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## Homework Statement

L1 and L2 are two lorentz trasformation.

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

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- Thread starter martyf
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- #1

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L1 and L2 are two lorentz trasformation.

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

- #2

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

- #3

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what is the dot product?

- #4

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- #5

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I have: [tex]\Lambda [/tex] [tex]^{\mu}_{\nu}[/tex] and [tex]\Lambda\widetilde{} [/tex] [tex]^{\mu}_{\nu}[/tex] : lorentz trasformations.

Show that [tex]\Lambda\overline{}[/tex] [tex]^{\sigma}_{\rho}[/tex] = [tex]\Lambda\widetilde{} [/tex] [tex]^{\sigma}_{\mu}[/tex] [tex]\Lambda [/tex] [tex]^{\mu}_{\rho}[/tex] is a lorentz trasformation

- #6

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

- #7

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

- #8

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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 ... ?

- #9

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x[tex]^{2}_{0}[/tex] - r [tex]^{2}[/tex]= ([tex]\Lambda[/tex] [tex]^{0}_{\nu}[/tex] x[tex]_{0}[/tex])[tex]^{2}[/tex] -( [tex]\Lambda[/tex] [tex]^{\mu}_{\nu}[/tex] r[tex]_{\mu}[/tex])[tex]^{2}[/tex]

- #10

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is it right?

- #11

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[tex](x^0)^2 - r^2 = (\Lambda^0{}_\nu x^\nu)^2 - (\Lambda^j{}_\nu x^\nu)^2[/tex]

Where j goes from 1 to 3. More compactly:

[tex]g_\alpha_\beta x^\alpha x^\beta = g_\mu_\nu \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta x^\alpha x^\beta[/tex]

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

- #12

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Can I demostrate :

[tex]g_\alpha_\beta x^\alpha x^\beta = g_\mu_\nu \Lambda^\mu{}_\alpha \Lambda^\nu{}_\beta x^\alpha x^\beta[/tex]

with \Lambda =L3= L1 L2 ?

- #13

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Yes you can.

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