NHarhan

Since you're on this forum, I assume you watch Isaac Arthur videos, which also makes me guess you like educational YouTube videos. Thus, I wonder whether you've seen these YouTube series by physicists about relativity:

Don Lincoln from Fermilab: www.youtube.com/watch?v=BhG_QZl8WVY&list=PLCfRa7MXBEspw_7ZSTVGCXpSswdpegQHX
PBS Space Time: www.youtube.com/playlist?list=PLsPUh22kYmNAmjsHke4pd8S9z6m_hVRur (also www.youtube.com/watch?v=fHRqibyNMpw&list=PLsPUh22kYmNCLrXgf8e6nC_xEzxdx4nmY&index=4 points out that Hub's photon clock is indeed similar to how the interactions in matter lead to our perception of time.)
For detailed video explanations of the math, you might check out: www.youtube.com/watch?v=ev9zrt__lec&list=PLkyBCj4JhHt_pz8HUG7rbMeKFsStae10k

I don't really understand general relativity yet, at least not on a mathematical or deeply conceptual level, but the math for special relativity really isn't that hard. It's mostly algebra, though sometimes people use more advanced types of math with it, as they always do. Play around with the math and Minkowski diagrams for a while (maybe with an online graphing calculator tracking lots of points, like I did). Also, on the subject of math, especially linear algebra, which is useful here because A: the Lorenz transformation is a linear transformation, and B: the equations of General Relativity use tensors, which linear transformations are an example of, I highly recommend 3Blue1Brown videos. Also, my actual introduction into the special relativity and how it actually worked that helped me start to understand it was a 50s or 60s edition of a little book Einstein wrote called "Relativity: The Special and the General Theory" (from "Über die spezielle und die allgemeine Relativitätstheorie") designed to teach the concepts without anything more than high school math, although it does take a lot of thinking, and the way I learned from it is that I read the first half a few times and messed around with the math; and I've read the whole thing once or twice, without really managing to fit it all in my head. My dad was probably right when he said that Einstein's book probably wasn't the best introduction to relativity, although I did like the way it started out explaining basic philosophy of math and physics and then explained the logic that lead to the development of the theory (or at least logic that could lead to it). I've also never seen the discussion of the relativity of rotating objects anywhere else yet, although I know it was a big area of research for relativity physicists in the first half of the twentieth century.

EDIT:
Firstly, since I like Minkowski diagrams, here are a couple in case you don't know what I'm talking about:

en.wikipedia.org/wiki/Minkowski_diagram#/media/File:Lorentz_transform_of_world_line.gif
(The dots on the line in the linked animation are ticks of the object we're following's clock as we move our reference frame along its worldline at one tick per second of our time. You can see how these ticks look farther apart from each other in time when we look at parts of the object's path that are at different velocities from the current velocity. That represents the fact that objects moving on those paths would seem time-dilated to our object if it didn't accelerate to enter those reference frames. The other dots are events that happen at specific times and locations, not physical objects in space, as those would exist across multiple times and thus look like lines. The reason the description says "momentarily co-moving inertial frames" is because our time axis extends straight along the path of a constant-velocity object, when we could have instead used a reference frame where the entire path of the object was a straight line, which would distort everything else so that paths of non-accelerating objects would look like they were accelerating, although I think everything would just move down in straight lines as we scrolled across the worldline after we'd done this, since we'd only have to change our time coordinate as we scrolled rather than our velocity as well. This would be an accelerating reference frame—one with changing acceleration/gravity, in fact. Space-time coordinates in general relativity are based on something called "Gaussian coordinates" rather than Cartesian coordinates like we're used to. I think this ability to make any curve into a straight line if you want is related to the idea I've been told that General Relativity works just as well if you use curved coordinates on flat spacetime as if you use coordinates as straight as possible on fundamentally curved spacetime.)

Also, I want to point out a couple of things I don't see mentioned much about relativity.

