On the concept of simultaneity in relativity

This paper refutes Spavieri et al.'s claim that Wang et al.'s interferometric experiment disproves special relativity by demonstrating that their argument relies on circular reasoning and constitutes a logical fallacy.

The Gist

The Big Picture: A Battle Over "Now"

Imagine you and a friend are arguing about whether two events happening in different places happened at the exact same time.

• Your friend (Spavieri et al.) says: "Yes! Simultaneity is absolute. If two things happen at the same time for me, they happen at the same time for everyone, no matter how fast they are moving."
• The author (Lambare) says: "No, that's wrong. 'Now' is relative. What looks simultaneous to you might look like one event happened before the other to your friend. This isn't a bug in the system; it's a feature of how the universe works."

The paper is a defense of Einstein's Special Relativity against a claim that a specific experiment (the Wang Experiment) proves Einstein wrong.

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The Setup: The "Moving Conveyor Belt" Race

To understand the argument, we need to visualize the experiment described in the paper.

The Analogy: Imagine a giant, moving conveyor belt (the fiber optic cable) in a factory.

1. The Track: The belt is a loop. It has a bottom section moving one way and a top section moving the other way (or rather, the whole loop is moving, but the light beams go in opposite directions around it).
2. The Racers: Two light beams are sent around this loop in opposite directions. One goes with the flow of the belt, the other goes against it.
3. The Finish Line: They meet back at the start.

The Result: The scientists found that the two beams did not arrive at the exact same time. One got there slightly faster than the other.

The Critics' Argument (The "Paradox"):
The critics looked at this and said: "Wait a minute! If the speed of light is always constant (as Einstein says), and the belt is moving, the math doesn't add up. It looks like the light beam traveled a shorter distance than it should have. It's as if a chunk of the track is missing. They call this the 'Missing Path Paradox.'"

They argued: "Because there is a 'missing path,' the theory of relativity must be broken. Simultaneity must be absolute."

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The Author's Rebuttal: The "Time Travel" Illusion

Lambare says the critics made a classic mistake: They assumed the answer before they did the math.

Here is the simple breakdown of why the "Missing Path" isn't actually missing.

1. The "Two-Frame" Problem

The critics tried to look at the race from the perspective of a clock sitting on the moving belt. But here's the catch: The clock on the belt is not in a single, smooth state of motion.

• First, the clock is on the bottom section of the belt.
• Then, it has to turn a corner and go onto the top section.
• In physics terms, the clock changes its "frame of reference" (it switches from one moving coordinate system to another).

The critics tried to treat the clock as if it were in one single, unchanging world the whole time. But because the clock changes direction, it experiences a shift in how it perceives "now."

2. The "Time Gap" is actually a "Time Jump"

This is the most important part. The author uses a metaphor of jumping time zones.

Imagine you are driving a car (the clock) and you cross a border into a country where the clocks are set 1 hour ahead.

• You leave your house at 12:00 PM.
• You drive for 1 hour.
• You cross the border. Suddenly, your watch says 1:00 PM, but the local clock says 2:00 PM.
• You arrive at your destination.

If you ignore the time zone change, you might think you traveled faster than light or that you arrived before you left. But the "missing time" was just the time zone jump.

In the Wang experiment, when the clock moves from the bottom section to the top section, it effectively "jumps" in time relative to the light beam.

• The Critics' View: "The light beam didn't travel far enough to cover the whole track!"
• The Author's View: "The light beam did travel the whole track. But when the clock switched tracks, it realized that the light beam had already traveled a huge chunk of that distance before the clock even started its new leg of the journey."

3. The "Missing Path" is Just a Misunderstanding of "Simultaneity"

The "missing path" ($\delta d_\phi$) that the critics are so worried about? It's not a missing piece of the road. It is the distance the light traveled before the clock "woke up" to the new reality of the top section.

Because simultaneity is relative:

• In the bottom section's view, the light and the clock start together.
• In the top section's view, the light already started a long time ago.

When you add up:

1. The distance the light traveled in the bottom section.
2. The distance the light traveled in the top section.
3. PLUS the "head start" distance the light had because of the time jump (the relativity of simultaneity).

...You get the exact total length of the track. The "missing path" vanishes.

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The Core Logic: Why the Critics are Wrong

The author points out that the critics' argument is circular reasoning.

• The Critics say: "Relativity is wrong because it leads to a missing path."
• The Author says: "You only see a missing path if you assume that 'simultaneity is absolute'."

It's like saying: "I proved that gravity doesn't exist because I dropped a rock and it didn't fall. But I only dropped it because I assumed I was in a zero-gravity room."

The "Missing Path" only exists if you refuse to accept that "now" is different for different observers. Once you accept that time is flexible (relative), the math works perfectly, and the path is complete.

The Takeaway

• Intuition is tricky: Our brains are wired to think that if two things happen at the same time for us, they happen at the same time for everyone. The universe doesn't work that way.
• The "Paradox" is a teaching moment: The author argues that this "paradox" isn't a flaw in Einstein's theory; it's a great way to learn why relative simultaneity is necessary.
• The Verdict: The Wang experiment does not disprove Special Relativity. It actually confirms it, provided you understand that a clock changing direction experiences a shift in how it perceives time and space. There is no missing path; there is only a missing understanding of how time works.

