13 ms·

Here is another explanation. In Newtonian physics lots of common quantities are represented by vectors: position, momentum, force, the magnetic potential, etc.

Here is another explanation. In Newtonian physics lots of common quantities are represented by vectors: position, momentum, force, the magnetic potential, etc. Other important quantities are represented by scalars: time, energy, power, the electric potential, etc. 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". Each four-vector has three spatial components and one temporal component. So in the two lists I wrote above, each scalar quantity gets paired with the corresponding vector quantity: time & position; energy & momentum; power & force; the electric & magnetic potentials. In Newtonian physics a rotation in three-dimensional space will transform one component of your position vector into another (say, some of your x-component becomes a bit of y-component). In special relativity observers moving at different velocities are related by a similar kind of transformation of their four-vectors --- a bit of the time component mixes with the spatial component and vice versa. But what happens if you do this with velocity? Velocity is a vector. It's corresponding temporal quantity is a bit weird: change in time per change in time. But it's a little more subtle than that, because it's actually the change in coordinate time per change in proper time. Basically this is the ratio between how fast you observe a clock tick in an observer's reference frame relative to how fast they see it tick within that reference frame. 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 is why the clocks of moving observers tick more slowly.) And when you slow down, you just mix some of the spatial components into the temporal component. And when you're at rest the spatial components are all at zero, so the only component of your four-velocity is in the temporal direction. So at rest your four-velocity points directly in the future with a magnitude equal to the speed of light. This is what is meant by the statement that we travel through time at the speed of light.

