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Why can’t I go faster than the speed of light? (gravityandlevity.wordpress.com)
271 points by lanna 8 days ago | hide | past | favorite | 213 comments

I think a lot of people assume that this means you couldn't go more than a few (~100) light-years in your lifetime... But this is not actually correct. Counter-intuitively you can theorically go any number of light-years (essentially) in your lifetime, as long as you are able to approach the speed of light because when you do so the distance is dilated and hence you're covering far more ground within your reference frame (of course you'd be in the deep future from the perspective of anyone in our normal reference frame).

You can even do so without going near the speed of light if you build a very massive ship and utilize time dilation of gravity.

Arvin Ash has a cool video on this that is probably too complicated to describe in a HN comment:


As long as we are talking about how far you can go in one lifetime, how would human body react in a spaceship with that much mass? Wouldn't the gravity crush any humans to death. 10% of the sun's mass is more than 30,000 times earth's mass. Add to that the fact that the diameter is only 620 m, and the gravity becomes 1.4e13 g.

If we could somehow make it into a donut planet we could presumably sit in the middle of it experiencing no acceleration. Would the time dilation effects still occur then?

I don't think so. As I understand it, gravity and acceleration are equivalent, if you aren't experiencing any acceleration then time dilation won't occur. (Assuming we are not traveling near the speed of light)

One of the more mindbending things to wrap your head around is that gravity isn't a force (i.e doesn't not cause acceleration in the F=Ma sense.). You're simply on a path through spacetime warped by gravity.

Think about it: when you are in free fall you feel 0 acceleration. You appear to be accelerating relative to the ground-- but you're actually motionless in an "inertial reference frame". (Similar to how the astronauts on the ISS don't "feel" acceleration despite accelerating rapidly relative to the earth.)

The "force" of gravity is often modeled as "gravity pulling you down" and the ground "pushing you back up". This works mathematically, but isn't quite logically consistent.

In reality, on the ground you're in a region of warped spacetime, so you feel constant upward acceleration despite not actually accelerating. (Thinking of this another way, standing on earth feels identical to being in a far away spaceship accelerating at 9.8 m/s².)

This is also why time "speeds up" near more massive objects. (Separate from "acceleration".)

We're so used to gravity this it doesn't seem weird. But when you consider the fact free-fall is when you're not accelerating... well pondering that from many angles is what ultimately led Einstein to his model of relativity.

(This is me trying to condense what could be a 10 minute explanation into a few sentences, so apologies if it's not particularly clear.)

The Veritasium youtube channel has a great video about this: https://www.youtube.com/watch?v=XRr1kaXKBsU

Then what about just orbiting around it? It would just need to be big enough to not cause strong tidal forces.

The solar system is traveling in space and is quite massive but I haven’t been crushed to death yet.

You are far away from the masses. We are alive because we only experience 1g on earth. 10g is enough to kill a human. I don't even know what 14,000,000,000,000g would do, but I imagine we would get added to the neutron star.

1g towards Earth's center. Well, center of mass.

:adjusts bow tie:

Welllll…combination of that and angular momentum. Combined with the masses of the sun, and the moon, the other planets, and all the other masses of the universe.

Fortunately, orbits factor in, too. Everything that orbits is essentially in free fall, wellll…until they get tooo eliptical and the oscillating accelerations get really noticeable.

And it all affects time. Time is a bunch of wibbly-wobbly…stuff.

(( Ok, ok: Matt Smith might not have been that wordy as The Doctor. ))

Your username adds an especially eerie context to your comment.

I know this is true, but I never got my head around this, maybe someone can help. So if a photon is emitted from the sun, it's passing earth immediately? Then why does sunlight need 7 or 8 minutes to get to earth? And let's say, i travel one light-year at the speed of light, that should be instantaneous, right? The odometer would show 1 light-year, my watch would should 0 seconds and some decimals. How much would I have aged by the end of the journey?

If I travelled at one percent of speed of light, same distance, i suppose 100 years would elapse for me, how much would elapse on earth? Odometer would still show one light year?

And what if I left earth at 50percent of light speed , traveled 1 light-year away and did a turn and came back to earth at same speed. For me, it would be 1 year, but if I had a twin brother who was waiting on earth, would i now be a year younger than him? And how is this possible?

> So if a photon is emitted from the sun, it's passing earth immediately? Then why does sunlight need 7 or 8 minutes to get to earth?

From the photon's perspective, it passes earth immediately. From our perspective, it takes 7 or 8 minutes.

> And let's say, i travel one light-year at the speed of light, that should be instantaneous, right? The odometer would show 1 light-year, my watch would should 0 seconds and some decimals. How much would I have aged by the end of the journey?

You would have aged as much as your watch says you would have aged. Zero seconds.

>If I travelled at one percent of speed of light, same distance, i suppose 100 years would elapse for me, how much would elapse on earth?

I don't know how to do the math, but a very, very long time would have passed on earth.

> And what if I left earth at 50percent of light speed , traveled 1 light-year away and did a turn and came back to earth at same speed. For me, it would be 1 year, but if I had a twin brother who was waiting on earth, would i now be a year younger than him? And how is this possible?

Yes, you would be younger than him, and it's possible because that's just how relativity and time dilation work. It's even practically measurable in "real" life: http://www.leapsecond.com/great2005/tour/

> > If I travelled at one percent of speed of light, same distance, i suppose 100 years would elapse for me, how much would elapse on earth?

> I don't know how to do the math, but a very, very long time would have passed on earth.

Eh, at .01c, not so much: https://www.omnicalculator.com/physics/time-dilation?c=USD&v...

(=100.005y, not even two days extra)

Yeah, all interesting things only begin at roughly 0.5c and go crazy at 0.9c. Google images on “lorentz factor” to get the quick idea.


> From the photon's perspective, it passes earth immediately.

To go even further: to a photon's perspective, it is able to teleport instantly anywhere in the universe, because time doesn't elapse for it.

just curious... does "time" have a "speed" ?

Is travelling at the speed of light, actually travelling a a fraction of the speed of time?

> Is travelling at the speed of light, actually travelling a a fraction of the speed of time?

This is probably one of the more counter-intuitive simple calculations you can do in physics.

In special relativity, distance is given by:

  ds^2 = dx^2 + dy^2 + dz^2 - (c^2)dt^2
if X is your total distance in space, you have:

  dX^2 = dx^2 + dy^2 + dz^2
Which is just the standard Pythagorean theorem of Euclidean geometry.

Further, your velocity is given by dX/dt. If you are traveling at the speed of light, you have:

From which you can derive

  dX^2 = (c^2) dt^2

  dX^2 - (c^2) dt^2 = 0

  ds^2 = 0
In other words, the "distance" light travels in space time is 0.

> just curious... does "time" have a "speed" ?

It is not clear how to parse this question. Traditional "speed" is defined as distance over time. We can give this meaning for time itself by realizing that there is no single notion of time in relativity. As such, you could consider the line parallel to the time axis in the coordinate system of observer A. Since dt=0 in the coordinates of observer A, the speed of this line is not well defined. However, we could consider the coordinates of observer B. Assuming B is moving relative to A, he would see this line as being slanted, with both a time component, and a space component. As such, B could compute the speed of this line as dX'/dt', where X' is the total displacement along B's 3 spatial dimensions, and dt' is the displacement in B's time dimension. As such, B could meaningfully answer "what is the speed of A's time". Assuming I didn't mess up on the math, dX'/dt' turns out to be the velocity of A relative to B. This is a curious result that I have never seen before, but I can't really see any physical significance to it.

B could also compute dt/dt', where t' is the time axis in B's coordinate system. This computation seems more useful as it gives a direct measure of time dilation. Unsurprisingly, it also works out to be the Lorentz factor.

In special relativity, you have two notions of time:

- Time relative to an observer: Designating a non-accelerating massive object ("observer"), you get a coordinate system which assigns a time and distance to each event (point in spacetime). In this coordinate system, the observer moves along the time axis.

- Proper time: For an object taking any path through spacetime, you can measure the "subjective" time which has passed between two points on its trajectory.

The two notions coincide along the path of an observer: For each second of subjective time, the observer moves one second along the time axis in its coordinate system. Observer time moves at one second per second, if you will.

What you usually call "velocity" is distance/time in the coordinate system of some observer. For massive objects, this is always smaller than the speed of light. If you want, you can define another notion of speed, to illustrate the original commenter's point: distance in some coordinate system per proper time. This can be arbitrarily high, because at high velocities the proper/experienced time becomes shorter.

To get back to the speed of time: If you measure coordinate distance per proper time, it is only natural to also measure coordinate time per proper time. If you take earth as the observer and follow a spaceship, this is earth time per spaceship time. For a fast spaceship, time on board passes slower than on earth (time dilation), so reciprocally, the spaceship moves through (earth) time faster than 1s/s.

(Unfortunately, the notion of proper time becomes useless for massless particles moving at the speed of light: proper time along their trajectory is constant. They "do not experience time".)

Yes, one second per second.

I know this might seem like a joke answer but I love it because it's the truest answer.

The 'speed of light' is nothing but a coefficient between seconds and meters (or in general, between units of time and units of distance and is equal to 1 in any sensible measurement system) since the spacetime in GR is unified

We are all time travelers traveling at one second forwards in time.

It can be said that everything is traveling through spacetime "at c (the speed of light)". The math works out such that the faster you move through space, the slower you move through time and vice versa.

The faster you move through space, the faster you move through time as well, actually! That's because distance in spacetime is defined with a negative sign for time periods. The profound statement is now that the subjective time (which can be measured by a clock moving along with you) matches the theoretically-defined spacetime distance (which is constructed to be invariant under Lorentz transformations).

So it sounds like they did a decent job sticking to the science on the Interstellar movie?

Interstellar correctly demonstrates how general relativity will work.

Even the 5-dimension tesseract was a decent representation as well.

Matthew Mcconaughey using love to navigate through the 5-dimension is classic hollywood.

'love' might have been their interpretation.

Another one would be cause and effect, if that had already happened it means it must happen again... and so whatever he does there it will succeed in setting events in motion again... Interesting bit would have been how it all started.

Einstein can help: https://books.google.com/books/download/Relativity.pdf?id=3H...

His book was intended to help people understand these exact questions, without getting into any complex math.

Regarding "how is this possible?": This was experimentally verified in the 70s in the Hafele–Keating Experiment. Read more here: https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experim...

That Google link doesn't work.



> And let's say, i travel one light-year at the speed of light, that should be instantaneous, right?

Time slows down for the faster moving particles. And by time slowing down we mean all the particles in your body equally all start to move slower and more sluggishly in sync.

