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The text

First, look at the first paragraph of the second page. It says:

The things that you can see at some time are the inside of the light-cone at o.

I don't get it. Suppose an event happened outside the light cone. Light from the event will reach me at some time, right? Because... Light keeps moving. For example, say, an event happened at (-1s, 5 Light second). So 1s in the past and 5 Ls away. This is clearly outside of my light cone at o. But I will see this event once light reaches me from there after 5s.

Next, the last paragraph of the second page. I only understand the first two lines there. I've no idea what this part is about:

As the horizontal plane hits the hyperboloid containing q, what you will see is a point that turns into an expanding sphere about you. As you watch the sphere expand, while the physical points are moving away from you, the spacetime distance between your birth and the sphere remains same. The increase in time is canceling out the increase in spatial length in the Hyperbolic Pythagorean Theorem, equation (1), since the separation is constant.

When the horizontal plane hits $q$, I should just see the event $q$ happening, right? What expanding sphere is it talking about? I don't get any of the next lines.

The text is from A Geometric Introduction to Spacetime and Special Relativity by William K. Ziemer.

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  • $\begingroup$ web.csulb.edu/~wziemer/Papers/specialrelativity.pdf $\endgroup$ – JEB Jun 26 at 13:09
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    $\begingroup$ The first problem is just careless writing. "The things you can see at some time" means "the things you can see at some particular instant in time." At a later instant in time you will see more things because their light has now reached you. But translating the second paragraph into English is beyond me, sorry! $\endgroup$ – alephzero Jun 26 at 13:32
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As far as I can tell, there is some colloquial use of the word "see", and a disregard for things like pluperfect future conditional. When you're created at $o$, the future light cone contains all the points you can possibly visit.

I suppose you could say see, if for instance you consider your point at $(-1, 5)$. You will "see" that event at $(4,0)$, and the distance between $(-1,5)$ and $(4,0$) is zero, and the point at which you see it is indeed in the forward light cone of $o$. So the act of seeing $(-1, 5)$ is really looking at light at $(4,0)$, but that is a confusing interpretation, though valid.

Regarding the locus of with fixed magnitude $||r||$ from $o$: A lot of confusion in relativity (at least on the internet) comes about from the concept of "see"-ing. Does it refer to what you physically see using light rays with delays and Doppler shifts, or is it what you would observer using all the apparatuses available to you in your thought experiment that has perfect rulers, clocks, and reconstruction of events? It is usually the latter.

The locus of points with time-like separation from $o$ equal to $||r||$ is not something you "see" very well. You cross it at $(r,0)$ and then it expands away from you at greater then the speed of light forever. Its significance is that it contains all the points in which your age is $r$ for some reference frame.

Note that the complementary space-like locus of points is something you never see. Though it is moving slower than the speed of light, it is always just out of reach of your light cone throughout your stationary life. It contains all the points that are simultaneous with your birth and a distance $r$ from it, in some frame.

Regarding your horizontal plane: that is your definition of "now". (IMHO, understanding what "now" is everywhere and how it changes with velocity is the most important single part of special relativity). When your horizontal plane passes $q$, that means you say "q is happening now", but you don't know about unil later, because at the time it happens, it is outside your light cone. Again, this is the problem with considering "what you see" versus "what you consider your reality based on complete knowledge allowed by thought experiments".

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  • $\begingroup$ Does the hyperboloid become a sphere when my "now" plane intersects it? Why does it expand? And what do you mean by 'seeing the locus of points'? The locus lies in spacetime, while eyes can only see 3D space. $\endgroup$ – Ryder Rude Jun 26 at 14:47
  • $\begingroup$ @RyderRude BE careful - your eyes don't "see 3D space" at all. They just respond to photons that hit them. Mixing up thought experiments with reality can be a $\endgroup$ – alephzero Jun 26 at 15:04
  • $\begingroup$ @alephzero Okay, let's assume I can observe points on 4D spacetime. I still don't understand why I'd observe the locus of points, having constant distance, as expanding with speed greater than light. What does it mean? And what's the sphere? $\endgroup$ – Ryder Rude Jun 26 at 15:15
  • $\begingroup$ like I said, seeing doesn't mean looking it, it means "this is what happens in your reference frame". Distance in spacetime is $x^2-ct^2 = r^2=$constant, so as $t$ increases so does $||x||$. Note that the slope of the hyperbola is always less than $c$, meaning that the place where it intersects fixed $t$ moves outward faster than $c$. $\endgroup$ – JEB Jun 26 at 22:39

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