General confusion with four velocity and four acceleration

[HJ Farnsworth]

Greetings,

I have been trying to teach myself SR and GR and have been going the Bernard Schutz's First Course in General Relativity to do so. I thought I was understanding all of the concepts perfectly because I have been flying by most of the exercises without having any trouble. However, there are some exercises in the middle of chapter 2 that I have been completely unable to solve, so any conceptual advise would be appreciated. Here are parts of the problems:

19)
A body is said to be uniformly accelerated if its acceleration four-vector a has constant
spatial direction and magnitude, 4-acceleration vector dot 4-acceleration vector = alpha².
If a body is uniformly accelerated with alpha=10m/s², what is the body's speed after time t?

In trying to solve this, I started by converting alpha to natural units, but don't really know where to go from there. Digging around on the Internet, I've found that I should try to use u=alpha*tau, where u is the proper velocity given by v=tanh(u). I can sort of see how I would solve the problem using this, but I'm not sure how to prove that u=alpha*tau, so I don't want to use it. For that matter, I found on the Wikipedia article for proper velocity that if object B is traveling as observed by Observer A, then proper velocity is A's distance divided by B's time: u=dx_obs/dt_obj. I decided to check if this was consistent with v=tanh(u), and found that it wasn't - the equation I got was u = v/sqrt(1-v²). My questions for this problem are thus a) how do I prove that v=tanh(u)? b) how do I prove that u=alpha*tau?

20) In some IRF, the worldline of a particle is described by equations:
x(t) = at+bsin(omega*t), y(t) = bcos(omega*t), z(t) = 0, where |b*omega|<1. Find the 4 velocity U and 4 acceleration a.

Basically, I'm having trouble doing this because the the regular 3-velocity isn't constant. Having never done a problem like this before, I'm not sure how to go about it. So I'm just looking for some general tips.

Thank you very much for any help you can give.

-HJ Farnsworth

PS - does the Latex Reference editor generally work for people? Whenever I try to use it, it does stuff that it obviously shouldn't do, like putting individual Greek letters on their own line, or deleting the symbol I use for a vector when I try to put an arrow over it, and then putting the arrow on its own line. Does this website have a better tool for entering equations anywhere?

 

[Meir Achuz]

A body is said to be uniformly accelerated if its acceleration four-vector a has constant
spatial direction and magnitude, 4-acceleration vector dot 4-acceleration vector = alpha².
If a body is uniformly accelerated with alpha=10m/s², what is the body's speed after time t?

To solve the problem, use the fact that if the particle starts from rest then
$${\bf a}$$ is parallel to $${\bf v}$$, where I use boldface to denote three-vectors. Then, by calculating
$$a^{\mu} a_{\mu}$$, you can find
$$\alpha=\gamma^3{\bf a}=(d/dt)(\gamma{\bf v})$$. This can be integrated to find $${\bf v}$$ and then
$${\bf x}$$.

If my latex has worked, you can use 'quote' to see how I did it.

 

[Bill_K]

Let u be the 4-velocity. Split u into its space and time components, u = (a, b, 0, 0). Then u.u = 1 implies

(1) a² - b² = 1

Let a = du/dτ be the 4-acceleration, where τ is the proper time. a = du/dτ = (da/dτ, db/dτ, 0, 0), and a.a = - α² implies

(2) (da/dτ)² - (db/dτ)²= - α²

The solution of (1) and (2) is a(τ) = cosh (ατ), b(τ) = sinh (ατ)

 

[Mentz114]

Clem, I don't know if you intended to start a new line with every expression, but if you use itex and /itex instead of tex and /tex, you get inline tex.

 

[HJ Farnsworth]

Thanks very much for the responses.

I think I understand problem 19 fairly well at this point. However, I'd still like to know how the equations v=tanh(u) and u=$\alpha$$\tau$ are derived for their own sake, since they both seem like highly relevant equations, especially the former. I could be mistaken in thinking that the u in v=tanh(u) is the proper velocity - that would explain why the equation I got is not equivalent to v=tanh(u). If this is so, what is the u? And does anyone know how to derive those two equations?

Also, any advice for #20 would still be helpful, I think it's the sort of thing where if I saw how to go about a problem like that once, I would gain a strong grasp of the concepts used.

By the way, the itex thing is basically what I was looking for as far as latex, so thank you for that as well.

Thanks again for any help you can give.

-HJ Farnsworth

 

[Mentz114]

Also, any advice for #20 would still be helpful, I think it's the sort of thing where if I saw how to go about a problem like that once, I would gain a strong grasp of the concepts used.

You are looking for the 4-vector $u^\mu=( dt/d\tau, dx/d\tau, dy/d\tau,0)$ which has a norm of -1. Using $dx/d\tau = (dx/dt)(dt/d\tau)$ you can get $dx/d\tau, dy/d\tau$ which should be enough to find $u^\mu$.

 

[Chestermiller]

I've been writing up some notes on spectial relativity, and the second chapter of my notes is devoted to providing a better mechanistic understanding of the fundamental geometric structure of 4D space-time, and how frames of reference are traveling through 4D space-time. I've been very unhappy with how special relativity is taught in the various texts that I've worked with, and feel that it is much easier to understand than the way it is presented. If you are interested in seeing what i have written so far, please send me your email address at chestermiller@mindspring.com, and I will email you a copy. This is a work in progress, but I'm sure you will find the second chapter very illuminating.

 

[HJ Farnsworth]

Hi Chestermiller,

I would be very interested in looking at your text. My email address is sjha_@msn.com (that's an underscore after the sjha). I sent you an email at about the same time that I wrote this post – if you don’t get it for some reason, let me know.

Thank you very much for your response and offer.

-HJ Farnsworth