This is because it takes more energy to accelerate a particle as it approaches the speed of light. So if you had a pendulum clock moving almost at the speed of slight, that velocity of the pendulum at rest would be at X m/s, but if the whole system is already moving super fast the extra X m/s would take too much energy. So since the energy is constant the relative speed of the pendulum just becomes much slower.

This is a superb description.

If you were watching their lives out the window of your spaceship you’d see them in fast forward vv. they’d see you in slow motion. Everything is relative.

The main ”paradox” found by experimention is that the speed of light is a constant, regardless of your velocity. The only way this could be true, if you do a thought experiment, is if time was dialating for you. As for the physical and mathematical “why is this happening”, that’s where Einstein comes in.

It’s an exponential. At 0.5c 100 years to you would be 115 years elasped to an observer, 0.9c 229 years, 0.999c 2,236 years, etc.

Here’s an online calculator for the dilation effect https://www.omnicalculator.com/physics/time-dilation

> If you were watching their lives out the window of your spaceship you’d see them in fast forward vv. they’d see you in slow motion. Everything is relative.

Both observers, looking at one another, would see the other moving near c. Neither would know who was ‘actually’ moving. Yet, you assume there would not be a symmetry in their respective views of the other’s passage of time.

Explain why.

In simpler terms, a twin in a c-speed rocket could very well assume he was still and the earth was moving away. He should expect to find a younger twin when the earth ‘returned.’ Yet the examples only have the earth twin age, so to speak, and not the rocket twin.

It’s still a hard topic related to the rotational “symmetry”.

https://m.youtube.com/watch?v=cPEwkMHRjZU (7 minutes)

It doesn’t answer your question directly, but the tricky part of a twin paradox is not that the earth twin observes his brother in a slow motion (the space twin also does that, thus a paradox). It’s because a space twin actually changes direction by acceleration at some point B, and at that time he skips over a big part of earth’s timeline. The video above only addresses why it’s NOT the earth twin who changes direction by acceleration, which you’re reasonably questioning. The universe somehow knows who is really “steering” and what remains more or less inertial. The pendulum example at the end may give a hint on why.

Edit: also, the space twin doesn’t have to experience any additional acceleration from the “engines” - looping around some gravity well would work too. E.g. an entire trip could be that the space twin goes to the orbit around the earth, gets slung away by a quickly passing blackhole, loops around a distant blackhole and returns, all in a complete free fall.

The symmetry is broken in your twin example because the travelling twin had to accelerate to depart, accelerate to turn around, and accelerate to stop again at earth.

It has nothing to do with acceleration. You can play games with acceleration and start to get "wrong" answers with the twin paradox.

What actually matters is who travels the longest World Line a.k.a. the longest path in 4-d spacetime https://en.wikipedia.org/wiki/World_line . That is all.

A person on a non-curved geodesic spacetime path ages more quickly than a person on a curved non-geodesic spacetime path

I asked this in physics when I first heard the twin experiment. The answer comes from more complicated relativity based on acceleration and depends entirely on which reference frame you meet in. If you travel back to earth you age slower, if the twin catches up to you they aged slower....

It's not intuitive but time actually warps and there is no true concept of "simultaneous" in a special relativity world.

> there is no true concept of “simultaneous” in a special relativity world

This did my head in. Wikipedia has a great explanation.


The one who experienced the force of acceleration is the one who shifted their inertial frame w/re to the reference.

This TEDEd video works through the thought experiment, with messages sent from each twin to the other at 1-year interval (in the sender's frame of reference).


because light doesn't experience time, and the closer you get to the speed of light, the less time you experience. So what OP didn't say was that, while YOU can travel many lightyears in your lifetime, everyone you know will long be dead.

Also, it doesn’t take too much acceleration to make that trip. Comfortable earthlike 1g (~ 10m/s/s) is enough to build up a decent speed in a very reasonable time. Energy is the issue though.

It's a fantasy as long as you are still living under the tyranny of the rocket equation.

I don't think anybody thinks such a ship is possible with chemical rockets.

It's worse than that. It's not possible with any type of rocket. If you are accelerating by pushing mass in an opposite direction then you'll never build a ship that can accelerate at 1G for years at a time.

Unless we invent a reactionless drive the idea of traveling between solar systems remains a pipe dream.

Just need Astrophage /s

Ha, got the Hail Mary ref.

Yep. At 1g you could literally go to the edge of the visible universe in less than 50 years (assuming you are targeting the edge as defined at time of departure).

If this is true (I'm too stupid to do the math), then consider my mind blown!

It is technically true, but requires all kinds of miracles, not the least of which is avoiding every trace of matter/antimatter between here and there.

Oh, and all light from the universe is now gamma radiation focused intensely ahead of the spacecraft cooking the whole thing.

An atomic nucleus sitting in deep space becomes an apocalyptic collision at 99.99% light speed.

You need to spend equal amounts of energy and time on both acceleration and deceleration. It doesn’t help to go fast if you can’t stop at where you are going.

Depends what the cargo is.

True - if it's a warhead, the acceleration is a bonus.

Well, if you just want to destroy a planet at near light speed, they would never see it coming and deceleration is unnecessary. Scary thought though.

Check out Pohl Anderson's "Tau Zero" which examines precisely this scenario. A starship using bussard ramjet needs to find an "empty" section of space free of interstellar hydrogen so they can fix their engines and decelerate without getting fried. This requires them to continue accelerating so that time dilation effects will allow the trip within the crew's lifetime. That's the setup, not the plot.

To put it another way, time travel machines are very much theoretically possible, and in fact are “only” an engineering problem (an extremely hard one, however). But, only forward travel is allowed: time travel machine could take you as far forward as its engineering would allow it, but there is no going back.

At 1G constant acceleration (both speeding up and slowing down) you can make it on a short vacation to the Andromeda galaxy and back in a lifetime (or 5 million years from the perspective or Earth).

It looks like about 57 years.

Assuming constant acceleration to the 1/2-way point, flip, deceleration, and using http://www.projectrho.com/public_html/rocket/slowerlight3.ph... :

  Time elapsed (in starship's frame of reference, "Proper time")
  T = (c/a) * ArcCosh[a*d/(c^2) + 1] (given acceleration and distance)
  year = 365.25*24*3600; c = 3E8; a=9.8; d=1.25*1_000_000*(c * year)
  from math import acosh
  T = (c/a) * acosh(a*d/(c*c) + 1)
  => 14.3 years each quarter 
  => 57 years round trip
The speed at flip would be 99.99999999993978% c - good thing intergalactic space is mostly empty.

Bad news CMB shifts into infrared, visible, uv, xray and then hard gamma.

Oh, what a shame. I was about to install some rockets on my RV and head off to Andromeda. Guess I don't need to put my newspaper subscription on hold now.

since we are talking insane speeds and energies required to reach them why not just say you are in a bigger ship with more radiation shielding a dozen or so meters of water in the hull and a layer of lead a foot thick aught to stop most of it

FWIW, the Project Rho page I linked to also gives the derivation of the relativistic rocket equation. When I plugged in numbers, assuming an exhaust velocity of the speed of light (ie. impossibly high), I get a mass fraction of about 2,000,000 to reach top speed. Then another 2,000,000 to slow down.

Assuming the fully loaded RV weighs 5 tons, this means at max velocity the rest of the ship weighs 10 megatons, which is 10M cubic meters of water, or a cube 200m on each side.

That sounds like plenty of material, right?

The same Project Rho page links to https://arxiv.org/ftp/physics/papers/0610/0610030.pdf which calculates that at the relatively slow 0.995c "the penetration depth of protons of this energy will be ~40 m in water and ~10 m in titanium".

For 99.99999999993978% c, even 10M cubic meters isn't going to be enough.

They are only theoretically possible if you allow for negative mass and energy--not an engineering problem so much as a "need to find exotic matter"

Basically people ran the EFE "backwards" to see what matter distribution makes the wanted curvature. You get either negative mass-energy or the bubble doesn't travel ftl iirc.

Gp is talking about forward-only time travel though, you don't need anything exotic for that at all, just lots of energy and an efficient way to turn it into thrust.

The Alcubierre drive “beats” the speed of light, for the effects described 0.7-0.9c are more than good enough (if slightly unpractical).

We can’t do that either for now, but is way easier on the feasibility scale.

> The Alcubierre drive “beats” the speed of light

The Alcubierre drive is only a thought experiment that requires "exotic matter" (aka fairy dust) to work.

Papers have been published that have found ways to create warp drives without the need for exotic matter: https://arxiv.org/pdf/2006.07125.pdf

I know, I was making reference to this

> They are only theoretically possible if you allow for negative mass and energy--not an engineering problem so much as a "need to find exotic matter"

Basically people ran the EFE "backwards" to see what matter distribution makes the wanted curvature. You get either negative mass-energy or the bubble doesn't travel ftl iirc.

No, for forward time travel you don’t need any sort of exotic matter. You just need a really fast rocket.

It just means you can't within the lifetimes of the people you left behind. For that you need more exotic contraptions (wormholes, warp drive, etc).

If they were to wait a year and then come find you far away, would it net out and you’d be the same age?


> I think a lot of people assume that this means you couldn't go more than a few (~100) light-years in your lifetime... But this is not actually correct

It his, it just depends on the observer.

An external, stationary observer will never see you go further than 100 light years, but yourself? Assuming you are able to make your ship go any arbitrary speed approaching c, you could be traveling billions of light years.

It's just that when you stop (if you manage to stop), the universe around you will have aged billions of years, while you will only be a few years older.

>Why can’t I go faster than the speed of light?

I assume people know how bad the penalty can be for going merely 10 MPH over the speed limit.

Don't worry about an individual photon except as part of an image in this example.

The light from the source hits the subject and is reflected toward your eyes at the speed of light.

That's why they call it the speed of light, and radio signals do it too between their source & receiver.

So now imagine you could travel faster than the speed of light to a planet a number of light-years away and you are going to get there from here.

Once you leave Earth orbit you will be able to accelerate up to and beyond c in the safest most gradual way directly toward your destination.

While still in orbit you look down on the traffic in your hometown, and everything is still moving at normal speed no matter how far it is down there, since you are a steady distance away from what it is you are looking at.

As you accelerate away from Earth and approach the speed of light itself you're beginning to catch up with the light that was reflected off your home planet quite a bit earlier than the light which is simultaneously being seen by those back in orbit.

So at half c you look out the window and it looks like everyone back in your hometown is moving at half speed. But naturally time marches on down there. You just can't be so sure any more. At that speed if you left 2 years earlier you will only be able to know additional things about your home which happened no more recently than 1 year ago at that point.

Of course you can't communicate with them about this in real time because of how long it takes the radio signal to get back & forth so you don't bother.

