An explorable explanation of relativistic spacetime | Hacker News Reader

An explorable explanation of relativistic spacetime

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150 pointsmxfh57 comments

This is quite nice, although I think I'm too familiar with to topic to know whether it's actually useful to someone trying to learn this stuff :)

Although I think that "explorable explanation" is a bit of a grand claim for this sort of thing. The explanations (the connections, reasons, mechanisms, etc.) are given in the text, which isn't any more "explorable" than a book; whilst the "explorable" parts (the graphs/diagrams) aren't really "explanations" since they have no depth: the relationships are hard-coded, so the user can only adjust some parameters and get told what the "answer" is for that parameter.

Many years ago I tried to make a more in-depth "explorable explanation" of some concepts from thermodynamics, using the Smalltalk-based EToys system. I did this by setting up many small simulations, e.g. of "atoms" bouncing around in containers, of pistons and thermometers, of falling weights, etc. The behaviour (scripts, methods, etc.) of each object could be inspected, and mostly involved simple stepped-integral scripts like those in games (update position based on velocity, update velocity based on acceleration, etc.). "Measurements" would be taken of the aggregate behaviour, e.g. of a piston's position, and plotted against some other parameter (time, pressure, velocity, etc.), to produce graphs which (hopefully) demonstrate the relationship described by the text. These were "explorable" since the user had complete control to inspect and modify the dynamics of the simulations and see what happens.

Unfortunately I ended up abandoning the project once it maxed-out the RAM of my OLPC XO-1, since I was no longer able to load and save it :(

As a kid there was a military air base in the neighboring city, so fighter jets passed our home regally. It was always fascinating that the sound and the plane where on different locations. You could always hear them but it was hard to spot the actual plane in the sky. Now imagine if they jets where going faster then light and that light was a bit slower so we could see them. That would mean the visible position would be different then the physical position of the plane ...

There are some weird edge cases in physics where you can move faster than the speed of light. We don't have any means, yet, to move faster than light in a vacuum, but light slows down as it enters materials (0.75c in water, for example). One effect is Cerenkov radiation, which is light emitted when a charged particle moves through a medium faster than light does. You see this in nuclear reactors. It's basically a 'photonic' boom that happens in the wake of the moving charge.

Your scenario works for things that are far enough away that the travel time of the photon is significant. Another poster mentioned the Sun. While that example is wrong (the Sun moving), it's true that if the Sun was whisked out of existence, we wouldn't know about it for 8 minutes. An example of this is supernovae in very distant stars. We see the flash of the explosion many years after it happened.

Or you know, just things far away moving at normal speeds, like also happens with sound. The sun for example is not physically in the place it appears to be due to the 8 minutes of light travel time and the fact it moves about one diameter in two minutes.

You've got two mistakes here. First, the sun appearing to move a diameter in two minutes is due to the rotation of the Earth, not the relative motion of the Earth to the sun (that's obviously a year). This can happen even if the sun and the Earth are fixed in space (with the Earth still rotating about its center). So this certainly does not make the sun look like it is in the 'wrong' spot.

Secondly, even the motion of the Earth in its orbit around the sun does not cause an 8 minute delay in the apparent location of the sun. The location the sun appears is determined by the direction that its light rays are travelling when they hit Earth. Even with the Earth travelling through space, the light rays that it hits at every point are always pointing back to the exact physical center of the sun as if there Earth were not moving. Yes, those light rays left 8 minutes ago from the sun, but the ones that Earth hits now are the ones that were heading towards Earth's current position, not the ones heading to Earth's position 8 minutes ago. Hence they are pointed from the sun to the Earth's current position.

If you can't visualize this, then imagine a alien sitting in its space ship hovering fixed in space (relative to the sun) just next to Earth's orbit as Earth passes by. Both you and the alien shoot a bullet such that its velocity is going exactly back along a light ray from the sun. Clearly the aliens' bullet will reach the sun. But then so too must your bullet: both you and the alien matched the same ray of light so your bullets are actually travelling in exactly the same direction from the same starting point. So in that sense the ray of light that you see is pointed right at where the sun is right then.

Ok fine: The sun for example is not physically in the place it appears to be due to the 8 minutes of light travel time and the fact it moves about one diameter across the sky in two minutes.

I just feel that if you talk about where the sun appears to be you can infer from context that I talk about actual sky positions and motions, not the inferred global state of the solar system, but here we are.

