The Spacetime Continuum. Detailed Description

In thinking about the Universe, modern cosmology demands that we consider not only the objects it contains. As well as thinking about the radiation content of the Universe (which we shall cover in the next chapter) we must also consider space itself.

Kepler’s laws of planetary motion were the first to describe how objects move through space. They were kinematic in nature because they made no attempt to explain why the planets moved. Newton went one stage further by considering the force which was causing the motion, namely gravity.

Newton’s laws of motion led him to theoretically derive his universal law of gravitation which gives the magnitude of the gravitational force, F, between two masses, m, and m, which are separated by a distance, r.

G is the universal constant of gravitation and takes the value, 6.67 x 10^-11 Nm²/kg². This work of Newton’s not only ushered in the era of theoretical science but also proved that planetary motion was caused by the familiar terrestrial force of gravity. It was a triumph of unification and, as we shall see in the next chapter, the task of unifying nature’s diverse phenomena continues to this day. Newton used his equations to deduce that orbits could be circles, ellipses, parabolae or hyperbolae. These are known as conic sections because their shapes can be obtained by slicing through a cone (see Fig. 1. 7).

Fig 1. 7. All orbits are represented by one of the conic sections. Different conic sections are obtained by slicing a cone at different angles. Closed orbits are circles and ellipses, obtained by slicing the cone at zero or shallow angles to the symmetry axis. Open orbits are the parabolae and hyperbolae, obtained by slicing the cone at steep angles to the symmetry axis. (Adapted from Kaufmann, W.J., Universe, W.H. Freeman, 1987.)

Another great unification took place early in this century between space and time. This was made possible by the concept of spacetime which was developed by Hermann Minkowski, instead of thinking of space as being three-dimensional, with time as a separate property, spacetime postulates a four-dimensional framework in which the time ordinate can be given the same units as distance by multiplying it by the speed of light.

In 1915, Einstein built upon spacetime and presented the general theory of relativity. This was also an extension of Einstein’s previous work - the special theory of relativity - and helped to explain gravity. It finally gave astronomers a tool with which to study the Universe on its largest scale.

In 1929 a very important conclusion was reached, based upon a set of interesting observations. Hubble’s work on galaxies yielded the result that all the galaxies, apart from those in the Local Group, were receding from the Milky Way. The further away the galaxy, the faster it was receding.

The receding motion of the galaxies was described by Hubble’s equation, which related the recessional velocity of a galaxy, v, to its distance, d:

V = Hd (1.4)

H is known as the Hubble constant and has a value which is thought to lie somewhere in the range 50-100 km/s/megaparsec.

The general theory of relativity could be made to agree with this conclusion if one ,major caveat were placed upon the explanation. Although it appeared as if it were the galaxies which were moving through space, in actuality it was the space between the galaxies which was expanding. So the galaxies were being driven apart in the same way that raisins in a dough mixture are moved away from one another when the bread rises (see Fig. 1. 8).

Fig. 1. 8. The Universe is thought to be in a state of uniform expansion. This is similar to a rising dough mixture which contains raisins. Because the dough between the raisins expands, the further the raisins are from each other the faster they will move apart

Interestingly enough, when Einstein had derived the equations which define the spacetime continuum, the equations did not allow the Universe to be static; it had to be either expanding or contracting. Since, at the time, Hubble had yet to prove the expansion of the Universe, Einstein had recast his equations for presentation so that they included a new constant which would hold the Universe static. When Hubble presented his results, Einstein immediately removed his cosmological constant and called it the greatest mistake of his life!

General relativity provides us with a way to visualise how gravity is created, by explaining the way mass interacts with spacetime. The theory states that massive objects distort the spacetime continuum into curves in much the same way as a heavy ball might deform a rubber table top (see Fig. 1. 9).

Fig. 1. 9. The spacetime continuum can be visualised as a rubber sheet onto which are placed heavy objects. These objects (stars, galaxies etc.) deform the rubber sheet, causing it to curve. Any object which strays too close rolls down the curve and touches the object causing the curvature. This is the general relativistic analogy to gravity. (Adapted from Berry, M., Principles of Cosmology' and Gravitation. Adam Hilger. 1989.)

Anything which gets too close rolls down the curve as if it were being attracted to the massive object. Thus, an intimate relationship between space itself and matter exists. The curvature of the spacetime continuum caused by mass in turn creates gravity, which tells other masses how to move through it.

One of the universal paradoxes which cosmologists have had to deal with is that, although gravity is by far the weakest of the four fundamental forces of nature, it dominates the Universe on its largest scale. The strong nuclear force and the weak nuclear force only operate over the distance of an atomic nucleus, whilst the electromagnetic force cancels out over large distances.

With nothing left in competition, gravity and the motion it causes can sculpt the Universe on all but the smallest of scales (see Fig. 1. 10). Indeed, it is gravitational forces 4vhich are responsible for every structure we have so far discussed in this chapter. As far as the future of the cosmos is concerned, gravity has also sealed the fate of the Universe. All that remains is for the cosmologists to work out exactly what that fate is!

Fig. 1. 10. The vantage point from which we observe the Universe at large is anything but static Our Frame of reference is a superposition of many different millions Our overall velocity can be measured with reference to the cosmic microwave background radiation because of the Doppler shill that the velocity induces

Cosmology is a rich field of study which encompasses all scales of the Universe. The Big Bang itself is understandable only by thinking in terms of quantum theory, the method we have for understanding the Universe on its smallest scale. The long-term future of the Universe is understandable only if we use general relativity: a theory for understanding the Universe on its largest scale. Cosmology is, literally, universal in content and scope. It is the science from which all others are born, and we gain access to it by taking our minds on a journey from here towards the edge of the Universe.