(For historical purposes, this thread is preceded by (in reverse chronological order)
)
In the SE question, the OP drew some
clock-faces to go along with "events", which inspired me to draw (near the end of the thread) a spacetime diagram in Geogebra with those events and the associated clock faces pasted on the diagram. This prompted me to modify my Desmos spacetime-diagrammer to also show associated clock-faces. And so, here we are....
But, for completeness, some definitions first.
A "spacetime diagram" is essentially a "PHY 101 position-vs-time graph" (when and where stuff happens)
that one draws on a blackboard or sheet of paper.
(The following statements in this paragraph arebased on Geroch's General Relativity from A to B.)
A "point" on the graph is called an "event",
which physically represents (e.g.) "the snapping of one's fingers",
idealizing a very-short-lived occurrence in a very-small-volume.
Two such occurrences mark "the same event" if they occur "at the same space at the same time",
regardless of what caused the marking.
When "clock faces that reads 1 second emits a light flash", that is one event.
Just as one can place a sheet of graph paper in various orientations on the Euclidean plane
and then use that graph paper to assign coordinates (x,y) to describe events,
the use of a differently-oriented sheet of graph paper would assign a different set of coordinates (x',y') to the describe the same events.
In spite of these different descriptions, there are certain quantities called "invariants" (like the square-distance between two points using the Pythagorean theorem ) that have the same numerical value regardless of the orientation of the graph paper: That is, ##{(x_B-x_A)^2+(y_B-y_A)^2} = {(x'_B-x'_A)^2+(y'_B-y'_A)^2}##, which is an example of an invariant in Euclidean Geometry. This invariant reveals the importance of "the circle" in Euclidean geometry.
Similarly, on a spacetime diagram in special relativity, (t,x)-coordinate descriptions of events by different inertial observers lead to different coordinate-assignments of various events. However, the quantity called the square-interval ##{(t-B-t_A)^2-((x_B-x_A)/c)^2} = {(t'_B-t'_A)^2-((x'_B-x'_A)/c)^2}## is an example of an invariant in special relativity (developed by Minkowski in 1907 who realized that one can "do geometry" on a position-vs-time graph that encodes Einstein's results from 1905, in analogy to ordinary geometry done on a Euclidean plane). This invariant reveals the importance of "the hyperbola" in Minkowski spacetime-geometry. (Later, in retrospect, it was realized by some that one can consistently do suitably developed "Galilean geometry" on the PHY 101 position-vs-time graph. Had Galileo worked with Descartes on studying Euclid, they may have developed a flat nonEuclidean geometry that is now called Galilean-spacetime-geometry. )
We can represent "the history of a point-particle" (when and where it was)
by a continuous curve in the position-vs-time (spacetime) diagram called its worldline
(although timeline might be a better, more-descriptive term).
"When" is determined by the particle's wristwatch [in the spirit of Taylor and Wheeler's Spacetime Physics], which always flows steadily increasing.
"Where" is determined by wherever the particle is.
In Galilean relativity and in special relativity, this curve never intersects itself
and is subject to the associated causal structure. For special relativity,
at each event on the worldline, the tangent-vector to the worldline must always point
into the interior of the future lightcone of that event.
So, an important fact is that
at each event on the particle worldline, the wristwatch reads a unique time.
The wristwatch is never stuck, never reading two values at the same event, never changing its direction and rate (as read by the particle, its owner), never jumping around.
If you specify a time-reading, there is exactly one event on the particle worldline corresponding to that time-reading.
Analogously, on a highway, the milemarker-values change, never two differently-valued milemarkers at the same place on that highway, never changing its direction and rate (as read by the odometer of a car following the highway), never jumping around.
the actual wristwatch-reading is not necessarily equal to the t-coordinate assigned by other observers.
Minkowski would call the wristwatch-reading the particle's "proper-time"
(eigenzeit meaning "own time", "proper time" as in "property"--not "proper" as in proper-vs-improper).
Bondi would call the wristwatch-reading the particle's "private-time" (as opposed to "public time").
So, here are some diagrams from the Desmos script,
drawn by inertial-observer Alice, with inertial-observer Bob traveling with (3/5)c with respect to Alice.
- FIG 1 (Using the event labeling from an earlier thread)
The event E1 "Alice's wristwatch reads 1 second" is simultaneous-according-to-Alice with
the event E2 "Bob's clock reads 0.8 seconds".