You keep going and reach full light speed which finally matches the rate the images are being reflected away from the Earth at, so now look back home and everyone appears to be standing still on Earth, as expected. Even though as far as you know they are still carrying on like normal.

OTOH, approaching the destination planet at the speed of light, that's pretty fast, but you have to realize their alien traffic is actually only moving half as quickly as it looks from your craft, whilst you are speeding so rapidly in their direction. Don't let that fool you, the aliens are only half as advanced as they look.

If you want to really see something, go faster than the speed of light and the planet you are approaching will be moving more than twice as fast as normal, and looking Earthward all you can see would be things moving backwards.

One thing that's happening is that you are always seeing images of these two planets where the light source originated from two different suns.

Once the distances get far enough, it's possible to launch a mission to a destination that actually no longer existed any more for quite some time before launch, only who knew?

In that case the earlier you make your reservations the more unwise it could be.

If distances are dilating, why isn't that /less/ ground?

The distance doesn't dilate; time does. The distance contracts.

E.g., to a photon moving at 1c, the whole universe has a contracted length of 0 meters, and it crosses the whole universe instantly. To us, observers at <1c, the whole universe has a non-contracted length of <a lot> and the photon takes <a long time> to cross the whole universe. By the time the photon's 0-second journey across the entire universe has finished (whatever that means), we're all extremely old. :D This is the time dilation meme of slowly-aging space travelers but taken to the extreme.

So if I was in a space battle and the enemy 'jumps to light speed'(1) to make a quick escape ... they would actually be easier to target with a laser because they 'slow down' from my perspective?

(1) 'light speed' as in the speed of light, not as in a sci-fi context of hyperspace jump/FTL jump.

> E.g., to a photon moving at 1c, the whole universe has a contracted length of 0 meters, and it crosses the whole universe instantly.

Since the universe expands with >1c, I wonder if the photon actually crosses the whole universe. And if not, how it would look like from the prespective of the photon?

Thanks for the clarification

It basically depends on your perspective. To someone on earth watching the spaceship speed up to the speed of light, the spaceship looks like its contracting. To the person on the spaceship, the rest of the universe looks like it is approaching the speed of light, and hence, contracting.

Yup. If Musk builds a continuously accelerating rocket, he could die at the end of the universe instead of on Mars.

I'm pretty sure this doesn't work out because you have to slow down.

So thinking about it, if you are traveling at that speed because of your inertial reference frame it is equivalent that everyone else around you is moving at (or near) the speed of light and they are moving slowly through time. This is the classic twin paradox and there is a resolution to it, which is that you can't instantaneously turn around.[0] Or in our case, we have to turn around to slow down.

[0] https://www.youtube.com/watch?v=0iJZ_QGMLD0

It does work out, in fact you could go arbitrarily far in space (and, mandatory, in external time) if you had infinite energy to spend. The acceleration and deceleration phase is almost negligible.

Edit: the faster you go, the slower your own (inertial) time passes. That means the external time passes faster, and the factor grows to infinity the closer you get to C.

In fact, subjectively there is no speed limit. As you go faster, anything around you ages faster, but you yourself won't encounter any speed limit.

But in your inertial reference frame the people on Earth are moving at (near) the speed of light. So they are the ones that should be staying young. Or similarly the planet you are traveling to is actually speeding towards you and you are staying still. This is why the twin paradox is a paradox, because of the reference frames.

You're right that the apparent symmetry is broken by acceleration(s!), and to show that I'd point to Michael Weiss's twin paradox equivalence principle analysis at https://www.desy.de/user/projects/Physics/Relativity/SR/Twin... rather than rewriting it.

There is a subtlety not explicitly raised in the writeup, mainly that in General Relativity metrics do not superpose cleanly, in the sense of getting another solution to the Einstein Field Equations. We do not worry about this in the ultrasimplified twin-paradox model where the spacetime is flat in the sense that the Riemann tensor vanishes everywhere. However, if we want to consider the behaviour of gravitational waves with amplitudes outside of the weak https://en.wikipedia.org/wiki/Linearized_gravity limit, we are in a world of calculational pain.

Physicalizing this subtlety, if our travelling twin is travelling in our neighbourhood of the galaxy, it is probably in for a bumpy ride due to gravitational waves from nearby binary stars https://news.berkeley.edu/2021/02/22/binary-stars-are-all-ar... . We cannot easily extract how bumpy by adding in the uniform pseudogravitational field proposed by Weiss. On the other hand, we probably cannot quantify the effects of gravitational waves at all by simple adapatation of the other strictly Special Relativity analyses at the related Weiss link, https://www.desy.de/user/projects/Physics/Relativity/SR/Twin... (which lists among other the resolution in the minutephysics youtube link you provided above).

You're also right that the problem is one of reference frames. We are not obliged to use that of one twin as the spatial origin. In principle any will do, but some choices have advantages driven by features deliberately excluded from the Special Relativity twin paradox.

Let's consider the "(s!)" tacked on at the end of acceleration. We have not only that of the travelling twin's spacecraft engine, but also that which drives the expansion of the universe.

From within our galaxy we observe a highly spatially homogeneous and isotropic arrangement of extragalactic luminous matter (and cosmic radiation, locally) without distortions in the shapes of distant spiral galaxies that imply a spatially non-flat universe. The metric expansion of this, retaining bulk isotropy, gives us a preferred foliation (Wald's 1984 textbook develops this pp 92-93, but alternatively we could use Weyl's principle). Each twin is free to use a "cosmic fluid" observable (like the dipole-free temperature of the cosmic microwave background, which expands adiabatically), even while accelerating, to determine the https://en.wikipedia.org/wiki/Scale_factor_%28cosmology%29 . For example, each twin could consider the dipole pattern dT/T = v/c where T in the twin's proper time. Each twin can thus determine whether it is the relativistic traveller or not, even if it only wakes up occasionally and only long enough to look at a snapshot of the CMB. The travelling twin thus sees a clear breaking of the Copernican principle along the direction of its travel. Or more precisely, with respect to the bulk flow of matter and radiation in the universe, the non-travelling twin can conclude that it is effectively a Eulerian or comoving observer, while the relativistically-travelling twin cannot.

Moreover, the twins (and any third party) can use "cosmic fluid" observables to determine the scale factor when the twins are together at the start of the travel, and when they (or at least one and the other's remains) are together again at the end.

In this approach there is no paradox at all, there is only the consequences of one twin with a worldline with sections where the proper time is at a higher tilt to the cosmic time than the other twin's. We also avoid the difficulties in attaching a pseudogravitational field to a spacetime where there are gravitational waves of reasonably large amplitude, or relativistic stars and other massive compact objects.

We head into the land of apparent paradox by stripping out evidence of an expanding universe. We must also eliminate evidence of the aging of galaxy clusters through gravitational collapse (including the rate of star formation and the change in abundance of heavy elements). Indeed, we have to arrive in a setting in which neither twin can determine that it has departed from a point at which some reasonable generalization of the Copernican principle applies.

Indeed, the usual formulation of the apparent paradox gets rid of everything but the twins, so that one cannot even use Rindler/Unruh-like observables in flat spacetime, and this really emphasizes the "Special" in Special Relativity.

In that setting, as I said above, relying on the equivalence of being in uniform acceleration (even if it's instantaneous) and being immersed in a uniform (pseudo)gravitational field, is a reasonable way to eliminate the apparent paradox.

There is a related "love triangle" Special Relativity problem where there are three parties: stay-at-home (S), early-outbound-passer (E), and late-inbound-passer (L). None of the parties ever experience any acceleration: they remain eternally in uniform motion, with E & L travelling relativistically.

At our origin, S and E synchronize observe their identical atomic wristwatches coincidentally agree that it is "0". Light-years away, E and L come very close to one another and exchange timestamps showing that coincidentally their identical atomic wristwatches agree. Finally, L and S come very close to one another and compare timestamps from their identical atomic wristwatches. All the wristwatch times are identical to those at the three points in the diagram of the "instant turnaround" version of the twin paradox, we've just turned the travelling twin into two unrelated travellers on different trajectories.

The argument is that this "love triangle" is resolved because E & L are different travellers in uniform motion, so all parties must combine the times acquired in two different reference frames (E's and L's) to compare with the times acquired in S's reference frame. The further argument is that this duplicates the "instant turnaround" version of the twin paradox if we can have the travelling twin change direction without acceleration.

Firstly, we can still solve this with a pseudo-gravitational field popping up at the moment E & L exchange timestamps. It's no more of a coincidence than the identical timestamp when S & E are close.

Secondly, it's not clear that the paradox remains interesting in this case, because there is no expectation that S & L should be the same age when they are close to one another again. They aren't twins. Unless we add in accelerations, there is no way by which S, E, and L could all have been born at close to the same location in spacetime.

Thirdly, it's unclear that there can be an instant turnaround without acceleration. A couple flavours have been explored here and there.

One involves a slingshot around a star to change directions from away to towards the stay-at-home twin. In this picture the travelling twin is always in free-fall. But here we are substituting real gravitation (that of the star) from pseudo-gravitation. We've moved from everywhere-flat Minkowski space -- the spacetime of Special Relativity -- to something closer to Schwarzschild spacetime, which is only asymptotically flat. Moreover, we are using the near region of Schwarzschild to accomplish the slingshot.

Another substitutes the open flat Minkowski space with one in which there is a compact spatial dimension that curls back on it self. A universe with the geometry of a cylinder with infinite height and small circumference, or a torus, or a sphere would do. The cylindrical case has been explored recently : https://doi.org/10.1119/10.0000002 with comparisons to Minkowski space (the spacetime of Special Relativity), §IV (Conclusion) being pithy. Again, I see this as trying to substitute pseudo-geometry with real geometry, an adapted clock-comparison recipe, and a highly privileged frame for the traveller, in order to avoid a non-gravitational acceleration opening the door to a pseudogravitational field arising in the ultrasimplfied and thus strictly Special Relativity problem.

The pseudogravitational field approach comes from Einstein in 1918: https://en.wikisource.org/wiki/Translation:Dialog_about_Obje... which was fun to read.

Finally focusing on the latter part of my comment that I'm self-replying to (mostly for my own benefit), we have only done away with one acceleration by the returning twin. We still have the effects from the behaviour of matter in the expanding universe with which to clock S, E and L, removing the remaining paradox if we somehow contrive to have S, E & L expecting to age similarly. If we are abandoning Special Relativity in order to avoid acceleration by the returning without invoking outright magic, why only do it along one spacelike dimension, or by importing a very finely tuned third traveller?

But motion is all relative. Who is considered to be moving faster?

The simplest way I could ever wrap my head around this, is to understand that everything travels through spacetime at the same speed. There is no travelling faster or slower, just crossing the graph at a different angle.

