General Relativity and Standard Cosmology (Part 1)

The Universe Before the Big Bang: General Relativity and Standard Cosmology

The physical description of the current Universe, and of the forces that stars, galaxies, and clusters of galaxies exert upon one another at the cosmic distance scales corresponding to their huge spatial separations, requires the use of general relativity, i.e., Einstein’s gravitational theory. Some may ask why we do not just use Newton’s gravitational theory. Why would it not be possible, for our Universe, to build up a consistent, Newtonian-type cosmology based upon the universal law of gravitation that we learnt at school, according to which the mutual attraction between two bodies is directly proportional to the product of their masses and inversely proportional to the square of their relative distance?

The answer to this question is very simple. It is based on the fact that Newton’s theory is not a relativistic theory, and is thus valid only for sufficiently low velocities and energies. Strictly speaking, Newton’s formulation is valid as long as the kinetic and potential energies of the bodies under consideration are small compared to the energy associated with their rest mass. This implies, in particular, that in order to apply the Newtonian theory correctly to a given system the associated potential energy per unit mass (the so-called gravitational potential) has to be much smaller than the square of the speed of light.

This requirement is certainly fulfilled by the gravitational forces that we ordinarily experience. This condition is satisfied, for instance, by the force exerted by our planet (the Earth) on us, and on its satellite (the Moon); it is satisfied by the forces by which the Sun holds onto its planets; it is even valid for the mutual forces holding together stars and galaxies. However, the above condition is not satisfied if we take into account the whole Universe accessible to our observation.

In fact, if we compute for the Universe the quantity that could represent the equivalent of the gravitational potential – multiplying Newton’s constant G by the total mass of the Universe (i.e., by the sum of the effective masses associated with all its cosmic components), and dividing by the radius of the portion of space containing these masses – the result we obtain is of the order of the square of the speed of light itself. Hence the need to resort to a fully relativistic theory of gravitation in order to formulate cosmological models able to consistently describe the dynamics of the Universe as a whole.

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In the past, various attempts have been made to generalize Newton’s theory in order to turn it into a relativistic theory of gravity. For instance, starting from the formal analogy between the Newtonian gravitational force (amongst masses) and the Coulomb force (amongst electric charges), attempts were made to describe gravity in terms of a four-dimensional vector field similar to the vector potential appearing in the relativistic theory of electromagnetism. These attempts failed for a fundamental reason: as is well known, electric charges of the same sign are mutually repulsive, while electric charges of opposite sign are mutually attractive (the same phenomenon also occurs with the poles of a magnet). In the gravitational case, however, there are no negative masses (to the best of our present knowledge), so that all masses have the same sign; nevertheless, we all know that masses attract each other, while for a vector-type gravitational theory they should repel each other, as happens in the case of electromagnetism.

Another approach attempted to describe gravity by using a relativistic scalar field to represent its potential, under the assumption that the gravitational potential energy retains its Newtonian form even when the typical speed of a body becomes relativistic. This attempt was also unsuccessful, for a number of reasons. In particular, the type of motion it predicts for the planets is inconsistent with observations. For instance, according to this scalar theory of gravity, the secular drift of the point closest to the Sun in Mercury’s orbit (called the perihelion) would be much smaller – about one sixth of what is actually observed.

The correct path was actually taken by Einstein, nearly one century ago, when he came up with the idea of describing gravity in terms of a geometrical tensor field – a radically different approach to those used to describe the other known forces – and of formulating a consistent theory of gravity by extending and generalizing the fundamental principles underlying special relativity. Actually, this is the reason why his theory is called general relativity. In this theory, the well-known postulate of special relativity asserting that the physical laws are the same in all inertial frames generalizes into: the physical laws are the same in all reference frames, regard- less of the coordinate transformations connecting them.

