Solving for Vector u in Terms of Vector η

[ehrenfest]

Homework Statement

$$\mathbf{\eta} = 1/ \sqrt{1 - u^2/c^2} \mathbf{u}$$ represents the proper velocity in terms of the ordinary velocity. Find vector u in terms of vector eta.

Then find the relation between proper velocity and rapidity.

Homework Equations

The Attempt at a Solution

Scalar u is the magnitude of the vector u, so I cannot just bring the denominator to the other side, right? Do I have to get scalar eta in terms of scalar u?

 

[George Jones]

Do I have to get scalar eta in terms of scalar u?

The other way round. Find scalar u in terms of scalar eta.

 

[ehrenfest]

The other way round. Find scalar u in terms of scalar eta.

Yes. That is what I meant. Those velocity magnitude's so I am not sure how to do that...

 

[dextercioby]

Remember that both $\mathbf{u}$ and $\mathbf{\eta}$ are ordinary vectors in R^{3} in which the norm of a vector is given by a scalar product. So compute $\left \langle \mathbf{\eta},\mathbf{\eta}\right\rangle$.

 

[ehrenfest]

$\left \langle \mathbf{\eta},\mathbf{\eta}\right\rangle$.

$\left \langle \mathbf{\eta},\mathbf{\eta}\right\rangle = \eta_x^2 + \eta_y^2 + \eta_z^2$

But we have

$$\sqrt{1 - u^2/c^2} \mathbf{\eta} = \mathbf{u}$$ so how do you get that scalar u in terms of the scalar eta?

Let me check my understanding of what u and eta are. The usage of these two terms implicitly uses an inertial reference frame S and an object in motion with respect to S. vector u is the displacement measured in S per unit time measured in S and vector eta is the displacement measured in S per unit time measured by a clock attached to our object. All good?

 

[dextercioby]

$$\left\langle \vec{\eta},\vec{\eta}\right\rangle =\left\langle \vec{u},\vec{u}\right\rangle \frac{1}{1-\frac{u^{2}}{c^{2}}}\Rightarrow \eta ^{2}=\frac{u^{2}}{1-\frac{u^{2}}{c^{2}}}\Rightarrow u^{2}=\frac{\eta ^{2}}{1+\frac{\eta ^{2}}{c^{2}}}$$

$$\vec{\eta}=\vec{u}\frac{1}{\sqrt{1-\frac{u^{2}}{c^{2}}}}\Rightarrow \vec{u}=\vec{\eta}\sqrt{1-\frac{u^{2}\left(\eta \right)}{c^{2}}}=\vec{\eta}\sqrt{1-\frac{1}{c^{2}}\frac{\eta ^{2}}{1+\frac{\eta ^{2}}{c^{2}}}}=\vec{\eta}\frac{c}{\sqrt{c^{2}+\eta ^{2}}}$$

 

[ehrenfest]

Thanks. But is that paragraph I wrote correct? I am very confused about proper velocity. It is supposedly proper distance divided by proper time. But what is proper distance? Isn't the distance an object moves in a reference frame attached to it always 0?

 

[ehrenfest]

Let me check my understanding of what u and eta are. The usage of these two terms implicitly uses an inertial reference frame S and an object in motion with respect to S. vector u is the displacement measured in S per unit time measured in S and vector eta is the displacement measured in S per unit time measured by a clock attached to our object. All good?

I think I got it. Also, is the relation between proper velocity and rapidity the following:

tanh theta = eta/ sqrt(c^2 + eta^2)

?