Space & Time

maxresdefaultSpacetime is a new and unique substance in physics. It has 4 dimensions: 3 in space and 1 in time. Yet it is much more than just a taping together of time and space. This article will examine, in basic terms, its nature and implications for cosmology.

Space is, on the whole, intuitive. Its 3 dimensions can all be given in kilometres, and they are interchangeable. If a map tells you to go 2 km North and 2 km East, you will know that it doesn’t matter in which direction you go first. And for a shortcut you might instead go North-East.UntitledNow add a time dimension, let’s say that your train leaves in 20 minutes. Is there any way of adding this dimension to the diagram above? Not really. Our intuition doesn’t allow time and space to be co-represented, let alone reconciled. 1 metre and 1 minute are not just two different things but two different kinds of things. However, in General Relativity, distance and time are, in a sense, interchangeable. This is a drastic step, and not simply an addendum, for each dimension now represents something different than before; x is no longer Newton’s x, since it relates to time in a way that the length of a football pitch intuitively does not. Yet Einstein’s idea has been vindicated, and precisely by its most absurd predictions. Gravitational lensing, the idea that light is deflected in a gravitational field despite having no mass, was observed in 1979, and as recently as 2016 gravitational waves were found, emitted by orbiting masses pulling on the fabric of spacetime like ripples on a pond. Where our intuition falls short, Einstein’s relativistic spacetime triumphs.

Having established therefore what it is not, what actually is spacetime? In basic terms, spacetime is the substance of which the Universe is made and through which everything in it moves. It is no longer a stage on which events happen but instead a key player helping to shape the scene. Spacetime tells matter how to move and matter tells spacetime how to curve. This relationship is formalised in Einstein’s Field Equations, where the left-hand side describes curvature and the right-hand side matter.2

Untitled 10This may appear to be only 1 equation, but it is in fact 10. The subscripts μ and v (pronounced ‘mew’ and ‘new’) each take the values of x, y, z and t. One term so far left undefined is Λ (capital lambda). This is the cosmological constant, which Einstein introduced in 1917 and later called his ‘biggest blunder’. His motivation for inserting it was that, earlier in 1917, the astronomer Vesto Slipher had observed interstellar redshift, a clue that the Universe might be expanding. Einstein’s equations didn’t allow for this. In fact they described a cosmos dominated by gravity and doomed to collapse back on itself with nothing to provide resistance, let alone expansion. The cosmological constant was costless and perhaps face-saving, hence his later regrets, but he turned out in fact to be correct. Λ has now been measured to be ~10⁻²⁵ kg m⁻³, and physically it represents the dark energy opposing gravity and causing the Universe to expand.

But what exactly is expanding? Cosmology is changed utterly by the introduction of spacetime. The Big Bang is normally visualised as an inflating balloon, but this is a gross misrepresentation. It is spacetime – not space – that is expanding; that new and unfamiliar substance. Whereas a balloon expands in x, y and z, spacetime expands in x, y, z and t, and as we’ve seen those first three dimensions aren’t the same as before. A balloon cannot travel faster than the speed of light, yet spacetime can. The reason we are limited to an observable Universe is because the fabric on which the rest of it lies carries it away from us faster than c. Indeed, spacetime is a very different kind of creature than the space with which we are familiar. And we should be careful not to oversimplify it. ‘What is it expanding into?’ is a very good question for the high school teacher’s 3-dimensional balloon, but is – it turns out – nonsensical when applied to a 4-dimensional Universe. For the time being the great cosmological questions remain unanswered, but it is worth noting that the questions themselves have changed.

Dark Energy


In the 20th century as galaxies further afield were resolved astronomers found that the rate of cosmic expansion has increased over time; the Big Bang is, in fact, speeding up. This article will briefly cover the evidence leading Adam Riess et al. to this conclusion in their 1998 Nobel-winning paper, as well as its profound implications on the future of the universe.

