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The spacetime diagrams below might be helpful. …

UPDATE (to address the comment by the OP): A spacetime diagram is useful to clarify the ideas. … Using my "spacetime diagrammer" https://www.desmos.com/calculator/dha8izuxz1 , with event $P(x,t)=(0.5,1)$, Alice (in RED) says that distant event P is simultaneous with tick-1 on her clock... …

I'll add vector-signs and hat-signs, but I'll only work in (1+1)-Minkowski spacetime with $(+,-)$-signature. Can you quote the source (the "tutorial")? Maybe I am misunderstanding the situation. …

So, one important value of the structure of Minkowski spacetime is that it can be used in more general situations than it was initially conceived for. …

Eq.10 is incorrect. Observe that: $$\frac{cT_2}{cT'}=\frac{(1+\beta)cT}{cT'}=\frac{1}{k}\tag{10}$$ implies $$ \begin{align} cT &=\frac{1}{(1+\beta)k}cT'\\ &=\frac{1}{(1+\beta)\sqrt{\frac{1+\beta}{1-\b …

Here is a spacetime diagram [drawn on rotated graph paper] of the back of the stick showing two reference frames, one for the back of the stick (through the origin event), and the other for the front … On a spacetime diagram drawn by another inertial observer who meets [the back of the stick at] event O and travels with velocity (-3/5)c with respect to the stick, the stick-ends' frames appear as shown …

(update2: So, it seems the length of path in a SPT-diagram is equal to the spacetime-interval between the endpoint-events (i.e. the proper-time for the inertial astronaut experiencing the endpoint-events … By constast, here's my variation of the standard Minkowski spacetime diagram: https://physics.stackexchange.com/a/507592/148184 (go to the "Clock Effect" section halfway in) as my answer to What is the …

Can we say that it's the time part that's involved in defining the space part and the space part in defining the time part? A displacement 4-vector has a time-component that has nothing to do with …

You seem to be confusing (or improperly comparing) the spacetime-displacement 4-vector of an event that is represented in a different frame by an equal-Minkowski-magnitude 4-vector with different spacetime-components … (addendum: These unequal-Euclidean-magnitude spatial-displacement vectors are the spatial-components of the equal-temporal-Galilean-magnitude spacetime-displacement 4-vectors related by a Galilean-boost …

For that worldline, all observers agree on the readings of the clock on that worldline. ) The Spacetime [square-]interval is like the squared-distance between two points in the plane (taken along a straight … In Minkowski spacetime, the square-interval from chronologically related events A to B is the square of the proper time along the inertial worldline from A to B. …

For completeness [especially for those without easy access to the reference], it might be good to include the other assumptions. Each admissible observer presides over a 3-dimensional, right-handed, …

Note that Galilean physics is all you need to answer this question. If you draw the position-vs-time graph of a plane moving with velocity v, you have a line with slope v. If you reorient the axes so …

Yes. A hypersphere is a hypersurface that is not a hyperplane. Hyperspheres and hyperplanes are examples of hypersurfaces. It's akin to a sphere and a plane being hypersurfaces in three dimensions.

Write $\Delta s_{OQ}^2=\Delta t_{OQ}^2-\Delta x_{OQ}^2$, where we interpret the spacetime-displacement $\Delta s_{OQ}$ and $\Delta t_{OQ}$ is the adjacent side and $\Delta x_{OQ}$ is the opposite side. …

In special relativity, one uses a spacetime diagram, with the underlying Minkowski metric. … The time dilation problem is essentially relating "the spacetime interval (the elapsed proper time) along a straight worldline segment from event A to event B" with "the temporal-component of the spacetime-displacement …