You have to be careful when talking about 'speed' when you make time a dimension in your geometry. Traditionally, speed is a measure of how much of your space-time curve is in along the time axis.

For any space-time path, you can consider the coordinate system as an observer traveling that path would see it, in which that observer would see itself traveling through time at a rate of one second per second. Geometrically, if you were to draw where on the path the observer's clock ticks, the distance between ticks as measured along that path is constant regardless of the path.

The speed of light limit says something different. It limits what paths a physical observer can take.

Condsider a 1+1 dimensional universe (or our 3+1 universe with a test particle moving along a single spatial dimension).

Pick a non accelerating observer to construct the 'stationary' coordinate system. Plot spatial coordinates along the horizontal axis, and the time coordinate as the vertical axis. Pick units such that the speed of light is 1.

A particle moving at a constant velocity will follow a straight line. If the line is vertical the particle is stationary. If the line is at an angle, the speed of the particle is the inverse of the slope of the line. The speed of light limitation says that this line cannot be shallower than 45 degrees.

In more analytic terms, the distance metric for our 2 dimensional spacetime is given by ds^2 = dt^2 - dx^2. The speed of light limitation says that ds^2 cannot be negative for any path a particle actually takes.

In other words all particles must must have at least half of their travel be along the time dimension.

To me, this was why the statement that "B would see the objects growing further apart while A would see them getting closer together", if FTL was allowed, was not a very satisfying answer.

So given what you've described, that means that forces applied to bodies need to have a time component equal in magnitude to the spatial components. Forces must always exist along that 45° line. The limit of the force required to continue to rotate that spacetime velocity out of the time component and into the spatial components goes to infinity as the vector approaches that 45° line.

The fact that forces are unidirectional is the unexplained part. If they weren't, then we could rotate that vector further, and start traveling backwards in time. Then wouldn't B expect to see the objects moving apart, while A sees them moving closer together?

To me, the impossibility of the disparity in observations is a consequence of, dependant on, no-FTL, not an explanation thereof.

Addendum: I don't understand why you said 45° instead of 90°. I thought objects traveling at the speed of light would experience infinite time dilation, and thus be observed as having 0 passage of time.

> Addendum: I don't understand why you said 45° instead of 90°. I thought objects traveling at the speed of light would experience infinite time dilation, and thus be observed as having 0 passage of time.

Think about what the diagram shows: Every (s[pace], t[ime]) coordinate pair on the spacetime diagram shows an observation of a particle. So in natural units, a photon's wordline is given by s=t or s=-t (traveling in one or the other direction). If you draw that, it's a 45° line. It also gives you the light cone of the observer at (0, 0).

A horizontal wordline would be something moving at infinite speed, not the speed of light, as it is observed at every place at the same time.

And the reason for this, as we all know, is because the universe runs on distributed computers with no global state, where each computer simulates its own local physics and connects only to other nearby computers, and thus the so-called speed of light is simply the emergent rate that data may travel across the network.

It's funny, as a programmer i had similar interpretations. Universe is like a processor and the c is the frequency limit.

There was a talk with lawrence krauss I think, where he explained that at galaxy scales, everything is distributed, there's not one reality but an infinity since no point in space can be aware of far points in the universe.

I also wonder if there would be ways to tweak C.

This short story might tickle your fancy.


(I also highly recommend their book "There Is No Antimemetics Division" if you're at all interested in SCP Foundation)

Good story! What is SCP Foundation?

Nice one, thanks for sharing.

No, has to be more subtle than that.

The rate the simulators transmit data are irrelevant to our timeframe (since we’re a part of the computation).

I believe it’s more like a centralized interpreter with decentralized (NP) evaluation strategies. The limit reflects the minumium size of a recursive expression.

I meant NTM*, that is to say, assuming many-world interpretation where the way your conscious experience of your life makes quantum measurements by (somehow) following the shorest accpeting computation of the NTM. Your life is thus meaningful as everything you experience from birth to death is literally the solution.

Consciousness (at least in this world) can perchance then be understood as the artifact of a feedback loop constructed by a (meta-circular) interpreter that converts the topologies of molecules into qualia, which then makes adjustment to the reflective tower (with physiological-changes-in-brain&body as correspondence).

Time perception then is the same consequence that follows relativity ie a direct result of this speed limit that reflects the minumium size of a recursive expression (but concerning only the viscosity and turbulence of neurotransmitters in relative to eletric signals?)

Yes, the magnitude of something's 4-velocity vector is always c^2 (set it to 1 in your unit system) by definition. It's not as simple as a R^n vector because you have a time coordinate of opposite sign and you switch coordinate systems by Lorentz transformation, but it's the right intuition. (It's complicated further in the presence of gravity, where you need an enriched differential geometry notion of a velocity vector since the spacetime is curved.)

Hm, but if I can make my angle orthogonal to the time axis, isn’t that like instantaneous movement?

Exactly. At one end of the scale, you're not travelling through space (or practically-zero on a relativistic scale), and you're experiencing the full affects of time.

At the other end of the scale, you're travelling through space (or practically-C on a relativistic scale), and you're not experiencing time.

So the whole theory of zipping around space and coming home to find you've barely aged, is just spending more time at a higher "angle" than everyone else.

It is instantaneous from the perspective of the object’s internal clocks.

Traveling at the speed of light results in infinite time dilation. Which means that from the perspective of an outside observer, no time at all is passing inside the spaceship.

What happens to fields (magnetic, gravitational, electrostatic, etc) in that situation? E.g. can a photon 'feel' the surfaces it would ultimately interact with?

One way I like to think of this is the term 'sun-kissed'. From the perspective of the photon, the sun is actually giving you a kiss on a summer day.

'feel' the surfaces it would ultimately interact with

I would like to see the map of the universe at different potential speeds (or thrusts). E.g. you choose a point in space nearby the sun. At thrust zero you only see hot sun everywhere, because there you go anyway. But at greater thrusts the sun turns into a circle and you start to see sections of the “sky” where you could land, given the thrust is constant. Some areas would be still black because of blackholes, orbits and event horizon. I always wanted that simulation but never found it. It would be much more interesting than just looking around via reversed photons flying into your eyes.

That's correct. As you approach the speed of light, the elapsed time in your frame of reference approaches zero.

If you could move in space but not in time, then you would be in two places at the same time which would violate conservation of matter, right?

That depends on how you define 'conservation of energy'. Consider an arbitrary bounded volume of spacetime. Conservation of energy says that the net flow across the boundary of any such volume is 0. Under this definition moving in space but not time is not a violation, as the same amount of mass enters the volume as exits it. The only odity is that both events happen at the same time

That definition sounds broken, since energy is still conserved if I move something into or out of the bounds.

It stops being conserved if there's suddenly more or less energy inside the volume without the same amount crossing the boundary

The idea is that the volume we are talking about is a 4 dimensional volume of space-time and is bounded; not a three dimensional volume of space that extends to infinity along time.

Conservation of energy says that it is impossible for energy to enter this volume with that same amount of energy exiting the volume.

Consider what it would mean for this to be violated. For the sake of argument, assume that all particles must move forward in time by a non zero amount at all points along there path. Since the volume is bounded, any particle with an infinite path must eventually have a time coordinate beyond the largest time coordinated in the volume. Therefore, the particle must eventually exit the volume. If you were to work out the geometry more carefully, you could show with relative ease that the particle must exit the volume an equal number of times as it enters. If a particle were to enter the volume without exiting the volume, it would mean that said particle was destroyed within the volume. Similarly, if a particle were to exit the volume without entering, it would have to have been created within the volume. Both of these situations are possible if an interaction occurs within the volume, but the net energy of the particles leaving such an interaction, must be the same as the net energy of the particles entering the interaction.

Put another way, assume that all interactions obey the conservation of energy. If our original volume was V, we can construct a new volume V' from V by carving out sub volumes in which an interaction occurs. Since all such sub volumes obey the conservation of energy (by assumption), the net energy flow into and out of V' must be the same as for V. However, since no interactions occur withing V', all particles entering V' must exit V' an equal number of times, so the net energy flow of V' must be 0. Therefore the net flow of V must also be 0.

Yes, which is why a photon experiences zero proper time.

That proper time is zero everywhere along a photon's geodesic does not mean that the photon cannot evolve from point to point along it, and we can show this by taking advantage of total coordinate freedom.

We can parametrize (as in make parametric) arbitrary curves through spacetime however we like. Parametric representations of unique curves are generally nonunique.

Some of the infinite possible parametrizations of a chosen curve have useful properties, such as uniquely labelling every point on the curve with some monotonically ordering value and keeping the form of some set of equations reasonably simple.

For timelike geodesics, particularly in the Minkowski space of Special Relativity, proper time (being a Lorentz scalar) is a good option. However that is not true for all geodesics in Minkowski space (as you note, the proper time is everywhere zero on a null geodesic, and so a bad option), much less all curves through general curved spacetimes.

For null geodesics, following the logic of GP's question, we may wish to preserve the tangent vector under parallel transport; this requires the parametrization to be affine. Some gory details at https://en.wikipedia.org/wiki/Geodesic#Affine_geodesics and a brief useful summary at https://www.reddit.com/r/AskPhysics/comments/9aenid/what_act...

As is noted below the comment directly pointed to by the second link, labelling a timelike geodesic with proper time is choosing one specific affine parametrization on that geodesic, and that this choice is driven by convenience.

One of the neat outcomes of affine parametrization is that we can take a point on an affinely-parameterized null geodesic and look at the derivative with respect to the affine parameter there, and define a momentum k^{\mu} = \dot X^{\mu}. In a Lorentzian spacetime, with curvature, we can compare the momentum at two different points on the null geodesic, giving us the gravitational redshift between those two points of the photon's wavelength equiv. frequency.

They should make a movie called ‘Photon’ and track its life. A parody based on science.

The thing that I really have trouble wrapping my head around is where each "thing" begins and ends, ex. for astronomically large bodies that take light years to travel across, how does it move and how does information propagate through it? I assume the simplest way to think of it is for every atom to have its own bar in the graph, and every "thing" is kind of "wiggly" as information travels through it.

But then that sounds like I'm describing the speed of sound, no? Maybe I'm confusing two concepts.

What gets me about these classical ways of thinking about higher dimensions is collisions. Couldn't you bump into something going a different time rate and get deflected backwards if it was indeed analogous to classical movement?

Yeah, but if something goes at differrent time rate, it means it will have (typically very vastly) different space velocity. You can't have objects very near going at low space velocity and high time velocity, so this effect will not be noticeable.

Xkcd style explainer: When you have two things going at noticeably different time rates, you typically prepend "relativistic" [0] to all interactions. "Relativistic collision" sounds almost like "changes into huge amounts of plasma escaping from contact point".

