Axioms
Axioms
Posted Feb 22, 2021 13:46 UTC (Mon) by mathstuf (subscriber, #69389)In reply to: Axioms by ale2018
Parent article: An introduction to lockless algorithms
AFAIR, it's not possible to compute a one-way speed at all. You can only get a round-trip speed (light could be traveling at some factor based on its angle with a "universal axis" and we'd never be able to tell) because you can't communicate back about any time synchronization without being affected by the same limits. So, no, without an axiom that "the universe has no preferred axis for light travel", you get an infinite number of possible solutions for a speed-of-light gradient.
Posted Feb 22, 2021 18:44 UTC (Mon)
by ale2018 (guest, #128727)
[Link]
That's right.
Let me quote Leslie Lamport, from the 1978 article cited by Paolo Bonzini:
The references [1] and [2] are two introductory texts on relativity of those times.
Posted Feb 23, 2021 0:03 UTC (Tue)
by Wol (subscriber, #4433)
[Link] (31 responses)
Oh, and I believe it IS possible to calculate the speed of light based on pure logic ...
You'll have to look it up for yourself, but if you take a bunch of mathematical constants like e, pi, ln and so on, I think they somehow dictate a set of possible values for c, g, and other physical constants. It's too long ago, but I'm sure I've seen the physical constants defined in terms of the mathematical constants. And that's why astronomers are so unhappy with the cosmological constant. Because they can't define it in terms of anything else, it shouldn't exist ...
Cheers,
Posted Feb 23, 2021 0:38 UTC (Tue)
by mathstuf (subscriber, #69389)
[Link] (26 responses)
It'd also lay a blow to various theories of multiple universes which differ in the physical constants. Also, what is a "fundamental constant"? Is the radius of a proton a constant or a derived value? The size of a helium-4 nucleus? Ratio between the lepton sizes? Even physics is a *bit* arbitrary here.
Posted Feb 23, 2021 1:19 UTC (Tue)
by Wol (subscriber, #4433)
[Link] (25 responses)
Who said anything about fundamental? Both e and pi are mathematical constants, as is the golden ratio. They're all of equal importance.
Things like c, g, permittivity, that stuff, are all physical constants, which should be explainable in terms of maths.
Posted Feb 23, 2021 13:10 UTC (Tue)
by mathstuf (subscriber, #69389)
[Link] (7 responses)
But…why? What is this assumption based upon? Is it a bit of Platonic idealism?
Posted Feb 24, 2021 0:27 UTC (Wed)
by Wol (subscriber, #4433)
[Link] (6 responses)
Cheers,
Posted Feb 24, 2021 21:58 UTC (Wed)
by mathstuf (subscriber, #69389)
[Link] (5 responses)
If we wanted, we could say that the speed of light is pi/e Wol_lengths/Wol_time intervals. Doesn't mean that those units are any more useful than our numerical value in m/s.
Posted Feb 25, 2021 7:48 UTC (Thu)
by jem (subscriber, #24231)
[Link] (4 responses)
Posted Mar 16, 2021 21:09 UTC (Tue)
by nix (subscriber, #2304)
[Link] (3 responses)
Posted Mar 17, 2021 10:37 UTC (Wed)
by jem (subscriber, #24231)
[Link]
Well, I said derived, not defined.
Posted Mar 18, 2021 10:14 UTC (Thu)
by mgedmin (subscriber, #34497)
[Link] (1 responses)
a cubic *deci*meter (which is also one liter) of water.
Posted Mar 20, 2021 1:24 UTC (Sat)
by nix (subscriber, #2304)
[Link]
(And obviously you're right, unless water has suddenly become much, much denser than lead :) )
Posted Feb 23, 2021 15:24 UTC (Tue)
by rschroev (subscriber, #4164)
[Link] (16 responses)
Posted Feb 24, 2021 0:23 UTC (Wed)
by Wol (subscriber, #4433)
[Link] (1 responses)
I think the equation is G=g.m1.m2/d^2
G is 9.8m/s^2, while g is the gravitational constant, which is believed to be the same everywhere in the universe.
Cheers,
Posted Feb 24, 2021 0:47 UTC (Wed)
by rschroev (subscriber, #4164)
[Link]
Posted Feb 24, 2021 0:31 UTC (Wed)
by Wol (subscriber, #4433)
[Link] (11 responses)
It can *de*scribe it, yes. But I think you'll find it also *pre*scribes it and says "completely different constants and laws doesn't make logical sense".
Cheers,
Posted Feb 24, 2021 1:01 UTC (Wed)
by rschroev (subscriber, #4164)
[Link] (10 responses)
BTW there's a video of Richard Feyman's lecture about that relationship here: https://youtu.be/obCjODeoLVw
Interesting stuff, if you're interested in that kind of stuff.
