# S-4A-2     The Frequency Shift and the Expanding Universe

 Index The Sun S-1. Sunlight & Earth S-1A. Weather S-1B. Global Climate S-2.Solar Layers S-3.The Magnetic Sun S-3A. Interplanetary         Magnetic Fields S-4. Colors of Sunlight Optional: Doppler Effect S-4A-1 Speed of Light S-4A-2. Frequency Shift S-4A-3 Rotating Galaxies             and Dark Matter ------------------------   S-5.Waves & Photons Optional: Quantum Physics Q1.Quantum Physics Q2. Atoms Q3. Energy Levels Q4. Radiation from         Hot Objects Christian Doppler was born in 1803 in Salzburg, Austria, where his father was a stonemason. In school he excelled in science and therefore, even before finishing his regular school education, he moved to Vienna to study at the polytechnic institute, a university. He later became a teacher in Prague (now capital of the Czech Republic, then part of the Austrian Empire) and then a professor of physics, first in Prague, then in Vienna. He died of a lung disease at age 49.    The way he deduced what is now named the Doppler effect used reasoning similar to the one of Ole Roemer concerning eclipses of Jupiter's moon Io. Consider a single tone of sound spreading in air, a periodic fluctuation of pressure advancing in space. The duration T between two consecutive pressure peaks is the period of the wave: the shorter T is, the higher the pitch we hear. If the wave spreads with velocity v, the number of peaks arriving each second--the frequency f (also denoted by the Greek letter ν or "nu")--is   f = 1/T oscillations-per-second, and the larger f, the higher the pitch.    If the wave advances towards the listener with velocity v, the distance it covers in one second is also v, and those f oscillations may be imagined as spread out over a distance v as a "wave train" of one second--its front emitted at the start of the second, its rear emitted at the end.    Next imagine that whatever source produces the sound still oscillates f times per second, but now is approaching the listener with velocity u. Now the "wave train" is shorter. Its rear is only a distance (v–u) behind its front, because by the time it was emitted, the source has advanced a distance u.    There are still f oscillations in the train, but because its length is shorter, the time between each two peaks is shortened to T '= (v–u)/f The listener, hearing waves with peaks spaced only a time T ' apart, will sense a wave with peaks more closely spaced, as if its frequency increased to f ' = 1/T '= f/(v–u)

If the source is instead moving away with velocity u, the same calculation shows the observed frequency is

f '= f/(v+u)

and is therefore reduced. That, in essence, is the Doppler effect.

Doppler's prediction was tested in 1845 by the Dutch physicist Buys Ballot, who later also proposed Buys Ballot's Law by which hurricanes and other low-pressure weathr systems spin counterclockwise north of the equator, clockwise south of it. That was the time of the first railroads, and Buys Ballot arranged for a horn player to play a steady note as he passed a trained musician on the ground, who estimated the drop of the tone heard by him. The speed of the train was also measured, so that the predicted change could be tested. The whistles of today's fast trains, of course, produce a very notable drop in pitch as they pass the listener.
And at home...   A sturdy children's swing stands in our back yard, about a mile from a busy highway. Swinging on it idly one morning, I could hear a rumble rising and falling with each swing. Of course, the Doppler effect. The rumble came from highway traffic, much of whose noise is at frequencies too low for the human ear. Swinging towards the highway raised the effective frequency, making more of the noise audible. Swinging away from it lowered the frequency and reduced audibility.

## Light

Doppler also predicted a corresponding effect in light. The nature of light was not yet understood, but observations with narrow slits and thin layers, and later with diffraction gratings, suggested that it too spread as a wave. Doppler claimed that the above argument held for it, too. The wavelength of light is associated with its spectral color: red has the longest waves and violet the shortest ones. Of course, waves of the same family also exist beyond those limits, it is just that the human eye cannot detect them.

The nature of such "electromagnetic waves" is discussed elsewhere. Here let it just be said that Doppler's guess was accurate, and his effect is used in many ways--for instance, in radar "guns" used by the police to monitor the speed of cars.

Detecting the Doppler effect can be done by monitoring either the wavelength λ or the frequency ν, depending on the detecting instrument, but of course the same effect is measured. With light, which propagates in empty space with velocity c the two are related by

ν  =  c / λ

A previous relation gives the frequency actually observed as

ν'  =  ν c / (c–v)  =  [c / (c–v)] (1/λ)

or since ν' = 1/λ' with
λ' the sensed wavelength:
λ'  =  [(c–v) / c] λ

Since the velocity v of most objects is much smaller than c, the ratio is usually tiny, so that detecting the effect may require great sensitivity to shifts of λ or ν. Instruments able to detect such tiny shifts in visible light were developed in the late 1800s and proved especially useful for observing frequency shifts of double stars ("binaries"). These rotate around a common center of gravity, so that when one component approaches, the other recedes. The method actually works best when the two stars are very close: the telescope may then fail to tell them apart, but their Doppler shift is large, because orbital velocity is highest at close distances.