Anyway, the central fact in my argument is that the light that reaches us from the sun has travelled for 8 minutes, and so shows us now^1 where the sun would have been on the sky^2 8 minutes ago if we then had observed it by superluminal means.

Now, that there are no superluminal means to observe the sun with (as far as we know) is not crucial to the point because there are other situations where one mode of observation carries older information than other. That is the example of observing^3 the distant air plane with sound waves, and then with the supersonic means of light waves. The sound waves emitted at a given point in time reach you after the light waves emitted at that same instant, and so it looks^4 like the plane is at a different position than the sound is coming from. Since most of us rely mostly on sight, the sound is judged to be erroneous, but clearly we could just as well pick the sound position to be the baseline, which we do for instance if the plane is hidden by a cloud.

Also, in the spirit of your objection, you are almost completely wrong, in an exact technical way, when you say that

> the light rays that it hits at every point are always pointing back to the exact physical center of the sun

Because the sun is so close, it is in fact NOT a point, but it resolves to a disk, and since most of the disk is not on the one (infinitesimally thin) light ray that points back trough "the exact physical centre of the sun", most of the light does not point exactly there.

1: By which I mean a given instant in time, and due to how close the sun is and our reasonable velocity relativity of simultaneity is not important 2: Here "on the sky" means what position in an earth co-moving sky coordinate system 3: Again, I am talking about the motion of objects across the sky, in an earth co-moving sky coordinate system. The various and important ways that the example objects differ are neglected if they do not matter for the observed motion in the sky. 4: That is, you see optical waves with your eyes and infer from that a specific sky position different from the one reconstructed by your brain based on the sound waves detected by your ears.

>moves about one diameter in two minutes

Not in its own reference frame.

Uh, maybe the sun prefers to use the galactic standard of rest? Did you think of that? Maybe it likes to think of itself as an active and healthy star that takes care of itself and puts in the hours on the treadmill?

Anyway, I was talking about the motion on the sky, not in an arbitrary reference system.

If you are interested in this, check out the formalization of space time in geometric algebra. It kinda makes sense! (i.e. more sense than in other formalisms) https://en.wikipedia.org/wiki/Spacetime_algebra

I have to add that although there are quite a few great resources explaining special relativity, resources explaining general relativity, beyond focusing on describing the math behind it, are few and far between. Anyone's got a contradicting example?

General relativity is, fundamentally, the study of the metric tensor: it is the object that connects the geometric representation of space and time with mass and energy. Different metric tensors can produce wildly different results, from flat space to black holes to bubbles in space time that provide inertial reference frames that can travel "faster" than than the speed of light.

You could spend your entire life studying small variations in metric tensors. Some people do.

More simply:

1. Light travels from point A to point B via the shortest path.

2. In the absence of mass (aka where the Minkowski metric applies), the shortest path is a "straight line".

3. In the presence of mass (where a non-trivial metric exists), space itself curves. Light, which follows the shortest path between two points, must follow this curved space.

As a lower-dimensional analogy, think of what a "straight" line means on the surface of a sphere[1]. The shortest path between two points is, by definition, a "straight" line, but if the space itself that the path is embedded in is curved, then when viewed from a higher dimension (i.e. on the surface of the sphere it's straight for you, but if you were in space) it would appear curved.

The analogy fails when you attempt to think about it in higher dimensions, since humans have a very difficult time perceiving anything greater than the standard 3+1 spatial & time dimensions. But the math still holds.

Light will always travel in a straight line, it just so happens that gravity redefines what "straight" actually means.

1. http://mathworld.wolfram.com/GreatCircle.html

I learnt a lot from “Introducing Einstein's Relativity” by Ray d'Inverno and “A First Course in General Relativity” by Bernard Schutz. However, they are by no means leisurely reads. You need to be prepared to expand your mind (both your physical intuition and mathematical technique) and work very hard (you don't learn just by reading theory, you need to solve exercises and perform long calculations).

I’ve been trying to learn all this stuff lately. One thing that bothers me though, that perhaps someone here can help with.

In Minkowski diagrams you set things up to have rays as diagonal lines. But this way of illustrating things suggests that light moves in time. But they should really not be interpreted as that since they essentially define the limit of the time dimension.

So I was thinking that I would like to have another illustration of things where light moves in 90-degree angles instead to fix this.

Are there some specific, established ways to do this?

In any reference frame, light moves one (light) second of distance in one second of time. So the worldline of light should always be at 45 degrees in a minkowski diagram (with appropriate units).

