Chapter 17: Cosmology

Albert Einstein

Modern cosmology began with Albert Einstein. First, his special theory of relativity showed that time and space are supple and not adequately described by the rigid, linear measures proposed by Newton. Experiment shows us that the speed of light is a constant number, regardless of the motion of the observer. There are three bizarre consequences of the fact that we cannot measure our motion with respect to a beam of light: (1) Time slows down for a fast-moving object. (2) A fast-moving object shrinks in the direction of motion. (3) The mass of a fast-moving object increases, with the energy of motion converting into mass by the famous relationship E = mc². These effects are not visible in the everyday world where objects move slowly compared to the speed of light.

Special relativity deals with objects in constant relative motion. By contrast, general relativity deals with objects in non-uniform motion. Einstein was led to the general theory of relativity by pondering what seemed to be a coincidence in physics. The gravitational mass of an object — its response to a gravitational force — is identical to its inertial mass — the resistance it presents to a change in its motion. At first glance, these appear to be very different forms of mass. Imagine a large, smooth piece of iron at rest on a slick, icy surface. Gravitational mass dictates the force with which the iron presses down on the ice. Inertial mass is the resistance of the iron to a change in speed, either trying to speed it up or slow it down. The inertial mass has nothing to do with gravity since the motion on the ice is horizontal and gravity acts vertically. Nevertheless, inertial and gravitational masses are measured to be equal to an exquisite degree of precision: modern experiments find the difference to be less than 1 part in 10¹⁵. Einstein believed that this coincidence held the key to understanding gravity.

An analogy will show how difficult it is to distinguish between motion caused by gravity and motion caused by any other force. Suppose that you are trapped in a windowless elevator. Einstein showed that there is no way to distinguish between the motion of objects in an elevator at rest on the Earth's surface and in an elevator accelerating through distant space at a rate of 9.8 m/s². In each case, you would have your normal weight. He also realized that there is no way to distinguish between the motion of objects in an elevator floating freely in space and in an elevator in free fall toward the Earth's surface at a rate of 9.8 m/s². In each case, you would be weightless. In short, there is no measurable difference between acceleration caused by gravity and acceleration from any other force.

Gravity is just a convenient way to describe how the presence of mass causes an object to change its motion. Einstein "generalized" his special theory by showing how gravity could distort space and time. General relativity replaces Newton's force of gravity with the geometry of space itself. The familiar Newtonian idea of masses placed in smooth and uniform space is replaced with the counterintuitive idea of space that is distorted by the masses it contains. Matter curves space and light and particles follow the undulating paths dictated by the curvature.

In 1917, Einstein applied the equations of general relativity to the universe as a whole. He assumed that the universe was static because astronomers at the time believed the universe to contain only the enormous Milky Way, with stars milling around in it. (Ironically, the astronomer Vesto Slipher was already gathering spectra that would reveal the recession of the galaxies and disprove the static model.) No matter how Einstein solved the equations, they stubbornly indicated a dynamic universe, one that was either expanding or contracting. To force a static solution to the equations, he added an arbitrary term. Einstein later admitted that this adjustment was "the greatest blunder of my life." Because of it, he missed the chance to predict the expansion of the universe ten years before Hubble observed it.

Georges Lemaitre

Alexander Friedmann

In the 1920s, Russian mathematician Alexander Friedmann and Belgian mathematician Georges Lemaître independently solved the equations of general relativity and showed mathematically that the universe was expanding. Then, with Hubble's discovery of the redshift-distance relation, what had originally seemed to be a purely theoretical model was supported by observations. Galaxies are not moving apart through space in a large-scale version of the Doppler effect. Galaxies are being carried apart by the expansion of space itself. Galaxy recession velocities are indicators of a cosmological redshift. We can fall back on a simple analogy: the surface of a balloon with small beads glued on it to represent galaxies. (Keep in mind that this is a two-dimensional representation of a positively curved space that exists in three dimensions.) As the balloon is being blown up, its expansion reveals several relevant features of the expanding universe.

 

Determining the Hubble constant using supernova type 1a.

The raisin bread model of the universe explains how each galaxy can perceive every other galaxy in the universe as receding from it.

The balloon analogy for an expanding universe is accurate in the sense that the beads are carried apart by the stretching rubber of the balloon. In general relativity, the fabric of space is expanding, carrying the galaxies with it. The beads follow a Hubble law, with recession velocity proportional to distance. No bead is at the center of the balloon, and no bead is at the edge. Although the space is expanding, the beads remain the same size. In our universe, although galaxies and clusters are moving farther apart, their internal gravity keeps them from expanding in size. These regions of non-expanding space are well described by Newton's laws: the Solar System is not expanding, nor is your house. Note also that as the balloon expands, the curvature of the space decreases: think of the difference in curvature between a balloon the size of your fist and one the size of a house. We can even add to the balloon an analogy for the cosmological redshift. Imagine a wave of light (or any electromagnetic wave) drawn on the balloon while it is small. As the balloon expands, the wavelength is stretched or reddened. In the real universe, light travels through expanding space, and we see a redshift that increases with the distance that light has traveled.

