This isn't weird. The magnitude of any four-vector is frame invariant in relativity. Lorentz transformations in relativity are the analogue of rotations in ordinary Euclidean space, and have the same property of preserving the magnitudes of vectors.

> All you do by going faster is just mix some of the time component of your four-velocity into the spatial components.

I think this is a misleading way of putting it. In a frame in which you are moving, the time component of your four-velocity is larger than it is in your rest frame, not smaller. So it's not the case that some of the time component in your rest frame got "moved" to a space component.

What this means is that the length of a relativistic four-vector [ct, x, y, z] is sqrt((ct)^2 - (x^2 + y^2 + z^2)).

It is not the same as the Euclidian magnitude, i.e. NOT sqrt((ct)^2 + x^2 + y^2 + z^2)

The article does mention this, although with slightly confusing wording, but it should be emphasized that Lorentz invariant quantities need to be computed with the Minkowski metric.

https://www.euclideanrelativity.com/

Now, the same path is described by many different parameterizations. For example, if we pick q(r) = p(2r) that's the same path. The point varies twice as fast with r, but that has no physical meaning. The total path that is traced out in spacetime is still the same path. That is, the set {p(r) | r in [-∞,∞]} is the same as the set {q(r) | r in [-∞,∞]}, and it is only this set that has physical meaning.

In fact, we can also pick q(r) = p(r^3 + r) or some other parameter transformation.

We might look at the vector p'(r) tangent to the path. The length of this vector depends on the parameterization, but its direction doesn't. So p'(r)/|p'(r)| is a true physical property of the path, but the length |p'(r)| is just an artefact of the parameterization that we chose to describe the path.

In order to make the parameterization somewhat more canonical, we usually pick a parameterization where |p'(r)| = c. As it happens, the four velocity is defined to be p'(r) given this choice of parameterization.

So the length of the four velocity is c simply by definition. This statement has no physical meaning. The statement "we travel through time with the speed of light" is uninteresting. It tells you nothing about physics. It only tells you something about the conventions we use when we describe paths through spacetime. We could very well have picked the convention |p'(r)| = c/2, and then we'd have the statement "we travel through time with half the speed of light", or even |p'(r)| = 2c, and then we'd "travel through time with double the speed of light".

Given any path through spacetime whatsoever, we can pick a parameterization such that it travels through spacetime at the speed of light. So the condition that something travels through spacetime with the speed of light places no constraints on the path that it takes. The statement "we travel through spacetime with the speed of light" makes it sound like there are paths that do travel through spacetime with the speed of light, and there are paths that do not travel through spacetime with the speed of light, and physical particles go along paths of the first kind. This is wrong. There are no paths that do not travel through spacetime at the speed of light, because whether or not it does is a property of the parameterization that we choose to describe the path, not a property of the path itself.

Four-velocity is change of spatio-temporal position (Δx,Δt) in an inertial frame per unit of time Δτ measured by the object in motion. For a concrete example, think a race track with distance markers accompanied by (synchronized) stationary clocks, and a vehicle carrying its own clock. Three-velocity will be given by Δx/Δt (the distance along the track divided by the time on the stationary clock), whereas the spatial component of four-velocity will be given by Δx/Δτ (the distance along the track divided by the time on the vehicle's clock). The temporal component of four-velocity will be c·Δt/Δτ, ie proportional to the clock ratio.

Turns out no matter the speed of the vehicle, it will always hold that

    (c·Δt/Δτ)² - (Δx/Δτ)² = c²

However, once we've baked this relation into the geometry of spacetime, we can of course take the more abstract perspective described above and think about reparametrization-invariant dynamics, with choice of eigentime as parameter an insignificant way to fix an arbitrary gauge.

If you have a clock moving through spacetime along some path p(r), put tickmarks at regular intervals [1] of its arc length ∫|p'(r)|dr. Those tickmarks indicate when the clock ticks.

[1] e.g. choosing the length of an interval by matching it to a one second tick of a stationary reference clock.

In differential geometry, we traditionally parametrize curves by a parameter "t" and think of it as "time", so the parametrization allows us to "walk along" a curve. This is very intuitive geometrically of course.

But in relativity, we also have a "time" coordinate (or at least a timelike unit-length tangent vectors), which then completely oposes this geometric intuition about curves.

Of course now physicists decide to rename well-established concepts and start calling "arc length parametrization" by "proper time parametrization", which makes it sound like it is something special, while it, as far as I can tell, has no actual physical meaning.

Some vectors can vary in magnitude with time, yes. But rotating your coordinate system does not change the magnitude of any vector in Euclidean space. A Lorentz transformation is the spacetime analogue of rotating your coordinate system, and similarly does not change the magnitude of any vector.

The weird part isn't that two observers agree on the vector magnitude. The weird part is that one observer calculates the same value for any two objects. Frame-invariance isn't the relevant property.

That's not weird either, because the value being calculated has all object-specific information removed. See my response to andi999 downthread regarding rest mass.

