 MORE PRECISELY 25-1 Relativistic Redshifts and Look-back Time |
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| When discussing very distant objects such as quasars, astronomers usually talk about their redshifts rather than their distances. Indeed, it is very common for researchers to speak of an event occurring "at" a certain redshift—meaning that the light received today from that event is redshifted by the specified amount. Of course, because of Hubble's law, redshift and distance are equivalent to each other. However, redshift is the "preferred" quantity, because it is a directly observable property of an object, whereas distance is derived from redshift using Hubble's constant, whose value is not accurately known. (In the next chapter we will see another reason why astronomers favor the use of redshift in studies of the cosmos.)
The redshift of a beam of light is, by definition, the fractional increase in its wavelength resulting from the recessional motion of the source.
Let's illustrate this with two examples, rounding the speed of light cto 300,000 km/s. A galaxy at a distance of 100 Mpc has a recessional velocity (by Hubble's law) of 65 km/s/Mpc Unfortunately, while the foregoing equation is quite correct for low velocities, it does not take into account the effects of relativity. As we saw in Chapter 22, the rules of everyday physics have to be modified when velocities begin to approach the speed of light, and the formula for the Doppler shift is no exception. (More Precisely 22-1) In particular, although our formula is valid for velocities much less than the speed of light, when v = c, the redshift is not 1, as the equation suggests, but is in fact infinite. In other words, radiation received from an object moving away from us at nearly the speed of light would be redshifted to almost infinite wavelength. Thus do not be alarmed to find that many quasars have redshifts greater than 1. This does not mean that they are receding faster than light! It simply means that their recessional velocities are relativistic—comparable to the speed of light—and the preceding simple formula is not applicable. Table 25.1 presents a conversion chart relating redshift, recession speed, and present distance. The column headed "v/c" gives recession velocities based on the Doppler effect, taking relativity properly into account (but see Section 26.2 for a more correct interpretation of the redshift). All values are based on reasonable assumptions and are usable even for v Because the universe is expanding, the "distance" to a galaxy is not very well defined—do we mean the distance when the galaxy emitted the light we see today, or the present distance (as listed in the table), even though we do not see the galaxy as it is today, or is some other measure more appropriate? Largely because of this ambiguity, astronomers prefer to work in terms of a quantity known as the look-back time (shown in the last column of Table 25.1), which is simply how long ago an object emitted the radiation we see today. Researchers talk frequently about redshifts and sometimes about look-back times, but hardly ever of the distances to high-redshift objects. For nearby sources the look-back time is numerically equal to the distance in light years—the light we receive tonight from a galaxy at a distance of 100 million light years was emitted 100 million years ago. However, for more distant objects, the look-back time and the |
present distance in light years differ because of the expansion of the universe, and the divergence increases dramatically with increasing redshift. For example, a galaxy now located 15 billion light years from Earth was much closer to us when it emitted the light we now see. Consequently, its light has taken considerably less than 15 billion years—in fact, only about 8.8 billion years—to reach us.
As a simple analogy, imagine an ant crawling across the surface of an expanding balloon at a constant speed of 1 cm/s relative to the balloon's surface. After 10 seconds the ant may think it has traveled a distance of 10 cm, but an outside observer with a tape measure will find that it is actually more than 10 cm from its starting point (measured along the surface of the balloon), simply because of the balloon's expansion. Furthermore, the difference between the actual distance and 10 cm depends on the details of the balloon's expansion during the journey—the more rapid the expansion, the greater the disparity. In exactly the same way the present distance to a galaxy with a given redshift depends in detail on how the universe expanded in the past. We will see in Chapter 26 that the details of the past expansion are not well known, so the distance is not well determined. The distances given in the accompanying table are used consistently throughout this book, but realize that they are subject to considerable uncertainty. Finally, notice that, according to the table, the recessional velocity equals the speed of light—and the redshift becomes infinite—for objects that emitted their radiation about 10 billion years ago. We will examine the reasons for and implications of this fact in the next chapter.
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100 Mpc = 6500 km/s. Its redshift therefore is 6500 km/s ÷ 300,000 km/s = 0.022. Conversely, an object with a redshift of 0.05 has a recessional velocity of 0.05 
c. We take Hubble's constant to be 65 km/s/Mpc. The conversions in the table are used consistently throughout this text. Notice that the entries in the table agree very well with our earlier simple formulas for small redshifts (less than a few percent, corresponding to objects lying within 100 or 200 Mpc of Earth), they differ greatly for larger distances. 