Does Einstein’s original paper on relativity predict antiparticles and tachyons? In Einstein’s original paper on relativity (pages refer to this copy) page 149 he gets an initial form of the now known as Lorentz transformation
$$\mathbf{x’}=\phi(v) \Lambda \mathbf{x} \tag{1}$$
(I used current matrix representation as the lazy way), where $\Lambda$ is Lorentz transformation matrix and $\phi(v)$ is an unknown function of $v$.
He then pursues the value of $\phi(v)$ and gets
$$\phi(v) \phi(-v)=1 \tag{2}$$
In page 150, and finally
$$\phi(v)= \phi(-v) \tag{3}$$
In page 151, which led him into the conclusion that $\phi(v)=1$
However, in my point of view, that is not the only solution he could get. By combining $(2)$ and $(3)$ one actually gets
$$\phi(v)^2=1 \tag{4}$$
Meaning
$$\phi(v)=\pm 1 \tag{5}$$
It’s obvious that choosing the positive solution one just continues as Einstein did. However, the existence of the negative solution seems to indicate that for every particle moving right, there should be another one that moves into the same direction backwards in time (using negative solution and negative velocity), which

*

*seems to be one characteristic of tachyons;

*seems to predict anti-particles as they can be modeled as particles moving positively backwards in time (I’m not claiming people believe it or not).

Are these conclusions sound? Is the negative solution valid in this case? Or the positive solution is the only possible solution and equation $(4)$ is not allowed. Please notice that “not allowed” here means, it is contradicted by using any of the arguments inside this paper alone.
 A: There is an additional constraint which may not be explicit in Einstein’s paper. In the limit of zero velocity, $v\to 0$, the Lorentz transformation must tend towards the identity, $\mathbf x’ \to \mathbf x$. This requires that $\phi(0)$ have the value $+1$.
A: There are also the assumptions that I think we can safely infer were intended: $φ(0) = 1$ and $φ(v)$ is a continuous function of $v$. That rules out any sign flip.
A more careful study of the transform group, of course, leads to the result - now well-known - that two manners of sign flips may be included: one, $P$, for the spatial coordinates and one, $T$, for the time coordinate. These are the improper Lorentz transforms, but are treated as being part of the overall transform group. With the set up given in the paper, a combined $PT$ transform would be $φ(0) = -1$ and $v = 0$; but $P$ and $T$ by themselves can't be obtained. So, his treatment is not as general as it could be.
The restricted transforms are those continuously connected to the identity transform, and they all preserve spatial parity and time orientation: so no sign flips: no $P$ or $T$. That's what his treatment leads to.
The weak nuclear force is not $P$-invariant. The jury is still out on whether this is a fundamental violation of $P$ symmetry or whether $P$ is still a bona fide symmetry - albeit a "broken symmetry"; though the consensus tends to the idea that it is a fundamental violation. On the other hand, people like Penrose believe it's a broken symmetry. This would require a yet-to-be-discovered opposite-handed version of the weak force. There is circumstantial evidence (the layout and regularities in the spectrum of fundamental fermions) that supports that idea.
$C$ is charge conjugation (matter - anti-matter flip) and the CPT-theorem essentially asserts that $PT = C$. So, under that assumption $PT$ flips between the matter and anti-matter universe.
There are much better ways to arrive at the key results Einstein is seeking out that also show off a larger range of possibilities. The classical case in point is the classification of possible kinematic symmetry groups in:
Bacry, Henri, Lévy‐Leblond, Jean‐Marc (1968) Possible Kinematics. Journal of Mathematical Physics, 9. 1605-1614 doi:10.1063/1.1664490
https://www.mindat.org/reference.php?id=5749905
A reification of it, framed in terms of Lie algebra, with a uniform coordinate representation that spans all the kinematic groups, is presented here:
https://groups.google.com/g/sci.physics/c/FNUUEoRJL5M?pli=1
That include the 4D Euclidean group, the hyperspherical and hyperbolic groups, the Galilei group (actually: its central extension - the Bargmann group) the Poincaré, which the Lorentz transforms belong inside of, the Carroll group, the Static group, the deSitter and anti-deSitter groups and their non-relativistic versions (the Newton-Hooke and anti-Newton-Hooke groups), para-Poincaré, para-Galilei and para-Euclid. It's actually a 3×3×3 classification, but the members on opposite sides of the 3×3×3-cube coincide, which reduces the 27 members to 13 pairs and the 14th one in the center (the Static group).
Einstein did not have a grasp of the infrastructure of Lie groups or Lie algebras at that young age of his, even though he was replicating some of its methods in his paper. It was still new (only arising in the 1870's and 1880's), so it was probably not well-known enough.
Cosserat & Cosserat were early pioneers in that endeavor - which was 4 years after Einstein's paper.
Cosserat, E. and Cosserat, F. (1909) Théorie des Corps Déformables. Hermann, Paris.
