# Homework Help: S. Relativity question

1. Jul 12, 2010

### lol_nl

1. The problem statement, all variables and given/known data
(my own translation)

Light with a wavelength of 360 nm emitted from a quasar is observed as light with a wavelength of 120 nm on earth.
a) Calculate the speed of the quasar relative to the earth.

The universe is approx. 16 * 10^9 yrs old in the earth system. Assume that all the material exploded into the universe with the Big Bang. Throughout the years the separate pieces of material from the quasar merged to formed the quasar. Assume that all these separate pieces have been travelling with the same speed as the quasar, relative to the earth. Assume that light from the quasar has just reached the earth.

b) Calculate when the quasar was formed in the earth system.

Assume that the quasar has a lifetime of about 1.0 * 10^6 yrs in the system of the quasar.

c) Calculate how many years light from the quasar will be visible on earth.

2. Relevant equations
It's my first attempt at solving a more complex special relativity problem and I have a very basic knowledge of it. But I know the basic equations of time dilation, length contraction, doppler effect, relativistic speed that seem relevant.

3. The attempt at a solution

a) is simple and can easily be solved using the Doppler equation.

I'm getting stuck at b). I have drawn some diagrams and tried to understand the situation, but I can't make head nor tail out of it. Can you express the time expired since the BB in the quasar system using time dilation? If so, how can you possibly find out when the quasar was formed?

I only have a slight idea for c).
Using time dilation one can calculate the lifespan of the quasar in the earth system. Then just add that to the time when the quasar was formed. You'll need the answer from b). The time it takes the light from the quasar to reach the earth is the time between the formation of the quasar and now. So if you count that up you get simply the lifetime of the quasar in the earth system?

2. Jul 12, 2010

### hikaru1221

b/ In the Big Bang, material condensed at a very small "point" explodes, so everything starts to move at the same place. Take t=0 when the explosion takes places. Consider the Earth's frame. When the materials merge into the quasar, it has gone away from the Earth a distance $$d=vT$$, where T is the time when the quasar is formed. At this point, the quasar emits radiation towards the Earth, and it takes $$T_{light} = d/c = vT/c$$ for the radiation to come to the Earth. At the time when the light reaches the Earth, $$t=T+T_{light}$$, which is also the age of the Universe, as the problem assumes that light from the quasar has just reached the earth. You have v, c and the Universe's age, you can solve for T

c/ Now you have the lifespan of quasar in the Earth's frame, say $$T_{quasar}$$. Imagine that at t=T, the quasar, at a distance d, emits its first radiation, and at $$t=T+T_{quasar}$$, it, at a distance d', emits its last one. Calculate d' and solve for the difference in time taken for lights in the 2 times to reach the Earth.

3. Jul 12, 2010

### lol_nl

Thanks! I think I'm getting what you mean, thought putting it on paper is a little harder :).
I'll figure out how to use LaTeX so I can post some my calculations next time.

For b, the only thing I kept confusing about was the sign of v. I know that the quasar is moving toward the earth from a). Eventually I could see from my calculation that I would get a value of T greater than t if I took v as negative, so I took it as positive :).
Eventually I got a value of 7.1*10^8 years which is the same as that he answer sheet.

I thought I had c as well, but it turns out my answer is different from that of answer sheet. Turns out I misunderstood something.
Here's roughly what I did (rounded off answers):

Tquasar = (gamma) * Tquasar' (Tquasar' = 1.0 * 10^6)
d = 7.1 *10^8 light years
d - d' = v*Tquasar = 2.2*10^6 light years
(d' < d because the quasar is coming toward the earth)
Tlight2 = d'/c = 8.86 * 10^8 years
Te = T + Tquasar + Tlight2 - t = 5.56 * 10^5 years

Now the answer says 3*10^6 years.

It's a bit unclear if I write it down like this I guess. I'll get LaTeX soon.

4. Jul 12, 2010

### hikaru1221

Hmm now you mention it. I didn't notice the wavelengths. I thought the quasar should move away from the Earth, because the pieces also move away from the Earth. It should produce redshift instead. Quite weird! And the ratio of the wavelengths implies a very large speed!

I haven't checked the numbers yet, but I see something strange in your calculation:
You eventually got d'>d???

5. Jul 13, 2010

### lol_nl

Thanks. I found the solutions to the problems. C is actually much simpler if you consider the time in the quasar system. If you calculate the time when the quasar was formed in its own system and then write an expression for the time that passed in the quasar system in terms of the time that passed in the earth system, you can equate that to the quasar lifespan and then solve for the time that passed in the earth system. But thanks anyway :)!