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In other words, the physical equivalence of all inertial observers (i.e., those moving at constant speed) is extended to all observers, even to those whose motion is accelerated. This leads to what is known as the principle of general covariance, according to which the form of physical laws has to be invariant under any coordinate transformation – not only under the Lorentz trans- formations that connect inertial observers, and that represent the basic symmetries of special relativity. Strictly speaking, the principle of general covariance demands that the physical laws must be expressed as equalities between identical-rank tensors and that the latter, representing physical quantities like the energy, the force and so on, must be mathematical objects consistently defined with respect to generic coordinate transformations.

From a geometrical point of view, general covariance leads to a full-scale revolution in the geometrical structure of space-time: from the rigid structure of special relativity, which is of Euclidean type, we change to a deformable, generally curved structure, of Riemannian type (named after the mathematician Georg Riemann who first studied the geometry of such spaces). To visualize the differences, a table-top may be thought of as an example of rigid, two-dimensional Euclidean space, whereas an elastic net – which is flat when empty, but bends when massive objects are placed upon it – is an example of a two-dimensional (possibly curved) Riemannian space.

But why should general covariance, i.e., the physical equivalence between accelerated observers, require non-Euclidean geometry? A precise answer to this question would involve technical details that would take us beyond the scope of this book. To give an intuitive answer, it suffices to observe that in a Euclidean space the square of the distance between two points is given by the sum of the squares of the distances along the various axes of a Cartesian frame (as follows from a simple application of the well known theorem named after Pythagoras). If this is true in an inertial frame, then it remains true after performing coordinate transformations leading to any other inertial frame. However, this result is no longer valid, in general, when a coordinate transformation brings us to an accelerated reference frame. Here, to obtain the square of the distance between the two points, one needs to take the squares of the distances along the axes and, before making their sum, multiply them by non-trivial functions of the new coordinates which depend on the given transformation, and which represent the components of the so-called space-time metric.

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Thus, an accelerated observer measures a space-time geometry which is not of Euclidean type. Instead, it is a Riemannian geometry, characterized by a metric which looks different in different reference frames. This explains why, extending the class of physically equivalent observers to include accelerations, one should expect a generalization of the geometry, and one should be ready to accept the possibility that the space-time is not generally flat.

But what do the Riemann metric, or the curvature, have to do with gravity?

It is precisely the link between the space-time curvature and the gravitational interaction that is likely to represent the most novel feature of the theory of general relativity. In fact, in a curved space all test bodies tend to follow curved trajectories. Thus, their motion deviates from a straight line, as if they were subject to forces. Choosing a suitable metric (i.e., an appropriate space-time geometry), it then becomes possible to reproduce the gravitational forces, even in the Newtonian limit where the velocities are small.

In this way, one can directly incorporate the gravitational interaction into the space-time geometry. The latter is said to be flat (or Euclidean) in the absence of gravitational forces, while it is said to be curved when such forces are present. The equations of general relativity express this link between the space-time curvature and the gravitational properties of material bodies in a detailed fashion.

It is important to stress that this geometrization of forces works well in the gravitational case by virtue of the universality of gravity, i.e., the fact that all bodies “feel” gravity (and react to it) with the same intensity. Such universality is experimentally guar- anteed by the well-known equality between the inertial mass and the gravitational mass. In particular, it is just this universality that underlies the so-called equivalence principle, according to which the effects of a gravitational field are locally indistinguishable at a given space-time point from those produced by a properly chosen accelerated frame. As a consequence, it is always possible to eliminate the gravitational field at a given point by means of an equal and opposite acceleration.

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For other interactions, characterized by non-universal coupling constants, the geometrization of the corresponding forces would not be so effective and useful. Consider for instance the electromagnetic force. Since different bodies may have different electric charges, a geometric description would require associating different space-time geometries with the same electric field, depending on the test body upon which the force is exerted.

The geometrization of the gravitational field not only provides a new and elegant formalism for describing universal forces, but also has deep physical consequences and, in particular, predicts new gravitational effects (even in the low velocity regime) that are not present in the Newtonian theory of gravity. For the applications of this book, we particularly stress the slow-down (or time dilation) experienced by clocks in the presence of a gravitational field.