vnorilo 6y ago
This is great, thanks! This is the first time I hear it explained like this and it just somehow clicked immediately. 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.
- mjfisher 6y ago
  That's exactly it - it's not wildly incorrect at all.
TeMPOraL 6y ago
Thank you for clear explanation. I wonder how changes in 4-vector velocity relate to "energy & momentum" 4-vector? Does that vector also have fixed magnitude?
- mpcjanssen 6y ago
  There is indeed also a relation between momentum and (rest) energy describing the conservation of energy, In the rest frame this relation reduces to the famous E=mc^2.
  - CuriousSkeptic 6y ago
    So Feynman diagrams describe rotations?
- cygx 6y ago
  Does that vector also have fixed magnitude? 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 whereas three-momentum = gamma * mass * three-velocity
  - alexpetralia 6y ago
    Not sure if you will find this comment, but I'd be very curious to hear what the magnitudes of "four-force" and "four-potential" are as well! My layman's research suggest these are, respectively, "electric charge" and "current"?
    - cygx 6y ago
      That doesn't sound right to me. Note that four-force is a generic concept not limited to electromagnetism. Current is the source term of Maxwell's equations, which connect it to the potential via the d'Alembert operator.
cletus 6y ago
Yeah, I'm not a physicist but hearing a good explanation of the time-space four-vector being constant as an explanation for time dilation (among other things) was a real light bulb moment for me. 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.
pdonis 6y ago
> the really weird thing about this particular four-vector is that it always has a magnitude of exactly the speed of light 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.
- dheera 6y ago
  It should be stated somewhere here that relativity operates in Minkowski space and not Euclidian 4D space. 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.
  - tgvaughan 6y ago
    Just an aside: exploring the implications of relativity in a Euclidean space is the basis of Greg Egan's "Orthogonal" book series.
  - Retric 6y ago
    The early versions of Minkowski space used imaginary numbers which you prefer is mostly just notation.
  - michaelsbradley 6y ago
    It’s possible to formulate SR/GR in Euclidean geometry as well: https://www.euclideanrelativity.com/ https://www.euclideanrelativity.com/
- 6y ago
yamrzou 6y ago
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?
- jiggawatts 6y ago
  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.
  - rl3 6y ago
    >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.
jules 6y ago
What matters is your path through spacetime. The four velocity is just a tangent vector of this path. The magnitude of the tangent vector has no physical meaning, only its direction does.
moralestapia 6y ago
Thanks antognini, it made so much more sense now.
vchak1 6y ago
What is so special about the speed of light? As a thought experiment, if everyone on the planet was blind, would c have been replaced by the speed of sound?
- Chilinot 6y ago
  The speed of light is just the speed of any particle without matter. We just use "speed of light" since its simpler to understand and talk about rather than saying "speed of matter-less particles". 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.
  - bbojan 6y ago
    To be pedantic, a particle without mass. Matter also includes things without substance like physical fields. So a photon is matter. At least this is what I was taught in school many years ago...
    - davrosthedalek 6y ago
      To be pedantic, a particle without rest-mass...
    - Chilinot 6y ago
      Yes, that was a typo on my part.
- dkersten 6y ago
  As chilinot said, it’s the speed of any massless particle in a vacuum. A massless particle has nothing slowing it down, so it moves at the maximum possible speed. It’s actually the propagation speed of cause and effect, or put another way, how long it takes for a quantum event to affect whatever is in the adjacent point one Planck length away. It’s simply how quickly these things ripple forward when there is nothing slowing these ripples down. 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.
rubatuga 6y ago
Sounds like the time cube is relevant once more.
- Viliam1234 6y ago
  To be pedantic, it's time-space tesseract, but yes.
browsergap 6y ago
THat's probably the best fucking explanation in a HN comment I have ever read. Mind expanded. I actually feel like I get this stuff, which I didn't study at college, a little bit. I took chem... physics always seemed impossibly strange.
throwaway_pdp09 6y ago
> 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". > 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)
BunsanSpace 6y ago
> exactly the speed of light *causality The speed of light is variable, the speed of causality isn't.
Lambdanaut 6y ago
During a salvia trip I once saw the universe like this. I had been sitting on this conveyor-belt-like structure my entire life, which was the 4th component(time) in that 4-Vector. The conveyor belt moved me along perpindicular to space, which I have been observing my entire life from the constant speed of time. Only on this trip was I able to look "down" and realize that I was riding on time. 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.
- dreamcompiler 6y ago
  > 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. 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.
  - deleted 6y ago
    [deleted]
thr0w__4w4y 6y ago
That is a great explanation. I remember you, or somebody else here on HN, provided a similar explanation when this general topic came up within the last year or so... and I thought the same thing then -- finally this makes some sense to me. But then I'd lost the post and recently couldn't find it when a colleague could have benefited from it. Thank you.
zackees 6y ago
The funny thing about this theory and it’s speed of light proposition is that fails to explain why things are observed to travel faster than light. https://www.google.com/amp/s/www.sciencealert.com/faster-than-light-speed-in-jets-that-produce-gamma-ray-bursts/amp https://www.google.com/amp/s/www.sciencealert.com/faster-tha...
- mpcjanssen 6y ago
  Faster than light in a medium. The last part matters. There is currently no reason to doubt relativity.

vnorilo 6y ago

This is great, thanks! This is the first time I hear it explained like this and it just somehow clicked immediately. 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.

mjfisher 6y ago
That's exactly it - it's not wildly incorrect at all.

TeMPOraL 6y ago

Thank you for clear explanation. I wonder how changes in 4-vector velocity relate to "energy & momentum" 4-vector? Does that vector also have fixed magnitude?

mpcjanssen 6y ago
There is indeed also a relation between momentum and (rest) energy describing the conservation of energy, In the rest frame this relation reduces to the famous E=mc^2.
- CuriousSkeptic 6y ago
  So Feynman diagrams describe rotations?
cygx 6y ago
Does that vector also have fixed magnitude? 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 whereas three-momentum = gamma * mass * three-velocity
- alexpetralia 6y ago
  Not sure if you will find this comment, but I'd be very curious to hear what the magnitudes of "four-force" and "four-potential" are as well! My layman's research suggest these are, respectively, "electric charge" and "current"?
  - cygx 6y ago
    That doesn't sound right to me. Note that four-force is a generic concept not limited to electromagnetism. Current is the source term of Maxwell's equations, which connect it to the potential via the d'Alembert operator.

