On page 227 the title of section 10.1 emphatically proclaims
_Straight_ is a relative term
I would argue just the opposite: (1) There is a notion of "straight" defined by geometry, without reference to relativity. (2) Indeed relativity depends on a prior notion of straightness, not the other way around.
On Day One of relativity, i.e. Day One of modern science (1638) Galileo specified that it applied to uniform motion of the ship. That is, he depended on a prior notion of straight, uniform motion.
Later, Sir Isaac said:
"The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn."
Applying this to the example in figure 10.2: I would argue that the graph of y versus t was curved all along. If you plot in in such a way that the t-axis is grossly de-normalized and collapsed to a point, then you are looking at it from exactly the wrong angle, from the one angle in all the world where the _projection_ of the trajectory onto your field of view _appears_ straight.
In contrast, when we boost into a reference frame that is moving past the scene with some x-velocity, that is a rotation in the xt plane. That allows us to "look around the corner" and see that the curvature in the trajectory. The x-position becomes a proxy for t. I made a diagram showing x, y, and t together. Go to
https://www.av8n.com/physics/spacetime-welcome.htm#fig-straight-nonstraight-xyzt
and push the "launch graphics" button. To get something analogous to figure 10.2, rotate the scene so that you are looking straight down the t axis.
To repeat: The trajectory was strongly curved in the yt plane all along. Sometimes the projection of a curved thing looks straight, but only if you are looking at it from exactly the wrong angle.
I would want to simplify this before offering it to students in the introductory course, but the basic idea is true and important: We can define straight without reference to relativity, but not vice versa.
See item #201 for a catalog of issues related to special relativity.
On page 227 the title of section 10.1 emphatically proclaims
I would argue just the opposite: (1) There is a notion of "straight" defined by geometry, without reference to relativity. (2) Indeed relativity depends on a prior notion of straightness, not the other way around.
On Day One of relativity, i.e. Day One of modern science (1638) Galileo specified that it applied to uniform motion of the ship. That is, he depended on a prior notion of straight, uniform motion.
Later, Sir Isaac said:
Applying this to the example in figure 10.2: I would argue that the graph of y versus t was curved all along. If you plot in in such a way that the t-axis is grossly de-normalized and collapsed to a point, then you are looking at it from exactly the wrong angle, from the one angle in all the world where the _projection_ of the trajectory onto your field of view _appears_ straight.
In contrast, when we boost into a reference frame that is moving past the scene with some x-velocity, that is a rotation in the xt plane. That allows us to "look around the corner" and see that the curvature in the trajectory. The x-position becomes a proxy for t. I made a diagram showing x, y, and t together. Go to https://www.av8n.com/physics/spacetime-welcome.htm#fig-straight-nonstraight-xyzt and push the "launch graphics" button. To get something analogous to figure 10.2, rotate the scene so that you are looking straight down the t axis.
To repeat: The trajectory was strongly curved in the yt plane all along. Sometimes the projection of a curved thing looks straight, but only if you are looking at it from exactly the wrong angle.
I would want to simplify this before offering it to students in the introductory course, but the basic idea is true and important: We can define straight without reference to relativity, but not vice versa.
See item #201 for a catalog of issues related to special relativity.