The paradox of coincidence

Abstract

This article is intended to state a paradox arising from special relativity.

I will only use well known and accepted propositions of physics and geometry and very simple inferences to achieve the following conclusion :

Any point in space-time coincides with all points of its light cone (passed and future) .

Since coïncidence is invariant in a change of coordinate system, coincidence should also be such for us, in our observer coordinate system. This is obviously not the case.

My expectation is to be proven wrong.

For clarity, I will identify any proposition taken from physics or geometry by a "Proposition"

Development:

The notion of coincidence is fundamental to physics. It is by means of the coincidence that coordinates of an event can be defined in a coordinate system.

Proposition: Two events P (x,y,z,t) and P' (x',y',z',t') * do coincide if their 4 coordinates, expressed in the same referential, are identical: x=x' y=y' z=z' t=t'

Proposition: If two events coincide in a given referential, they coincide in any other referential.

* Events P and P' are indeed "points" of space-time, only defined by their 4 coordinates, they have no physical reality associated.

The question of the photon.

That question goes back to the first days of the special relativity:

- For an observer travelling together with a photon, what would be the measure of his own journey ?

To answer that question, let's go back to the 2 pillars of special relativity:

Proposition:
- Equivalence of all coordinates systems in uniform translation relative to each other

Proposition:
- Invariance of the speed of light.

From these 2 principles one can deduce that (voir explications wikipedia):

- The space-time interval Δ s between 2 events is invariant

with Δ s² = c². Δ t² - Δ x² -Δ y² - Δ z² (where c is the speed of light)

that could also be written: Δ s² = c².Δ t² - Δ l²

- Δ s² = 0 for two events located on the trajectory of a photon (such as Δ l=c Δ t)

Fig 1

Let's consider 2 points P' et P on the trajectory of a photon.

Let's compare the coordinates of points P and P' in a coordinate system attached to the photon as defined by figure 1 (the Z axis is not represented)

In such a coordinate system, all 6 coordinates x,y,z et x',y',z' are null as a result of the system definition. Therefore x=x', y=y', z=z' and Δ l² = 0

Since both points are on the trajectory of a photon, P and P' are such that Δ l = c Δ t

Then Δ s² = 0

And therefore c Δ t² = Δ s²+ Δ l² = 0

Since c is not null, Δ t = 0 which means also t = t'

Since x=x', y=y', z=z' & t = t' point P' coïncides with P what we could note P'=P

Proposition:
The arythmetical relation of equality of coordinates is symetrical ( x=x' -> x'=x), the relation of coïncidence of points is also symetrical (P'=P -> P=P')

then P coïncides with P'

This reasoning is independant of the choice of P and P' on the trajectory of the photon.

Proposition:
The arythmetical relation of equality of coordinates is transitive ( x=x' & x'=x'' -> x'=x''), the relation of coïncidence of the points is transitive (P=P' & P'=P'' -> P'=P'')

Conclusion : All points of the trajectory of a photon coincide

That conclusion is paradoxal: Since coïncidence is invariant in a change of coordinate system, coincidence should also be such for us, in our observer coordinate system. This contradict our representation of space and time which actually shows a distance and a time lapse (possibly huge) between emission and réception of the photon.

This paradox is often disregarded on the grounds that it would result from a use of the Lorentz's transformation "on the limit" with v=c. But there is no use of Lorentz's transformation in the above reasoning.

Fig 2: 5 photons converging towards P

The question of the light cone

In figure 2 above, N photons are converging towards P. Applying the above reasoning one could say that:

P1=P'1 , P2=P'2 , P3 = P'3 , P4 = P'4 , P5 = P'5 .......Pi = P'i

In figure 2, points Pi may be as near of point P as we want to, but remain spacially distinct from P. This is aimed to clearly show that P is not a physical event, a unique observable effect where all photons would be absorbed. P is only a point in space-time.

Fig 3: light cone

In figure 3 above, a coordinate system is associated with point P.

A light cone is associated to point P.

Generators of that cone are the possible trajectories of photons (such as Δ l = c. Δ t).

Therefore, Pi=P'i on any generator and since generators are concurrent in P:

the top of the cone coincides with all points of its surface.

That conclusion seams paradoxal too. It contradicts our intuitive representation. If we stand, as observer, at point P, it seems to us incredible that all "emission" events of all light beams coming to us coincide with the "reception" event here and now.

Figure 4 hereunder shows that for each point P'i of a light cone with top at P, one can associate a light cone of which P'i would be the top.

On the first cone P=P'i and P'i = P''j on the second light cone..

Since coïncidence is transitive, P= P''j

One can reiterate that construction from P''j onwards, to build an endless assembly of light cones. P will coincide with all points of this assembly.

fig 4: transitivity of coincidence

I will refrain here to propose any explanation or comment to this paradox.

I will only point again that all points (P, P', P'' etc...) referred to in this article are only geometric, connected to no physical reality.

Such a situation can therefore only be assessed and not observed.

Normalized space:

Space-time of relativity as well as newtonian space are normalized. Coïncidence (ds²=0) as defined above, relies on that normalization. The article "espace désordonné" reminds us that to represent a physical situation in a normalized space, the situation shall be normel, i.e. the belonging (∈) shall be equivalent to inclusion (⊂). Belonging shall be transitive.

In real situations, the law that rules the belonging to the compound reality "particle" relies statistically on the interdependancies between its component realities.

It is very unlikely that this very law, when applied to the simplicity of the photon, allows to state that all states of the photon belog to the "particle".
Therefore, the condition for normality are no more fulfilled, the coincidence particle/photon does not apply to the states of the photon: The distance P/Pi is not defined in the frame where the particle is presented.
Note: The continuity of space time, as a concept to represent reality, should take this paradox and its consequences into account.

I believe that, when splitting a given physical reality into simpler subsets, ther is a point where the conditions of normality are no more fulfilled, space-time can no more be represented as normalized. Components are no more "inside" the compound.

To paraphrase A; Badiou (french ontologist) this is the site where a presented situation narrows the void, a "hole" in space-time of a presented situation where it can interact with other situations, the site for event, a connetion open to interactions.