Is your issue that, loosely speaking, if you travelled from X to Y at the speed of light, you wouldn't experience any time elapsing as you travel? If so, remember that the Minkowski diagram relates to the observer's reference frame, not the light's reference frame.

Also note that the “light’s reference frame” is not a meaningful concept.

AIUI, "light has no reference frame" is the same as "light does not experience time" is the same as "light photon teleports from its birthplace (emission) to its death place (absorption), in its own reference frame".

“Reference frame” is a way of assigning a coordinate system so as to split spacetime into one dimension of “time” and three dimensions of “space”. Light doesn’t have a reference frame. Light doesn’t experience “space” either. Two points in spacetime along the path of the light are displaced by a “null vector”, i.e. the relativistic magnitude of the displacement is zero.

If you were to imagine photons having an internal clock, that clock never ticks. That's not the same as light not moving through time. Light does not experience proper time.

Any statement that combines the word "move" with the claim that it shows how light lacks time has already contradicted its goal.

I think it is better to imagine light as line-like objects that extend through time, not point-like objects that move through time, and do away with speaking of light as something that moves, if you wish to express that it lacks a clock (proper time).

Every particle has a life line (segment). Light particle's life line segment is perpendicular to the time axis, jumping across empty space (from birth/emission to death/absorption) at one constant time. If an atom emits an alpha particle (helium atom) and gamma particle (photon) at the same time, the alpha particle moves through space while its proper time changes (its "imaginary clock" ticks), whereas the gamma particle instantly arrives at its destination somewhere else in the Universe without experiencing any change in time, absorbed by another particle.

Since light has no mass, it has no impact on spacetime geometry, so it only exists at emission and absorption. There is no way to detect or affect it between those to moments in its life, so it makes sense to consider them exactly one moment. Lights line segment is a "dotted line" the line segment (exclusive of endpoints) between emission and absorption does not really represent any part of our Universe; it's purely virtual. The only way to affect a photon is to plan ahead and set up an apparatus before it is born. Once the photon exists (relative to any observer), it's already gone (absorbed) by whatever first observed it. (And nothing else can directly observe it. All of us can bounce lots of stuff off the moon to observe it, but only one person can bounce only one thing off a photon , which kills it)

But still, that line, in a minkowski diagram, represents a boundary, so it still seems more intuitive to me to eliminate the area beyond it completely from the diagram.

That boundary is in 45 degree angle for a reason.

Spacetime interval is lightlike when spacetime interval is zero (Δs²=0). Spacetime interval is zero when Δr²=c²Δt². In other words light moves exactly as much in time (t) and space (r).

If you want to see 'lights point of view' in spacetime intervals and not in the block universe model, every reachable spacetime location is zero distance away. The whole cone, not just the boundary is just one dot (light can travel into any point inside the lightcone using mirrors) and outsides are removed.

edit: the whole eqution is Δs² = Δr² - c²Δt². If we write all coordinates x,y,z,t open, the distance is Δs² = Δx² + Δy² + Δz² - c²Δt². Otherwise nice euclidean distance measure is screwed by that minus sign before time that makes spacetime hyperbolic.

For a single observer that area is effectively invisible; they cannot be affected or affect things in it (outside of their forward/backward light-cone).

However, other events can still exist in there. And their lightcones may intersect at a later point in time, so they are not totally uninteracting. Point (x1, t1) may never be able to affect (x2, t2), but it can always affect (x2, t2 + <waiting for a while>).

It’s not that the points outside the light cone don’t exist. It’s just that each point represents a “spacelike displacement” rather than a “timelike displacement”. If you like, you can think of any point outside the boundary as happening simultaneously to the spacetime point at the vertex of the cone, if you just pick the right reference frame.

A causal boundary, but not a real boundary. That is to say, events separated by that line can never meet, never interact, never observe each other, but they can still exist.

For a “real world” example see: https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates

They exist in the sense that the uncountable real numbers exist. The reals you can name are like your light cone. By symmetry it seems that the other reals/ spacetime points should exist, but there's no way to prove they actually do if you're not already "nearby". :-)

I think you’re talking about null vectors/geodesics, which define light, which from the viewpoint of any external observer does move through spacetime.

https://physics.stackexchange.com/questions/23058/why-is-lig...

https://physics.stackexchange.com/questions/188859/what-is-a...

Remember that it’s deceptive to imagine things like the POV of a photon, and better to think in the usual terms of external observation.

I'm not entirely sure what your objection is; light does move in time.

