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## Main Question or Discussion Point

I want to determine the orbits of the proper orthochronous Lorentz group [itex] SO^{+}(1,3) [/itex].

If I start with a time-like four-momentum [itex] p = (m, 0, 0, 0) [/itex]

with positive time-component [itex]p^{0} = m > 0 [/itex],

the orbit of [itex] SO^{+}(1,3) [/itex] in [itex] p [/itex] is given by:

[tex] \mathcal{O}(p) \equiv \lbrace \Lambda p \mid \Lambda \in SO^{+}(1,3) \rbrace [/tex]

Now the point is: how do you show that

[tex] \mathcal{O}(p) = \lbrace q \mid q^{2} = m^{2}, q^{0} > 0 \rbrace [/tex] ?

Essently, the question is: why does a Lorentz transformation [itex] \Lambda \in SO^{+}(1,3) [/itex] exist

such that two four-vectors [itex] p [/itex] and [itex] q [/itex] with [itex] p^{2} = q^{2} = m^{2} [/itex] and [itex] p^{0}, q^{0} > 0 [/itex] are related via [itex] q = \Lambda p [/itex] ?

In fact, it is possible to answer my question(s) by brute-force calculations. But I am searching for an "elegant way", e.g. with the help of group theory.

I already searched in the literature, but in most cases it seems to be "trivial" for the authors

and I see now explicit proof.

Does anyone know of anything like that?

If I start with a time-like four-momentum [itex] p = (m, 0, 0, 0) [/itex]

with positive time-component [itex]p^{0} = m > 0 [/itex],

the orbit of [itex] SO^{+}(1,3) [/itex] in [itex] p [/itex] is given by:

[tex] \mathcal{O}(p) \equiv \lbrace \Lambda p \mid \Lambda \in SO^{+}(1,3) \rbrace [/tex]

Now the point is: how do you show that

[tex] \mathcal{O}(p) = \lbrace q \mid q^{2} = m^{2}, q^{0} > 0 \rbrace [/tex] ?

Essently, the question is: why does a Lorentz transformation [itex] \Lambda \in SO^{+}(1,3) [/itex] exist

such that two four-vectors [itex] p [/itex] and [itex] q [/itex] with [itex] p^{2} = q^{2} = m^{2} [/itex] and [itex] p^{0}, q^{0} > 0 [/itex] are related via [itex] q = \Lambda p [/itex] ?

In fact, it is possible to answer my question(s) by brute-force calculations. But I am searching for an "elegant way", e.g. with the help of group theory.

I already searched in the literature, but in most cases it seems to be "trivial" for the authors

and I see now explicit proof.

Does anyone know of anything like that?