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For Moontanman: in the above, that 0.5c is as measured by the space station, and each rocket to the station; and the 0.8c is as measured by each rocket to the other rocket. For the space station, the closing speed of the rockets (which is not the speed of either rocket) is c.

I think you really need to get clear on the difference between measuring closing speed (which is the speed of a gap, a nothing), and the speed of a thing. And it depends a lot on who is observing. Say the fastest car in the World can do 500 km/h. Stick two of them on a track facing each other and run them, at top speed. The gap between (from the point of view of the track) them is decreasing at 1000 km/h. Hang on! That's faster than the car can go! But that 1000 km/h isn't the speed of either car according to the track. Sure, if the track considers it from the point of view of one of the cars, then the other is getting closer at 1000 km/h, but that reference point is moving according to the track. It's an illusion, if you like. Relativistic addition comes in (in this scenario) when you consider the point of view from one of the cars. Each car can consider itself as still, and the other car moving towards it. But note that the track is also moving towards it! The track is moving towards each car at 500 k/h, and the other car is moving towards it at 500 km/h relative to the track. And that's where you cannot (at relativistic speeds where it starts to matter) just add the 500 and 500. Each car will consider the other car approaching at 999.99999 km/h. The track considers the closing speed as 1000 km/h, and the cars consider the other is approaching at 999.99999 km/h. These are different numbers.

You are confusing yourself by being imprecise. You cannot make your mind up on whose point of view is involved. Who is the observer, one of the rockets? Or someone else? (And: As noted before, someone else (not in the rockets) can consider the rockets as approaching each other at more than c, closing speed can be as much as 2c.)

No they can measure speed relative to themselves, as in your first post. In your first post, rocket A (to give it a name) measures rocket B to be coming at 0.99999c. And vice versa.

The existence of the space station doesn't affect anything. But if that's what speeds are being measured against, it changes your scenario. You started off saying each rocket measures the other as going some speed. Well then that's just what they measure. But if you change the scenario so that each rocket measures some speed from themselves to the station, and the other rocket as at some speed relative to the station, then yes, each rocket needs to use the proper formula to determine the speed of the other rocket relative to themselves.

Then swansont gave you the answer.

Try rephrasing the question with a little more precision on who is measuring what.

Edit 2: darn, misread. I'll leave this here but as swansont points out I've misread your question. [ The formula is here: https://en.wikipedia.org/wiki/Velocity-addition_formula The "interesting" bit (to me) is the division by 1 + ( ( v x u' ) / ( c x c ) ) Where v and u' are small fractions of c, this ends up being a division by almost 1, i.e. at regular day to day speeds, it hardly makes a difference. But where v and u' are large fractions of c, that becomes close to a division by two, i.e. you can add to speeds close to c, and still get a speed close to c. As exchemist points out, you need to be careful who thinks what speed is what. edit: but don't mix this up with closing speed, that can be as much as 2c. e.g. someone considering themselves as still, who sees two rockets coming towards them from opposite directions, each at 99% of c, does see the gap between the rockets decreasing at close to 2c. But that's not the same as seeing a thing moving faster than c; neither that middle observer nor either rocket sees any rocket going over c here. ]

You may be looking for: from the equivalence principle, clocks at the tip and base of an accelerating rocket will measure different times.

I'd add: learning latex would be a good start. In the photo of math, it's not all oriented the same way as the forum, and being a photo is impossible to properly quote, or dissect for discussion. Nobody is being paid to review your posts, if you want feedback you need to make the interest high and the effort low. I will say your handwriting is very good.

Are you claiming the acceleration is a direct cause of the differential aging in the twins' paradox?

Oh sure, I actually agree here, I did note 'I personally go with "there is none"'. But it's like "proving a negative", hard to be absolute. (Back when I was busy (ha!) dropping out of University I went to exactly 1 philosophy lecture, and this was the topic.)

Lack of proof there is a god of some kind, isn't proof that this god doesn't exist. There are also plenty of reasons why reasonable people find comfort in some kind of belief, including upbringing (indoctrination?) and nervousness at the unknown. Not that I'm agnostic, I personally go with "there is none", but I think it's a bit too strong to say "delusion". It did sell some books, though. (dimreepr, is the incorrect "you're" in your signature ironic?)

No, because your math is gibberish. Yes. And you don't even need math to see it. You are missing the point. Top half of image: In the rest frame of the light clock (i.e. anything at rest with it: in the same train - in your unattributed image) the pulse of light is bouncing between A and B, travelling L back and forth. This is not about seeing the pulse of light, it's just doing what the pulse is doing. Bottom half of image: Considered from a different inertial frame, one where the train and the light clock are moving from left to right, the pulse of light makes a different path, travelling D back and forth. D is longer than L. But the speed of light was earlier shown to be invariant. For the same pulses of light to travel from A to B at the same speed, over different distances: it must be that time is relative.

Where it would be, is non-flat space. Edit: snap

Did they pay you, or did you pay them? Who reviewed it?

(For what it's worth, in English I've only ever heard "nickname", not "nick". YMMV.)

That's also how you ended your last thread on this topic. Nobody is disturbed. I remember many many years ago at school, sitting on the step of a building watching someone about 150m away bouncing a basketball on the footpath. Thanks to the brick wall of the building they were outside, I was also hearing the ball hit the concrete. They stopped bouncing the ball. And I heard one more bounce. Slightly jarring at first. The distance was such that (like watching a movie where they sync up visuals and sound and ignore reality) I was seeing and hearing the bounce at the "same time", but of course the sound of each bounce was getting to me later than the sight. None of this is new.

Did the Universe exist before brains?

I don't know about that, but if we treat 55 and 145 as lengths in metres, you could fit 450 average bananas between them.

Do authors pay to be published by them?

Web forums are more dangerous than the printed word, I'd say.

Nothing that big would have stayed secret. https://en.wikipedia.org/wiki/Atomic_spies

Can you use pipeclamps and bolts? Glue is pretty one-way. (Though drilling polycarbonate sheet for bolt holes can be tricky.)

As a rule of thumb (i.e. close enough), you'd block Proxima Centauri from all viewers on Earth at the same time with a disk the diameter of Earth, if the disk is next to Earth, or the diameter of Proxima Centauri if the disk is next to that star. Anywhere in between would be a ratio. (Give or take a little gravitational lensing, and assuming everything stays still ... etc ...)