A related effect is Doppler broadening of light. Characteristic frequencies or "spectral lines" emitted by atoms usually have very sharply defined wavelengths--but if the light originates in a hot gas, the wavelength range is broadened, because the emitting atoms are in constant motion. Some emit their light while moving towards the observer, causing it to be Doppler-shifted to shorter wavelengths; others emit it while moving away, causing an opposite effect. The net result is that the wavelength (=color) is less sharply defined, and the loss of definition can tell about the temperature in the region of emission (though other effects also cause broadening).

## The Expanding Universe

However, the most famous Doppler shift of light is probably the one observed in distant galaxies. Until 1925 it was widely held that the various diffuse "nebulae" seen through the telescope were clouds of dust and gas of one sort or another, located in the Milky Way Galaxy. In that year, however, Edwin Hubble announced a study of variable stars in such nebulae, which suggested that while some were "local," many were distant "island universes" similar to our own galaxy but at vast distances.

This led to another fundamental discovery. Astronomers had previously noticed that such nebulae emitted spectral lines of unidentified wavelength. It turned out that these actually were common emission wavelengths, severely Doppler-shifted because their sources were moving away from us, at speeds which could be an appreciable fraction of c. In fact, this "red shift" seemed a universal feature, present in all directions and increasing with apparent distance. It seemed to suggest that all distant galaxies are moving away from us--the more distant they were, the faster their motion.

Unless Earth is the center of the universe--which astronomers ceased to believe long ago--this meant that galaxies everywhere were receding from each other, and that an observer anywhere would observe the same effect, a velocity of recession growing with distance. Such behavior is impossible in ordinary 3-dimensional space and suggested instead that our 3-dimensional space is curved, as if it were embedded in 4 dimensions. In a similar way points on the 2-dimensional surface of an expanding balloon, being inflated in our 3-dimensional space, are constantly moving away from each other.

If space is expanding, we can look back and inquire (in various ways) when did that expansion start--when was the "size of the balloon" zero? Observations suggest it happened about 13.7 billion years ago. That was the "Big Bang" (Fred Hoyle's term). We cannot see light which is 13.7 billion years old, which might tell us what that beginning looked like, because the early universe was dense and not transparent (though theory can deduce a lot about it). The best we can do is see light created about 300,000 years afterwards, when it started getting transparent. That is the "primordial fireball" radiation, which by now (because the distance is so great, creating a huge velocity of recession) has shifted into the microwave range. So far, all observations have agreed with the "Big Bng" scenario.

Astronomers are obviously interested in details of the expansion. One of the tools has been the Hubble Space Telescope. Early in its career, it created the "deep field" image, a time exposure of 10 days which revealed much fainter (and more distant) objects than any before. Most were galaxies. A later "ultra deep field" image, extended that to 3 months and provided even more detail.

What about the energy of the universe? Starting out as a very hot "primordial fireball," it gradually expanded to cover an increasingly large volume, and as it did so, its heat went into overcoming the gravitational attraction of its parts. Was this initial heat energy sufficient to keep it expanding indefinitely? If not, some speculated, a time may come when the expansion would stop, after which gravity will gain the upper hand, pulling all mass (and space) together again. The universe would then end in a "Big Crunch," a mirror image of the Big Bang, a time when all mass will again collapse into a very small volume.

No one can predict the future, but one can look to the past for guidance: has the expansion rate slowed down, or has it remained steady? Astronomers can look at galaxies with Doppler shifts which give their distances as 2,4,6 or 8 billion light years--assuming the "expansion rate ("the "Hubble constant") was always the same is it is now. But they can also check the real distance, by looking at "Type 1 Supernovas" in those galaxies, which always produce nearly the same amount of light (after corrections are applied). By seeing how bright these supernovas appear to us now, astronomers can derive the actual distance.

The result found was unexpected: not only isn't the expansion rate slowing down, it is speeding up. In other words, the energy of the universe, far from getting used up to overcome gravity, is actually increasing. Astronomers refer to this addition as "dark energy" because it is not associated with any visible feature of the universe. They are still sorting out what it means.

Be warned that understanding this expansion, like the Big Bang itself, involves general relativity (the name given to Einstein's theory of gravitation), and the apparent "near-flatness of the universe." For more, read "The Origins of the Future" by John Gribbin, Yale University Press, 2006.

Questions from Users:
***     The Big Bang
***     Do absorption lines have a Doppler shift?
(The item below was written before this section
and may be read as a brief summary)
***       Confusion about the Big Bang
***     The Big Bang
***     Doppler shift from the Big Bang
***     The Hubble Constant
***     Dependence of cosmic redshifts on direction

Next Stop,
(S-4A-3) Rotating Galaxies and Dark Matter
or else
(S-5) Waves and Photons

Author and Curator:   Dr. David P. Stern
Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .

Last updated: 9 December 2006