. This is called the "red shift," because red light has longer wavelengths than blue light.
Evidence of an expanding universe is one of the most important astronomical observations of this century. Light received from a distant galaxy is "old" light, generated years ago at a time te when it was emitted. When this old light is compared to light generated at the time received tr, it is found that the characteristic colors, or spectral lines, do not have the same wavelengths.
The red shift was discovered by V. M. Slipher (1875 -- 1969), who measured shifts for more than 20 galaxies during 1912 -- 1925. The observed wavelengths from all distant galaxies are longer than the emitted wavelengths (assuming constant spectra), . This is called the "red shift," because red light has longer wavelengths than blue light.
The Doppler effect says that wavelengths from a transmitter that is moving away from us are longer.
The Doppler effect is familiar from the sound of a speeding truck approaching, passing and receding, eeeeeee - aaaaaaa. (Lower pitch means longer wavelength for the receding truck.) The Doppler shift is connected with the velocity of the emitter, ve, by the formula
Since all distant galaxies are red-shifted, they all are moving away from us. It seems they can all be moving away only if the universe is expanding. (If we were simply moving through the universe, we would be moving away from some that would appear red-shifted and toward others that would appear blue-shifted.)
In 1929 E. P. Hubble (1889 -- 1953) announced the empirical rule now called Hubble's law:
Older galaxies with large difference in time are more distant, because light travels at constant speed and we observe them all in our relatively short lifetime. Suppose we let t denote the time (in years, for example), with some specific time set as t=0. The time that the light is received tr is fixed at our lifetime, but the time that the light was emitted te varies from galaxy to galaxy (and te<tr.) The age of the light equals the difference tr-te and is proportional to D, because distance equals the rate times the time (in the appropriate units.)
Hubble's law means that the old expansion is faster than the new expansion. Explain why this is so. Notice that older light travels a longer distance D and produces a smaller ratio.
The math here is simple, but the scientific interpretation takes some thought.
It might help to rewrite Hubble's law by replacing D/c by the amount of time the light has traveled:
is closer to zero.
You want to show that if the light is very old (tr-te is large), then
is "very shifted" from
(so
is small compared to
), whereas if the light is not so old, then
is not so shifted from
.
The next problem uses the microscope approximation two ways and the continuity approximation of functions to show that Hubble's law is a "local" consequence of Doppler's law given above.
Problem 8.1
Show that Hubble's law follows from the "microscope" approximation by working through the following steps. We will use the function D(t) for the distance from earth to the galaxy emitting light as a function of time.
- 1. Show how to use the microscope approximation
for the function D(t) to give the two approximations
(Hints: First use x=te and, then use x=tr. Be careful with signs.)
- 2. All light travels at the speed c, so you can express
in terms of the distance D[te] and the speed of light, c. Do this and substitute the expression in the above approximations.
- 3. Divide both sides of the second approximation
by D[tr] to show that
- 4. Hubble's constant is defined as "the ratio of the speed at which a distant galaxy is receding from the earth to its distance from the earth." This is the ratio
and is currently estimated somewhere between 20 and 100 kilometers per second per million parsecs. A parsec is an astronomical unit equal to 3.26 light-years orkm. (The speed of light in vacuum is
km/sec, so light travels
km in one year.)
Consider the term
in the approximation of (3) for D[te]/D[tr]. If tr-te is small, we have
Which theorems in the main text justify this?- 5. Let
and substitute into the approximation of (3), to give the approximation
but not Hubble's law.- 6. Use the first approximation of (1) above,
, to show that
Do some additional substitution to see that
The velocity of the galaxy at the time the light was emitted is D'[te], so the expression above yields
where ve is the velocity of the emitter and c is the speed of light.- 7. Show that this gives Hubble's law in terms of the lambdas, H, D, and c by using the two expressions for D[te]/D[tr] and the Doppler law (at the beginning of the section) for the ratio of the lambdas.
Contemporary astronomers no longer believe that Hubble's constant is actually constant. Why would Hubble's law still appear to be true in (astronomically) short periods of time?
