A course correction – and let’s get started!

I have received some feedback from people that felt the posts were too technical. I am going to address this by constructing a simpler thread of posts on one topic that will start simpler and stay conceptual rather than  become technical.

I want to discuss the current state of Cosmology, given that it is possibly the field in the most flux these days. And the basic concept to understand in Cosmology is that of Cosmological Red Shift. So here goes…

The Cosmological Red Shift means this – when we look at far away galaxies, the light they emit is redder than we would expect. When you look at light of various colors, the redder the light, the longer its wavelength. Why would that be? Why would we perceive the light emitted by a galaxy to be redder than it should be?

To understand Cosmological Red Shift, you need to understand two things – the Doppler Shift and Time Dilation.

Let’s start with Doppler Shift.

If you listen to an ambulance approaching you on a road, then (hopefully, if it hasn’t come for you) speeding away from you on the road, you will hear the pitch, i.e., frequency, of the siren go up, then go down. Listen to this here .

Why does this happen?

Sound is a pressure wave. When the instrument producing a sound vibrates at a certain rate (frequency), it pushes and pulls on the air surrounding it. Those pushes and pulls are felt far away, because the fluctuations in density of the air propagate (see the video). The air isn’t actually going anywhere as a whole – this is why when you have waves in the ocean, the ocean isn’t actually sending all the water towards you, its just the disturbance coming towards you. So these pressure variations hit your ears and that’s how you hear something – the eardrum vibrates the little bones in the ear, which set up little waves in the cochlear fluid that then create electrical signals that go to your auditory cortex and voila, you hear!

Now, waves are characterized by wavelength (\lambda), frequency (\nu) and their speed (\bf{v}). There’s a relation between these three quantities

\bf{v} = \lambda \nu

Sal Khan has a nice video describing this formula in some detail. Let’s try and understand this – wavelength (\lambda) is the distance between the successive positive crests of the wave, frequency (\nu) is the number of crests shooting out of the emitter per second, then (\lambda \nu) is the length of wave coming out of the emitter per second as measured by the emitter. That’s how far the first crest traveled in one second, i.e., the speed of the wave.

Now what happens if the emitter is moving away from you – think of the pressure waves like compressions of a spring, as in the video link above. If the emitter is moving away, that’s like the spring being extended while it is vibrating – ergo, the wavelength is increased in proportion to how fast the emitter is running away from you (call the emitter’s speed v_{emitter}). The formula is

\lambda_{observed} - \lambda_{emitted} = \frac {v_{emitter}} {\bf{v}} \lambda_{emitted}

Aha – this makes sense, so the sound that I hear when an ambulance is driving away from me has a longer wavelength – so it has a lower frequency – it has a lower pitch. If the ambulance is driving towards me, so v_{emitter} is negative in the above formula, then we hear shorter wavelength sound, which has a higher frequency, i.e., a higher pitch.

As an example, if the emitter flies away as fast as the speed of sound in the air, then the observed wavelength should be \bf {double} the emitted wavelength. In the simple picture of the emitter shooting out wave crests at a rate \nu per second, the emitter shoots out one crest, then shoots out another crest after a time interval \frac {1}{\nu}, by which time it has moved a distance \frac {\bf {v}} {\nu} which is indeed one wavelength! So the distance between the crests in the eyes of the observer is twice the emitted wavelength.

Whew!  So that is the Doppler effect. If something is moving away from me, I will hear the sound it emits will seem to be of lower pitch. Since light is also a wave, if a galaxy were moving away from me, I should expect to see the light looks like that of lower frequency – i.e., it looks redder.

When we specialize this to the case of light, so we replace {\bf{v}} by c, the speed of light. There is an additional effect that we need to think of, for light.

Onwards – let’s think about  \rightarrow Time Dilation.