[0] https://what-if.xkcd.com/1/

Right, this is like the 2d projection of it which makes it easier for our 3-space brains to comprehend.

Wow, this is the first time I have understood why nothing can go faster than the speed of light and why time dilation is a thing.

A very easy to read article, give it a try.

Also if you are curious why a moving charge creates a magnetic field watch this Veritasium video: https://www.youtube.com/watch?v=1TKSfAkWWN0

Frankly though, I not sure why they couldn't just use a simpler correction: postulate that moving charge only creates magnetic field with regard to objects moving relative to it. Which is also basically true?

This way, the stationary observer will sure notice that there's F_B, but will also know that the charge moving alongside the (also moving) rod does not experience it.

That's the thing though, in the article the charged object does not move relative to the rod. The rod, object, and person B are stationary relative to each other. I was actually really surprised to read that two charged objects moving with the same velocity relative to each other generates a magnetic field from A's perspective.

Two long charged rods moving with the same velocity relative to each other are two parallel wires carrying current, the second of which is a typical and easy-to-calculate example of electromagnetic attraction. In fact, that situation is so prototypical that it is used to define the Ampere in terms of what current is required to produce a certain force between parallel wires. [0]

[0] http://www.physics.louisville.edu/cldavis/phys299/notes/mag_...

In this scenario, rods are stationary but they have current running in them - a different situation. Can it be translated to the relativistic one?

Maybe what they really wanted to say is "there is a speed c at which the charge of a moving rod is indistinguishable of current running through stationary rod"? Now that would make a lot of sense.

It's not different as far as electrodynamics is concerned. The motion of the charge is what matters; whether or not the neutral part is stationary or moving is of no concern.

My favorite general relativity paradox: A runner is running with a 20m pole at a sufficient velocity so that a stationary observer sees the pole having length 10m. The runner runs through a barn which is 10m long. And two people outside shut the doors simultaneously so that the runner is completely enclosed in the barn for a moment.

But from the runner's perspective, the barn has length 5m long. What does she see happen when the doors are closed on her?

Since the two doors are not in the same location in space, there is no absolute "simultaneously".

What one frame of reference sees as a simultaneous closure of the doors looks like non-simultaneous closure of the doors in another frame of reference.

Simultaneity means that two events coincide in time and space.

Different time and/or space means: not simultaneous!

> And two people outside shut the doors simultaneously

No, they don't. I'd phrase it differently, e.g. “each door is shut for an instant when the end of the pole is just inside the door”. (Also, make it clear that the barn is long enough to fit a 10m pole between the two shut doors.)

When you put it as “shut the doors simultaneously”, it's more troll physics than a paradox.

Meta-puzzle: figure out how to still imply that the runner is completely enclosed for an instant, without incorrectly insinuating that it is so from her perspective.

From the frame of reference of the runner the pole will first hit the door at the end of the barn (unless it opens in time), and the door at the front of the barn will only close just after the back of the pole has passed the door.

If the doors actually don't open in time then the resolution for the fact that the pole doesn't physically fit is that nothing can be rigid at relativistic speeds. So either the door or the pole or both will deform (violently) to allow the pole to fit inside the barn.

At these high speeds and tiny distances, very little would probably be noticeable to a human observer. If any matter collided it would be a catastrophic release of energy evaporating the human.

But I think over longer distances there would be a time delay for the light of the explosion to reach the individual, so they would probably the light of a visually warped pole colliding with visually warped barn doors very briefly before disappearing into an explosion.

This is a special relativity paradox.

D'oh! I know better than to make that mistake and yet "their" it is.

How do the two people at opposite ends of the barn agree on when to close the doors as a runner approaches at relativistic speeds?

Also, relevant xkcd (what-if) https://what-if.xkcd.com/1/

Most likely from the runner's perspective the world turns white as they dissolve to plasma prior to reaching the barn.

The beginning is the best line, that no one really knows "why" and what I have said to even other physicists who don't really understand it. A lot of the counterintuitive consequences of the paradox of the moving charges detailed here are a fact of nature but we don't really know why it must be that way. For example, in a modern particle physics starting point for qed, we force the equations to be lorentz invariant from the start but that is a starting assumption. But that's it, there's no real deeper answer to "why" beyond it being a experimentally observable fact, and the only resolution of these observations is that lorentz transformation and thus things like time dilation and length contraction must happen.

Small relativity related tidbit: I hate when people say the phrase "the faster you go, time slows down for you." This is a problem because it implies that the moving observer notices their own time dilation which is reverse of the case. Of course, every non-accelerating observer is in their own rest frame, so it doesn't make sense to say "time slows down for you," because you are your own reference and there is no other frame to base your measurements on (I mean that was the whole point of relativity, there is no universal rest frame). Instead, when you measure the rate of change for other reference frames moving relative to you, their clocks move slower when measured by your clock. So the actual phrase should be something like "the faster others go, the slower their time appears to you."

Q1. Does the future already exist? I realize this might be purely philosophical, but if I leave Earth, and fly through space near c kph, and return home a few hours later to find everyone aged 50 years... are they the "same" people, or are they a future-instance of the people left? To clarify, theoretically I could leave earth, and return home in exactly 1 hour (from my ref frame), and basically make people whatever age I want by varying my speed. Since the same 1 hour passes for me no matter the age for them I chose, it seems like I'm selecting an already existent future from a stack, not fast-forwarding the present I left.

Q2. How do black holes exist if time stops inside a black hole? That is, they continue to move through spacetime, even though spacetime is not moving within the hole? I accept that a black hole can form, and can stop time, I'm just curious how something that stops time continues to persist in the present. The way I'm visualizing this, is like a lava lamp about to bleb off some goo from the top [1]. As the object gets more and more dense, it curves and drags spacetime more and more, until eventually its density passes the Schwarzschild radius and blebs off.

[1] https://m.media-amazon.com/images/I/615deDvfDkL._AC_SS450_.j...


> basically make people whatever age I want by varying my speed.

No more so than making the Earth change shape by moving your physical location.

> Since the same 1 hour passes for me no matter the age for them I chose, it seems like I'm selecting an already existent future from a stack, not fast-forwarding the present I left.

Yeah these are more philosophical questions rather than a testable hypothesis. It’s an interesting way of viewing the world, but if you’re talking about a future “existing” scientifically and materially in concert with the present, then you’d need to devise an experiment to be able to enter or interact with that future beyond just waiting for that future to exist.

Otherwise time travel isn’t even necessary for this thought experiment. You could say the morning already exists and fall asleep and wake up and boom it’s morning time. But in reality all of the ticks of time happened between when you fell asleep and when you woke up, you just weren’t able to observe all of them at the same speed as someone who stayed awake all night.

You can even make this clear in the "leave Earth in a fast ship and return" scenario - have the people on Earth send you a birthday card by radio every year that you're away, on your birthday in their reference frame. You'll receive those ticks of time as you travel (most of them on your way back).

I suggest stop giving time a privileged meaning, and think of it as just one of four dimensions. John Wheeler starts his book on Special Relativity[1] by talking about surveying.

You and I survey a building (plot of land, whatever) separately because we want to check each other's work. At then end we look at our data and disagree on every single coordinate except the origin. Ah, you're a crappy surveyor, I conclude. No, you are, you reply.

Then for whatever reason I ask what distance you compute for the distance from the origin to the corner of the desk. 3.183 meters. Huh, exactly same number I compute. Okay, what about the distance from the far corner of the room to the southmost window? 18.45 meters. Me too!

A bit more chatting and it turns out you used true North as the y-positive direction, and I used magnetic North. Opps. Our frames are just rotated in respect to each other.

Now, is it freaky and weird that I say the speaker is 2.78 meters in the y direction, and you say it is 2.619? No, we are just using different frames of reference. There is no 'reality' to any given y direction or coordinate. It is arbitrarily chosen, as our the units (I could use yards instead of meters, and have an entirely different number yet).

OTOH, what is real, and invariant, are distances. That's a physically real thing. hence we always comput the same distance between any two points, despite using different coordinates for our (x,y) tuple. If you want to be mathy about it we say the metric is s^2 = x^2 + y^2. Pythagoras, in other words, in a Euclidean space.

Well, we don't live in space, we live in spacetime, where time is a dimension. When we move at different speeds relative to each other our 4-D coordinate systems are rotated relative to each other. That includes time. So, if you rotate yours relative to mine, travel for awhile (time and space!), well, you will end up with different coordinates for x, y, z, and t. It's no odder than if you and I travel 'North' in your car, but you use true North and I use magnetic we end up in different places on the globe.

In 4D space what is 'real' is not coordinates or time, but events, and what is constant is the interval between events. Just like what is 'real' in 2D Euclid space is not some arbitrary y-direction, but the distance between two objects. Distance is invariant in 2D space, event intervals (space and time) are invariant in 4D Minkowski space (the space we live in absent of gravity).

There's a bit of handwaving in there, but that's pretty much the physics; any 7th grader can do it. The main part that will lead to bad conclusions is that the metric in Minkowski space uses a negative number for time; so s^2 = x^2 - c^2 t^2. That's hyperbolic, so if you use intuition from Euclidean space you may conclude that in some instance distance will contract when it expands, or vice versa.

So, finally, to your Q1, if I travel magnetic North, is the position I reach on the "same" Earth as the one where you use true North? Feels like a weird question that misses the point, right? Same Earth, just a different location than you expected because my frame was rotated wrt yours.

Note that every experiment we have ever carried out bares this out. Accelerate a clock, bring it to Earth, that clock is younger (I'm ignoring general relativity's effects here, but the experiments don't). Measure how long a very short living particles live that are created by other particles crashing into our atmosphere, and they live exactly as much longer as SR would predict. Etc. did that clock "select" a different version of you? No, it just travelled a different path in 4D spacetime than you, and hence ended up at different coordinates. To go deeper into that I'd have to introduce "proper time", but since the ending x,y,z are the same are your x,y,z, can you see that intuitively it must be the t coordinate that changed?

[1] https://www.eftaylor.com/spacetimephysics/ This book is released under CC, free to download and share, and utterly fantastic. All you need is junior high math to master the material.

Wheeler also postulated that all electrons had indistinguishable properties because they were the same electron (moving back and forward through time). So in some sense Wheeler is treating time like a stack. (cant remember if Feynman convinced him otherwise).

That said your comment makes sense. Thanks for posting!

Regarding Q2, spacetime curvature is the grid itself. Think of it as "space" in the simple term where objects including black hole travel through. Black hole's gravity warps the surrounding spacetime. Think of it as the "space" is stretched out by the black hole's gravity.

Regarding time, there's no universal global coordinate for time in General Relativity. Time is based on each observer's time coordinate. When time stops inside a black hole, it means time appears to stop in an outside observer's time coordinate.

