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I think spacetime diagrams are considered hard or too mathematical because
that was Einstein's first impression of them, and
many introductory physics textbooks take a historical/pseudo-historical development of relativity
[focusing on the ether, michelson morley, Einstein 1905, and then the classic effects (what I sometimes consider the classic traps and pitfalls) , and then appeal to experimental verification]
but very little on the more modern spacetime view of (that mathematician) Minkowski,
which might help develop intuition.
It's refreshing to see a new intro textbook (maybe not so new anymore) like Tom Moore's
Six Ideas that Shaped Physics, Unit R. http://www.physics.pomona.edu/sixideas/
(I like to point out that
the position-vs-time graphs of PHY 101
has an underlying non-euclidean geometry:
the Galilean geometry.
If you draw the clock effect/twin paradox scenario in Galilean physics, you get a triangle
where the kinked worldline of the traveler has the same length
(where arc-length is proper-time, measured with a wristwatch carried by that observer)
as the stay at home twin. That is, no time difference along the paths,
However, that triangle doesn't satisfy the triangle inequality.
In addition,
a t=const line is [Galilean-]perpendicular to all worldlines, regardless of their velocities (slopes)... which is the condition of "absolute time" and "absolute simultaneity".
That doesn't sound Euclidean.
However, we have been taught to read that ordinary position-vs-time graph a certain way.
So, we are generally unaware of it to complain about its nonEuclidean features
...but enter the spacetime diagram for special relativity... )
(Last quote, then I'll get off my soapbox.
http://aapt.scitation.org/doi/10.1119/1.17728
"Lapses in Relativistic Pedagogy" by Mermin
makes some good points
Now... back to the ideas for this thread.
As others have said, and as some of us have done,
draw a spacetime diagram.
(Since you don't usually solve a geometry problem with rotation matrices,
it might be good to try to solve special relativity problems using methods other than just the formulas for Lorentz transformations, length-contraction, or time-dilation.
Try trigonometry.)
that was Einstein's first impression of them, and
many introductory physics textbooks take a historical/pseudo-historical development of relativity
[focusing on the ether, michelson morley, Einstein 1905, and then the classic effects (what I sometimes consider the classic traps and pitfalls) , and then appeal to experimental verification]
but very little on the more modern spacetime view of (that mathematician) Minkowski,
which might help develop intuition.
It's refreshing to see a new intro textbook (maybe not so new anymore) like Tom Moore's
Six Ideas that Shaped Physics, Unit R. http://www.physics.pomona.edu/sixideas/
(I like to point out that
the position-vs-time graphs of PHY 101
has an underlying non-euclidean geometry:
the Galilean geometry.
If you draw the clock effect/twin paradox scenario in Galilean physics, you get a triangle
where the kinked worldline of the traveler has the same length
(where arc-length is proper-time, measured with a wristwatch carried by that observer)
as the stay at home twin. That is, no time difference along the paths,
However, that triangle doesn't satisfy the triangle inequality.
In addition,
a t=const line is [Galilean-]perpendicular to all worldlines, regardless of their velocities (slopes)... which is the condition of "absolute time" and "absolute simultaneity".
That doesn't sound Euclidean.
However, we have been taught to read that ordinary position-vs-time graph a certain way.
So, we are generally unaware of it to complain about its nonEuclidean features
...but enter the spacetime diagram for special relativity... )
(Last quote, then I'll get off my soapbox.
http://aapt.scitation.org/doi/10.1119/1.17728
"Lapses in Relativistic Pedagogy" by Mermin
makes some good points
)Mermin said:...Lorentz transformation doesn't belong in a first exposure to special relativity. Indispensable as it is later on, its very conciseness and power serve to obscure the subtle interconnnectedness of spatial and temporal measurements that makes the whole business work. Only a loonie would start with real orthogonal matrices to explain rotations to somebody who had never heard of them before, but that's how we often teach relativity. You learn from the beginning how to operate machinery that gives you the right answer but you acquire little insight into what you're doing with it.
Now... back to the ideas for this thread.
As others have said, and as some of us have done,
draw a spacetime diagram.
(Since you don't usually solve a geometry problem with rotation matrices,
it might be good to try to solve special relativity problems using methods other than just the formulas for Lorentz transformations, length-contraction, or time-dilation.
Try trigonometry.)