The first method of finding the distance to another galaxy was provided by Henrietta Leavitt in 1912 when she, having tracked thousands of stars in the Magellanic cloud, found a relation between the period and luminosity of Cepheid variable stars. Distinguishable by the signficant He²⁺ content in their spectra, this transparent gas ionises at high temperatures to form the opaque He³⁺, making the star dimmer. A resulting expansion following this temperature increase causes the star to cool and contract, reverting to He²⁺, upon which this process repeats and the star oscillates. By measuring the period and relative brightness of Cepheid variables their distances, and those of the galaxies to which they belong, can be found. Another way of finding the distance to another galaxy is to measure the brightness of Type 1A supernovae. These are the result of white dwarfs gobbling up nearby stars, a process that the Pauli exclusion principle predicts must reach a critical instability at ~1.44 solar masses. Type 1A supernovae therefore share the same luminosity, making them a reliable standard candle, whose distance relates to their relative brightness. Generally, these and other methods are used in combination in order to improve precision and minimise distance uncertainties.

The velocity of a galaxy can be found by measuring the redshift of its spectrum. Elementary peaks are compared with their known laboratory wavelengths, and redshift z is defined as:Screen Shot 2017-06-06 at 14.28.06Where λ₀ is the known wavelength of a peak and Δλ is by how much it is shifted in a measured spectrum. The velocity relates to z by:
Screen Shot 2017-06-06 at 14.28.10The galaxies Hubble observed had, on average, a positive redshift, and the further away one was the redder its spectrum, and so the faster it was receding. More profoundly, Hubble showed that this would be the case at every point in space, meaning that the redshift was not purely a Doppler effect but is caused the expansion of space between the galaxies while the light is in transit. Hubble defined a rate of constant expansion H₀, which is presently defined as 67.8 km per second per megaparsec, and relates to velocity by:Screen Shot 2017-06-06 at 14.53.47Since c and H₀ are taken to be constants, the graph of this equation is simply a straight line. Observations agree with this on the relatively small scales Hubble observed in 1929, but as galaxies are mapped at ranges up to 12,000 megaparsecs, ~90% of the radius of the known universe, Hubble’s straight line bends, as shown below, indicating a much lower expansion rate in the past.
hubbleacc-2This is precisely the opposite of what is expected in a matter-dominated universe. In such a universe the force of gravity is sufficient to eventually slow down any expansion and cause a contraction. To overcome this force requires energy, and thus dark energy is introduced, with the word ‘dark’ conveying no more information than its being unknown. So the question of whether the universe will expand forever or eventually stop and collapse in on itself is no more than a question on the ratio of matter to dark energy. This cosmic density term, aptly given the letter Ω, is conveniently defined such that if Ω > 1 gravity prevails and the universe will collapse in on itself while if Ω < 1 dark energy prevails and the universe expands forever. Current measurements put a value on Ω at almost exactly 1, predicting that the universe will eventually slow down but never quite stop. Further research is underway to more precisely measure Ω and reduce what are considerable uncertainties.

For now the acceleration continues. It is predicted that within a few million years many galaxies now visible will have vanished from the night sky, since they will be receding from the Milky Way faster than the speed of light. In several billion years all distant galaxies will have vanished, leaving astronomers stranded in a much smaller universe, resembling the one in which they thought they lived at the start of the 20th century.

Riess, Adam et al. (1998). “Observational evidence from supernovae for an accelerating universe and a cosmological constant”. Astronomical Journal. 116 (3): 1009–38 [PDF]
Mazzali, P. A. et al. (2007). “A Common Explosion Mechanism for Type Ia Supernovae”. Science. 315 (5813): 825–828 [PDF]
Michael Richmond. (2000). Estimating Distances to Far-away Galaxies. [HTML]
Kirshner, R. P. (2003). “Hubble’s diagram and Cosmic expansion”. National Academy of Sciences. [HTML]
Peebles, P. J. E. (2003). “The cosmological constant and dark energy”. Reviews of Modern Physics. 75 (2): 559–606. [PDF]
Riess, Adam et al. (2004). “The Expanding Universe: From Slowdown to Speed Up”. Scientific American. 290 (2). [HTML]
De Bernardis, P. et al. (2000). “A flat Universe from high-resolution maps of the cosmic microwave background radiation”. Nature. 404 (6781): 955–9. [PDF]
Krauss, L. M.; Scherrer, R. J.  (2008). “The End of Cosmology?” Scientific American. 82. [PDF]

