Proving L3 is a Lorentz Transformation

[martyf]

Homework Statement

L1 and L2 are two lorentz trasformation.
show that L3=L1 L2 is a lorentz trasformation too.

Homework Equations

The Attempt at a Solution

 

[phsopher]

I could be wrong but I think it would be enough to show the invariance of the dot product under L3.

 

[martyf]

what is the dot product?

 

[phsopher]

The scalar product, or inner product or whatever it's called of 4-vectors. In Minkowski space it's $$r_1 \cdot r_2 = c^2 t_1 t_2 - x_1 x_2 - y_1 y_2 - z_1 z_1$$ or with the signs reversed depending on the convention. But if you haven't yet done the Minkowski notation then you could write L1 and L2 as Lorentz transformations with some velocities and show by applying them consecutively that L3 has the same form.

 

[martyf]

I write better :

I have: $$\Lambda$$ $$^{\mu}_{\nu}$$ and $$\Lambda\widetilde{}$$ $$^{\mu}_{\nu}$$ : lorentz trasformations.

Show that $$\Lambda\overline{}$$ $$^{\sigma}_{\rho}$$ = $$\Lambda\widetilde{}$$ $$^{\sigma}_{\mu}$$ $$\Lambda$$ $$^{\mu}_{\rho}$$ is a lorentz trasformation

 

[George Jones]

What is the definition of Lorentz transformation given in your notes and/or text?

 

[martyf]

a trasfonmation that not change the space-time distance of a point to the origin.

 

[George Jones]

a trasfonmation that not change the space-time distance of a point to the origin.

Can you write a definition in terms of mathematics, i.e., [tex]\Lambda^\mu{}_\nu[/itex] is a Lorentz transformation iff ... ?

 

[martyf]

...if :

x$$^{2}_{0}$$ - r $$^{2}$$= ($$\Lambda$$ $$^{0}_{\nu}$$ x$$_{0}$$)$$^{2}$$ -( $$\Lambda$$ $$^{\mu}_{\nu}$$ r$$_{\mu}$$)$$^{2}$$

 

[martyf]

is it right?

 

[phsopher]

You don't have the indices right, your right hand side depends on on $$\nu$$ whereas the left hand side doesn't. It should be:

$$(x^0)^2 - r^2 = (\Lambda^0{}_\nu x^\nu)^2 - (\Lambda^j{}_\nu x^\nu)^2$$

Where j goes from 1 to 3. More compactly:

$$g_\alpha_\beta x^\alpha x^\beta = g_\mu_\nu \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta x^\alpha x^\beta$$

Where g is the metric. Use the fact that L1 and L2 satisfy this to show that L3 satisfies it as well.

 

[martyf]

L1 and L2 are two generical lorents tranformation. I must demostrate that their product is also a lorentz transormation.
Can I demostrate :

$$g_\alpha_\beta x^\alpha x^\beta = g_\mu_\nu \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta x^\alpha x^\beta$$

with \Lambda =L3= L1 L2 ?

 

[phsopher]

Yes you can.