Archive for General Theory of Relativity

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:

cosmo

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}

and

\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.

Getting the Measure of Space

Posted in The Universe and Stuff with tags , , , , , , , on October 8, 2014 by telescoper

Astronomy is one of the oldest scientific disciplines. Human beings have certainly been fascinated by goings-on in the night sky since prehistoric times, so perhaps astronomy is evidence that the urge to make sense of the Universe around us, and our own relationship to it, is an essential part of what it means to be human. Part of the motivation for astronomy in more recent times is practical. The regular motions of the stars across the celestial sphere help us to orient ourselves on the Earth’s surface, and to navigate the oceans. But there are deeper reasons too. Our brains seem to be made for problem-solving. We like to ask questions and to try to answer them, even if this leads us into difficult and confusing conceptual territory. And the deepest questions of all concern the Cosmos as a whole. How big is the Universe? What is it made of? How did it begin? How will it end? How can we hope to answer these questions? Do these questions even make sense?

The last century has witnessed a revolution in our understanding of the nature of the Universe of space and time. Huge improvements in the technology of astronomical instrumentation have played a fundamental role in these advances. Light travels extremely quickly (around 300,000 km per second) but we can now see objects so far away that the light we gather from them has taken billions of years to reach our telescopes and detectors. Using such observations we can tell that the Universe was very different in the past from what it looks like in the here and now. In particular, we know that the vast agglomerations of stars known as galaxies are rushing apart from one another; the Universe is expanding. Turning the clock back on this expansion leads us to the conclusion that everything was much denser in the past than it is now, and that there existed a time, before galaxies were born, when all the matter that existed was hotter than the Sun.

This picture of the origin and evolution is what we call the Big Bang, and it is now so firmly established that its name has passed into popular usage. But how did we arrive at this description? Not by observation alone, for observations are nothing without a conceptual framework within which to interpret them, but through a complex interplay between data and theoretical conjectures that has taken us on a journey with many false starts and dead ends and which has only slowly led us to a scheme that makes conceptual sense to our own minds as well as providing a satisfactory fit to the available measurements.

A particularly relevant aspect of this process is the establishment of the scale of astronomical distances. The basic problem here is that even the nearest stars are too remote for us to reach them physically. Indeed most stars can’t even be resolved by a telescope and are thus indistinguishable from points of light. The intensity of light received falls off as the inverse-square of the distance of the source, so if we knew the luminosity of each star we could work out its distance from us by measuring how much light we detect. Unfortunately, however, stars vary considerably in luminosity from one to another. So how can we tell the difference between a dim star that’s relatively nearby and a more luminous object much further away?

Over the centuries, astronomers have developed a battery of techniques to resolve this tricky conundrum. The first step involves the fact that terrestrial telescopes share the Earth’s motion around the Sun, so we’re not actually observing stars in the sky from the same vantage point all year round. Observed from opposite extremes of the Earth’s orbit (i.e. at an interval of six months) a star appears to change position in the sky, an effect known as parallax. If the size of the Earth’s orbit is known, which it is, an accurate measurement of the change of angular position of the star can yield its distance.

The problem is that this effect is tiny, even for nearby stars, and it is immeasurably small for distant ones. Nevertheless, this method has successfully established the first “rung” on a cosmic distance ladder. Sufficiently many stellar distances have been measured this way to enable astronomers to understand and classify different types of star by their intrinsic properties. A particular type of variable star called a Cepheid variable emerged from these studies as a form of “standard candle”; such a star pulsates with a well-defined period that depends on its intrinsic brightness so by measuring the time-variation of its apparent brightness we can tell how bright it actually is, and hence its distance. Since these stars are typically very luminous they can be observed at great distances, which can be accurately calibrated using measured parallaxes of more nearby examples.

Cepheid variables are not the only distance indicators available to astronomers, but they have proved particularly important in establishing the scale of our Universe. For centuries astronomers have known that our own star, the Sun, is just one of billions arranged in an enormous disk-like structure, our Galaxy, called the Milky Way. But dotted around the sky are curious objects known as nebulae. These do not look at all like stars; they are extended, fuzzy, objects similar in shape to the Milky Way. Could they be other galaxies, seen at enormous distances, or are they much smaller objects inside our own Galaxy?

Only a century ago nobody really knew the answer to that question. Eventually, after the construction of more powerful telescopes, astronomers spotted Cepheid variables in these nebulae and established that they were far too distant to be within the Milky Way but were in fact structures like our own Galaxy. This realization revealed the Cosmos to be much larger than most astronomers had previously imagined; conceptually speaking, the Universe had expanded. Soon, measurements of the spectra of light coming from extragalactic nebulae demonstrated that the Universe was actually expanding physically too. The evidence suggested that all distant galaxies were rushing away from our own with speed proportional to their distance from us, an effect now known as Hubble’s Law, after the astronomer Edwin Hubble who played a major role in its discovery.

A convincing theoretical interpretation of this astonishing result was only found with the adoption of Einstein’s General Theory of Relativity, a radically new conception of how gravity manifests itself as an effect of the behaviour of space-time. Whereas previously space and time were regarded as separate and absolute notions, providing an unchanging and impassive stage upon which material bodies interact, after Einstein space-time became a participant in the action, both influencing, and being influenced, by matter in motion. The space that seemed to separate galaxies from one another, was now seen to bind them together.
Hubble’s Law emerges from this picture as a natural consequence an expanding Universe, considered not as a collection of galaxies moving through static space but embedded in a space which is itself evolving dynamically. Light rays get bent and distorted as they travel through, and are influenced by, the changing landscape of space-time the encounter along their journey.

