Each depth is written as a self-contained route. Choose one without needing to read the other two, or use Read all for a continuous article.
Intuition
A wave and a message to one receiver are not the same quotient
A source emits a marked electromagnetic disturbance. In a chosen external frame, let the front propagate at \(c\).
A receiver is ahead of it. If the receiver moves away, the front takes longer to reach it. If the receiver moves towards it, reception occurs sooner.
Nothing about that statement requires the wave to have propagated at anything other than \(c\).
The message is defined by two events
A specific message begins at emission and ends at reception. Moving the reception endpoint therefore changes the message transit time.
That is different from asking how fast the wave front propagates through the chosen coordinate frame.
The Argument
1. Wave trajectory
Let the source emit at \(x=0,t=0\). In the selected frame, the marked front follows
Hence
2. Receding receiver
Let the receiver begin a distance \(L\) ahead and recede at speed \(v\):
Reception occurs at the intersection
Therefore
3. Define the receiver-dependent message rate
If the operational question is how quickly the original source–receiver separation \(L\) is completed by this message, define
Then
For approach,
The numerator is the initial endpoint separation, not the physical distance travelled by the wave front.
4. The wave itself still travels at c
For the receding receiver, the wave actually travels
Hence
Both statements are simultaneously true because they answer different questions.
5. Local measurement is a third question
A receiver can perform a local light-speed measurement with its own clocks, rulers or phase measurements. Standard special relativity assigns the result \(c\).
Path II does not replace that local result with \(c\pm v\). It asks whether the local wave measurement and the emission-to-moving-receiver message rate should be silently treated as the same physical quantity.
Deep Notes
The distinction comes entirely from which distance and which pair of events are used.
1. Emission and reception
Let event \(A\) be emission and event \(B\) reception. The electromagnetic disturbance connects them causally, but the source event itself does not travel.
2. Wave speed
Following the front through the chosen frame gives
3. Message completion
For a receding receiver,
Thus
Using the original endpoint separation as the operational message distance gives
For approach, the same geometry gives \(c+v\).
4. Why this is not a new light-wave speed
The physical front travels farther than \(L\) when the receiver recedes and less than the naive fixed-endpoint geometry when the receiver approaches. Dividing the actual front path by the same coordinate travel time returns \(c\).
The receiver-dependent quantity belongs to the emission–reception task, not to the intrinsic propagation law assigned to the front in that coordinate frame.
5. The canonical distinction
The rest of Path II examines what happens when these quantities are collapsed into one phrase and then used to construct distant time.