How do you measure the distance to a galaxy? There are three basic problems:
We will first work out the solution to the first two problems.
The second issue above shows that one should really include time as a distance parameter: The distance to a galaxy is not defined unless the time of the measurement is specified. We therefore define distances between 'events' rather than 'things'. An event is defined as a point in space-time with coordinates (t,x,y,z) (e.g. a photon leaving a galaxy); the distance between events (e.g. a photon leaving a galaxy and arriving at its destination) is given by (Dt,Dx,Dy,Dz).
The numerical value assigned to the distance between events should be independent of the observer, although each individual coordinate difference (e.g. Dx) will clearly depend on velocity and position of the observer. Different observers will only agree on two things:
To define a 'distance' in a way that all observers agree on, we should start with these two points. The Minkowsky metric is the only way to do this. it calculations the separation between two events (t,x,y,z), but treats the time t and spatial x,y,z cooordinates separately:
| (67) |
ds is the newly defined Minkowsky distance. This metric The metric consists of a time part (c dt) and a space part: it calculates the distance a photon can travel during the time interval between two events, and subtracts the actual distance. If the photon filed to make it to the second event, the answer is negative, otherwise positive. The value of dt, dx, etc., depends on the position and velocity of the observer. For instance, assume that two photons reach you from opposite directions at the same time. To you, the events they came from appeared to happen simultaneously, that is, d t = 0. But if you stood some distance in the direction of one of th photons, that one would have reached you earlier and the other one later. Thus, here d t ¹ 0.
Any two events connected by a photon path have zero distance: the time parts and the spatial part cancel. In other words, along the trajectory or geodesic of a photon, ds = 0. Every observer will agree on this! Every observer will also agree on whether eq. (67) is positive (eventas are causally connected) or negative (unconnected).
Figure 1: The distance between two events, A (at the origin) and B. The vertical axis indicates the direction of time, and the horizontal axis represents space (squashed into a single dimension rather than the usual three). The dashed line indicates a possible path between A and B, never exceeding the speed of light. The path of a photon is also indicated: traveling at a single speed in in a single direction. The lightcone is defined as the region inside the possible photon paths. Within this cone events are causally connected to the origin, outside they are not. The Minkowsky distance d s2 between A and B is indicated.
Figure (1) illustrates this further. It shows the location of one event, A, with coordinates (t,x) (the other spatial coordinates are a little difficult to draw!). A photon traveling from A to event B reached the correct x-position too early. This means that the two events are causally connected: information has been able to reach B from A. Two possible photon paths are indicated: they travel in opposite directions. The surface connecting all possible photon paths is called the light cone: all events within this cone are causally connected to A. Distances inside the light cone are called time like: you can travel to the place of the event and wait a certain amount of time for something to happen. Distances outside the light cone are called space like: the best you can do is travel until the time of the event, but finding yourself still some distance away (a bit like the start of a lecture). Note that these distances are taken from the origin: if you want to know the distance from another event C to B, you should draw a light cone with C at its origin.
The Minkowsky distance d s2 is in essence the spatial separation between B and the nearest photon path. This is indicated in Figure 1. A time-like separation corresponds to ds2 > 0; separations with ds2 < 0 are space-like.
We can now define the proper time t as
| (68) |
which gives a definition of time (or rather, time difference) that all observers can agree on. For an observer at rest with respect to a clock, d x = 0 and therefore dt = dt. For any event on the light cone, d s = 0 and therefore dt = 0.
In polar coordinates, eq. (67) is written as:
| (69) |
Polar coordinates are normally better suited for describing spherically symmetric systems, such as the Universe.
The Minkowsky metric can be used to derive the Lorentz time dilation. Take two observers, 1 and 2: 1 is at rest with respect to a clock, and 2 is moving with uniform velocity v. The Minkowsky distance between two successive ticks on the clock (say one second, if the clock happens to be Swiss-made), as measured by the respective observers is:
| (70) |
However, we know that d s must be the same for any observer. So we find:
| (71) |
The time between ticks seems longer for the moving observer, or, in other words, a moving clock is perceived to run slower. If a photon is emitted such that its frequency corresponds to the number of clock ticks per second, c d t is equal to its wavelength. It follows that
| (72) |
This effect is unlike the usual Doppler shift, because it is independent of the direction of the motion. In the Hubble flow the velocity is directed away from the observer, and the Doppler shift still has to be added:
| (73) |
We have now derived the relativistic equation which we have already used.
Excercise: Derive the spatial Lorentz contraction from the
Minkowsky metric.
The Minkowsky metric solves our first two problems, on how to define distances in such a way that every observer will agree on them. However, it is valid for flat space only. If the curvature of space is taken into account, the metric becomes the Robertson-Walker metric:
| (74) |
where polar coordinates are used. As before d s is an invariant: every observer will find the same value no matter at what velocity they move while carrying out their measurement. We have introduced the parameter
| (75) |
where r is the distance to a galaxy, and S is the scale factor, or 'radius' of the Universe. s defined in this way is a co-moving coordinate: as the galaxy moves away from us because of the expansion of the Universe, s remains the same. By definition, we assign ourselves to be s = 0, the centre of the coordinate system. The polar angles q and f also are measured with respect to us. As the expansion of the Universe is purely radial, these angles will not change for any particular galaxy.