But moreover, the rest mass isn't the issue here. Momentum in classical mechanics also takes mass into account but doesn't have the "weird" property that two objects of the same rest mass always have the same momentum magnitude.

If you think for stationary examples it just means that the eigenzeit has a different pace, so two clocks next to each other going at a different pace. But this is ruled out by another definition, such that clocks are references by light clocks. And now it is also clear why this invariance is c.

It's a unit vector, and physicists working with relativity habitually set C=1 to make the equations simpler. That being said, it isn't really a vector -- it's just a direction. The "equal to C" bit is a mathematical artifact of both vectors being normalized.

Because this 4-vector removes the factor of rest mass, which distinguishes objects. Putting back the rest mass converts 4-velocity to 4-momentum, which does not have the same magnitude for all objects--the magnitude is the rest mass.

The interesting thing was that I had this feeling that the other elements of the vectors were all also conveyor belts, moving in a single direction. X, Y, Z(and an infinite of other more subtle scalars) were each their own belts, with entities of a very different hyperdimensional sort riding along them. Essentially, they were each moving at a constant non-changing rate along a single dimension(X, Y, or Z) and because of this, they used time as we do a spatial dimension.

Think about it like this.

Imagine that a trillion years ago you were stationary, and time didn't exist. SUDDENLY you and all the stuff nearby you got pushed in space along the X axis at a speed of 100000000000 km/hr.

You kept moving along the X axis at a relatively constant rate with the stuff around you, such that the X-axis became a constant for you, while the other spatial dimensions were your degrees of freedom.

In this scenario, time is the X axis for you.

I noticed that there were entities moving perpendicular to our concept of time, treating our X-axis as their conveyor belt moving them along at a constant rate.

All a drug trip of course, and I don't know much about physics, however, this experience has shaped how I concieve of time and space.

Those entities exist: We call them photons (and a few other things). Photons move through space at a constant rate, c, and their clocks never move. If a photon were conscious it might ask "What is this time you speak of?"

It gets weirder: Because photons don't experience time they can't "remember" being in Place A "before" they were in Place B, so they might also ask "What is this space you speak of?" Photons are always everywhere they need to be, from their point of view.

I recommend watching it!

For what it's worth, time is distinguished in physics. It isn't just "another dimension" that "happens" to be used as time, but if you could just twist correctly it would become a spatial dimension. It is truly a "time dimension". The simplest way of looking at it is in the Minkowski metric, where distance is SqRt(x^2 + y^2 + z^2 - t^2), note the minus sign on the "t" which distinguishes it.

If you're really up for a solid math-based mind-screw, Greg Egan worked out the shape of physics in a universe that works on essentially Einsteinian relativity, but with two spatial dimensions and two temporal dimensions: https://www.gregegan.net/DICHRONAUTS/DICHRONAUTS.html It turns out that A: sensible things can be said about this and B: it looks nothing like what I expected two temporal dimensions to look.

(And if that's not enough, he's also got a universe of four spatial dimensions: https://www.gregegan.net/ORTHOGONAL/ORTHOGONAL.html)

I am looking forward to this next Christopher Nolan movie.

That is, I can imagine time as a single axis so that you could go back and forth. (Ooops, about to be eaten by a ghost, let's rewind.) But I can't imagine what it means to go northwest in time.

Disclaimer: my ideas come before having done my homework on the link above, so I need to read Greg Egan's stuff really. He's such a wonderful person and I need to dedicate more brainpower to his work.

In Pacman terms, I guess you could program it so that going up or down with the second joystick changes the terms of the game slightly. Maybe the maze configuration changes or in one timeline the ghosts bounce off you when you run into them but the walls themselves zap you.

In a more nuanced conception of the timeline axis, I suppose y-points would have to be branched or braided off x-points. Maybe that's where the third dimension comes in.

I'm sure Egan has worked it out with more rigor. I checked out the link for one of his books above but I admit I blanched when I got to the synopsis:

Seth is a surveyor, along with his friend Theo, a leech-like creature running through his skull who tells Seth what lies to his left and right.

I've got enough on my plate at the moment with the dissolution of civil society. I'm not sure I'm ready to entertain leech-like symbionts confronting the disintegration of time itself.

Or be under gravity.

Saw this in a salvia trip once

Edit: Why the downvotes? This was clearly supposed to be a joke.

> 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?

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.

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.

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.

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.

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!

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.

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 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.

Notice I said irreducible, not irreversible. Yes, you can reverse the computation; but there's no known way to "shortcut" the forward process to predict the outcome of the series I mentioned any faster than simply letting it run. Letting it run is not prediction in the sense I mean here. By "predict" I mean foretell in advance what the system will do without needing to let it run.

> 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.