This is because the spacetime curvature inside a black hole has been stretched so much that the stretched curvature approaching infinite long. See the pulled down funnel [1]. An object traveling in no faster than light speed in the infinitely long curvature takes forever and its time appears frozen to an outside observer.

But it's an illusion to the outside observer. The observer is seeing the light imprint emitted from the object, not the real object itself. See the diagram below.

    BH     << o.     EH.     .   .  . .......... >> P
BH is the black hole singularity and o is the falling object toward BH. EH is the event horizon. P is the observer. The dots are the light photons emitted from the object along the way. The photons are traveling away from o and BH toward P. When the photons reach P, P can measure o's movement and time.

Imagine the object o has a blinking light beacon at its tail, blinking every second by its clock, i.e. on for half of a second and off for half of a second. You can see the light every second and measure the object's time.

Since the speed of the light is constant, the photon traveling away from BH will take longer and longer to move across the stretched out space, as shown by the spaced out dots between EH and P. As o approaching EH, P will see the blinking light slows down because the photons take longer to move across the stretched space. P would conclude that o's time is slowing down. Passed EH, the photon is not coming out because the space is stretched longer than it can cover in its constant speed. At EH, the photon can still come out but at a very slow pace because the space is stretched matching the speed of light. The beacon is not blinking because the light stream been stretched to infinitely long. Object o appears frozen as its time has stopped by P's measurement of the blinking rate. But what P sees is the long stream of photons stretched out when o approaching EH. The object o has long gone in its forever falling across the infinitely stretched spacetime. Only its light imprint before EH is being observed as frozen.

[1] https://www.sciencenews.org/wp-content/uploads/2017/05/05121...

Q1b. Are you the same person you were yesterday?

I think yes, and the answer is more concrete for "oneself", since the molecules that make-up oneself are all accelerating and decelerating with you/as you in your own reference frame.

I mean, apart from cells and etc dying and being replaced as we age. ;)

Truly, the Ship of Theseus was inside us all along. (Which is super convenient, considering how often we step into the same river twice!)

What would happen if you had a very large rotor of let's say 50.000km in diameter. And let it rotate at 1r/s. The inner part would only move at very low speed and the outer part at near lightspeed. What kind of time dilation effects would be seen? If you would sit on the tips for a few days and then stop the rotor, travel back to the center, you would see a very old one right?

I think veritasium discusses that as well. Essentially at relativistic speeds there are no chemical bonds that can hold and the device shreds itself.

Wow thank you ! I have read the whole article but I couldn't wrap my head around it. I found the "rod story" confusing then I followed your link (and also watched the related Youtube video).. This really makes things clearer. I am really not into maths but this version seems much easier to understand and very logical.

Looks to me like the short answer to the question posed in the headline is that matter is composed at an atomic or subatomic level of energies similar to light and so it would be quite inconvenient to try to propel the mass as a whole faster that its constituent parts can go. A similar principle may well be true for straight-line inertia, which might be caused by the same as the force that makes gyros stay on course. In general the scientific discussions seem to touch these ideas but approach theories in reverse of intuition so it's hard to know if they align.

Nobody postulated that the speed of light is constant because it was obvious: light just like every other wave has a constant speed in the medium though which it propagates. The fact that nothing can go faster is a consequence of the Lorentz transformation, so nothing had to be postulated. The important thing that Einstein recognized is that the Lorentz transformations apply for all physical phenomena and not only the Maxwell equations and that started a revolution on physics.

This is very interesting. Does it mean we can observe the constant speed phenomenon for other waves e.g waves in water or sound waves or waves in a spring or a string?

That will be very comprehensible

A bit all over the place, the author assumes "It is even less well-known that the rule “nothing moves faster than the speed of light” is a consequence of the laws of electricity and magnetism" but at the same time "This is also a well-known property of electricity and magnetism" and then drops some equations.

Most people for whom this property of electricity and magnetism is well known, also know that the speed limit from general relativity comes from Maxwell's equations (which are on themselves a compilation of other previous rules).

I also dug deep into this problem a while back, but without proper physics background couldn't get too far. The speed of light comes directly from the Vacuum permeability and Vacuum permittivity, since light is an electromagnetic wave. If these were different, then the speed of light would be different. Both of these seem to be values of the behavior of vacuum in our universe (and possibly electron's [1]).

It is Maxwell's equations gives us that the speed of electromagnetic radiation needs to be c ^ 2 = 1 / (e * u) [2].

[1] https://en.wikipedia.org/wiki/Fine-structure_constant

[2] https://en.wikipedia.org/wiki/Electromagnetic_wave_equation#...

The author's assumptions were spot on for me. I am familiar enough with E&M to know the equations and roughly understand relatively but did not know where the intuition for a universal speed limit comes from.

People can disagree with your assessment, but it’s pretty sad that you write the only comment in this thread that actually engaged with the content of the linked article, and it’s downvoted.

I highly recommend watching https://youtube.com/playlist?list=PLoaVOjvkzQtyjhV55wZcdicAz...

They do such a good job at using video to explain Special Relativity.

In the 70s, I remember reading "The Quincunx of Time", which made great use of a key distinction (for the purposes of some classic 70s scifi):

It's not that nothing can move faster than the speed of light, it's that nothing can accelerate to a speed faster than the speed of light.

If you could bring particles into being that already moved faster than light, they would not violate our understanding of relativity or the rest of physics. Hence ... the tachyon. https://en.wikipedia.org/wiki/Tachyon

Light isn’t limited in the speed it can travel. Mass is. Light can’t exist without matter. The fundamental limit is mass versus all of the forces which act as drag.

Why are gravitational waves limited by the same speed limit? There is no mass traveling in that case, I suppose.

What we think of as the speed of light could also be thought of as the rate at which change propagates throughout the universe. The rate at which cause and effect travel.

Two runners overtake the field at incredible speed, only to cross the finish line at exactly the same time. Would also raise eyebrows.

I wouldn’t dare to hazard a guess here, but I do think it’s worth remembering that matter is always in motion.

This is an excellent question, which predates the theory of General Relativity by a few years, and has only been astrophysically verified after 1974 (Hulse-Taylor) or even later. As far as I am aware, a complete theoretical answer in General Relativity itself is still elusive.

I'll largely stick with the theory, which I guess is what you are interested in.[1]

The topic is in Part VIII (Chapter 35) of Misner, Thorne & Wheeler's Gravitation ("MTW"), which is the gold standard reference/textbook for General Relativity.

I will later try to briefly summarize the section.

Instead, first, at the root of my answer is the non-linear nature of the Einstein Field Equations, which imply that gravity self-gravitates. This is usually side-stepped by textbooks, which instead proceed to linearize the Einstein Field Equations, that is, they consider the weak limit of gravitation. This is usually to split a metric with a gravitational wave into some minimally-or-even-non-dynamical background and the wave-part, using the former to define the speed of the latter.

For strong gravitational waves, or a very dynamical background, we cannot do this. I think this regime is best studied in a vacuum solution, i.e., where there is only gravitation that self-interacts, although most of the work in this regime appears to have the aim of resolving questions about the distribution of matter in the very early universe (e.g. Misner's mixmaster). An interesting exact solution of the Einstein Field Equations is the https://en.wikipedia.org/wiki/Kasner_metric which can generate singularities and other features formed by gravitational-wave interactionsn. That is, there is manifestly non-linear gravitational self-interaction in the highly-dynamical Kasner chaos. (This is the worst case for the linearized treatments in textbooks).

The Kasner metric can be applied to a Lorentzian manifold (3 space, 1 time dimension), and so is consistent with our universe's causal structure. An interesting feature of this solution is that it engages only two constants in the Einstein Field Equations: c and G. The only velocity scale that we can construct from any combination of these two constants is c itself.

This is highly suggestive that in a universe like ours gravitational radiation must propagate at c.

Again, this is just an argument that there may be an answer within the theory of General Relativity itself, without treating the speed of gravitational waves as a postulate.

This argument is made without regard to the obvious craziness in a Kasner chaos universe compared to our own. General Relativity admits complete gravitational solutions for all sorts of craziness, including universes with any number of space and time dimensions as long as there are at least two total, stress-energy which is distributed very differently from ours (including negative energy, or energy that pops in or out of existence without a cause), and so forth. It is very General. (Special Relativity is very Special: it's defined on -- and only on -- a gravity-free (flat, no gravitational waves) spacetime of exactly 3 spatial and 1 timelike dimension.)

Returning to MTW, they conclude that the propagation speed instead can be no greater than c.

The authors develop an exact vacuum plane-wave solution in §35.9, where the only thing in the spacetime is single large pulse of gravitational radiation in otherwise totally empty flat spacetime, and a set of "test particles", which are well defined probes in General Relativity defined so as to not perturb the solution. They proceed to compare this solution to that of an electromagnetic plane wave in Special Relativity, and arrive at a more physical viewpoint where the gravitational plane wave, if it has anything like a physical source (e.g. a pair of masses in mutual orbit; they return to this in Ch. 36) should be more like a set of "ripples in the spacetime curvature ... propagating on a very slightly curved background spacetime ... The most striking difference between the background and the ripples is not in the magnitude of their spacetime curvatures, but in their characteristic lengths". There is a characteristic length of this background spacetime, determined by its (much much larger) radius of curvature.

They then grind out effective stress-energy tensors, which would couple with any matter in a non-vacuum environment. The argument is that anything in the stress-energy tensor must in certain causal structures (like the one in our universe) must propagate at no more than c.

Their treatment is the basis for a particular type of graviton (Ex 35.16), the scattering and redshift of gravitational waves (.17, .18), and various ways to express them without resort to an effective stress-energy tensor.

In these approaches, c is the limiting speed of gravitational radiation, but gravitational radiation may move slower than c in non-vacuum. In vacuum, the pure pulse may in some circumstances develop a trailing edge that propagates at less than c, but when this can even happen the effect is weak when the wavelength is short compared to the background length scale, or when the amplitude of the pulse is small. Perhaps gravitational astronomy can hope to find someday a high-enough amplitude wave to put this to the test.

For clarity, though, the linearization-is-good-theory claim is on solid footing. LIGO uses the linearized equations and have found agreement between the speed of gravitational waves they've detected to their theory to something around nineteen decimal places. A good multimessenger signal, which seems inevitable, will almost certainly improve that. There is no good reason to expect the speed of gravity in a full solution to the Einstein Field Equations to differ. One can make the same argument about the march of results from post-Newtonian expansions (as below) too. The only "wiggle" room is that cosmic inflation is probably much more dynamical, and the gravitational waves are of much greater amplitude.