The expanding universe has space curvature. The best way to measure the curvature is through the deflection of light. If light and all other forms of electromagnetic energy have an equivalent mass, given by E = mc², then light should respond to space curvature just as particles do. A light beam sent across an elevator that is accelerating through space is deflected by a tiny amount because during the time it takes to cross the elevator, the elevator has moved. However, Einstein showed that this situation is indistinguishable from an elevator at rest on the Earth's surface. The same amount of deflection is predicted due to the gravity of the Earth! Mass curves space and both radiation and particles follow the trajectories dictated by the curvature.

Astronomers know that space can be locally curved. The observation that starlight is deflected around the edge of the Sun by 1.8 seconds of arc (only 0.1% of the Sun's angular diameter!) was a dramatic confirmation of the general theory of relativity. Gravitational lensing occurs because a galaxy or a cluster warps space and causes the distortion of light from a background quasar or galaxy. In the extreme example of a black hole, space is so highly curved that it is "pinched off," and matter and radiation are trapped within an event horizon.

General relativity allows for the possibility that space is globally curved by all the matter and energy in the universe. To understand this, it is useful to explore the difference between Euclidean and non-Euclidean geometry. Newton's gravity relied on the familiar three-dimensional geometry of Euclid. In Euclidean geometry, space is flat and two-dimensional surfaces have no curvature. The sum of the angles in a triangle is 180°, and parallel lines or beams of light will never meet. Euclidean space is flat.

Examples of curved spaces in everyday life

In the late 1800s, mathematicians in Germany, Italy, and Russia became fascinated with types of geometry quite different from Euclidean geometry and that of everyday experience. None of those mathematicians dreamed that their esoteric work would be applied to the field of cosmology. Two classes of non-Euclidean space exist: spherical and hyperbolic. Positively curved geometry in our analogy is like the surface of a sphere. The sum of the angles in a triangle is greater than 180°, and parallel lines or beams of light converge, so this spherical space is also called a "closed" space. Slightly less familiar is negatively curved geometry, which in our analogy is shaped like a saddle or a hyperbola in two dimensions. The sum of the angles in a triangle is less than 180°, and parallel lines or beams of light diverge, so this hyperbolic space is also called "open" space.

Here is a summary of the three types of space curvature: Euclidean (or flat), spherical, and hyperbolic:

• Euclidean: zero curvature, infinite volume, the sum of angles = 180°, parallel lines stay parallel

• Spherical: positive curvature, finite volume, the sum of angles > 180°, parallel lines converge

• Hyperbolic: negative curvature, infinite volume, the sum of angles < 180°, parallel lines diverge

It helps to use a two-dimensional analogy for curved space because experience and intuition help in understanding the two-dimensional situation, whereas the three-dimensional situation is difficult to grasp without mathematics. The surface of flat or open space is infinite. Just imagine a sheet that continues forever. (In three dimensions this corresponds to an infinite volume.) By contrast, the surface of a closed space is finite and so is its volume. The Earth's two-dimensional surface, for example, is just such a finite closed space. However, the Earth is also unbounded: it has a definite area but you can travel in one direction forever without coming to an edge. By analogy, we can imagine the universe as a finite, closed space in which the galaxies stretch into space in every direction, but where there is nevertheless no edge. This analogy answers the age-old question of Archytas and the other Greek thinkers: the universe can be finite and unbounded.

How can we prove that this analogy truly represents the universe? Everyday experience gives us no clue as to whether or not space is curved. Similarly, out in the desert or on the ocean, the planet we live on appears to be flat. No local surveying technique would show any departure from Euclidean geometry. However, observations over a large distance can indeed measure the curvature. If you were traveling along the Earth's equator from East to West, you could make a right turn (90° angle) and you would be traveling directly toward the North Pole. At the North Pole, if you made another right turn from your direction of arrival, you would be traveling back down toward the equator. At the equator, if you made a third right turn, you would be traveling along the equator again and would arrive back where you started. The sum of the angles in your triangular journey is 270° — proof that Earth does not have a flat surface! Also, if you travel straight in any direction on the Earth's surface and travel far enough, you will eventually return to your starting point. We cannot duplicate these experiments in the three-dimensional universe, but astronomers have invented clever ways to try and measure space curvature.