Dark Matter


To approach the topic of dark matter is to approach the frontier of modern physics. This article presents briefly the evidence for this hypothetical substance, after a short overview on why it is widely considered to exist.¹

The first person to suggest the presence of dark matter was the Dutch astronomer Jacobus Kapteyn in 1922.² By this time telescopes had grown powerful enough to resolve galaxies, providing a testing ground for general relativity. Yet, while Einstein’s theory acquired overwhelming evidence closer to home, galaxies fell far short of its predictions. To account for this, Vera Rubin and Kent Ford published in 1980 a model for a new substance with the required mass and which did not emit light, that had to be six times more abundant in the universe than visible matter.³ The evidence leading to such a conclusion is summarised as follows:

  • First, the rotation of galaxies indicates a spread of mass throughout rather than a central concentration. Kepler’s law predicts that the average velocity of stars orbiting the centre of a galaxy is given by:Screen Shot 2017-05-31 at 23.20.46Where r is the distance from the centre. Instead of dropping, v is observed through optical spectroscopy to be constant.⁴ By equating the gravitational and centripetal forces the implications of this on galactic density are revealed:Screen Shot 2017-05-31 at 23.29.16Screen Shot 2017-05-31 at 23.26.21Screen Shot 2017-05-31 at 23.26.35Where G is the Newtonian constant, and M the mass of a galaxy whose derivative with respect to distance is dM/dr. In words, constant velocity requires almost-constant density meaning large amounts of matter must exist on the very edges of galaxies, far more than shows in optical imaging.
  • Second, and on an even grander scale, galaxy clusters appear to have many times more kinetic energy than their visible mass allows. For a galaxy of mass m moving in a cluster of mass M a distance r from the centre, its kinetic and potential energies will be:Screen Shot 2017-05-31 at 23.53.38Screen Shot 2017-05-31 at 23.53.44But for a stable system, the centripetal and gravitational forces must be equal, returning expressions for K and P:
    Screen Shot 2017-05-31 at 23.55.27Screen Shot 2017-05-31 at 23.55.33Screen Shot 2017-06-01 at 00.02.12.pngSo the potential should be equal to twice the kinetic energy, yet for observed clusters, the kinetic energy appears several times greater than the potential, implying an unseen source of extra gravitational energy.⁵
  • Third, gravitational lensing. Einstein showed that gravitational fields can deflect light, where the angle of deflection is:Screen Shot 2017-05-31 at 22.23.43Where d is the distance between the foreground and background galaxies, found by measuring their redshifts. The results again imply several times more mass in a galaxy cluster than is visible, as would be explained by dark matter.⁶
¹ Kroupa, P.; et al. (2010). “Local-Group tests of dark-matter Concordance Cosmology: Towards a new paradigm for structure formation”. p. 14. (PDF)
² Kapteyn, Jacobus Cornelius (1922). “First attempt at a theory of the arrangement and motion of the sidereal system”. p. 1. (PDF)
³ Rubin, Vera C.; Ford, W. Kent, Jr. (February 1970). “Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions”. (PDF)
⁴ Corbelli, E. & Salucci, P. (2000). “The extended rotation curve and the dark matter halo of M33”. p. 1. (HTML)
⁵ Richmond, M (2004) “Using the virial theorem: the mass of a cluster of galaxies”. (HTML)
⁶ Taylor, A.N.; et al. (1998). “Gravitational Lens Magnification and the Mass of Abell 1689” (PDF)