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.
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.)
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 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
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.
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.
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.
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".