Technical Summary

1. Problem Statement

The paper addresses a claim made by Spavieri et al. (Ref. [1]) regarding the Wang interferometric experiment (a modified Sagnac interferometer using linear motion). Spavieri et al. argue that this experiment disproves the Special Theory of Relativity (STR). Their core argument is that the experiment reveals a "missing path" and a "time gap" ($\delta t$) when analyzed using the Lorentz transformation. They conclude that simultaneity must be an absolute concept independent of the observer's state of motion, otherwise, the spacetime continuity in the fiber optic medium is violated.

Lambare identifies this claim as a logical fallacy based on circular reasoning: Spavieri et al. assume the falsity of relative simultaneity to prove that relative simultaneity leads to inconsistencies.

2. Methodology

Lambare employs a rigorous analysis of reference frames and the Lorentz transformation to deconstruct the "missing path paradox."

• Experimental Setup Analysis: The Wang experiment involves a fiber optic conveyor (FOC) moving at linear speed $\nu$. Two light beams travel in opposite directions. The phase shift is measured in the Laboratory (LAB) frame.
• Reproduction of the Counter-Argument: Lambare first reproduces the calculations by Spavieri et al., which involve:
  • Defining two inertial frames, $S''$ (resting with the lower fiber section) and $S'$ (resting with the upper fiber section).
  • Calculating the time and distance traveled by a photon in these frames.
  • Showing that the sum of distances calculated via the Lorentz transformation ($L' + L''$) is less than the total proper length of the fiber ($2\gamma L$), creating a "missing path" $\delta d_\phi = 2\gamma L \beta$.
• Relativistic Resolution: Lambare re-evaluates the scenario by strictly applying the Lorentz Transformation (LT) and the concept of relativity of simultaneity.
  • He defines two key events: $C_A$ (clock at point A) and $P_B$ (photon at point B).
  • He demonstrates that while these events are simultaneous in frame $S''$ ($t''_{CA} = t''_{PB} = 0$), they are not simultaneous in frame $S'$.
  • Using the LT, he calculates the time coordinate of event $P_B$ in frame $S'$, showing it occurs in the past relative to the clock's frame switch ($t'_{PB} < 0$).

3. Key Contributions

• Identification of Circular Reasoning: The primary contribution is exposing that the "paradox" only exists if one assumes absolute simultaneity. By assuming simultaneity is absolute, one forces the events $C_A$ and $P_B$ to be simultaneous in all frames, which contradicts the Lorentz transformation. Thus, the "proof" of inconsistency relies on the very premise (absolute simultaneity) it seeks to establish.
• Resolution of the "Missing Path": Lambare provides a geometric and temporal explanation for the "missing path" ($\delta d_\phi$). He demonstrates that the "missing" distance is actually the distance the photon traveled before the clock switched reference frames, a fact that is only visible when accounting for the relativity of simultaneity.
• Clarification of Non-Inertial Frames: The paper highlights that the paradox arises from naively applying "inertial intuition" to a clock that undergoes a change of inertial frames (non-inertial motion). The correct analysis requires stitching together descriptions from two distinct inertial frames ($S'$ and $S''$) using the LT.

4. Results

• Mathematical Proof of Consistency:
  • The "missing path" $\delta d_\phi = 2\gamma L \beta$ is shown to be exactly equal to the distance the photon traveled in frame $S'$ prior to the clock's arrival at the transition point, calculated as $c|t'_{PB}|$.
  • The total distance covered by the photon is reconstructed as:
    $$L_{total} = L'' + L' + \delta d_\phi = 2\gamma L$$
  • The total time of flight is reconstructed as:
    $$T_{total} = |t'_{PB}| + T = \frac{2\gamma L}{c} = (T)_E$$
  • This matches the expected time $(T)_E$ derived from the invariance of the speed of light, proving there is no time gap or missing path.
• Refutation of Spavieri's Reply: Lambare addresses Spavieri's rebuttal, arguing that Spavieri's claim that adding the time interval $t'_{PB}$ is an "ad hoc" adjustment is incorrect. The interval is a direct, necessary prediction of the Lorentz transformation. Furthermore, Spavieri's insistence that the total time the photon was in motion must be equal to the clock-registered time T ignores the fact that, in S', the clock changes frames after the photon had already done the same, so it is unjustified to require that it should register the whole time the photon was in motion.

5. Significance

• Defense of Special Relativity: The paper reinforces the validity of the Special Theory of Relativity against claims that it fails to explain interferometric experiments involving moving media.
• Pedagogical Value: It serves as a didactic refutation of similar anti-relativistic arguments. It illustrates that apparent paradoxes often stem from intuitive misunderstandings of simultaneity rather than genuine logical inconsistencies in the theory.
• Conceptual Clarity: It emphasizes that distant simultaneity is not an objective physical reality but is observer-dependent. The paper argues that the "missing path" is an artifact of ignoring this relativity, and that accepting relative simultaneity resolves the apparent contradiction in spacetime continuity.

In conclusion, Lambare demonstrates that the Wang experiment is fully consistent with Special Relativity and that the claim of a "missing path" is a logical fallacy resulting from the incorrect assumption of absolute simultaneity.