- --

[1] The non-theory answer is that the behaviour of orbits in known astrophysical systems which are very post-Newtonian (think black holes, or extremely fast-moving galaxies containing predictable spectra from hydrogen or light curves from supernovae) are consistent with a speed limit on any gravitational interaction, and that the speed limit is very close to c. It turns out to be hard to measure the speed limit exactly. See e.g. Will @ https://arxiv.org/abs/astro-ph/0301145 which discusses light from quasars being gravitationally lensed by Jupiter and how it would look different under theories that admit a propagation speed c_{gravity} different from c_{electromagnetism}. One could also compare bimetric theories of gravitation in which in the early universe gravitational self-interaction propagates differently from electromagnetism, with a view to resolving some questions in the distribution of galaxies in our sky. These generally have to decay the additional metric (meaning gravitational radiation propagates like electromagnetism) in the very early part of the universe or we get something very different from the cosmic web of galaxies that we observe.

This is fascinating, thank you! You’re absolutely correct, the single GR course i took linearized the EFE for the section on gravity waves and the concept that gravitation self-gravitates did not really come through. In hindsight maybe that should have been obvious though, since all waves have this property.

I have a probably very dumb question. If speed is relative, when we say something travels at... I don't know... 50% the speed of light, that speed is relative to what? how do you know it is 50% and not 53%?

How do we know we're not already moving at 99% the speed of light (like our observable universe as a whole having that speed )?

I love this stuff, but it is so counter intuitive for the average human.

When we say something travels at 50% the speed of light, it's always relative to some inertial, approximately inertial, reference frame, often implied.

So eg. if you hear about the particles at the LHC travelling at near the speed of light, it's implied that's relative to the Earth.

The speed limit comes in here: no-matter what nearby object you look at, its speed relative to you always be less than the speed of light. (Here, nearby means something like "within the local group of galaxies" - objects a long way away can be receding faster than the speed of light due to the expansion of space in between you and the object). You can accelerate as much as you like for as long as you like and when you stop accelerating and check, that'll still be the case.

"that speed is relative to what"

To you. There is no such thing as absolute speed. You say cosmic rays are crashing into the earth at 99% the speed of light. The cosmic ray says it is sitting still and you (and the Earth) are approaching it at 99% the speed of light. Some other particle moving at 70% relative to you will give a different speed for the cosmic ray. All of you are right.

No matter of fast you are moving, you will agree with a “stationary” observer about the speed of light. What you won’t agree on is the color of the light. Apparent frequency shifts to maintain the speed of light. The color shift is how we know how fast stars are moving relative to us.

Higgs field is present everywhere and mediated by Higgs boson, so just measure speed relative to Higgs field. Or you can measure speed relative to CMB.

My favorite way to think of it is that the speed of anything is measured in distance per unit time. You can’t change the constant that is the speed of light, but what if you could change what “time” means? The constant holds, but time doesn’t; it varies among observers. A second from the perspective of observer A is not identical to a second from the perspective of observer B.

Freaky, but true.

I still don't get this. I simply can not accept that the speed of light is that same for all observers, irrespective of their respective speeds vs the light source. It is obviously a cornerstone of Einsteins theory (I forget which; either the special or general relativity), but to me this is simply not logical. To state that the speed of light is the same only because the relative time is variable for respective observers is unconvincing, which requires that things like blueshift/redshift to be explained by logical gymnastics, and not the straightforward reason that the speed of light for all observers is a direct function of the relative speed vs the light source. But what do I know.

It's a consequence of the time dilation. You are not stationary towards the photon to an outside observer, so to them it looks like the photon moves away from them faster than it moves away from you. And from the obserer's perspective, your clock is ticking slower. That's the key. The amount that your clock is ticking slower is such that you, the one moving, would calculate the speed of the photon the same as the stationary observer, because you would do so from your own slower clock.

Thanks, that sounds like the explanations I've seen too, and that I hope to understand some day. Atm I'm at the point where I gladly accept that time may seem to dilate, but not that it actually does. Maybe I just need to dive deeper into it than I've been willing to do so far.

You may want to investigate how GPS works. They have to incorporate the effects of time dilation to get accurate positioning data. I don't know if this is the correct direction to get the answer for which you are searching but it is a concrete example with the maths worked out. http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps....

If you don't know, every experiment we have ever carried out to test this tells us it is true.

I wrote a long answer to someone else; I'm not going to retype it. But what is invariant in the 3D space you think you live in is distances. The distance of you from the kitchen is computed to be 8 meters no matter what we choose to say the x and y directions are, right? If we choose different directions for y, you may say the kitchen is 6 meters from you in the y direction, I say it is 3.7 meters in the y direction. No mystery, our frames are rotated. But we both get the same value for distances. Rotating a map doesn't change the reality of how far you are from the kitchen.

Well, you don't actually live in 3D space, you live in 4D space, where time is a dimension. So what is invariant is not distance (3 of the 4 dimensions), but intervals (time and distance between two events). when you travel at a significant fraction of light speed, you are rotating that 4D 'map'. If you rotate frames, you get different values for x, y, z, and t. But the intervals are still constant.

The intervals are the same for all observers. It is just the individual coordinates (x,y,z,t) that vary for different observers, just like (x,y) are different for you and me wrt your kitchen, but the distance we compute are the same.

edit: in summary, wouldn't you find it bizarre if I rotated the floor plan to your house and suddenly got a different distance from your couch to the kitchen? That would be absurd! Well, when you rotate in space time intervals are unchanged. IOW, the speed of light is unchanged. It would be bizarre if by a simple rotation you got a different value! You just don't 'see' (literally and figuratively) that time is a dimension, and the map rotation as non-relativistic speeds is so tiny that you don't realize that t changes along with x, y, z. But it does. It would be bizarre if it didn't.

well, sure looks like you've grasped it. I'll certainly ponder it. But for the record it did not immediately help me in trying to solve the central riddle of the axiom: that the speed of light from a single light source is supposedly the same for all its observers irrespective of their respective speeds relative to the source. I.e. that the observed speed of light coming from our sun would be observed to be e, and an extremely fast spaceship travelling away from the sun would observe the speed of the suns light to be e, too.

Here is one aspect of the conundrum: a certain photon travelling at the speed of light from the sun reaches the earth, which is relatively stationary vs the sun, in approx 8 minutes. It will obviously reach our rocket (which at the moment the photon was fired from the sun was at the same distance as the earth from the sun) some time later, depending on its relative speed to the sun. However, once the photon hits the spaceship it supposedly has the same speed as the speed of light hitting the earth.

I realize this axiom seems to have been experimentally proven, and that I probably just have not found the right key for me towards the understanding of it. Looking forward to that day, which might also lead me to understand how to fit the fact that the blueshift/redshift observed in light allows us to actually determine relative velocities. As well as understand how the expansion of space is a different kind of speed than normal speeds, allowing for the speeds greater than light that it does.

In the meantime I find some solace in the fact that I'm not the only one to find this bizarre - hence that whole linked article as I understand it. But thanks for your effort.

> but to me this is simply not logical.

It is not logical if you really believe that speeds add linearly (that is, if you are going 5mph past an observer and throw a ball 5mph, that the result is the ball moving 10mph).

Speeds don’t actually add linearly like that, but they come very very very very close to doing so for all speeds humans are used to dealing with.

So, we all have very deeply held gut feelings that speeds should add linearly. Once you let go of that, it becomes much easier to understand many of the things that don’t feel logical about relativity.

Ultimately, it is logical, but from a different set of axioms than most humans tend to have.

But we don't really think that speeds add linearly. if you drive 80 kph North, and I drive 60 kph East, no one would say we are separating at 140 kph. we would use Pythagoras to get the right number.

We just don't realize we live in a 4D space where time is one dimension, so we think driving in opposite directions is a special case where we can add the speeds. but there is the fourth dimension, and we have to use an equation very similar to Pythagoras to get the right value.

I know you must know this, this is more for the benefit of the reader.

> To state that the speed of light is the same only because the relative time is variable for respective observers is unconvincing

Not sure what you mean by relative time?

You are right, 'relative' was not the correct term there.

I've been trying to understand this since high school. No Eureka moment but I think every year I understand just a little bit more.

I always feel like this questions is really asking "What is time?". This book gives a new perspective on time which I think is crucial for understanding the nature of it: https://www.harvard.com/book/the_janus_point/

Can a thing with no mass ever go less than the speed of light? Does it have a concept of time?

Sure, light is slowed down when it travels through different substances. That's why we have refraction.

Is it really slowed down? I always assumed that it just need to take a longer way through matter than through vacuum.

If you just extend time dilation beyond zero (I.e. negative time) then speed faster than light is possible. The constraint described in the article will still be maintained. Also, notice that negative time would be the perception of external observer.

PBS Spacetime has done some fantastic episodes on this including this one https://www.youtube.com/watch?v=msVuCEs8Ydo

On traveling at the speed of light I also highly recommend to watch this Veritasium video https://youtu.be/vVKFBaaL4uM

Interesting to know that if we would have a 1g acceleration spaceshil with almost unlimited energy onboard we can cross a large part of our galaxy in less than 70 years.

Thanks. Fun to know that the speed of light prevents the universe from splitting up. Cool. Also I was not aware that movement create magnetism.

This example has a flaw. Skateboard man sees object stationary, but the cable is moving. There should be magnetic force too.

wouldn't it go backward in time if it goes faster than light? Light travels in null-time from point a to point b (light-perspective of course) - anything slower, needs time to go from a to b. if i travel faster than light, i can - but i travel to b before i start at a. Maybe because this is not possible.

How much you will travel back in time by using a teleport (instant travel, infinite speed)?

If anyone wants to play with a speed of light, there [still] is a fun web game called “velocity raptor”.

Velocity is a relative number. So let's say your are going 0.9 c but I'm going at 0.8 c now according to me you can go 0.7 more at least. But you're already going 0.9 and adding 0.7 will put you above speed of light. You can say you're already travelling faster than light relative to something out there... So what the hell are we talking about here?

there's a vertical asymptote in e=mc2, with near infinite energy required to accelerate something to near c. however, does it prevent something from existing thats already (or always has been) traveling faster than c?

The best analogy for this I've read is that your motion through spacetime is constant. Think of a quarter circle dial with an arrow that goes all the way from 0 degrees (no relative movement in space) to 90 degrees (relative speed of light). The faster you go through space, the slower you go through time. The effects of this are barely noticeable until you get near the speed of light. This effect is time dilation.

This, I believe, is the general relativity view of spacetime.

What I like most about this is that it highlights the importance of understanding the domain of a function. The above shows how any object with rest mass has a space velocity domain of [0, c).

People really don't want this to be true so latch on to any wild theory that would seem to bypass this cosmic speed limit, be it wormholes, FTL drives or whatever. Pretty much all of these theories rely on taking an equation with a domain over real numbers and plugging in negative values for things like mass.