Still learning ;)

But IIUC, could you send a clock along 45-degree angles in a minkowski diagram it would not measure any time.

But perhaps there is time there relative to an observer at 0-degrees? But what does that even mean then? If that observer sent a photon towards a mirror, would the photon have aged when returned to the sender?

You can make your statement more accurate by saying that a clock will measure vanishingly little time as you get sufficiently close to the 45 degree line. But the 45 degree line isn't achievable.

(Note I've made the same inaccuracy further up the page. Sometimes it's just easier to drop the epsilons and speak informally.)

Physicists are precise people :)

Mathematicians are precise people. Physicists just assume their series converge :-)

Can someone explain that to me. When the observer is in the train, he sees the right tower future? Or maybe another way of saying it, the left and right tower time are not the same? Or yet another way of saying it would be, two points of an observed moving object are not at the same position AND not at the same time? Two points of a moving object got different delta positions and delta times?

From a modern perspective the relation of the time axis to the x axis is roughly the same as the relation between the x axis and the y axis. Saying that two events happened at the same x coordinate (delta x is 0) is a statement that depends on the observer. Another observer who is rotated with respect to the first will see the events at different x coordinate (delta x is not 0), because some of the x axis has been rotated into the y axis. Similarly, saying that two events happened at the same t coordinate (delta t is 0) depends on the observer. Another observer who is moving with respect to the first will see the events at different t coordinate (delta t is not 0), because some of the t axis has been "rotated" into the x axis.

This notion of "rotation" only differs from ordinary rotation by a minus sign. If you rotate the xy plane around the origin then points move in a circle x^2 + y^2 = const. If you rotate in the xt plane then points move in a hyperbola x^2 - t^2 = const.

Physically you "rotate" in the xt plane by accelerating in the x direction. This causes points to move along that hyperbola, which means that some delta x manifests as delta t and vice versa. We don't see that in ordinary life because we only move with respect to one another at incredibly low speeds, which corresponds to very small "rotations".

Imagine a world where all creatures are facing along the x axis with only very small deviations. These creatures may think that the x coordinate has physical meaning independent of the y and z axes. They might think that there is something special about their x axis. Similarly, we are all moving with roughly the same velocity through the universe. We initially thought that there was something special about our velocity, namely, we thought that we were at rest with respect to the aether.

edited 10 thousand times

Very good explanation, thanks for your time.

Still something that's unclear to me. Let say I'm observing a star moving away from me at speed of light. Will I see the star? If yes, it means that, those photon-things, have a property which I haven't. Some "move at speed of light" type of property. So now, we have two points on a line, the "value" of a photon (speed of light) and my "value" (speed x), do we? What about that? We cant call that speed because we are saying speed is relative. How do you call that property then? And what is the other bound? Cant be x right?

Or another solution is that, light depends on the observer. Ok got it. 100% got it. Light both has a speed (speed being always dependent from the observer) and both has a property I havent. The key is that light has a special property I havent, but it isnt about speed. Am I correct?

You can't move with the speed of light. If you move at 99% of the speed of light away from the star you do see the star but it will look red because the light waves get stretched out.

Photons indeed have the special property that they move at the speed of light. This speed is special. If A is moving relative to B and they both measure the speed of the same light ray they both measure the same speed. This is not the case if they measure a particle C going at less than the speed of light. In some sense the speed of light acts like an infinite speed in the sense that "c plus v = c" just like "infinity+x = infinity". But the plus in this equation is relativistic velocity addition: v plus w = (v + w)/(1 + vw/c^2) where c is the speed of light. You can see that if you set w=c you get v plus c = (v + c)/(1 + vc/c^2) = (v + c)/(1 + v/c) = c(v + c)/(v + c) = c. The speed of light has this property but the speed is definitely finite approximately 300,000,000 m/s.

If you move at 99% of the speed of light away from the star

What is the light ray speed I measure in that case? 1% speed of light?

The speed of light ray (in vacuum) is always 100%. Other things keep changing in a way that keeps the speed of light constant from everyone's perspective.

Thank you that helps.

Question: if I move away from A near speed of light, will I be seen moving near speed of light in all other frames?

What about another person walking around in your spaceship?

Thank you.

So the fact that you always see light at speed of light whatever is the frame, isnt because light is going at speed of light but because light is light. What's different between a photon and my spaceship is not so much that my spaceship cant reach a full 100% speed of light, but that youll always see a photon at speed of light because of some "magic" property which cause isnt speed. Is that correct?