1) When you see popular explanations of how gravity is really just curved spacetime, they often show planets making a depression in a sheet representing space. First of all, it's important to note that the geometry would work exactly the same if planets created hills rather than depressions, since all that matters is how the curvature affects "straight lines" on the surface; it has absolutely nothing to do with objects falling down the slope created by bending space, which would involve some external force of gravity in some other dimension space was embedded in. Of course, 3D space is embedded in another dimension — time, and I'm told that time curvature is really what causes gravity (though that sounds like a rather imprecise statement), so maybe the popular explanation isn't too far off in that respect. I don't actually understand it properly*, so I won't claim that.

*It does sort of make sense to me, though, in that time is required for things to fall. Imagine a "Minkowski space-time diagram", where the x-axis is one direction of space is distance along a particular line in space and the y-axis is time. (Usually, either measure the x-distances in light-time units (i.e., divide it by the speed of light) or measure the time in distance units (i.e., multiply it by the speed of light).) In flat spacetime, i.e., no gravity, two objects floating next to each other will look like vertical lines, with position staying the same at every point in time. (If they were moving relative to your coordinate system, but at rest relative to each other, they would look like parallel lines.) However, under if gravity were pulling them together, then their lines would have to cross at some point. General relativity says that rather than the lines of the the objects ("worldlines") curving to meet each other, spacetime curved so that they meet each other, sort of like if you had drawn your Minkowski diagram on the equator of a sphere and the worldlines of the two objects meeting at the pole like longitude lines (although obviously not exactly like that in most cases, since all the latititude lines on a sphere meet at the same time, so that would bring everything in a finite universe together into one point at the same time, like the Big Crunch). What I don't understand is how a massive object like a planet, which would look like a line on a Minkowski diagram, can create curvature that would do this. I strongly suspect that once I do know, then I will also understand why such curvature (gravity) causes time-dilation.

2) The effect of the Doppler shift looks like time dilation, but is actually a totally different effect on top of time dilation: If a rocket car is driving towards you at half the speed of sound blaring ridiculously loud music, that music will sound to you like it's playing at double speed. Imagine each beat of the song moving away from the car and towards you at the speed of sound for one 4-beat measure before it almost hits you. (What a jerk!) (Just to explain a measure to non-musicians, this should be "1, 2, 3, 4, almost-hit" from the car's perspective). When beat 1 reaches you, the car will already be half-way closer to you, and therefore emitting beat 3. In general, the car will always be half the distance away when you hear a sound as it was when that sound was emitted. The following table will be easier to follow than any explanation I can write of the other beats times:

("Passby" would be beat one of the next measure.)
(If you consider that the car is moving at constant speed, then you can see that the distance the car is at is also a good coordinate for the time events happen at.)
(Also note that I've defined a unit of distance that has a remarkably convenient length for this problem, no need to calculate what that is, which would depend on the tempo of the song and the speed of sound in the air.)
(Sure would be nice if there were a way to insert tables and I knew it, but alas no.)

As you can see, this adds up to the song sounding twice as fast from your perspective, since the beats are all scrunched up in front of the car and hit you in fast succession.
Afterwards, it will be moving away from you at half the speed of sound, and the effect will be one of making the music sound like 3/4 speed to you (so a measure is 150% as much time):

*This is what was called "passby" before — beat 1 of the second measure.
**= of measure 3
(I'm using negative units just to emphasize that the car is now moving in the opposite direction from the table above.)

This is because the beats have to travel back towards you at the speed of sound and each beat is emitted from a further distance away. If you imagine where the waves are in space, they are stretched out further apart from each other behind the rocket car, just as they are scrunched up in front of it. (The math is more confusing on this one because the sound-waves are moving back towards you at the speed of sound while the car is only moving away from you at half the speed of sound. If it were moving away at the speed of sound, the tempo would be halved.)