Posted Feb 24, 2021 18:28 UTC (Wed)
by Wol (subscriber, #4433)
[Link] (9 responses)
Cheers,
Posted Feb 24, 2021 21:54 UTC (Wed)
by mathstuf (subscriber, #69389)
[Link] (3 responses)
Posted Feb 24, 2021 23:22 UTC (Wed)
by Wol (subscriber, #4433)
[Link] (2 responses)
We think it may be four - that's what Einstein thought but that has a whole bunch of problems. If it *is* four, I believe that means string theory is correct as it's the explanation for relativistic singularities.
Or it could be ten or eleven. Anything betwen five and nine inclusive just doesn't work because we get an explosion of infinities - infinity itself isn't a problem, but there are different sorts of infinity and for reality to work they need to cancel out. For those dimensions they don't. (These universes, if I remember correctly, define mass as the fifth dimension ...)
(That's why I was moaning about computers crashing when you divide by zero. If you declare zero and infinity as non-numbers for which arithmetic doesn't work, you're in trouble. If you say "to make arithmetic work, they swap places on division", then you can do this sort of maths and come up with something that makes sense.)
At the end of the day, we have loads of maths that describes what we see. And that *constrains* what is a plausible universe. We have a local maximum or minimum, don't know which. By adjusting some values, we can force others to impossible values. Plausible universes must have all these values at maximum or minimum, not off the scale or impossible or at some non-equilibrium value.
Cheers,
Posted Feb 24, 2021 23:36 UTC (Wed)
by Cyberax (✭ supporter ✭, #52523)
[Link]
Posted Feb 25, 2021 0:05 UTC (Thu)
by nybble41 (subscriber, #55106)
[Link]
To say that division by zero is undefined is mathematically correct and not just an arbitrary computer limitation. There is no proper answer to the question "What number, when multiplied by zero, gives the non-zero product X?", at least not in any system that would uphold basic idioms such as the product of a number and its reciprocal being equal to one. "Infinity" times zero is not equal to any particular finite number X, so that isn't a solution. Depending on the particular forms of the equations which gave rise to the infinity and the zero (or infinitesimal) the product could be another infinity or any real number; it depends on how you phrase the question. $\lim_{x \to 0+} x ln \frac{1}{x} = 0$, but $\lim_{x \to 0+} x \frac{1}{x} = 1$. In both cases you're multiplying an infinitesimal ($\lim_{x \to 0+} x$) by an infinity ($\lim{x \to 0+} ln \frac{1}{x}$ or $\lim{x \to 0+} \frac{1}{x}$, respectively) but the specific form of the equation changes the result.
Posted Feb 24, 2021 22:24 UTC (Wed)
by Cyberax (✭ supporter ✭, #52523)
[Link] (4 responses)
If you want to read something mind-blowing, try the "Clockwork Rocket" series by Greg Egan. It's accompanied by a thesis-sized exploration of its (fictional) physics: http://www.gregegan.net/ORTHOGONAL/ORTHOGONAL.html
Posted Feb 24, 2021 23:26 UTC (Wed)
by Wol (subscriber, #4433)
[Link] (3 responses)
Cheers,
Posted Feb 24, 2021 23:34 UTC (Wed)
by Cyberax (✭ supporter ✭, #52523)
[Link]
You can also construct a quantum field theory for such a universe, it also would work just fine.
Posted Feb 25, 2021 15:16 UTC (Thu)
by mathstuf (subscriber, #69389)
[Link]
Posted Mar 8, 2021 14:59 UTC (Mon)
by LtWorf (subscriber, #124958)
[Link]
Physics tries to model measurements.
So for example you measure a planet that is orbiting, make an equation, and see if tomorrow the equation and the position are the same (within a certain range of precision).
Before Galileo saw that Jupiter had satellites, they had perfectly fine equations that predicted where everything would be in the sky. The problem arose because new data could not fit the model.
You can absolutely model an orbit of a planet using 3 dimensions, or you can model it in an higher space with an equation of a lower degree. Both work. We can't really know which is "exact" if both work.
You can keep adding dimensions and make equations that work, but we don't really know what the "truth" is.
Posted Feb 24, 2021 1:24 UTC (Wed)
by SiB (subscriber, #4048)
[Link] (1 responses)
Posted Feb 24, 2021 8:53 UTC (Wed)
by mpr22 (subscriber, #60784)
[Link]
The first formally adopted definition for the metre itself, proposed in 1791 by the French Academy of Sciences and adopted in 1793 by the National Assembly, was one ten-millionth of the distance from the North Pole to the Equator along the Paris meridian.
