Chapter 2

Special Relativity

The foundations of modern cosmology were laid during the second and third decade of the twentieth century: on the theoretical side by Einstein's theory of general relativity, which represented a deep revision of current concepts; and on the observational side by Hubble's discovery of the cosmic expansion, which ruled out a static Universe and set the primary requirement on theory. Space and time are not invariants under Lorentz transformations, their values being different to observers in different inertial frames. Nonrelativistic physics uses these quantities as completely adequate approximations, but in relativistic frame-independent physics we must find invariants to replace them. This chapter begins, in Section 2.1, with Einstein's theory of special relativity, which gives us such invariants.

In Section 2.2 we generalize the metrics in linear spaces to metrics in curved spaces, in particular the Robertson-Walker metric in a four-dimensional manifold. This gives us tools to define invariant distance measures in Section 2.3, which are the key to Hubble's parameter. To conclude we discuss briefly tests of special relativity in Section 2.4.

2.1 Lorentz Transformations

In Einstein's theory of special relativity one studies how signals are exchanged between inertial frames in linear motion with respect to each other with constant velocity. Einstein made two postulates about such frames:

1. the results of measurements in different frames must be identical;
2. light travels by a constant speed, , in vacuo, in all frames.

The first postulate requires that physics be expressed in frame-independent invariants. The latter is actually a statement about the measurement of time in different frames, as we shall see shortly.

Lorentz Transformations.

Consider two linear axes and in one-dimensional space, being at rest and moving with constant velocity in the positive direction. Time increments are measured in the two coordinate systems as and using two identical clocks. Neither the spatial increments and nor the time increments and are invariants-they do not obey postulate (i). Let us replace and with the temporal distances and and look for a linear transformation between the primed and unprimed coordinate systems, under which the two-dimensional space-time distance between two events,

is invariant. Invoking the constancy of the speed of light it is easy to show that the transformation must be of the form

where

Equation (2.2) defines the Lorentz transformation, after Hendrik Antoon Lorentz (1853-1928). Scalar products in this two-dimensional -space are invariants under Lorentz transformations.

Time Dilation.

The quantity in Equation (2.1) is called the proper time and the line element. Note that scalar multiplication in this manifold is here defined in such a way that the products of the spatial components obtain negative signs (sometimes the opposite convention is chosen). (The mathematical term for a many-dimensional space is a manifold.)

Since is an invariant, it has the same value in both frames:

While the observer at rest records consecutive ticks on his clock separated by a space-time interval , she receives clock ticks from the direction separated by the time interval and also by the space interval :

In other words, the two inertial coordinate systems are related by a Lorentz transformation

Obviously, the time interval is always longer than the interval , but only noticeably so when approaches . This is called the time dilation effect.

The time dilation effect has been well confirmed in particle experiments. Muons are heavy, unstable, electron-like particles with well-known lifetimes in the laboratory. However, when they strike Earth with relativistic velocities after having been produced in cosmic ray collisions in the upper atmosphere, they appear to have a longer lifetime by the factor .

Another example is furnished by particles of mass and charge circulating with velocity in a synchrotron of radius . In order to balance the centrifugal force the particles have to be subject to an inward-bending magnetic field density . The classical condition for this is

The velocity in the circular synchrotron as measured at rest in the laboratory frame is inversely proportional to , say the time of one revolution. But in the particle rest frame the time of one revolution is shortened to . When the particle attains relativistic velocities (by traversing accelerating potentials at regular positions in the ring), the magnetic field density felt by the particle has to be adjusted to match the velocity in the particle frame, thus

This equation has often been misunderstood to imply that the mass increases by the factor , whereas only time measurements are affected by .

Relativity and Gold.

Another example of relativistic effects on the orbits of circulating massive particles is furnished by electrons in Bohr orbits around a heavy nucleus. The effective Bohr radius of an electron is inversely proportional to its mass. Near the nucleus the electrons attain relativistic speeds, the time dilation will cause an apparent increase in the electron mass, more so for inner electrons with larger average speeds. For a 1s shell at the nonrelativistic limit, this average speed is proportional to atomic units. For instance, for the 1s electron in Hg is , implying a relativistic radial shrinkage of 23%. Because the higher s shells have to be orthogonal against the lower ones, they will suffer a similar contraction. Due to interacting relativistic and shell-structure effects, their contraction can be even larger; for gold, the 6s shell has larger percentage relativistic effects than the 1s shell. The nonrelativistic 5d and 6s orbital energies of gold are similar to the 4d and 5s orbital energies of silver, but the relativistic energies happen to be very different. This is the cause of the chemical difference between silver and gold and also the cause for the distinctive color of gold [2].

Light Cone.

The Lorentz transformations [Equations (2.1), (2.2)] can immediately be generalized to three spatial dimensions, where the square of the Pythagorean distance element

is invariant under rotations and translations in three-space. This is replaced by the four-dimensional space-time of Hermann Minkowski (1864-1909), defined by the temporal distance and the spatial coordinates , , . An invariant under Lorentz transformations between frames which are rotated or translated at a constant velocity with respect to each other is then the line element of the Minkowski metric

The trajectory of a body moving in space-time is called its world line. A body at a fixed location in space follows a world line parallel to the time axis and, of course, in the direction of increasing time. A body moving in space follows a world line making a slope with respect to the time axis. Since the speed of a body or a signal travelling from one event to another cannot exceed the speed of light, there is a maximum slope to such world lines. All world lines arriving where we are, here and now, obey this condition. Thus they form a cone in our past, and the envelope of the cone corresponds to signals travelling with the speed of light. This is called the light cone.

Two separate events in space-time can be causally connected provided their spatial separation and their temporal separation (in any frame) obey

Their world line is then inside the light cone. In Figure 2.1 we draw this four-dimensional cone in -space (another choice could have been to use the coordinates ). Thus if we locate our present event to the apex of the light cone at , it can be influenced by world lines from all events inside the past light cone for which , and it can influence all events inside the future light cone for which . Events inside the light cone are said to have timelike separation from the present event. Events outside the light cone are said to have spacelike separation from the present event: they cannot be causally connected to it. Thus the light cone encloses the present observable universe, which consists of all world lines that can in principle be observed. From now on we usually mean the present observable universe when we say simply 'the Universe'.

Figure 2.1 Light cone in -space. An event which is at the origin at the present time will follow some world line into the future, always remaining inside the future light cone. All points on the world line are at timelike locations with respect to the spatial origin at . World lines for light signals emitted from (received at) the origin at will propagate on the envelope of the future (past) light cone. No signals can be sent to or received from spacelike locations. The space in the past from which signals can be received at the present origin is restricted by the particle horizon at , the earliest time under consideration. The event horizon restricts the space which can at present be in causal relation to the...