The Physics of Energy, page 87
(21.23)
where the Greek index μ takes values 0, 1, 2, 3 for the four space-time coordinates. Points in this space are known as events, with specific coordinates in space and time. The separation Δ in space-time between a pair of events and is defined to be
(21.24)
where is known as a four-vector in analogy to the 3-space vector defined earlier. The speed of light is the same for all observers if the transformations from one inertial coordinate system to another leave invariant. Thus special relativity generalizes the concept of a rotation from space to space-time in such a way as to guarantee that all observers agree on the velocity of light.
The coordinate transformations that leave the speed of light invariant mix space and time in much the same way that ordinary Euclidean rotations can mix the x and y coordinates through a rotation around the z-axis. Thus, for example, two events that are simultaneous for one observer appear to occur at different times to an observer moving at a constant velocity relative to the first. Among the consequences of these transformations is the fact that the speed of light is the ultimate limit on the speed of any observer or object. This principle and many other surprising consequences of special relativity such as time dilation have been verified by many experiments. The appearance at sea level of muons created by cosmic rays high in the atmosphere is one example already encountered in §20 (see §20.5.2 and Problem 20.18).
Because all observers agree on the invariant interval between two events, it can be used to classify the causal relation between events. (21.24) can be positive, negative, or zero. If is zero, the events P and are on the path of a light ray and light from one event can influence the other. If is positive then there is a reference frame in which the two events occur at the same time and at different places (Problem 21.8). Since no signal can propagate faster than the speed of light, neither of these events can influence the other. This is reassuring since it is also possible to show that the time ordering of such events can be reversed in different reference frames, which would make the notion of causality problematic. Such events are termed causally disconnected. If is negative, then there is a reference frame where the two events occur at the same place and at different times (Problem 21.9) and the earlier one can influence the later. In this case the events are termed causally connected. Figure 21.1 illustrates these concepts.
Figure 21.1 Space-time with the z-direction suppressed. The paths of light rays, for example , emanating from O are shown in blue. Points O and are causally connected; the line connecting them represents a signal traveling at . O and are causally disconnected; a signal connecting them would have to travel at .
A convenient way of characterizing the invariant interval in eq. (21.24) is to generalize the concept of a metric tensor introduced in eq. (21.21). If the metric tensor of special relativity is defined by
(21.25)
then the invariant interval between two points separated by the space-time four-vector can be written
(21.26)
The metric tensor thus gives a way of characterizing distances in space-time.
Einstein’s Special Theory of Relativity
The special theory of relativity combines space and time into a single four-dimensional vector space (space-time). The laws of physics, including in particular the speed of light in a vacuum, are the same for observers moving relative to one another at a fixed velocity in space-time. The conserved energy and momentum of a system of mass m satisfy the relativistic relation .
The quantities that appear in physical laws can be classified by the way that they transform from one reference frame to another frame traveling at a constant velocity relative to the first. Some quantities, such as mass and electric charge, are invariant – they are the same to all observers. Others, such as momentum and energy and electric and magnetic fields, appear differently in different reference frames. Since energy and momentum are related to translations in time and space, it is perhaps not surprising that they together form a four-vector
(21.27)
This four-vector transforms from one coordinate system to another in the same way as the coordinate four-vector . Therefore the quantity
(21.28)
is the same in all reference frames. Substituting from eqs. (21.25) and (21.27) we find
(21.29)
This is the origin of Einstein’s famous relation (for an object at rest), which has been referred to in several previous chapters.
Other quantities of interest transform between reference frames in more complex ways. A magnetic field in one reference frame appears as a combination of electric and magnetic fields to an observer moving relative to the first frame. This fact was key to understanding the concept of electromagnetic induction as described in Box 3.6. In special relativity the electric and magnetic fields are encoded in the electromagnetic field strength tensor , which is antisymmetric in its indices (). An antisymmetric matrix has six independent components, which in this case form the spatial vectors E and B. In particular, and . The transformation rules for four-tensors are straightforward, but algebraically more complicated than the rules for four-vectors. One consequence is that the quantity , like , is the same in all reference frames. Thus although a magnetic field in one reference frame may transform to a combination of electric and magnetic fields in another frame, there is no reference frame in which only an electric field appears.
The metric tensor, , which can be regarded as a convenience and a notational simplification in special relativity, plays a central role in Einstein’s theory of gravity, the general theory of relativity, to which we now turn.
21.1.5 General Relativity
Einstein’s general theory of relativity provides a geometric description of classical gravity that goes beyond Newton’s gravitational force law. While not used elsewhere in this book, it is not possible to discuss the origins of energy in the early universe without referring to some aspects of the general theory of relativity.
