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Thank you for a clear explanation. I'm a total layman here, so forgive my questions if they don't make sense:

> One of the deep insights of special relativity is that these scalar and vector quantities are actually unified into a single, new entity called a "four-vector"

On which basis was this pairing between position and time "decided"?

> But the really weird thing about this particular four-vector is that it always has a magnitude of exactly the speed of light

Why is that? Is it by definition?

> it's actually the change in coordinate time per change in proper time

How does this translate to what we call "time" in our daily lives?

4 comments

It's a tautology. You cannot observe time slowing down, because anything that would affect the "speed of time" affects your brain also in the same proportion.

Hollywood movies are not how physics works. The plucky protagonist can never look down at their hand and see a stopwatch run slower or faster! There is no such thing. If some sort of science-fiction field existed that would affect a clock, it would affect the protagonist also.

If the clock runs at half speed, your brain runs at half speed, so you see it ticking forward at twice the rate, and 2.0 * 0.5 = 1.

Now, of course, I tell a small lie: There can be differences in the tick rate over some distance. But these differences are necessarily small. Big differences in the rate of temporal flow create gravity, as per Einstein's General Relativity. To get an observable difference in time flow over the space of, say, a meter would require a fantastically huge gravitational field that would instantly crush the poor scientists into a pancake.

Conversely, in the gentle field of Earth's gravitational field, if you hold an atomic clock in your hand a meter from you head, and an atomic clock in line with your head, then yes, you can see a tiny difference in the tick rates, on the order of nanoseconds per century or somesuch (I'm too lazy to work it out).

However, as the distance decreases between the clock and the observer, the difference gets smaller. Extrapolating down to atoms and the like the conclusion is inevitable: time locally always ticks forward at one second per second.

> Conversely, in the gentle field of Earth's gravitational field, if you hold an atomic clock in your hand a meter from you head, and an atomic clock in line with your head, then yes, you can see a tiny difference in the tick rates, on the order of nanoseconds per century or somesuch (I'm too lazy to work it out).

This is why GPS satellites need time correction. GPS satellites contain atomic clocks, and they are further from the center of the Earth than you are. Therefore they experience slightly less gravity than you do. Therefore their clocks run slightly faster than atomic clocks on Earth's surface. That's the General Relativity correction.

But GPS satellites also move a lot faster than you do, because they are in orbit. That makes their clocks run slower than clocks on the ground. That's the Special Relativity correction.

Both corrections have different signs but they do not cancel each other out.

>If the clock runs at half speed, your brain runs at half speed, so you see it ticking forward at twice the rate, and 2.0 * 0.5 = 1.0

Consider changes in gravity affect cognition.

Likewise, if the brain has any quantum mechanisms, then that 1.0 might not be exactly 1.0.

> Consider changes in gravity affect cognition.

That is literally what it does. It slows down cognition. In the exact same way that it slows down clocks. The "clock in your brain" and the "watch in your hand" are all operating on the same physical and chemical principles. It's all just atoms and electromagnetic interactions at the lowest levels.

> Likewise, if the brain has any quantum mechanisms

Again, this is a pop-science misunderstanding of all of physics, not just Quantum Mechanics (QM).

The rules of QM either apply to everything or nothing. The laws of the universe do not begin or end at the edge of the laboratory bench top. Similarly, there are no special rules that apply to just the human brain.

The rules that govern galaxies, particle beams, atomic clocks, mechanical watches, and human observers are the same. They all tick along at the same rate. One second per second, locally at least.

> The rules of QM either apply to everything or nothing. The laws of the universe do not begin or end at the edge of the laboratory bench top. Similarly, there are no special rules that apply to just the human brain.

Well, for now we don't really know how the rules of QM apply to macroscopic objects. The exact physical interpretation of wave-function collapse (if any) is still a matter of speculation, with all options still being on the table - maybe there is no collapse (e.g. many worlds theory), maybe there exists a collapse when interacting with large enough systems (measurement - the Copenhagen interpretation), maybe collapse is a physical process that happens at precise scales (Roger Penrose seems to believe something like this), maybe the wave function is a physical wave (pilot wave theory) and there is no collapse in another way.

> Well, for now we don't really know how the rules of QM apply to macroscopic objects

Sure we do, "we" just refuse to acknowledge this, keeping a 100 year old debate alive for no good reason.

Macroscopic objects follow microscopic rules. That's that. There's no further debate. There can be none.