cletus 6y ago

Yeah, I'm not a physicist but hearing a good explanation of the time-space four-vector being constant as an explanation for time dilation (among other things) was a real light bulb moment for me. 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.

pdonis 6y ago

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

dheera 6y ago
It should be stated somewhere here that relativity operates in Minkowski space and not Euclidian 4D space. 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.
- tgvaughan 6y ago
  Just an aside: exploring the implications of relativity in a Euclidean space is the basis of Greg Egan's "Orthogonal" book series.
- Retric 6y ago
  The early versions of Minkowski space used imaginary numbers which you prefer is mostly just notation.
- michaelsbradley 6y ago
  It’s possible to formulate SR/GR in Euclidean geometry as well: https://www.euclideanrelativity.com/ https://www.euclideanrelativity.com/
6y ago

yamrzou 6y ago

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?

jiggawatts 6y ago
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.
- rl3 6y ago
  >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.

jules 6y ago

What matters is your path through spacetime. The four velocity is just a tangent vector of this path. The magnitude of the tangent vector has no physical meaning, only its direction does.

moralestapia 6y ago

Thanks antognini, it made so much more sense now.

vchak1 6y ago

What is so special about the speed of light? As a thought experiment, if everyone on the planet was blind, would c have been replaced by the speed of sound?

Chilinot 6y ago
The speed of light is just the speed of any particle without matter. We just use "speed of light" since its simpler to understand and talk about rather than saying "speed of matter-less particles". 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.
- bbojan 6y ago
  To be pedantic, a particle without mass. Matter also includes things without substance like physical fields. So a photon is matter. At least this is what I was taught in school many years ago...
  - davrosthedalek 6y ago
    To be pedantic, a particle without rest-mass...
  - Chilinot 6y ago
    Yes, that was a typo on my part.
dkersten 6y ago
As chilinot said, it’s the speed of any massless particle in a vacuum. A massless particle has nothing slowing it down, so it moves at the maximum possible speed. It’s actually the propagation speed of cause and effect, or put another way, how long it takes for a quantum event to affect whatever is in the adjacent point one Planck length away. It’s simply how quickly these things ripple forward when there is nothing slowing these ripples down. 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.

rubatuga 6y ago

Sounds like the time cube is relevant once more.

Viliam1234 6y ago
To be pedantic, it's time-space tesseract, but yes.

browsergap 6y ago

THat's probably the best fucking explanation in a HN comment I have ever read. Mind expanded. I actually feel like I get this stuff, which I didn't study at college, a little bit. I took chem... physics always seemed impossibly strange.

throwaway_pdp09 6y ago

> 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". > 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)

BunsanSpace 6y ago

> exactly the speed of light *causality The speed of light is variable, the speed of causality isn't.

Lambdanaut 6y ago

During a salvia trip I once saw the universe like this. I had been sitting on this conveyor-belt-like structure my entire life, which was the 4th component(time) in that 4-Vector. The conveyor belt moved me along perpindicular to space, which I have been observing my entire life from the constant speed of time. Only on this trip was I able to look "down" and realize that I was riding on time. 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.

dreamcompiler 6y ago
> 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. 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.
- deleted 6y ago
  [deleted]

thr0w__4w4y 6y ago

That is a great explanation. I remember you, or somebody else here on HN, provided a similar explanation when this general topic came up within the last year or so... and I thought the same thing then -- finally this makes some sense to me. But then I'd lost the post and recently couldn't find it when a colleague could have benefited from it. Thank you.

zackees 6y ago

The funny thing about this theory and it’s speed of light proposition is that fails to explain why things are observed to travel faster than light. https://www.google.com/amp/s/www.sciencealert.com/faster-than-light-speed-in-jets-that-produce-gamma-ray-bursts/amp https://www.google.com/amp/s/www.sciencealert.com/faster-tha...

mpcjanssen 6y ago
Faster than light in a medium. The last part matters. There is currently no reason to doubt relativity.