What is the detection rate of neutrons in Bill's frame?

[Ken Miller]

I'm working through an introductory book on special relativity (by Resnick and Halliday), and am having trouble with one of the end-of-chapter problems.

Problem Statement:
Sue and Jim are two experimenters at rest with respect to one another at different points in space. They “fire” neutrons at each other, each neutron leaving its “gun” at a relative speed of 0.6c. Jim fires neutrons at a steady rate of 10,000 neutrons/second. State the rate that would be reported by a 3rd person (Bill) who is in a frame chosen so that Sue’s neutrons are at rest in it.

Relevant Equations:
One or more of: Lorentz transformation, relativistic velocity addition, relativistic doppler effect.

Attempt at a Solution:
Define frame S as that of Jim and Sue.

Define frame S’ as that of Bill.

Then in S, the neutrons are fired from the same location and at times separated by 100μs. So

Δt = 100μs and Δx = 0.

Using the Lorentz transformations:

Δt’ = γ(100μs) = 125μs.

But this means that the frequency as seen by Bill is 8000/s. But the book says 12,500/s, and I’m sure it’s correct. After all, when a source approaches you, the frequency of anything it “throws” or radiates towards you is higher than if it remains motionless with respect to you.

I suspect that the problem is similar to that of a researcher on Earth who is measuring the frequency of radiation coming at Earth from elsewhere. The frequency will be higher if the object radiating is approaching earth. But I don’t think it’s simply a Doppler effect (besides, if I use the Doppler effect equation, I still don’t get the answer).

Any help appreciated.

 

[Ibix]

The question is ambiguous. Does it refer to the rate at which neutrons are fired by Jim, or the rate at which neutrons pass Bill? Which one have you calculated? How could you calculate the other?

Spoiler: Hint

In SR you can never go wrong writing down coordinates and Lorentz transforming.

I think the stated answer is wrong in either interpretation. I don't have a copy of Halliday and Resnick, but one former poster and mentor on here said that you would be better with literally any other textbook for learning relativity. On the evidence of this question (ambiguous with a wrong answer), I'd tend to agree.

 

[Ken Miller]

Thanks for your reply.

I believe that I calculated the rate at which Jim fires neutrons, but as seen by an infinite array of clocks in Bill's frame. I had trouble thinking of how to calculate the rate at which neutrons pass Bill, but now I'm wondering if that is a simple Doppler-effect problem. That is, if $\nu_0$ is the rate at which Jim fires neutrons (measure in Jim's frame) and $\nu$ is is that rate that neutrons pass Bill, measured in Bill's frame, then:

$$ \nu = \nu_0 ~ sqrt {(\frac {1 + \beta} {1 - \beta} ) } $$
So
$$ \nu = (10,000/sec) ~ sqrt {(\frac {1 + 0.6} {1 - 0.6} ) } $$
or
$$ \nu = 20,000/sec. $$
Is that right?

 

[Ibix]

I believe that I calculated the rate at which Jim fires neutrons

Agreed.

but as seen by an infinite array of clocks in Bill's frame.

Better said, it's the launch rate, rather than the detection rate. But yes, I believe you understand what you've done.

I'm wondering if that is a simple Doppler-effect problem.[...]$$ \nu = 20,000/sec. $$
Is that right?

No. You can certainly approach it this way, but what does the derivation of the formula you are using assume about the signal (neutrons in this case, light in the usual case) traveling from source to detector? Will you have to derive a revised Doppler shift formula?

I'd say it would be easier to apply the standard recipe for solving SR problems:

1. Write down the (x,t) coordinates of one or more events in whatever frame it's easiest to work in
2. Transform to the frame you need to answer in

If you want to know the detection rate, what two events will tell you this?

Incidentally, \sqrt{} does square roots in LaTeX: $$\nu=\nu_0\sqrt{\frac{1+\beta}{1-\beta}}$$