As already known within the theory of special relativity, the relative flow of time is different for observers who are not moving with the same velocity. A moving clock, in particular, is seen to tick more slowly if compared with an identical clock at rest. In a curved space, beside the slowing down due to the relative velocity, we may also measure a relative slowing down between two clocks at rest, provided they are placed at different space-time positions.

This phenomenon occurs because the Riemann metric which defines the distance between two space-time points, taking into account the possible curvature, deforms (with respect to the Euclidean case) not only the spatial intervals but also the temporal intervals. Now, suppose that between a given pair of points there is a difference in the gravitational field, and thus in the metric which describes the associated geometry. It follows that the relative distortion of the Euclidean time interval is also different. The comparison between these different intervals then leads to a relative slowing down of the clocks between the two different space-time points. In particular, one finds that the more the ticking of the clock is slowed down with respect to a clock in a flat space, the more warped the space is, i.e., the more intense is the gravitational field to which the clock is subjected.

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For a periodic signal (e.g., an electromagnetic wave) propagating through a curved space, the time dilatation effect will thus produce an increase (with respect to the flat space) in the measured period, with a corresponding stretching of the wavelength (which is proportional to the period) and a decrease in the frequency (which is inversely proportional to the period).

This effect is commonly known as redshift – with reference to the spectrum of visible light, where a shift towards lower frequencies corresponds to a shift toward the red end of the color spectrum we know from the rainbow. This redshift is a peculiar example of a physical effect associated with the geometrical description of the gravitational field. Its experimental validation, as well as tests of other effects – such as the deflection and slowing down of light and electromagnetic signals in the presence of gravity, the correct prediction of the shift in Mercury’s perihelion, etc. – have marked the success of general relativity as a consistent and phenomenologically viable relativistic theory of gravitation, which both completes and enriches Newton’s theory. The gravitational redshift, in particular, leads to important applications in a cosmological context, as will be illustrated shortly.

In fact, assuming that general relativity is valid on length scales corresponding to cosmological distances, it is possible to formulate a relativistic description of the Universe which is consistent with current astronomical observations. This description serves to make predictions about the future evolution of our Uni- verse and also to piece together its past history, not to mention the possibility of gaining information about the very birth of the Uni- verse. Apart from the equations of general relativity, this extraordinary theoretical framework, known as the standard cosmological model, is based upon two further important assumptions.

The first assumption is that, on sufficiently large distance scales, the Universe and all its components can be described using a spatially isotropic and homogeneous geometry, which is to say a spatial geometry that does not admit either preferred directions (isotropy) or privileged points (homogeneity). This is equivalent to assuming that, at any given time, the spatial sections of the Universe can be described as spaces of uniform curvature (positive, negative or null). This is quite an oversimplification, but nevertheless a useful hypothesis, which allows one to describe the geometry of the Universe in terms of the well-known Robertson–Walker metric, containing only two parameters: the scale factor R, which is in general time-dependent – and which we shall call, for simplicity (but somewhat improperly), the radius of the spatial part of the Universe – and the uniform curvature of the spatial sections of the Universe.

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The other assumption concerns the physical properties of the particles and macroscopic bodies populating our Universe. It is assumed that, on large scales, they behave as a perfect gas with two main components: radiation, whose pressure is exactly equal to one third of its energy density, and non-relativistic matter, with zero pressure. (Recently, it has been found that, apart from these two components, other cosmic components seem to play a fundamental role, as discussed further in Chap. 9.) Moreover, the radiation is assumed to be in thermal equilibrium, i.e., characterized by a black-body frequency distribution (exactly like the electromagnetic waves present in a hot oven), and to evolve in time adiabatically, in such a way as to keep the entropy constant, in agreement with the laws of classical thermodynamics. The presence of this back- ground of cosmic radiation is interpreted as a relic of the Big Bang, i.e., the big explosion from which the Universe and space-time itself were born.

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General Relativity and Standard Cosmology (Part 2)