The event E3 "Bob's clock reads 1 second" is simultaneous-according-to-Bob with
the event E4 "Alice's clock reads 0.8 seconds".
Although the results "the other clock reads 0.8 when mine reads 1.0" are the same,
we have four distinct events:
a pair simultaneous-according-to-Alice (E1 and E2) and another pair simultaneous-according-to Bob (E3 and E4).
These distinct have four different clock-faces: 0.8 and 1 on Alice's red clock and 0.8 and 1 on Bob's blue clock.
On a spacetime diagram, following Minkowski definition of "normal" or "perpendicular",
simultaneity is determined by the tangent to the "circle" where the radius (the inertial-observer's worldline) meets the "circle".
On a spacetime diagram decorated with light-clock-diamonds, simultaneity-according-to-an-inertial-observer is along the direction of that inertial-observer's light-clock-diamond's spacelike diagonal.
- FIG 2: "the exercise" from my last posts in https://www.physicsforums.com/threads/how-does-reciprocal-time-dilation-work.1085952/page-3
What event on Bob's worldline does Alice regard as simultaneous with E?
What event on Bob's worldline does Bob regard as simultaneous with E?
The event "Bob's clock reads 0.80" is simultaneous-according-to-Alice with
the event "Alice's clock reads 1".
The event "Bob's clock reads 1.25" is simultaneous-according-to-Bob with
the event "Alice's clock reads 1".
(Drag the T_Bob event from T_Bob=1 to T_Bob=1.25.)
- FIG 3:
Although we had (from FIG 1)
the event E1 "Alice's wristwatch reads 1 second" is simultaneous-according-to-Alice with
the event E2 "Bob's clock reads 0.8 seconds"---by a calculation (e.g. radar, as shown below in FIG 4),
what Alice [optically] sees (views) is
the image [signal] of
the event "Bob's clock when it read 0.5 seconds".
(This is essentially the Doppler effect,
##k=\frac{T_{\rm reception\ period}}{T_{\rm emission\ period}}=\frac{1.0}{0.5}=2##,
which is expected for ##v=(3/5)c## used in ##k=\sqrt{\frac{1+(v/c)}{1-(v/c)}}##.)
- This is why, when referring to simultaneity or the assignment of (t,x)-coordinates,
"views" or "sees" is possibly ambiguous.
- FIG 4:
The event E1 "Alice's wristwatch reads 1 second" is simultaneous-according-to-Alice with
the event E2 "Bob's clock reads 0.8 seconds"---by a calculation
by a radar measurement of E2.
For Alice to measure (assign coordinates) to event E2,
she must send a signal at ##t_{send}=0.4## and
wait for its echo to be received at ##t_{receive}=1.6##.
So Alice assigns to E2:
##\Delta t=\frac{1}{2}\left( t_{receive}+ t_{send} \right)=1.0## and
##\frac{\Delta x}{c}=\frac{1}{2}\left(t_{receive}-t_{send} \right)=0.6##.
- FIG 5: Of course, by the Relativity Principle,
Bob will have the same measurements of Alice.
- I think FIGS 1-5 exhaust what we can do with two inertial observers.
- FIGS 6 and 7: It may be enlightening to introduce Carol, an inertial-observer in the median frame
(akin to the direction along the angle-bisector).
##v_{Carol}/c=\tanh\left( \frac{{\rm arctanh}(v_{Alice}/c) + {\rm arctanh}(v_{Bob}/c)}{2} \right)##.
In Carol's frame, Alice and Bob are traveling in opposite directions with the same speed.
In Carol's frame, Alice and Bob clocks read the same time,
although this time is smaller than Carol's clock time.
- According to Carol,
the event when "Carol's clock reads 0.2828s" is simultaneous-according-to-Carol
with the event "Alice's clock reads 0.2667" and with the event "Bob's clock reads 0.2667",
as these events are on a line parallel to the spacelike diagonal of Carol's light-clock-diamonds.
- According to Carol,
the event when "Carol's clock reads 1.0606" is simultaneous-according-to-Carol
with the event "Alice's clock reads 1" and with the event "Bob's clock reads 1",
as these events are on a line parallel to the spacelike diagonal of Carol's light-clock-diamonds.
Admittedly, the graphics are constrained by what can be done easily in Desmos.
But I think there is enough interactivity to help further develop intuition and "spacetime thinking".