This needs some knowledge of Einstein’s ideas about Special Relativity. I am going to give you a lightning introduction, but not much detail. I might write a post with some details later, but there are excellent popular books on the subject. Several years before Einstein, the Scottish physicist James Maxwell discovered the equations of electromagnetism. People had discovered universal laws of nature before – for instance Issac Newton discovered the Law of Gravitation, but Maxwell’s equations had a puzzling feature. They included a constant which wasn’t a mass, length or time, but was a speed! Think of that. If there was a law of nature that included the speed of your favorite runner (say how quickly that dude in “Temple Run” runs from the apes), how strange that would be. How fast does someone run, you ask. Well, it depends on how fast the observer is going! You must have seen this on the highway.  When your car has a flat and you are standing, ruing your luck, on the side of the highway, you think the cars are zipping past you at 50, 60,…80 miles per hour. When you are in one of the cars, traveling at, say 50 miles per hour, the other cars are moving \bf {relative \hspace {2 mm} to \hspace{2 mm} you} at 0, 10,…30 miles per hour only. That’s natural. How can a physical law, a universal law of nature, depend on a speed! The world is bizarre indeed!

Einstein discovered exactly how bizarre. It turns out if you want the idea of a universal constant that is a speed (for light) to make sense, ALL observers need to agree on the actual speed of light, regardless of how fast they are traveling, along or against or perpendicular to the ray of light. For that to happen their clocks and rulers need to get screwed up, in just the right way to allow for this. Suppose you have two observers that are moving relative to each other, at a constant speed in some direction.  Einstein derived the exact equations that relate the coordinates (x,y,z,t) that the first observer assigns to a moving object to the coordinates (x',y',z',t') that the other observer ascribes to the same object. It’s high school algebra, as it turns out, but the relation implies, among other things that a moving clock ticks slower than a stationary clock, {\bf when \hspace{2 mm} the \hspace{2 mm} clocks \hspace{2 mm} are \hspace{2 mm} compared \hspace{2 mm} at \hspace{2 mm} the \hspace{2 mm} same \hspace{2 mm} point \hspace{2 mm} in \hspace{2 mm} space}.That, by the way is how the twin paradox sorts itself out – the twins have to meet at some point in order to compare their ages, so one has to turn his or her rocket around.

When you use the formulas of relativity, if the emitter is flying away at speed v_{emitter} relative to the observer, the emitter’s clock will seem to run slower than the observer’s clock (from the observer’s point of view). Since the frequency of the emitted wave essentially is a “clock” for both, we will obtain (and this needs a little algebra and some persistence!)

\nu_{observed} = \nu_{emitted} \sqrt{1 - (\frac{v_{emitter}}{c})^2}

Using our previous relation connecting frequency and wavelength, this means the wavelengths are related as below

\lambda_{observed} = \lambda_{emitted} \frac{1}{\sqrt{1 - (\frac{v_{emitter}}{c})^2}}

so when we combine the two effects – Doppler and Relativity, which operate on the same emitted light, but successively, we multiply the two effects, we get the final observed wavelength

\lambda_{observed} = \lambda_{emitted} \sqrt{\frac{1 + \frac{v_{emitter}}{c}}{1 - \frac{v_{emitter}}{c}}}

We see that if something is moving away from us, i.e., v_{emitter} is positive, the observed wavelength is longer than the emitted wavelength, i.e., it is red-shifted. If the moving object emits light of a certain color, the stationary observer of this light sees it to be redder than the emitted color. So here’s the upshot – if you observe light from some object that is redder than you’d expect from that object, one strong possibility is that it is receding away from you. That’s how modern cosmology got started!

A note about terminology; astronomers define a quantity called “red-shift” – denoted by the letter z to define this wavelength difference. It is defined as the relative change in wavelength

z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}}

z is a “dimensionless” number – it is a ratio of two lengths. z=0 corresponds to you and me, things that are in the vicinity of each other. The moon isn’t receding away from us (if it is it is immeasurable), neither is our sun, so they all have a z=0. In fact, the entire Milky Way galaxy, our home, is at red-shift z = 0. We really have to leave the vicinity of our local group of large galaxies (that includes principally Andromeda and the Large and Small Magellanic clouds) to start seeing red-shifts exceeding 0. Conversely, the largest red-shifts we have seen are for distant quasars and intense galaxies – red-shift of about 11. Think of what that means – the 21 cm emission wavelength of neutral  hydrogen would be shifted by 232 cm – almost 7 feet! For people constructing prisms and other apparata for telescopes, this is a ridiculously (physically) large apparatus you need. More on this later!

 

Leave a Reply