Question from Rindler's Introduction to Special Relativity

In summary, the problem involves two identical particles moving in opposite directions along parallel lines with equal distance from the origin. The sum of their gamma functions is equal to twice the gamma function of their common velocity. By using hyperbolic identities and manipulating the equations, it can be shown that the separation between the particles must be greater than 2 times the distance travelled by each particle in the given time, in order to maintain the relativistic speed limit.
  • #1
learningphysics
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I'm stuck on this problem in the "Relativistic Particle Mechanics" section, number 26. I had no trouble with the first part... but the second part I'm stuck.

"Two identical particles move with velocities +-u along the parallel lines z=0, y=+-a in a frame S, passing x=0 simultaneously. Prove that all centroids determined by observers moving collinearly with these particles lie on the open line-segment x=z=0, |y|<ua/c"...

I had no trouble here. But now:

"Also prove that, keeping the same total (relativistic) mass and angular momentum, two such particles cannot move along lines closer than 2ua/c without breaking the relativistic speed limit."

My basic idea was to use the equation for conservation of relativistic mass leading to:

[tex] \gamma (v_1) + \gamma (v_2) = 2*\gamma (u)[/tex]

And conservation of 3-angular momentum which leads to:

[tex] \gamma (v_1)*v_1*r_1 + \gamma (v_2)*v_2*r_2 = 2*\gamma (u) * u * a[/tex]

to try and show the required inequality, but haven't been successful. I'd appreciate any help. Thanks.
 
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  • #2
You've solved the problem, but you're stuck on the algebra. The problem is all those nasty gamma functions. To get rid of them, try dividing your second formula by your first formula. That is, divide left side by left side, right side by right side.

Carl
 
  • #3
Hi Carl. I still don't see it. After dividing out, I can cancel the [tex]\gamma (u)[/tex], but still have the other gamma functions.

So I get:

[tex]\frac{\gamma (v_1) * v_1 * r_1 + \gamma (v_2) * v_2 * r_2}{\gamma (v_1) + \gamma (v_2) } = u * a[/tex]

but I'm not sure what to do here. I'm probably missing something very simple.
 
  • #4
Often, gamma [tex]\gamma[/tex] factors may be more easily manipulated if you use rapidities [tex]\theta[/tex].

[tex]v{\color{red}/c}=\tanh\theta[/tex]
[tex]\gamma=\displaystyle\frac{1}{\sqrt{1-(v/c)^2}}=\cosh\theta[/tex]
[tex]v\gamma{\color{red}/c}=\displaystyle\frac{v{\color{red}/c}}{\sqrt{1-(v/c)^2}}=\sinh\theta[/tex]

Once written in terms of rapidities, apply hyperbolic-trig identities and then do algebra.
 
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  • #5
learningphysics said:
I'm not sure what to do here. I'm probably missing something very simple.

Try showing that [tex]\gamma (v_1) = \gamma (v_2)[/tex].

Carl
 
  • #6
Thanks Carl and robphy. I see that I was over-generalizing the problem. I thought that r1 and r2 could take any values... however this isn't the case since we could take both particles having the same velocity (u) and position (a), and keep the same total relativistic mass and 3-angular momentum (about the origin) while their separation is 0.

So instead assume that the particles are moving in opposite directions along opposite parallel lines each at an equal distance r from the origin (which is what I believe the problem expects).

So we have:
[tex]\gamma (v_1) + \gamma (v_2) = 2 * \gamma (u) [/tex]

and

[tex](\gamma (v_1) * v_1 + \gamma (v_2) * v_2)*r = 2 * \gamma (u) * u * a[/tex]

Divide the two equations:

[tex]\frac{\gamma (v_1) * v_1 + \gamma (v_2) * v_2 }{\gamma (v_1) + \gamma (v_2) } * r = u * a[/tex]

Using sinh and cosh and solving for r:

[tex]r = \frac{cosh(\theta_1) + cosh(\theta_2)}{sinh(\theta_1) + sinh(\theta_2)} * u * a/c[/tex]

Using the hyperbolic identites for sum of 2 cosh, and sum of two sinh...

[tex]r = coth((\theta_1 + \theta_2)/2) * u * a/c [/tex]

[tex]coth((\theta_1 + \theta_2)/2) > 1[/tex] so

[tex]r > ua/c[/tex]

so the separation of the two particles is > than 2ua/c.
 

1. What is special relativity?

Special relativity is a theory in physics that explains the relationship between space and time. It was proposed by Albert Einstein in 1905 and is based on two main principles: the principle of relativity and the principle of the constancy of the speed of light.

2. What is the principle of relativity?

The principle of relativity states that the laws of physics should be the same for all observers in uniform motion. This means that the laws of physics should not depend on an observer's frame of reference or their relative motion.

3. What is the principle of the constancy of the speed of light?

The principle of the constancy of the speed of light states that the speed of light in a vacuum is always constant, regardless of the observer's frame of reference or their relative motion. This means that the speed of light is the same for all observers, regardless of their perspective.

4. How does special relativity differ from classical mechanics?

Special relativity differs from classical mechanics in several ways. In classical mechanics, time and space are considered absolute, while special relativity shows that they are relative and can be affected by an observer's motion. Additionally, special relativity introduces the concept of spacetime, where time and space are intertwined. It also predicts phenomena such as time dilation and length contraction, which are not observed in classical mechanics.

5. What are some practical applications of special relativity?

Some practical applications of special relativity include the development of GPS technology, which relies on precise time measurements and takes into account the effects of time dilation. Special relativity also plays a crucial role in nuclear energy and particle accelerators, as it helps scientists understand the behavior of particles at high speeds. It also has implications in the fields of astrophysics and cosmology, helping us understand the behavior of objects in the universe.

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