composed another piece of music after that day in 1926. I find that unbearably sad, but at least a lot of his work is available and now fairly widely played. Alongside the pieces I have mentioned, there are literally hundreds of songs, some of which are exceptionally beautiful, and dozens of smaller works including piano and violin sonatas.

Although they both lived in the same part of America for many years, I don’t think Charles Ives and Albert Einstein ever met. I wonder what they would have made of each other if they had?

If you believe in the multiverse, of course, then there is a part of it in which they do meet. Einstein was an enthusiastic violinist so there will even be a parallel world in which Einstein is playing the Ives’ Violin Sonata on Youtube…


Astronomy Look-alikes, No. 96

Posted in Astronomy Lookalikes with tags , on October 7, 2015 by telescoper

Heavens above!

It has been drawn to my attention that there is a remarkable similarity in visual appearance between planetary astronomer Albert Einstein and the creator of the theory of general relativity Jean-Pierre Bibring. I wonder if, by any chance, they might be related?





That Fishy Saying of Einstein…

Posted in The Universe and Stuff with tags , , , on March 10, 2014 by telescoper


There are two interesting things about the above Einstein meme that has been doing the rounds. The first is that there’s absolutely no evidence that I can find that Albert Einstein ever said the words attributed to him; that’s also true for the vast majority of Einstein quotes, in fact.

The other interesting thing (and I risk being labelled a pedant here) is that there are species of fish, such as the Mangrove Rivulus, that really are able to climb trees…

My talk at “The Origins of the Expanding Universe”

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , , , , , , on October 9, 2012 by telescoper

You may recall that I gave a talk recently at a meeting called The Origins of the Expanding Universe in Flagstaff, Arizona. I put the slides up here. Well, the organizers have now put videos of the presentations online so you have the chance to see mine, warts and all.

I was relieved when I saw this on Youtube that the organizers were kind enough to edit out the embarrassing bit at the start when my laptop refused to talk to the data projector and I had to swap to another one. Sorting all that out seemed to take ages, which didn’t help my frame of mind and I was even more nervous than I would have been anyway given that this was my first public appearance after a rather difficult summer. Those are my excuses for what was, frankly, not a particularly good talk. But at least I survived. Better is the end of a thing than the beginning thereof.

Origins of the Expanding Universe Conference – My Contribution

Posted in History, The Universe and Stuff with tags , , , , , on September 19, 2012 by telescoper

For those of you interested in such things, here are the slides I used in my talk at the Origins of the Expanding Universe conference. I spoke about the events on and after 29th May 1919, when measurements were made during a total eclipse of the Sun that have gone down in history as vindicating Einstein’s (then) new general theory of relativity. I’ve written quite a lot about this in past years, including a little book and a slightly more technical paper. This was a relevant topic for the conference because it wasn’t until general theory of relativity was established as a viable theory of gravity that an explanation could be developed of Slipher’s measurements of galaxy redshifts in terms of an expanding Universe.

Einstein and your Gas Bill

Posted in History, Television, The Universe and Stuff with tags , , , , , on October 11, 2011 by telescoper

Taking refuge in my office this lunchtime for a sandwich and a cup of coffee I turned to the latest edition of Physics World and came across an funny little story about a physicist (who is completely new to me) with the splendid name of Fritz Hasenöhrl.

The news story relates to a paper on the arXiv, part of the abstract of which I’ve copied below:

In 1904 Austrian physicist Fritz Hasenohrl (1874-1915) examined blackbody radiation in a reflecting cavity. By calculating the work necessary to keep the cavity moving at a constant velocity against the radiation pressure he concluded that to a moving observer the energy of the radiation would appear to increase by an amount E=(3/8)mc^2, which in early 1905 he corrected to E=(3/4)mc^2

Since I’ve been doing a bit of dimensional analysis with first-year students, I’m a bit surprised that the authors of this paper read so much into the fact that Hasenöhrl’s formula bears a superficial resemblance to Einstein’s most famous formula E=mc^2, probably the best known and at the same time worst understood equation in physics. In fact any physicist worth his or her salt no matter how incorrect their reasoning would have to get something like E =\alpha mc^2, with \alpha some dimensionless number, simply because the answer has to have the correct dimensions to be an energy.

Expressing energy in terms of the basic dimensions mass M, length L and time T is probability easiest to do when you think of mechanical work (force×distance). Since Newton’s laws give a force equal to mass×acceleration, a force has dimensions MLT^{-2}, so work (a form of energy) has dimensions ML^{2}T^{-2}. Now try to make this out of a combination of a mass (M) and a velocity (LT^{-1}) and you’ll find that it has to be mass×velocity2. You can’t get the dimensionless constant this way, but the combination of m and c must be the way it is in Einstein’s formula.

Anyway, all this suddenly reminded me of a day long ago when I appeared on peak-time television in the consumer affairs programme Watchdog, explaining – or, rather, attempting to explain – the physics behind the way gas bills are calculated. Apparently someone had written in to the programme asking why it was that they weren’t just being charged for the volume of gas that had flowed through their meter, but that the cost involved a complicated calculation involving something called the calorific value of the gas.

The answer is fairly obvious, actually. The idea is that to make competition fairer between different forms of energy (particularly gas and electricity) the bills should be for the amount of energy you have used rather than the amount of gas. Since the source of fuel varies from day to day so does its chemical composition and hence the amount of energy that can be extracted from it when it is burned. Gas companies therefore monitor the calorific value, using it to convert the amount of gas you have used into an amount of energy.

On the programme I was confronted by the curmudgeonly Edward Enfield (father of comedian Harry Enfield) who took the line that it was all unnecessarily complicated and that the bill should just be for the amount of gas used, rather in the same way that petrol is sold. When I tried to explain that the way it was done was really fairer, because  it was really the energy that mattered, it quickly became obvious that he didn’t really understand what energy was or how it was defined.  He didn’t even get the difference between energy and power. I suspect that goes for many members of the general public.

It was all a bit tongue-in-cheek, but I enjoyed the sparring. Eventually he came out with a question about why energy was given by E=mc^2 rather than mc^3 or something else. So I launched into an explanation of dimensional analysis and why mc^3 couldn’t be an energy because it has the wrong dimensions. His eyes glazed over. The shoot ended. My splendidly erudite and logically rigorous exposition of dimensional analysis never made it into the broadcast programme.

My brief career on BBC1 was over.