Garbage in, garbage out.

I mean if mass can be negative, why limit yourself to real numbers? Why not a complex number for mass?

As to why this is the case and how to reconcile it with a quantized view of spacetime... is beyond my pay grade.

You can go faster than the speed of light. Its slowing thats the hard part!

You can’t go faster than light for the simple reason that it would defeat the very many performance optimisations necessary to keep the universe running.

I prefer the question, “why can’t light go faster?”

That you can’t go faster than the speed of light is a direct, simple consequence of everybody agreeing on the speed of light. It's a real-life Zeno paradox.

Alice and Bob are in some spaceships by a long racetrack in space. Alice fires a very brief pulse of light down the racetrack, maybe we see the rays that don't go straight through successively illuminate some rings around the track. Alice challenges Bob to race another light pulse, Bob revs his engine.

As the countdown hits zero, Bob accelerates to speed c/2 relative to Alice, then checks the reflected light from these rings only to find out that the light pulse is still traveling at speed c away from him. So he drops a beacon at his current speed then accelerates to speed c/2 relative to that, but no dude: the light is still moving at c away from him. Bob realizes that he can never win, so tries to instead measure the speed of Alice, who he expects to be moving at speed c away from him, only to find that she is instead moving at speed 0.8 c away from him.

Now the question is, how can this be? Consider a much slower spaceship. When Alice fires the light pulse and Bob starts moving forward at a slow speed, Alice sees this bubble of light expanding in kind of a uniform sphere centered on her. (Say it reflects off of space dust instead of a track.) Since they were at the same position when the light Bob also sees an expanding bubble of light, with himself at the center. The weird stuff about time dilation and length contraction does not apply at low speeds, if Bob goes at c/1,000 say, then these are only one part per million.

They both basically agree on how far this bubble of light is from Bob, in the directions perpendicular to the motion. The motion is parallel to the bubble in those directions, and to first order those parallel lines will not get any closer or further away. (This is why I want Bob to move at a slow speed!) They only disagree along the motion. Alice thinks the light is receding from Bob at speed c–v ahead of him, at speed c+v behind him: Bob sees the light recede at speed c in both directions.

So they come back together to repeat the experiment and Alice decides to force the contradiction. Alice puts a clock ticking out every millisecond out at distance 1 light-second, but it will start at the moment she fires the pulse, stopping when the pulse hits it, showing 1000. She puts one of these in the direction Bob will travel, and in the opposite direction. Surely he must agree that the light started from here and that it intersected those two clocks when they both said 1,000.

Bob agrees that the clocks look synced up and films all of this with a high-speed camera to make sure that there is no funny business, and they repeat the experiment.

Right when the light hits, Bob accelerates at his usual 1,000,000 gees for 0.03 s, to get his final speed of c/1,000. (I need to rewrite this to make the numbers more reasonable LOL.)

Here's where something weird happens, and it is entirely contained to those first 30 ticks of both cameras, as Bob looks at them in his high-speed camera footage. Bob corrects for Doppler shift like you do, and agrees that these clocks appear to be ticking during the other 970 ms at one tick per ms, his camera has maybe microsecond resolution and not the nanosecond resolution you need to see time dilation.

But during those first 30 milliseconds when Bob was accelerating, even after correcting for the Doppler shift, his best guess is that Alice artificially slowed down the clock behind him and artificially sped up the clock ahead of him. Because the clock ahead of him definitely ticked 31 times in those 30 ms, while the one behind definitely only ticked 29 times. So Bob says the light did hit these clocks when they said 1000 ms, but the clock ahead of him should have said 999ms at that time, and the light should have gone past it a bit by 1000ms, while the clock behind should have said 1001ms, and the light was actually not yet there at 1000 ms.

This anomalous Doppler shift is proportional to both the distance of the clock you're looking at, and your acceleration. It is also called the relativity of simultaneity, and it is the only new prediction of relativity, in that length contraction and time dilation are second-order consequences of it.

Lightspeed is infinite, it is the simultaneity which is slow.

"Nothing can go faster than light" is just a convention and is not experimentally confirmed.

IIRC there are a few things that go faster than speed of light (e.g. universe expanding).

"Spooky action at a distance" is also not known very well. That could also break the speed of causality law. While Bell's theorem hints that this is not the case, there are some exceptions to Bell's theorem.

Right now a lot in physics are just convention like energy conservation and symmetry.

We need another paradigm upgrade to understand these things.

AFAIK 'nothing goes faster than c' is wrong in the 'lies to children category'. The devil is in the details. In fact, plenty of things go faster than c.

As a standard example, take a wall at 1m distance and a flashlight. move the light spot at 1 m/s. If you put the wall at 2m, the same spot will now go 2 m/s. If you put it at 300 000 km, the spot will go at 300 000 km/s, slightly over light speed.

The problem is more one of information: For any action you take, no consequence can happen outside your light cone. All information you generate travels at c at max.

None of the examples you gave violates this: Even if the universe expands at more than c, nothing you do will have influence on things too far from you.

That's not the "same" spot, though.

It doesn't consist of the same photons; the spot in the "new" position consists of photons which have been traveling in a straight line from your flash light at (assuming vacuum) a constant rate of C (and then reflected back to reach your eye again, also at a constant rate of C); they haven't traveled to or from the previous "spot" position at all, much less exceeded the speed of light at any point in the process of reaching the new position.


That's correct in both the superluminal and the subluminal case, and more or less the point of the example. The spot moves superluminal, but the particles creating it don't. This makes it impossible to use it for superluminal information transfer.

AFAIK Expansion of the universe is comparable. It pushes things away not by speed but by putting extra 'new' space between things.

Your sibling comment makes a variant of the same point:c is the maximal speed of information or causality. Light in a vacuum is just one of the things capable of reaching it.

That is a reasonable argument.

And what you point out is actually the speed limit of causality, instead of light. It just happens that light travel really fast.

Though the article wants to discuss the speed of light as you mentioned, so I focused on speed of light.

Hmm not sure if troll, but energy conservation is definitely not just some convention. It’s a fundamental thermodynamic law. Spooky action at a distance does not violate speed of light information propagation either. The particles have to be entangled before they’re set off in opposite directions. Only once one is observed does the other also collapse, but it doesn’t mean you can communicate faster than the speed of light because you had to prepare the information when the particles were together IIRC, it’s been a decade since I studied quantum information theory.

Why do you feel the need to say somebody is a troll?

Here: https://phys.org/news/2017-01-violations-energy-early-univer...

Here: https://phys.org/news/2015-02-space-faster.html - we are trying hard to reconcile this and categorize universe expanding as something else (e.g. not a movement that has speed because time itself is a dimension or something). But this is still up for debate, tbh.

Another example was one where we said CP symmetry was true (it was a law like a lot of things in physics) until it was violated by a weak nuclear force experiment.

And now we are holding the fort at CPT symmetry as the law.

In Physics, the evidence of anything is kinda light. A lot of reasonable extrapolations has been made. Still they are extrapolations (e.g. intelligent guess).

Even the big bang itself is just an extrapolation from the "theory" that the universe expanding.

To be fair, it is difficult to find good evidence because we can't dial back time, can't go observe things on Neptune, can't measure gravity at the subatomic scale, and etc. So, we have to work with what we can experimentally observe.

Our tools are getting better, and this is where the physics paradigm shift will come from.

You say like these are 100%. It is just a theory that we currently hold according to the little evidence that we have.

Failing to recognize that is straight up unscientific.

Apologies, I wasn’t trying to call you a troll, more that your first comment seemed to me like it might have been made in jest, clearly it wasn’t.

The articles you’ve linked to are interesting and there are clearly many scientific discoveries to be made by studying the early formation of the universe which will test our current models. However I don’t feel like convention is the right word for laws like the conservation of energy, even if there are some difficulties with tying up these theories and new experimental evidence from events at the scale of the Planck length.

Convention to me would mean something that has been accepted just because it’s always been done that way and people didn’t really bother to question why, but I don’t think that’s the case here. But we’re verging on pedantry now so no point going down that route any further.

>Convention to me would mean something that has been accepted just because it’s always been done that way and people didn’t really bother to question why.

You're wrong. The Thermodynamic laws are sort of axiomatic, meaning you really can't explain why energy is conserved. It's just experimentally shown to be maybe true, but no one knows why energy is conserved or has actually proved it to be true. It is totally "convention" as you defined it.

The caveat here is that entropy is not axiomatic. Entropy occurs as a consequence of probability. Probability is the real axiomatic assumption of the universe and entropy is a byproduct.

The thermodynamic laws were established before people fully understood the true nature of what was going on with entropy so these laws are sort of a hodge podge of axioms and derived theorems. From a temperature perspective these assumptions work so the laws still have their use. But the laws of thermodynamics aren't some elegant grouping of fundamental laws of the universe. It is a set of rules that are grouped arbitrarily.

>But we’re verging on pedantry now so no point going down that route any further.

I find this attitude rude. You called him a troll than apologized then gave your final answer and dismissed any further discussion as "pedantry." Like wow, you get the last word and shut down anything else he has to say? You were rude to assume he was in jest and you're being rude again by saying any further discussion after your final statement is pedantry.

Either way I disagree with you. It's not pedantry. This discussion is about convention and the conservation of energy. Your statement is wrong.

Thank you for writing this and introducing the word axiomatic. I was using the word convention as something that is defined and unquestionable. It's an assumption we base all of our physics equation on. Axiomatic might capture the meaning better.

This means, and I assume, we can also base all of our equations on say "energy is not conserved and always increase by 1 Joule", though the physics equations might be much more complex.

Pro-science folks are too enthusiastic about current science to the point that they become unscientific. Pointing out that these theories/laws/conventions might become invalid in the future is getting downvoted.

Newsflash: physics theories/laws/conventions are getting invalidated all the time.

>This means, and I assume, we can also base all of our equations on say "energy is not conserved and always increase by 1 Joule", though the physics equations might be much more complex.

We can but this won't match with observations. We assume energy is conserved only because our current observations show that it has been conserved thus far.

Referring to the OP article, he's basically writing that the c being the absolute speed limit is not axiomatic. It's not something we just assume to be true. It is a theorem derived from the assumption (aka axiom) that the universe cannot produce inconsistent events.

Of course you can go faster than light, way faster. When it comes to physics there are no limits how fast, far or deep you can go. People constraint themselves with articifial limits and it's not necessary a bad thing. But real limits doesn't exist.

I'v heard someone saying space is a big ball and outside of it is nothing. I imagine nothing as a black space but that actually is something. So there is something after all. And however weird or normal these other places are, it goes like this infinitely.

> Of course you can go faster than light, way faster.

OK. How?

I don't know

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