That's the Doppler effect with sound. It affects tempo of songs just as much as it affects pitch of sound waves. The Doppler effect with light also affects tempo of songs played on radios as much as it does the frequency they are transmitted at, but is different for two reasons. A: Sound moves relative to the air: if you were in the rocket car driving past someone standing still who was playing ridiculously loud music, you would actually get different results (1.5x speed on approach and .5 speed afterwards). This is not true with special relativity, since part of the whole premise of relativity is that all reference frames are equally valid. B: Special Relativity, which is actually based on this principle combined with the principle that light moves at a constant speed relative to everyone, has time-dilation and length-contraction affects that are separate from the affects of classical Doppler shift.

One interesting effect of this is that, if we ever intercepted a transmission from billions of light years away, in the early universe, everything would be in super slow-motion from the combined effect of the Doppler shift and time dilation, (although these are already combined in the redshift, and what is called "relativistic Doppler shift" includes time dilation.) On the flip side, one might note that time dilation of any kind looks like Doppler shift, so one should also expect redshifting of light coming from incredibly dense objects like neutron stars. I've never heard about this specifically, but I assume it must be something well known to astronomers who study such things.

3) In special relativity, your local time is simply the time you experience. If you have a clock, the time coordinate for any event you observe can be determined by the reading on your clock. However, if you look at a Minkowski diagram, your reference frame gives time coordinates for places other than your location. This seems like an obvious thing, but it's actually pretty weird if you think about it, and is arguably where all the weirdness of special relativity comes from. The way Einstein defines this time coordinate is rather interesting: If you observe light from an object one light second away, then whatever you see is happening one second in the past. This idea can be extended to all times. Also, it technically isn't talking about observing light, it's talking about any information reaching you at the "speed of light" or maximum speed of information.

The thing about this definition is that it isn't actually based on assumption about how fast information moves through space; it's more like a semi-arbitrary definition of how to relate time and space, since there isn't any better way to define how time in one location is related to time in another location. This definition is really at the core of special relativity. When people say that observers moving at different velocities disagree about the order of events in distant places, they mean that their time-coordinates, as determined by subtracting the light-time distance as measured by them from the time they observed the signal, do not come in the same order according to different observers. Arguably, our real notion of time comes from the causation, which can only happen when there is enough time for information to pass from one place to another. The above definition of the "time" that events far from the observer happen is convenient because it makes a coordinate system where everything that could have affected an observer is in a hyper cone spreading back in time before the present and everything the observers present could affect in the future is in a hypercone of exactly the same angle in front of the observer.

On the more practical and less philosophical side, this definition allows you to see how speed affects time and space (under these definitions): For example, if you are floating in space and see two bright lights on either side of you at almost exactly the same time, a person flying past you at a significant fraction of the speed of light, away from one and toward the other, will see the one in front of them significantly before they see the one behind them, and the difference in time compared to the difference in distance will be such that they conclude it happened first. (Put another way, it will have happened "first" in their coordinate system, even though the events happened "at the same time" in your coordinate system, all based on the above definition of times at distant locations being related to distance and the speed of light.) On a Minkowski diagram, you can investigate the times that different observers see things at by drawing 45 degree (i.e., slope 1) lines, which is what things moving at the speed of light looks like to any observer. This will allow you to see things like how how time time dilation is the same both ways (if 2 people are moving at different velocities, each person will judge the other's clock as moving slow). If you draw Minkowski diagrams on curved surfaces (or on "conformal" projections of them) you can even see general relativity effects, although I'll have to get back to you on how well that works for me.

