How Do Lorentz Transformations Relate Time-like Four-Momenta in SO^{+}(1,3)?

[parton]

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?

 

[dauto]

Isn't that the very definition of the group SO+(1,3)?

 

[WannabeNewton]

I agree with the above. It's trivial according to those authors because it's the definition.

 

[SophusLie]

Isn't that the very definition of the group SO+(1,3)?

No, it is not the definition of [itex] \mathrm{SO}^{+}(1,3) [/itex]. It is:

[tex] \mathrm{SO}^{+}(1,3) = \lbrace \Lambda \in \mathrm{GL}(4, \mathbb{R}) \mid \Lambda^{t} \eta \Lambda = \eta, \mathrm{det} \, \Lambda = 1, \Lambda^{0} \, _{0} \geq 1 \rbrace [/tex]
The elements of this group will not change [itex] p^{2} [/itex] and the sign of [itex] p^{0} [/itex] by definition.

So we can say that the set
[tex] \lbrace q \in \mathbb{M} \mid q^{2} = m^{2}, q^{0} > 0 \rbrace [/tex] remains invariant under Lorentz transformations.

But this does not necessarily mean that for any two fourvectors [itex]p, q [/itex] with [itex]p^{2} = q^{2} = m^{2}[/itex] and [itex] p^{0}, q^{0} [/itex] there exists a transformation [itex] \Lambda \in \mathrm{SO}^{+}(1,3) [/itex] with [itex] q = \Lambda p [/itex]. In other words this would mean that [itex] \mathrm{SO}^{+}(1,3) [/itex] acts transitivly on [itex] \lbrace q \in \mathbb{M} \mid q^{2} = m^{2}, q^{0} > 0 \rbrace [/itex]. And you need to show this in order to proof that
[tex] \mathcal{O}_{p} = \lbrace q \in \mathbb{M} \mid q^{2} = m^{2}, q^{0} > 0 \rbrace [/tex]

Actually I also don't see how to do this without using brute force. I think it is not that trivial.

 

[sardar]

I can provide a response to this content by first explaining the context of the problem. The Lorentz group, denoted as SO^{+}(1,3), is a mathematical group that describes the symmetries of special relativity. It is composed of all the Lorentz transformations that preserve the space-time interval, which is a fundamental concept in special relativity. The proper orthochronous Lorentz group, SO^{+}(1,3), is a subgroup of the Lorentz group that includes only those transformations that preserve the direction of time and have a positive determinant.

Now, to determine the orbits of this group, we can start with a time-like four-momentum vector, p = (m, 0, 0, 0), where m is the mass of the particle and the time-component p^{0} = m is positive. The orbit of SO^{+}(1,3) in p is then given by the set of all Lorentz transformations, \Lambda, multiplied by p. This can be denoted as \mathcal{O}(p) = \lbrace \Lambda p \mid \Lambda \in SO^{+}(1,3) \rbrace.

The question then arises, why does a Lorentz transformation exist that can relate two four-vectors, p and q, with the same mass, m, and positive time-components? The answer lies in the fact that the Lorentz group is a continuous group, meaning that it is a group of transformations that can be described by a continuous set of parameters. In the case of SO^{+}(1,3), these parameters are the six elements of the Lorentz matrix.

Therefore, there exists a continuous path of Lorentz transformations that can connect any two points in the orbit \mathcal{O}(p), including the point q = (m, 0, 0, 0). This is known as the orbit-stabilizer theorem, which states that there is a one-to-one correspondence between the elements of a group and the cosets of its subgroup. In this case, the elements of the Lorentz group correspond to the cosets of the subgroup SO^{+}(1,3).

In conclusion, the existence of a Lorentz transformation that can relate two four-vectors with the same mass and positive time-components is a consequence of the continuous nature of the Lorentz group