As we discuss briefly in our recent CKM matrix paper 2305.00668, the answer to all the above questions is yes! Just as gravitation is the geometry of our familiar 4D spacetime, the weak force is the geometry of the second 4D spacetime. Indeed, the weak force is a spacetime symmetry masquerading as an internal symmetry. Symmetry breaking of a 6D spacetime in the very early universe gives rise to these two 4D spacetimes, one curved by gravity and the other curved by the weak force. The size of the universe in the other spacetime is of the order of the range of the weak force. A beam of light going through the universe in this other spacetime would be back to the starting point in just 10^-24 seconds! This could help us understand quantum nonlocality and the EPR paradox. When Einstein said that if quantum nonlocality (assuming QM is complete) is true then special relativity must go, one way to interpret Einstein is to propose that our universe has two 4D spacetimes.
In the twistor picture of our spacetime, null lines are more fundamental than spacetime points. Consider two null lines (t-x) and (t+x). A point arises as the intersection of these two null lines. 4D spacetime can be arrived at by overlaying these two null lines on a complex plane (y + iz). Consider the 2x2 matrix
t-x y+iz
y-iz t+x
Its determinant gives the 4D line element.
To get the second 4D spacetime we overlay these two null lines on an independent complex plane (a+ib) and make a new 2x2 matrix
t-x a+ib
a-ib t+x
The determinant now gives the line-element of the second 4D spacetime.
Between them the two 4D spacetimes are labeled by six real numbers (t, x, y, z, a, b) which can be used to obtain a 6D spacetime.
We can try to visualise the second spacetime by thinking of a torus in which the horizontal circle is much much larger than the vertical circle. The horizontal circle is space of our spacetime, the vertical circle is space of the other spacetime. If we are at location A on the torus then a galaxy at location B on the larger circle is correspondingly at location B' on the smaller circle, and B is identified with B' : the smaller circle is a scaled down version of the bigger circle and in one to one correspondence with it. A photon starting from our location will reach the galaxy B much faster along the smaller circle.
Mathematically, the two spacetimes result because of the group theory relations SL(2, H) ~ SO(1,5) and SL(2,C) ~ SO(1,3) where H are the quaternions. And because the Clifford algebra Cl(3) is the direct sum of two copies of Cl(2). Each Cl(2) generates a 4D Lorentz algebra...one is for our spacetime. The other is for the second spacetime whose three rotations are the weak isospin rotations, and the three boosts give Lorentz transformations along the vertical circle of the torus. Cl(3) is the algebra of complex split biquaternions and one can make a 6D spacetime from it.
If there is indeed a second 4D spacetime in our universe, which obeys the laws of special relativity, has its own light cone, is microscopic and accessible only to quantum systems, it can offer a neat solution to the EPR puzzle. Along the other spacetime, the photon arrives causally at B much before than along A, and this looks nonlocal from our perspective. One could call this a much more believable version of ER=EPR.
The trillion rupee question is: can the second spacetime be used for communication? Can we talk to someone on Andromeda in real time? Perhaps by sending weak force waves, analogous to gravitational waves, along the second spacetime.