Einstein’s theory provides the theoretical foundations needed to construct a coherent framework for the interpretation of observations of the most distant astronomical objects, but only at the cost of demanding a radical reformulation of some fundamental concepts. The idea of space as an entity, with its own geometry and dynamics, is so central to general relativity that one can hardly avoid asking what it is space in itself, i.e. what is its nature? Outside astronomy we tend to regard space as being the nothingness that lies in between the “things” (i.e. material bodies of one sort or another). Alternatively, when discussing a building (such as an art gallery) “a space” is usually described in terms of the boundaries enclosing it or by the way it is lit; it does not have attributes of its own other than those it derives from something else. But space is not simply an absence of things. If it has geometry and dynamics it has to be something rather than nothing, even if the nature of that something is extremely difficult to grasp.

Recent observations, for example, suggest that even a pure vacuum of “empty space” possesses “dark energy” energy of its own. This inference hinges on the type Ia supernova, a type of stellar explosion so luminous it can (briefly) outshine an entire galaxy before gradually fading away. These cataclysmic events can be used as distance indicators because their peak brightness correlates with the rate at which they fade. Type Ia supernovae can be detected at far greater distances than Cepheids, at such huge distances in fact that the Universe might be only about half its current size when light set out from them. The problem is that the more distant supernovae look fainter, and consequently at greater distances, than expected if the expansion of the Universe were gradually slowing down, as it should if there were no dark energy.

At present there is no theory that can fully account for the existence of vacuum energy, but it is possible that it might eventually be explained by the behaviour of the quantum fields that arise in the theory of elementary particles. This could lead to a unified description of the inner space of subatomic matter and the outer space of general relativity, which has been the goal of many physicists for a considerable time. That would be a spectacular achievement but, as with everything else in science, it will only work out if we have the correct conceptual framework.

 

Dyson on Eddington

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , , , , , , on April 10, 2012 by telescoper

I’m grateful to George Ellis for sending me a link to a book review written by Freeman Dyson that appeared in a recent  edition of the New York Review of Books. I was particularly interested to read the following excerpt about Arthur Stanley Eddington. I have been intrigued by Eddington since I wrote a book about his famous expeditions (to Principe and Sobral) in 1919 to measure the bending of light by the Sun as a test of Einstein’s general theory of relativity; I blogged about this on its ninetieth anniversary, by the way, in case anyone wants to read any more about it.

Although I read quite a lot about Eddington, not only during the course of researching the book but also afterwards, as there are many things about his character that fascinate me. He died long before I was born, of course, but whenever I meet someone who knew him I ask what they make of him. Not altogether surprisingly, opinions differ rather widely from one person to another as his character seems to have been extremely contradictory. He doesn’t seem to have been very good at small talk, but was nevertheless a much sought-after dining companion. He was a man of great moral integrity, but at times treated his colleagues (notably Chandrasekhar) rather shamefully. He was a brilliant astrophysicist, but got himself hooked on his peculiar Fundamental Theory which was a dead end. He remains an enigma.

Anyway, this is what Dyson has to say about him:

Eddington was a great astronomer, one of the last of the giants who were equally gifted as observers and as theorists. His great moment as an observer came in 1919 when he led the British expedition to the island of Principe off the coast of West Africa to measure the deflection of starlight passing close to the sun during a total eclipse. The purpose of the measurement was to test Einstein’s theory of General Relativity. The measurement showed clearly that Einstein was right and Newton wrong. Einstein and Eddington both became immediately famous. One year later, Eddington published a book, Space, Time and Gravitation, that explained Einstein’s ideas to English-speaking readers. It begins with a quote from Milton’s Paradise Lost:

Perhaps to move
His laughter at their quaint opinions wide
Hereafter, when they come to model heaven
And calculate the stars: how they will wield
The mighty frame: how build, unbuild, contrive
To save appearances.

Milton had visited Galileo at his home in Florence when Galileo was under house arrest. Milton wrote poetry in Italian as well as English. He spoke Galileo’s language, and used Galileo as an example in his campaign for freedom of the press in England. Milton had witnessed with Galileo the birth struggle of classical physics, as Eddington witnessed with Einstein the birth struggle of relativity three hundred years later. Eddington’s book puts relativity into its proper setting as an episode in the history of Western thought. The book is marvelously clear and readable, and is probably responsible for the fact that Einstein was better understood and more admired in Britain and America than in Germany.

As a student at Cambridge University I listened to Eddington’s lectures on General Relativity. They were as brilliant as his books. He divided his exposition into two parts, and warned the students scrupulously when he switched from one part to the other. The first part was the orthodox mathematical theory invented by Einstein and verified by Eddington’s observations. The second part was a strange concoction that he called “Fundamental Theory,” attempting to explain all the mysteries of particle physics and cosmology with a new set of ideas. “Fundamental Theory” was a mixture of mathematical and verbal arguments. The consequences of the theory were guessed rather than calculated. The theory had no firm basis either in physics or mathematics.

Eddington said plainly, whenever he burst into his fundamental theory with a wild rampage of speculations, “This is not generally accepted and you don’t have to believe it.” I was unable to decide who were more to be pitied, the bewildered students who were worried about passing the next exam or the elderly speaker who knew that he was a voice crying in the wilderness. Two facts were clear. First, Eddington was talking nonsense. Second, in spite of the nonsense, he was still a great man. For the small class of students, it was a privilege to come faithfully to his lectures and to share his pain. Two years later he was dead.