I agree, with a minor caveat about language: you can describe a system which may display complex emergent behavior in terms of its underlying rules, but doing so is not guaranteed to give you useful information about (i.e., allow you to "predict", in the sense described above) the emergent behaviors. In contexts like these, to "reduce" typically means to describe in terms of a lower-level formalism while preserving predictive ability, AFAIK. In other words, the low-level formalism provides full information about the behavior of the entire system, such that you don't need to observe the system to know what it will do. In the Life example this is not the case.

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

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.

> 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...

Yes, absolutely agreed. My point was simply that we don't know how QM extends to macroscopic objects, not that there must be some hard boundary (though we can't exclude the possibility that there exists some hard boundary at some level of energy, just as we know that the Standard Model doesn't describe matter at certain high energies).

Until we have some unification of GR and QM, we can't say for sure that QM describes the macroscopic world, just as we can't say that GR describes the behaviors of particles. Most likely we will at some point find such a model, and find out exactly how QM applies to large systems - perhaps with some limits to uncertainty, similarly to how c acts as a limit to speeds.

Sure it does - gravitational lensing, etc.

As a programmer, I wholly endorse this notion.

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

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.

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.

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.

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.

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

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.

It's also why I'm basically convinced that any sort of FTL (including wormholes and compressing space), time travel or cheating causality is essentially impossible, no matter what results you contrive by putting things like negative mass and/or energy into various equations.

How this all gets reconciled with quantum mechanics is another matter. Part of me wonders if the problem isn't just that space and time are ultimately discrete (at the Planck unit level) and our equations describing it are continuous. But again, I'm no physicist.

I now have this (probably wildly incorrect, total layman here) vision of the time dimension of velocity being "a normal" simultaneously to all the spatial dimensions and acceleration as rotation of the four-vector away from the time axis...

sorry for any terminology errors.

> But the really weird thing about this particular four-vector is that it always has a magnitude of exactly the speed of light. No matter how fast you go, the magnitude of your four-velocity does not actually get larger. All you do by going faster is just mix some of the time component of your four-velocity into the spatial components.

This seems strongly to be confusing the map and the territory. The maths is a model, the description of vectors has not link to reality but happens to model reality in some useful way (reality and the model correspond to some useful approximation), but it isn't reality.

so

> at rest your four-velocity points directly in the future with a magnitude equal to the speed of light

may just be an artefact of the maths. IDK though. (and no offence intended, just my POV)

It's magnitude is called 'mass' (or 'rest mass'/'invariant mass' if you want to differentiate it from the concept of 'relativistic mass', which probably should be retired).

This means we have

    four-momentum = mass * four-velocity

    three-momentum = gamma * mass * three-velocity

My layman's research suggest these are, respectively, "electric charge" and "current"?

This was a gigantic problem, an experiment contradicting one of the most fundamental laws of nature as we knew them at the time - Galileo Galilei's principle of relativity.

Note that this observation has nothing to do with our eyes's ability to perceive light. The same observation will not happen with sound waves; and it will hold even for frequencies of light that we can't directly observe with our bodies, such as radio waves.

As others note, it was later discovered that this is not a special property of light itself. It is in fact a special property of the universe, and it applies to any particle without mass; the photon happens to be the only massless particle that we can directly observe, so it was the one which gave the name to the physical quantity.

That's wrong. Simultaneity is ill-defined in relativity.

The correct example is, "if Alice fires a light beam at you from a moving train and Bob fires a light beam from you from the platform, you will measure the Alice photons as going equally fast as the Bob photons.

However, you're right, your formulation is more precise, and actually possible to measure.

Another way to put it is that when you’re in a moving bus and you throw a ball towards the front of the bus, the ball is moving the speed you threw it plus the speed of the bus, but, when you shine a light, the photons from the light are moving the same speed as someone off the bus! That seems super weird.

Every other massless particle moves at the same speed. “Sound” is not a massless particle, it’s propagation of the compression of matter, so therefore moves much, much slower and would not replace c. It has nothing to do with what we can observe and rather to do with its properties. Personally, I find referring it to as “the speed of light” is confusing since it rarely has anything to do with light/photons other than that light happens to move at that speed.

Said like that, it reeeeeaaallly makes you think "we live in a simulation", doesn't it ^^ ?

Hell, why not take it a step further and say that the simulation is relative to an observer (which can be an inanimate thing, of course) as an optimisation because why bother simulating what isn’t seen or interacted with by an observer?

[1] https://www.youtube.com/watch?v=fHRqibyNMpw

Light (in a vacuum) goes as fast as the speed of cause and effect. And it just so happens that the speed of light is pretty easy to measure (as compared to other possible things).

So if we were all blind, we would still be affected by this speed.

The speed of sound is the speed of air-molecules bouncing into each other (~300m/s). Since air-molecules have matter, they dont travel at the speed of light.

At least this is what I was taught in school many years ago...

The experimental "proof" of the frame-independence comes from the michelson morley experiment. (Scare-quotes, because you can't prove things in physics, only disprove)

https://www.google.com/amp/s/www.sciencealert.com/faster-tha...