## A53

Inverting now this equation by writing it as v H0r, we obtain in the lowest approximation where h r . To the same approximation one also obtains H0 h l - (1 - ft) z2 6 h l - (1 - ft) r2 6c2r2 , (A55) where 2 is the redshift parameter. As is seen, and it is confirmed by experiments, Ho depends on the distance it is being measured it has physical meaning only at the zero-distance limit, namely when measured locally, in which case it becomes h 1 r. It follows that the measured value of Hq depends...

## The galaxy cone

The invariant equation (2.1), describing the distribution of galaxies in the Universe at any cosmic time, has a very simple geometrical interpretation. It enables one to present the locations of galaxies as a cone in the dual space of distance and velocity. One then has a galaxy cone, similar to the familiar light cone in special relativity (see Figure 4.1 of Chapter 4). The symmetry axis of the cone coincides with the x axis which extends from tc to +TC. Figure 2.1 The galaxy cone in...

## The Lorentz transformation

In the last section we presented the (nonrelativistic) Galilean transformation of spatial coordinates. We are now in a position to generalize this transformation to the relativistic case, where both the time and the spatial coordinates are involved. 4.3.1 Measuring rods and clocks We notice that the assumptions of the existence of measuring rods and clocks are not independent of each other. This is so since a light signal, which is reflected back and forth, may provide an ideal clock,...

## B13 Fivedimensional manifold of space time and velocity

If we add the time to the cosmological flat spacevelocity line element, we obtain ds2 c2dt2 - (dx2 + dy2 + dz2) + r2dv2. (B. 13) Accordingly, we have a five-dimensional manifold of time, space and velocity. The above line element provides a group of transformations 0(2,3). At v const it yields the Minkowskian line element (B.ll) at f const it gives the cosmological line element (B.12) and at a fixed space point, dx dy dz 0, it leads to a new two-dimensional line element The groups associated...

## A6 Physical meaning

To see the physical meaning of these solutions, however, one does not need the exact solutions. Rather, it is enough to write down the solutions in the lowest approximation in r_1. One obtains, by differentiating Eq. (A.33) with respect to v, for > 1, where a2 (Q l) 2 and A and B are constants. The latter can be determined by the initial condition r (0) 0 B and dr (0) dv t Aa c, thus This is obviously a closed Universe, and presents a decelerating expansion. whose solution, using the same...

## Bll Cosmic coordinate systems The Hubble transformation

We will use cosmic coordinate systems that fill up spacetime. Given one system x, there is another one x' that differs from the original one by a Hubble transformation x' x + t v, ti constant, (B.l) where v is a velocity parameter, and y and z are kept unchanged. A third system will be given by another Hubble transformation, x x' + t2v x + (ii + t2)v. (B. 2) The cosmic coordinate systems are similar to the inertial coordinate systems, but now the velocity parameter takes over the time parameter...

## Relative cosmic time

In cosmology one is not interested in comparing quantities at two reference frames moving with a constant velocity with respect to each other. Rather, one is interested in comparing quantities at two different cosmic times. For example, one often asks what was the density of matter or the temperature of the Universe at an earlier cosmic time t as compared to the values of these quantities at our present time now (t 0). The backward time t is the relative cosmic time with respect to our present...

## Four dimensions in classical mechanics

It is well known that classical mechanics is based on a four-dimensional manifold of three-dimensional space and the time. However, there is an essential difference between the concepts of space and time in classical mechanics and the four-dimensional spacetime of special relativity. In classical mechanics the three-dimensional subspace with constant t is absolute and is independent of the inertial coordinate system. This means one has a separate three-dimensional space, along with a...

## B41 The geodesic equation

As usual the equations of motion are obtained in general relativity theory from the covariant conservation law of the energy-momentum tensor (which is a consequence of the restricted Bianchi identities), and the result, as is well known, is the geodesic equation that describes the motion of a spherically symmetric test particle. In our five-dimensional cosmological theory we have five equations of motion. They are given by We now change the independent parameter s into an arbitrary new...

## Info

In general a set of four quantities Va, which transform like the components of the coordinates xa under the Lorentz transformation, is called a four-vector. One can extend the definition of vectors to tensors of any order in the Minkowskian spacetime. The event described by the position four-vector xa is just an example of such quantities. A scalar is then a tensor of order 0, whereas a vector is a tensor of order 1. Under a Lorentz transformation a scalar is left invariant, a four-vector Va...

## Inadequacy of the classical transformation

We first do it classically, and for simplicity it is assumed that the motion is one-dimensional. Denoting the coordinates and velocities in the systems K and K' by x, v and x', v', respectively, then x x tv, v' v, y' y, z' z, where v was assumed to be constant. The x's and v's in these equations represent the coordinates and velocities not for just one particle but for as many as one wishes, with t the same for all of them. The above transformation does not satisfy the equation of expansion of...

## Cosmology and special relativity

It thus seems that not only light propagates in a constant velocity in vacuum but the Universe also expands with a constant fashion whose proportionality constant is the Hubble constant. But there is an essential difference between them light propagation is expressed in terms of space and time, whereas the Universe expansion is by space and velocity at a fixed and every specific cosmic time. Thus we are dealing with what might be called the dual space in cosmology as compared to the ordinary...

## Problems

5.1 Given two inertial systems K and Kwhere the latter is moving relative to K with the speed v. As judged by an observer in K, the clocks in the system K' go slower than those in K. On the other hand, as viewed by an observer in K', the clocks in the system K go more slowly than those in K'. Show that there is really no contradiction between the above two observations. Solution The solution of the problem is left for the reader. 5.2 Find the timelike component of the acceleration four-vector...

## Proper time

When a particle moves in the ordinary three-dimensional space, it describes a path in the Minkowskian spacetime, called a world line. The four-vector dxa represents the infinitesimal change in the position four-vector xa, and it is a tangent vector to the world line. The square of dxa, namely r)apdxadxP, is a scalar, and it is denoted by ds2. Accordingly we have ds2 - r apdxadx13 c2dt2 - dx2 - dy2 - dz2. The physical meaning of ds can best be understood if we evaluate it in an inertial...