Curvature and Einstein’s Equations In Newton’s theory of gravity, objects at a distance attract one another with a force proportional to . In electromagnetism, the apparent action at a distance of Coulomb’s law is replaced by the local action of the electric and magnetic fields, as discussed in §3. Einstein’s theory of general relativity plays a similar role for gravity, but requires a more profound extension of the underlying physics. The basic idea of Einstein’s theory of gravity is that space-time itself is curved. Matter and energy produce local changes in the geometry of space-time, and this geometry in turn affects matter.
It may be difficult to imagine what it means for space-time itself to be curved. The situation is somewhat similar to that of a person walking in a vast desert landscape on our planet, where the ground is flat and appears to form part of an infinite plane. Although Earth’s surface is curved, the geometry of the surface looks locally like Euclidean two-dimensional space, where one can impose an infinite Cartesian coordinate system with distances measured by . In fact, however, the surface of Earth is not flat. A circle of radius r on Earth’s surface can be measured to have a circumference that is smaller than , with the deviation from this Euclidean result increasing as the size of the circle increases.
The curvature of space-time is conceptually similar to the curvature of Earth’s surface, though more difficult to grasp since our space-time is (as far as we know) not embedded in any higher-dimensional flat space. A massive object such as Earth produces localized curvature in space-time in its vicinity (Figure 21.2). This curvature in turn affects the trajectories of moving objects (such as the Moon). While this may seem like a very complicated story with which to replace the simple Newtonian notion of gravitational attraction, careful calculations show that the motion of the planets and other astronomical objects precisely follow the predictions of general relativity, which increasingly differ from those of Newtonian gravity as space-time curvature becomes more pronounced. For example, the orbit of Mercury – the planet closest to the Sun and therefore subject to the greatest space-time curvature – gradually precesses over time in a way that is correctly explained by general relativity and not by Newtonian gravity.
Figure 21.2 In Einstein’s theory of general relativity, the force of gravity arises due to curvature in space-time. A massive object such as Earth generates curvature in the local metric describing space-time distances. The consequence of this curvature is that objects moving through space-time in the vicinity of the massive object follow curved trajectories – like the elliptical orbit of the Moon around Earth (Credit: BBC)
Curvature in space-time is described by local changes in the notion of distance. The constant metric (21.25) that measures space-time distances in special relativity is replaced in general relativity with a metric tensor that varies depending on the position in space-time. The general metric tensor describes local distances between infinitesimally separated points and through
(21.30)
Curvature is then a symmetric two-index tensor that can be defined in terms of (second and products of two first) derivatives of the metric tensor. Einstein’s equations of general relativity take the schematic form
(21.31)
where G is Newton’s constant, and the energy–momentum tensor (also known as the stress–energy tensor) encodes the energy and momentum densities, and pressure and stress of matter and radiation, which are sources of curvature in space-time. The component describes the energy density, ; describes the density of momentum in the spatial direction for ; and for describes the flux of the i th component of momentum in the j th direction. For an isotropic distribution of matter, the off-diagonal components of vanish and each of the three diagonal components equals the pressure for . Note that in general relativity, unlike in most other physical systems studied in this book, energy enters as an absolute, not a relative quantity. The curvature of space-time determines a definite zero point for the measurement of energy.
Classically, energy in general relativity can be understood in terms of curvature. By looking at the metric of space-time on a large sphere, for example, surrounding a region containing matter, it is possible to compute the total energy within the region in terms of the curvature of the surrounding space-time. This gives a sensible way of thinking of energy in general relativity as long as gravitational effects are classical and the system considered is isolated (so that space-time approaches a flat geometry far away from the system). The nature of energy in general relativity becomes more subtle when quantum effects and the global structure of space-time are considered.
The Cosmological Constant When Einstein first formulated his general theory of relativity in 1916, the prevailing assumption was that the distant galaxies visible from Earth were, on average, moving neither toward nor away from Earth or one another. Einstein’s theory, like Newton’s, predicts that massive objects such as stars and galaxies attract and should accelerate towards one another. This is not consistent with a static universe. To resolve this apparent inconsistency, Einstein modified his equations by including a cosmological constant Λ as an additional source of curvature, replacing eq. (21.31) by
(21.32)
The cosmological constant Λ has units of m-2. Note that because , acts like a constant, positive energy density throughout space-time that only interacts with gravitational fields. The cosmological constant is thus often referred to as dark energy. Unlike matter and radiation, however, which contribute positively to the pressure (), the cosmological constant gives a constant negative pressure throughout space-time, that in the absence of other effects would cause the material in the universe to accelerate apart. Note that here the fact that the zero point of energy is not arbitrary in the context of general relativity is essential: the cosmological constant cannot be defined away by shifting all energies by a constant. With this term included, Einstein’s equations give a solution in which, by tuning Λ, the galaxies in the universe could maintain fixed distances from one another (though this solution would be unstable).