I can't even begin to describe how absurd it is to argue anything else. It's like... saying with a straight face that software doesn't "really" follow the rules of boolean algebra if it has enough lines of code. That somehow once a program gets "big enough", it can transcend truth tables and somehow go analog or something.

It's like a mathematician saying that really big equations, the kind that span several pages stop following the rules of algebra.

Get it? It's just... insane. The rules of the universe are the rules for all things in it. They apply to everything, at all scales, at all times.

If they don't, they're not the rules!

That's the point. They are not the rules.

QM and GR are approximations. The real rules are unknown to us. It just happens that "in the small", QM seems to be a good approximation that is consistent with experimental results. And "in the large" the same holds for GR. Neither of them works for everything though.

That just means we need to find a better approximation. That's not insane.

Bits are not an approximation of software "in the small". They are the real building block. We know that because we made them. First made them, then observed how they behave. But QM is a theory created by first observing. Physics is a natural science trying to understand what we observe. Math and CS are not. The objects they care about are conceived by us and observed second.

> Get it? It's just... insane. The rules of the universe are the rules for all things in it. They apply to everything, at all scales, at all times.

This is nowhere near as certain as you make it out to be. Take Conway's Life. With 4 simple "rules of the universe", you can create a series which to the best of our knowledge cannot be predicted from those rules. Clearly this series is a "thing" within the simulated universe, and clearly it emerges from the universal rules, but the rules don't give us any insight about it! The only known way to "predict" it is to let it run: in other words, it's irreducibly complex.

Causality can look different at different scales. It's well established that simple low-level rules can generate extreme if not irreducible complexity. This doesn't mean the bottom-level rules don't apply everywhere; it just means their descriptive/predictive utility is not necessarily preserved when you zoom out. Is a prime-finding algorithm best described by the mechanism of the computational substrate? Any number of substrates could suffice. We can best predict its output by thinking at a higher level of abstraction.

> you can create a series which to the best of our knowledge cannot be predicted from those rules

You can absolutely reversibly compute conways game of life state, and you can compute it forward as well (after all, it is a game). That's prediction.

> It's well established that simple low-level rules can generate [...] irreducible complexity.

Simple rules can create a very complex emergent system, but that doesn't mean it cannot still be reduced to it's component rules. That's what makes them rules and not just guidelines.

> It's like a mathematician saying that really big equations, the kind that span several pages stop following the rules of algebra.

Incidentally, Greg Egan wrote 2 short stories on basically that premise: Luminous and Dark Integers.

Of course he did! 8)

Have you ever observed an interference pattern for tennis balls? Have you ever been unable to place a stationary object in space because you knew it's velocity?

Furthermore, leaving behind direct observations which could perhaps be waved away with discussions of probabilities, we have one big problem: there is no gravity in QM and we have no idea how to account for it or curved spacetime in QM.

So for now, we have one working model for the macroscopic world (general relativity) and one for the microscopic world (QM), but the two are mathematically incompatible, they can't be simultaneously true, and we have not yet found an experiment which contradicts either of them.

Interference patterns have been observed for buckyballs, and superposition has been observed for MEMS springs consisting of many thousands of atoms. Electromagnetic interference effects can occur with radiation that has kilometer-long wavelengths. Similarly, quantum encryption and key exchange have been performed over many kilometers.

> there is no gravity in QM

The weakness(es) of any one particular theory doesn't in an way disprove that macroscopic objects follow the same rules that microscopic objects do.

Just because MySQL is bad doesn't mean that the relational model is false, or that databases are pointless.

Just because the current theories of GR and QM aren't easily extended to all regimes doesn't meant that there is some sort of hard boundary where the rules change. Our theories of the world don't affect how it works. The boundaries of our theories are not boundaries of the world.

You can't fall of the edge of the world because the maps only go so far...

> It's like... saying with a straight face that software doesn't "really" follow the rules of boolean algebra if it has enough lines of code.

As a programmer, I wholly endorse this notion.

>I can't even begin to describe how absurd it is to argue anything else. It's like... saying with a straight face that software doesn't "really" follow the rules of boolean algebra if it has enough lines of code. That somehow once a program gets "big enough", it can transcend truth tables and somehow go analog or something.

It literally does sometimes, just ask anyone that's programmed software designed to be resistant to bit flips caused by cosmic radiation.

If large programs are vulnerable to cosmic radiation-induced bit flips, so are small programs, all the way down to individual machine instructions. The point is that the rules of the system are consistent across scales.