Figure 2: Measuring the distance from us to a galaxy in a curved space. The Universe is approximated by the surface of a sphere: note that we are not at the centre. Distances are measured from us. The square shows how we determine a distance if we assume that the Universe is flat: the circle (seen in projection as an ellipse) corresponds to a distance s. The dashed line indicates the seemingly straight path towards the galaxy, with length l: the curvature makes l much longer.
A galaxy moving with the Hubble flow will therefore remain at its co-moving coordinates (s, q, f). The geometry of the problem is schematically indicated in Fig. 2. The dashed line shows the path along which the distance is measured. The figure shows that s runs from zero to 2 p. The polar angles f and q (not indicated in the figure) are measured from 'us', and not from the origin of the circle.
Along the line of sight towards a galaxy, the polar angles do not change (the obvious bending of the line of sight (Fig. 2 is at right angle to all the measured angles!). Therefore we can take d q = d f = 0. With this simplification, the Robertson-Walker metric (Eq. (74) becomes:
| (76) |
(i) space We first concentrate on the spatial part, d l. To obtain the distance l, we integrate d l along the 'straight' line.
| (77) |
with S the scale factor at the particular instant. The integral has three solutions, one each for k=0,1,-1. Inverting the solutions from l = F(s) to s = F(l), we find:
| (78) |
The hyperbolic sine, sinh, is defined as:
|
For k=1, s < l. For l/S = p, s = 0: this is characteristic of a closed surface. For k=-1, s > l. In the case of k=0,-1, there is no limit to the maximum value of s. For k=1, s has a maximum value of s = 1. The last case is depicted in Fig. 2: it corresponds to measuring s as the distance l projected on the flat square.
Eq. (78) shows that s in a flat Universe grows linearly with l, which is not unexpected. In a Universe with positive curvature s grows more slowly (and can even decline) while with negative curvature it grows faster than for the flat Universe.
One can now also calculate the volume of space at a distance s. Draw a sphere around us with radius s and thickness d s. The total volume within this surface is
| (80) |
Comparing this to the thickness d l, which is given by:
| (81) |
we find a relation between d V and d l:
| (82) |
The term on the right is the volume within this thin surface for normal, 'flat' geometry. The term on the left gives the actual volume which may be larger or smaller. Now consider the number of galaxies within this volume. If galaxies do not get lost or new ones are created, we can set the number of galaxies per co-moving volume as constant. (The number of galaxies within a co-moving box will not change because the box is expanding at the same rate as the Universe.) Eq. (82) tells you that in a closed Universe, you will see fewer galaxies at large distances than you would in a flat Universe.
Excercise: The curvature of the Universe becomes important for k s2 » 1. Show that this corresponds to a distance
|
where the vertical bars indicate the absolute value.
(ii) time So far we have discussed the spatial part of the Robertson-Walker metric. But this is a bit incomplete: the more distance parts of the Universe we see as they were in the past, because information about these can only reach us at the speed of light.
Figure 3: Measuring the distance from us to a galaxy in an expanding space. The dashed line indicates a photon path, traveling from the galaxy to us.
Figure 3 shows the situation where the Universe expands. The present distance l is indicated: it is the same as in the previous section. The past Universe is indicated by the smaller circles. The photon path travels both in space (along the circles) and in time (the radial coordinate). Along this photon path, d s = 0.
| (83) |
The scale factor S is time-dependent. Bring the time-dependent quantities to the left, and integrate from the time of photon emission te to the present time t0:
| (84) |
where the right-hand side is the same as the integral for l/S0 above. We find
| (85) |
Figure 4: Photon paths in three Universes: one with rapid deceleration (positive curvature), one with slower deceleration (flat) and one with no deceleration (negative curvature). Only in the first Universe does the photon path curve in such a way to allow us to see ourselves in the distant past.
Figure 4 compares the photon paths in three different Universes. It shows how in a closed Universe, the photon path curves in such a way that our own galaxy may even be visible in the past. In the other two cases, the photon could not have originated from our own galaxy. The term S0/S(t) in Eq. (85) is, in the figure, the ratio of the radius of the outer circle to the radius of the circle at time t .
In Eq. (84), the right-hand side is independent of time. Now consider a photon emitted at time te + Dte and arriving here at time t0 + Dt0.
| (86) |
| (87) |
Using the redshift relation, we find
| (88) |
So not only do photons arrive redshifted, but they also arrive with a further time delay. If two photons are emitted 10 days apart, they will be received here redshifted, and 10 ×(1+z) days apart.
(iii) horizon The particle horizon is defined as the largest comoving distance s from which light can have reached us. The particle horizon is found by integrating along the path of a photon, for which ds=0, with the time of emission of the photon at the origin of the Universe: te=0.
| (89) |
The second integral can be evaluated if we know how S depends on t. The easiest solution is if we take a stationary Universe with a finite age (and forget the obvious contradiction), and take k=0
| (90) |
In this case, the particle horizon is the same as the Hubble radius. In general, this will not be the case!
The event horizon is defined as the largest comoving distance s from which light will ever reach us. It is of importance for closed Universes (with finite life time), or when describing black holes.