In the late 1920s, observations by the American astronomer Edwin Hubble showed conclusively that distant galaxies are receding from our own with a velocity proportional to their distance. With the realization that the universe is not static, Einstein’s equations were able to describe observed phenomena without a cosmological constant. Late in the twentieth century the cosmological constant re-emerged in two crucial aspects of modern cosmology. Indeed, the cosmological constant, which parameterizes the zero-point energy of space-time, is related to some of the most profound outstanding puzzles in modern physics.
An Expanding Universe Hubble’s observation that the universe is expanding suggested that the observed universe was smaller in the past; this line of reasoning led in time to the big bang theory, which proposes that all observed matter has expanded from an original state of very high temperature and density. In such an expanding universe, gravitational attraction acts to slow the expansion. More recently, however, evidence has suggested that very early in its history, the universe underwent a period of accelerating expansion, and furthermore that it is entering a similar phase at the present time. In both eras the accelerated expansion can be accounted for by the presence of a cosmological constant, or something functionally equivalent. In particular, over the last decade, careful observations of distant supernovae and other astronomical phenomena have given definitive evidence that the rate at which distant galaxies are retreating from our own is increasing over time, not decreasing as would be expected from the gravitational attraction between the material in the universe. This is analogous to observing a ball, thrown upward, which slows down and then speeds up again, moving ever faster away from Earth. The simplest explanation for the increasing expansion rate is a cosmological constant, i.e. dark energy.
In the presence of a cosmological constant, the nature of energy in the universe presents some profound puzzles. The simplest solution of Einstein’s equations for general relativity (21.32) arises if we assume that space is completely empty except for the dark energy associated with a nonzero cosmological constant. In this case, in an appropriate reference frame3 the metric is given by
(21.33)
i.e. . Here is a time-dependent scale parameter. In this frame the components of Einstein’s equation (21.32) become
(21.34)
The solution to these equations is
(21.35)
where is known as the Hubble parameter. Note that the left-hand sides of the equations in (21.34) describe the curvature, which entails two derivatives of g, so both curvature and Λ have units of 1/[distance], while H has units of 1/[time].
The physical interpretation of the solution (21.35) is that the scale of the universe is inflating at an exponential rate. In this coordinate frame, the distance between any two points in the universe increases exponentially with time. To imagine what this means, consider an analogy with an expanding balloon. If the universe is the surface of the balloon, with various galaxies scattered about the surface, then as the balloon expands, each galaxy moves away from each other galaxy. One interesting feature of the solution (21.35) is that it produces an effective “horizon.” Points separated by a distance of more than cannot be in causal contact, since the universe expands so quickly that a light ray from one point never reaches the other point (Problem 21.11).
The equations (21.34) can be modified to include matter and radiation. If we assume that the matter and radiation are distributed in a homogeneous (spatially uniform) and isotropic (rotationally invariant) fashion, we can characterize the energy density and pressure by . This leads to the Friedman equations
(21.36)
From the second of these equations, we see that positive energy and pressure density act to slow the rate of expansion, acting in the opposite direction to a positive cosmological constant.
As the universe expands, the relative contributions to the energy density from matter, radiation, and the cosmological constant change over time. For nonrelativistic matter, the total energy remains constant as the volume increases proportionally to , so the energy density of (nonrelativistic) matter scales as
(21.37)
For radiation (or relativistic matter), the wavelength of each excitation (e.g. photon) increases with a, so that the energy density drops more quickly than for nonrelativistic matter
(21.38)
For the cosmological constant, on the other hand, the energy density attributed to Λ is independent of a. Here, the difficulty in defining the total energy in an expanding universe begins to become apparent, since both and change in time.
Note that for any given initial conditions of matter and energy density, the fraction of the universe’s total energy that comes from any nonzero cosmological constant grows as the universe expands, and when Λ is the dominant term on the right-hand side of eq. (21.36) the equations simplify to eq. (21.34). Indeed, expansion of the universe according to eq. (21.35) is believed to be the mechanism responsible for the increasing rate at which distant galaxies are now moving away from our own.
Quantum Gravity While general relativity is well understood as a classical theory, despite a century of effort there is still no complete quantum theory of gravity that describes our universe. To appreciate the difficulty, note that in a quantum theory, quantities that can be precisely determined classically, such as position and momentum, become uncertain. In quantum gravity, the shape of space-time itself becomes indeterminate. Most of our physical theories use a fixed space-time background metric as a scaffolding on which to construct the theory, but this cannot be done for quantum gravity. Furthermore, the realm of quantum gravity is far removed from experimental exploration. In the past, experimenters have guided the development of new domains of physical understanding with critical experimental discoveries such as atomic spectra and the electron spin. The distance scale where the direct effects of quantum gravity are likely to be observed can be estimated by combining Newton’s constant with Planck’s constant and the speed of light c to form a quantity with units of length. The resulting unit is known as the Planck length and has the value m, far smaller than the distances, of order m, that can presently be probed at the highest energy accelerators. Thus quantum gravity must be studied primarily theoretically.