> The rules that govern galaxies, particle beams, atomic clocks, mechanical watches, and human observers are the same.

And we do not know what those rules are. We have 2 widely accepted guesses: quantum physics and general relativity that have each been incredibly succesful in their predictive power.

The problem is, we do not know how to make these theories consistent with each other. Either relativity is wrong on quantum scales, or QM is wrong on relatavistic scales. Probably both.

>That is literally what it does. It slows down cognition.

Yes, but I'm talking actual changes in the cognitive output. Look at all the physiological changes astronauts undergo. Those changes extend to their cognition, even if the effect of general relativity is small.

>Again, this is a pop-science misunderstanding of all of physics, not just Quantum Mechanics (QM).

There's an entire field called quantum biology. I'm not advocating the brain has quantum magic (in fact last I checked the theory was pretty much completely discounted). Rather, on the off-chance that it does, your example breaks down via way of the elements that comprise the brain's computation not being necessarily subject to the same relativistic effects due to potentially vast distances between some of those elements.

Or, without QM at all, consider that there's an extremely minute difference in relativistic effects even across the few inches that span your skull. By that alone, 1.0 is not 1.0.

Not trying to be pedantic here, merely food for thought.

I'm saying that GR and QM apply to all things, not that the brain "isn't quantum". It follows QM rules in the ordinary sense that electrons orbit atoms in the brain the same way as they do in other well established contexts. Similarly, the many-world interpretation is that the non-locality experiments make perfect sense if you include the human brain in the experiment and stop excluding it. That's simply a statement that we are not "gods", somehow standing apart from the Universe and observing it from the outside. We're in it, and interacting with it, the same as everything else.

The only really interesting QM "stuff" that seems to be going on is that some enzymes have incredibly high efficiencies, even when bathed in dilute reactants. There are hints that this may be some sort of inherently quantum mechanical process that cannot be understood classically. But this is a tenuous hypothesis at best, there's no hard evidence yet, let alone a good theory.

This is opening up all new avenues of thought in my head! So theoretically if we tested an astronaut on a simple mental recall test on Earth and timed then, then had them repeat the test on a trip to the Moon but timed them on the Earth the astronaut outside of Earth's gravity well should be observed completing the test faster (from Earth), but if they measured the test on the space craft, the results would be the same as when they completed the test on Earth.

Correct, but the effect would be too small to measure like that (human performance at tests is noisy).

If you made the difference in gravity more extreme the difference in measurement would be trivial to notice, but we don't have a way to achieve that in practice.

I find using the speed of light confusing as it has little to do with light other than that light happens to travel at that speed [1], but rather the speed of propagation of cause and effect. If you stay absolutely still, cause and effect still propagate at the same speed and so time ticks forward at this speed. If you physically move (have velocity), cause and effect still ticks forward at the same speed, but some of it appears to tick into moving you physically. You can still only propagate at the speed of propagation, but some of that propagation is happening on the physical position axis, so less happens on the time axis.

[1] because light is massless, so in a vacuum there is nothing slowing it down

> On which basis was this pairing between position and time "decided"?

Three partially correct answers of the same question:

1) Nobody decided it. It is just how the universe is.

2) You have the formulas where the space-something appear, and you randomly try propose to replace something with things you know, like mass, time, color, ... It is actually more easy, because using the units you know that something must be measured in seconds, so it discards a lot of possibilities.

3) You have the "classic" formulas where space appears https://en.wikipedia.org/wiki/Lorentz_transformation that were actually discovered almost 20 years before the work of Einstein. And you rewrite them in the equivalent of the "vectorial" notation. In the usual space it is like replacing (x,y,z) with a big X and never looking inside X. If you try it, the only way you can success is if you group (ct,x,y,z).

The trick is to look carefully at the calculation and try to think what you can arrange it to be able to rewrite the calculation in the correct form.

For example with momentum, you look carefully and realize you must add energy as the fourth component.

If you choose to group space and time, and you choose to group momentum and energy, then you must use the same choose in all the formulas. You can not choose a different grouping for each formula.

But sometimes it is not so easy. With speed you realize that it is too bad to be fixed, and you must define another speed-like-thing and then give another formula that transform the "real speed that you measure in a lab" and the "abstract speed-like-thing that you use in the calculation".

With the electric field, ... well you can't put it in a 4-vector, you must use something more weird that is a 4x4-matrix.

The reason for creating these pairings is that they make calculations easier. Maxwells laws of electromagnetism in particular become very beautiful when defined in terms of these 4-vectors.