It is speed, the speed of light. Particles that have zero mass go at the speed of light. This is a property of the particle. That there is a maximum speed is a property of spacetime however. Even if there were no massless particles we could still discover special relativity, for example by noticing that when you speed up particles they can't get past a particular speed. In the large hadron collider they can speed up particles to 99.9999991% of the speed of light.

Following this discussion, I've started to study this: https://oyc.yale.edu/physics/phys-200

What you see on the diagrams are pictures taken by the drone swarm which moves alongside the train. In other words the pictures show events happening on a plane of simultaneity.

Observer in the train can only see events which happened in their past light cone. So they definitely cannot see right tower's future.

('They' is used as gender neutral version of gender neutral 'he')

Why is the left tower "in the past"? Because it "goes further away" from the train? What is "going further away" determined by? Position delta? So if position delta T->pA > position delta T->pB, time offset T->pA < time offset T->pB?

It depends on what you mean when you say "left tower is in the past". Space-time intervals are defined for the pairs of events, which are points in space-time. Left tower is not an event.

In the bridge plane of simultaneity both towers lighted at -100 units of time.

In the train plane of simultaneity left tower lighted at -65, right one did at -150.

The values are determined by the Lorentz transformation which includes speed of the observer. So "what's in the past" depends on the positions of the events, the times of the events, and the speed of the observer.

Wow it's so cool. So I guess the left tower lighted is not an event of the past, it will just appears differently depending on the observation frame.

So let me rephrase my initial question.

Why is the left tower lighted before right tower is lighted when observed from the train frame? Because the train "goes further away" from the left tower? What is "going further away" determined by? Distance delta? So, is the following correct: let two events A and B occurring respectively at positions pA and PB in the same frame, a mobile at position pM, if delta_distance(pM, pA) > delta_distance(pM, pB), then follow that observed_time(A) < observed_time(B) assuming observed_time(A) == observed_time(B) in pA and pB frame? Please note, it's not delta_position but delta_distance I've written. Also, by delta, I mean difference. (edit: I get that "distance delta" is velocity)

This is amazing! I remember having to study and draw some of these diagrams in an undergraduate Philosophy of Space & Time class.

Explanations of special relativity, explorable or otherwise, are dime a dozen. Try explaining general relativity in this fashion. Try explaining cosmology in this fashion.

I always wondered how a database would know which transactions to commit first in a relativistic world?

You don't need relativity, this problem is already answered by distributed systems that lack TrueTime clock guarantees. Today have to make arbitrary and inconsistent choices. (Or "eventually consistent", but then you lose some level of resolution in your ordering of moments)

Related, this classic economic article: https://www.princeton.edu/~pkrugman/interstellar.pdf

Relativistic systems still behave causally; What observers can't agree on is how much time passes between events. The database is just another observer, so it would commit the transactions in the order that light-rays arrive. The requirement to preserve causality is why there is an upper bound on the speed of information, so that all observers can still agree on the order of events.

beautiful work. To anyone who is interested in really understanding relativity deeply without intimidating math, I recommend Tim Maudlin's book: "Philosophy of Physics: Space and Time".

> without intimidating math

While having nothing against the book or its author, I must point out that ultimately it is "math in which we trust" (even experimental science leaves room for interpretation); philosophy, on the other hand, gives us no choice but to trust the philosopher!

Not exactly sure what you mean. The foundation of philosophy is logic. Also the idea that "math in which we trust" is kind of bizzare, as mathematics is an axiomatic system. So really, it's the axioms in which we trust. But that also opens a whole can of worms, because some axioms are kind of weird and controversial (like the Axioms of Choice, Replacement, Regularity).

Not to mention that any for any system (> Peano Arithmetic) you can't have your cake and eat it, too; sound, complete, consistent: pick two.

I think the point was simply that there is no understanding of physics without the math, as the math is the only accurate description of the physics involved. Admittedly, "trusting math" was a poor phrasing.

We can trust math because it is trivially verifiable. Its relation to physics is somewhat complicated (e.g. verifying physics is not as trivial), and yes - you can understand a lot about physics without math - by combining direct observation with trusting the "philosopher".

Actually, it is “sound, complete, consistent, recursively axiomatizable: pick three”.

I think you have made the arrogant mistake that this is a fluffy pop-sci book. You are incorrect. Maudlin takes real physics very seriously. The book actually gives sufficient math. The point is that high level math normally associated with SR, GR (Riemannian geometry) is unnecessary for understanding the interesting philosophical consequences of relativity.

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