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Is there a name for two null lines that lie on the opposite sides of the null cone? Each line can be obtained from the other by reflection in the axis of the null cone (the time-axis). In terms of world-lines, this corresponds to two photons moving in the opposite directions. If there is not a standard name, what would you choose to call them as a pair?

Qmechanic
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Andrey Sokolov
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2 Answers2

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To expand on Qmechanic's point, we can demonstrate how null directions get moved around even in flat space by Lorentz transformations:

Given a Lorentz vector $X^a$ you can construct a 2x2 Hermitian complex matrix $$X^{AA'} = \frac{1}{\sqrt{2}}\left(\begin{array}{cc}X^0+X^3 & X^1+iX^2 \\ X^1-iX^2 & X^0-X^3\end{array}\right)$$ The Lorentz norm of $X^a$ is then just the determinant of this matrix. If $X^A$ is null, then $X^{AA'}$ factorizes into the product of a pair of length 2 complex vectors (spinors) $$X^{AA'}=s^A\bar{s}^{A'} $$ so the null direction is determined (up to a complex scaling factor) by the pair of complex numbers making up the spinor $s^A$. Now the space of pairs of complex numbers ($\mathbb{C}^2$) modulo a scale is just the Riemann sphere ($\mathbb{C}P_1$). Lorentz transformations preserving the norm of $X^a$ are represented by 2x2 complex matrices with unit determinant acting on $X^{AA'}$:$$X^{AA'} \rightarrow \Lambda^{A}_{\;B}X^{BB'}\bar{\Lambda}^{A'}_{\:B'} $$ These matrices, elements of $SL(2;\mathbb{C})$ act on individual spinors $$s^A \rightarrow \Lambda^{A}_{\;B} s^B$$ and since these spinors represent null directions, this is how the null directions get scrambled around by Lorentz transformations.

The "opposite null directions" you would like would be antipodal points on this Riemann sphere, but these are not preserved by the general $SL(2;\mathbb{C})$ (i.e. Lorentz) transformation (although an individual specific transformation will have fixed points). The easiest way to see where points on the Riemann sphere get sent to is to think of the mapping as a Mobius transformation.

Since it would have no invariant meaning to do so, there is no standard terminology for null lines corresponding to antipodal points on the Riemann sphere (which is the nearest concept I can envisage which corresponds to your "opposite null lines).

twistor59
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  • You are labouring in vain. I haven't asked whether these lines are invariant. Qmechanic already explained why there is no standard terminology, in flat spacetime, and suggested a term for these lines. You do not offer any alternatives. – Andrey Sokolov Dec 10 '12 at 03:06
  • +1 for the $CP(1)$ picture of the lightcone. If you enjoy reading this answer, see also this related Phys.SE post. – Qmechanic Dec 20 '12 at 11:00
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There isn't an official standard name for opposite null lines. Note that opposite null lines are not a coordinate-independent geometric (invariant) notion, and hence it is not a very useful concept. If two null lines happen to lie on opposite sides of the light-cone in one reference frame, then they may not lie on opposite sides of the light-cone wrt. a boosted reference frame. Conversely, two different non-opposite null lines wrt. one reference frame may be boosted in such a way that they become opposite null lines wrt. the new reference frame.

Qmechanic
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