@Stellarator

I think it's a little silly to put too much store in the definitions of life currently used by biologists. Those are designed specifically to differentiate between naturally occuring (i.e. not artificially human-made) living and non-living things on Earth, not for xenobiology. The modern definition of life that I heard says that all life-forms have cells, but if we found aliens who didn't, we would still call them life. I think behavior is much more important, and anything that could be grey goo would be a lot more like life then it would be like fire or crystals or even viruses. Viruses are completely inert until they come across a living cell that they react with and take over. Grey goo would have to be something that acted independently, rather than being dependent on other life forms; it would have to react in very complex ways to stimuli in order to be able to break down so many different types of compounds, and it would definitely eat, emit "waste", use energy to do work, and reproduce itself (though it might not grow). Also, there's no reason something like grey goo wouldn't have something like cells; after all, cells are just a way to separate different solutions from each other and from the awkwardly large outside environment so that organisms can control chemical reactions better. Why wouldn't we copy a simple idea from nature for such a thing? (Well, it might not be necessary if the nanites were super small I guess, except that living cells are already made of lots and lots of tiny nanites working together, and cells are more efficient for them and allow them independence.)

Of course, no one's going to intentionally try to make canonical grey goo except maybe as a super-weapon, or perhaps as some kind of mining technology (which might do the construction with the material itself), but a lot of the same points could apply to other nanotechnology. It also should be noted that the most advanced nanotechnology humans will be making will probably be very much like the products of life for a long time, since we will probably continue to take advantage of the preexisting nanotechnology in living cells for a long time, just with more and more alterations and supplements. We already use genetically altered bacteria to synthesize drugs, and a lot of biochemistry has been using similar techniques for decades. In a way, genetic engineering is nanotechnology; we do use macromolecules we've taken from inside organisms to copy DNA, and specially altered ones to cut it and put it into cells.

Just a general post, Isaac Arthur talks a lot about how Dyson spheres would have to emit all the heat they absorb, but is it not posssible for them to be built to only emit it in one or a few directions? For one thing, that's basically the idea behind a Shkadov thruster, but it's also a good way of hiding, if there were any reason to, and it might even be possible to use such waste-heat to communicate or even push starships without producing much heat at all that would have to be radiated in all directions. Of course, maybe I have some woeful misunderstanding of thermodynamics, but it seems like all you would need would be some very cold outer layer with devices made from mirrors and/or superconductors that could somehow move the waste-heat from however thick of a Dyson sphere you wanted to create (since the Matrioshka brain concept would also apply to to just making almost indefinitely thick Dyson swarms if you could stand the low power-per person/volume/etc.) a direct it into some kind of beam going in one direction, maybe even a highly columnated beam or laser. The thing about this is that only people who happened to be in just the right direction would detect anything.

There still couldn't be very many Kardashev 3 civilizations hanging around in this scenario (assuming nothing better than Dyson spheres were found), since we'd expect to have noticed at least one half-way converted galaxy by now if there were a lot of them. Also, if to many of these things were being made, we ought to be able to notice that more mass was visible in the earlier universe (i.e., far from us) and that a large amount of it had converted into dark matter as time advanced to our time (i.e. close to us), and I don't think any such difference between near and far has been detected. Also, this would definitely fall under the candidate of MACHOS if they were just stars in galaxies, and I think there are reasons to doubt that as a primary explanation for dark matter (although that doesn't mean it couldn't explain some of it). I'm really not sure whether we are currently able to to detect entire separate galaxies of dark matter yet, and maybe that could explain some dark flows, but I feel like we probably can, and if we can and haven't, that's another sign that galaxies converted to Dyson spheres must be quite rare still. (Note I'm using dark matter in the strictly cosmological/astronomical sense. Most of dark matter would still need some other explanation in any realistic version of this scenario.)

It has the same issue as most stay-at-home civilization ideas, it works best if you also assume they kill themselves off or devolve or freeze into an almost perfectly controlled and unevolving society/organism or something after a while, because otherwise this "government" would have to somehow keep this policy up for billions of years. They would presumably want to to something about their sun dying, too, though that certainly be fixed by star-lifting, and even if not, it would theoretically be possible to just make sure the entire society just moved to another star system or somehow just hung around and built a Dyson Sphere around the white-dwarf or something.

Of course, you've already given quite a few ways for a home-world to be wiped out if anyone did break such a policy before it could be made safe, and if it happened early enough, the world-destroyers would probably die themselves, too.