Theory of Dilated Locality

In Figure 6.10, we can see Alice’s original spins in point Y and Bob’s original spins in point X, while the avatars are all together in point Z.

Figure 6.10 Avatars, original spins and the distances between them.

Let us suppose that Alice now performs a quantum measurement on her particle. At the same time all of the following facts happen:

-Alice knows the spin of her particle,

-Bob has a reverse spin (i.e., opposite orientation) to Alice's, -quantum measurement destroys the entanglement,

-the avatars go from being 4 to being 2 and their values go from  to 1. This last case coincides with that one of two completely independent particles.

At the same time, a quasi-message of cancellation of entanglement passes from Alice to Bob through the plenipotentiary representatives of their original spins: the avatars, which are the ones that actually communicate since, as we know, this communication is instantaneous. As the original spins are not local to each other, they could never communicate instantaneously between them.

67

Let us also suppose that this message is initially transmitted at the speed of light, therefore, the time it takes to transmit such cancellation message of the entanglement between avatars will be,

0 0 ZZ avatars d t v c      (6.31)

But being the distance between avatars null dZZ 0, the speed v could be much slower than the speed

of light; however, the message would continue being instantaneous,

0 0 ZZ avatars d t v c      (6.32)

When we talk about avatars, there is no problem with the speed since they are local to each other, but Alice and Bob are not local to each other even though they are communicating instantaneously. Although the avatars are the ones who actually communicate, there will be a situation in which Alice communicates with her avatars, the avatars with each other, and Bob's avatars with him. If Alice and Bob are at a distance of Earth-Mars type, then the distance between Alice and her avatars is Earth- Mars divided by 2, so, how can the communication between Alice and Bob be instantaneous although it is the avatars who really communicate? In other words, the final effect is that Alice notifies Bob of the cancellation of the entanglement instantly even though both are mutually nonlocal. How can this be explained? Let us try to do it by adapting the equation (6.31) so that it can reflect this reality from the point of view of the non-locality of Alice and Bob, although it maintains the hypothesis of instantaneity, which we know it happens,

0 0 0 0 0 XY ZZ XY a vatars originals d d d t t v c v c v c  _{ }  _   _{ }            (6.33)

This equation simultaneously involves the avatars and the original spins’ points of view. In the third member of the equation (6.33), we replace the inter-avatars distance by the distance between Alice’s and Bob’s spins conditioned by the homothecy of the equation (6.4). In the fourth member of equation (6.33), we go down the Lorentz factor to the denominator so that it gets involved with the speed leaving in the numerator the distance between Alice and Bob alone. Consequently, for an instant- neous communication between Alice and Bob, like that one between avatars, the only possibility is that the homothecy also reaches the speed of light. As we know nothing exceeds the speed of light, so, what equation (6.33) is really telling us is that locality dilates like a balloon that is inflated, that is, we speak of an equivalent effect and not of an original reality.

Everything said so far can be seen in detail in Figure 6.11. In the upper part of this figure, we can see two entangled particles in the Alice’s and Bob’s hands at points A and B, respectively. We also see an intermediate point Z where we know the avatars are. The lower part of Figure 6.11 shows us the combined effect analyzed above which is interpreted as if we had two equivalent giant particles of a diameter equal to the distance between Alice and Bob (potentially infinite), superimposed and centered at point Z and whose original spins are the avatars. They seem to be one giant particle because they are superimposed, but they are, actually, two. All this speaks to us of equivalent behaviors, exclusively, given that:

- nothing exceeds the speed of light,

- there are no two equivalent giant particles, and by the way, - avatars do not really exist either.

Entanglement works as if the original spins behave like the avatars, and by behaving like them we can justify the extraordinary attributes of entanglement without crashing GR and QM. At the same time, the true equivalence between these theories is explained.

68 Figure 6.11 Theory of Dilated Locality.

Figure 6.11 resembles an interferometry effect, by which small things operatively arranged in an appropriate layout equate to a single larger thing. This happens in radio astronomy where a kilometer antenna is simulated thanks to small antennas, forming different patterns on land [168, 169] or in Synthetic Aperture Radar (SAR) where the movement of a small airborne or satellite antenna synthe- sizes an equivalent antenna of kilometers which never could be transported practically by plane or satellite [170]. Both the radio astronomy equivalent antenna and the SAR equivalent antenna do not really exist, however, we enjoy their benefits every day.

Besides, in Figure 6.11, the equivalent contraction of the distance between Alice and Bob is equiva- lent to the apparent dilatation of c. We can symbolize the dilatation of the locality of Alice and Bob as,

equivalent original

locality locality



Also, it is as if the past light cone of the original spins was dilated. Finally, all this leads us to an inevi- table conclusion,

Non-locality



All these meanings of Figure 6.11 explain why when we touch something from an entangled pair, we touch everything, which is absolutely consistent with the collective behavior of entanglement through the Bell’s bases of equation (2.11) where individualities are completely displaced.