This is question 5.8 from Rindler's Relativity: Special, General, Cosmological book, with(adsbygoogle = window.adsbygoogle || []).push({}); A, Uthe 4 acceleration and velocity;a, uthe 3 acceleration and velocity. It has 3 parts which I think I've answered correctly below, but need help with the last part.

So could you have a look and comment on it?

a) Prove that, in any inertial frame whereais orthogonal tou, [tex]\textbf{A}= \gamma^{2} (\textbf{a},0)[/tex], and conversely.

my answer:

[tex]\dot{\gamma}=\gamma^{3}/c^{2}\textbf{a}\cdot\textbf{u}[/tex]

[tex]\textbf{a}\cdot\textbf{u} = 0 [/tex]

[tex]\textbf{A}= \gamma d\textbf{U}/dt = \gamma d/dt(\gamma\textbf{u},\gamma c) = \gamma(\dot{\gamma}\textbf{u} + \gamma\textbf{a}, \dot{\gamma}c ) [/tex]

and so

[tex]\textbf{A}= \gamma^{2} (\textbf{a},0)[/tex] --- (1)

which proves the first part.

Given (1), then

[tex] \dot{\gamma}c = 0 [/tex] and so

[tex]\dot{\gamma}=\gamma^{3}/c^{2}\textbf{a}\cdot\textbf{u} = 0 [/tex]

which is only possible for [tex]\textbf{a}\cdot\textbf{u} = 0 [/tex]

Hence the converse is also true.

b) Deduce that for any instantaneous motion it is possible to find an inertial frame S_|_ in whicha, uare orthogonal to one another.

my answer:

In s_|_ , [tex]\textbf{A}= \gamma^{2} (\textbf{a},0)[/tex]

I need to show [tex]\dot{\gamma}c[/tex] inAcan transform to 0 in S_|_:

[tex]\gamma_{v}(\gamma_{u}\dot{\gamma_{u}} - v(\gamma_{u}\mathbf{a_{x}} + \dot{\gamma_{u}}\mathtbf{u_{x}})/c^{2}) = 0 [/tex]

[tex] v = \gamma_{u}\dot{\gamma_{u}}c^{2} / (\gamma_{u}\mathbf{a_{x}} + \dot{\gamma_{u}}\mathbf{u_{x}})[/tex]

[tex] v = \gamma_{u}^{4}\mathbf{a}\cdot\mathbf{u} / (\gamma_{u}\mathbf{a_{x}} + \dot{\gamma_{u}}\mathbf{u_{x}})[/tex]--- (2)

Hence there exists a v and therefore s_|_.

c) Moreover, prove that there is a whole class of such frames, all moving relative to S_|_ in directions orthogonal toa.

My answer:

The stationary frame can be rotated giving a newa_x, u_xin (2) and hence another v and s_|_. But I can't show they're moving relative to S_|_ in directions orthogonal toa.

Thanks for your help.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Acceleration 4 vector question

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