Posted Feb 23, 2021 1:06 UTC (Tue)
by mpr22 (subscriber, #60784)
[Link] (3 responses)
There is one particular physical constant, µ_0 (the permeability of free space) which was historically defined (as a consequence of the relationship between electrical current and induced magnetic fields) as exactly 4π × 10^-7 H/m ... but that value is no longer strictly valid (although it's close enough for the vast majority of practical purposes), because the ampere has been redefined in terms of the elementary charge and the second.
Posted Feb 24, 2021 1:00 UTC (Wed)
by SiB (subscriber, #4048)
[Link] (2 responses)
Some years ago in our coffee room at the institute there was a discussion about the possibility that the fine-structure constant α may not have been constant in the past after all. The question was raised, which of the constituents of α=e²/(4πε₀ħc) may be the reason. Well, the answer is e, if at all. α is a constant without units. It is pure physics. It defines the strength of electromagnetic interactions. The universe cares a lot about the value of the fine-structure constant α. In our experimental units it is the charge of the electron e that represents that strength, but in the end, it also just defines units of measure.
Theoretical physics defines the units differently than experimental physics. They mostly use c=4πε₀=ħ=k=1. Everything is measured in eV or eV¯¹.
From a theoretical point I am a bit unhappy about the new definition of the SI units. They fixed the value of e, although it indirectly represents a quantity the universe cares about. The cost is that µ₀ and ε₀ are now _not_ fixed by definition in the SI, as they were before, like Wol said.
Ideally, all units were defined by fixing the value of some fundamental constant. We are now almost there. The second is still defined by an artifact. At least it is an atom, so it can be reproduced everywhere in the universe. To fix that we need to fix the gravitational constant. But metrology defines units by what they can measure with the best precision. The gravitational constant does not apply, it has to catch up at least eight orders of magnitude in precision.
Posted Feb 25, 2021 3:46 UTC (Thu)
by NYKevin (subscriber, #129325)
[Link]
The universe may not care about the speed of light, but the stuff in the universe surely does. To borrow from your "theoretical physics normalizes everything to 1" example, the mass of the proton is much, much (numerically) smaller than the charge of the proton, and surely there's some physical significance of that? Or, alternatively, we can say that there's nothing wrong with the proton, and instead it's gravity which is too weak, but that's just reframing the same question in different units. I'm not sure you can ask that question without using units at all.
Ultimately, I suppose the "proper" framing of this question is some nonsensically complicated question about why the relevant quantum fields happen to interfere in exactly the way that they do to give the proton the properties that it has. But you can't even ask that question until you know what gravity is, at a quantum level, and to the best of my knowledge, nobody does.
Posted Feb 25, 2021 4:35 UTC (Thu)
by NYKevin (subscriber, #129325)
[Link] (3 responses)
Ultimately, science is about predicting the outcomes of experiments and observations. If you can't observe it, directly or indirectly, it is beyond the realm of scientific inquiry. The one-way speed of light falls into this hole. Speed is normally defined as distance divided by time, but when we're talking about relativistic speeds, we need to be clear about which time we mean.
So, let's imagine a specific setup. We'll send a pulse of light from point A to point B. So, what is the one-way velocity of the light? It is the distance between A and B, divided by the time it takes for the signal to arrive. But whose time?
It turns out that, not only is there no reasonable way for the scientists at A and B to communicate the fact that the pulse has arrived (short of sending a second pulse and thereby measuring the round-trip time instead), it is actually not meaningful for an event at A and an event at B to happen "simultaneously." There will always be an observer who thinks that A's event happened first, and an observer who thinks that B's event happened first, unless the events are in each others' light cones, in which case everyone will agree that one comes before the other. But you can't get everyone to agree that both events happen simultaneously.
This creates a problem, because no matter where you put the clock, it has to be synchronized with respect to two distant events:
1. The pulse being sent from A.
You cannot synchronize a clock with respect to an event unless the clock is physically present at that event, because simultaneity is otherwise relative to one's choice of reference frame. So the clock has to be at both A and B. You could imagine two clocks, synchronized with each other, but then you have to decide what you mean by "synchronized," because simultaneity is still relative. There is always a reference frame in which the clocks are not synchronized.
If the pulse were traveling slower than light, then you could move the clock with the pulse. This would give you a metric called the "proper time," and that actually does have some useful properties in this context. In particular, it is independent of reference frame or choice of coordinates, so that the one-way velocity of any slower-than-light object is perfectly well-defined and even measurable. But you can't do that with a pulse of light, because the pulse is traveling at the speed of light, and the clock can't go that fast if it has any mass. If the clock doesn't have any mass, then it cannot function as a clock in the first place, because massless particles do not experience the passage of time.
As a result, there is no operational definition of the one-way speed of light that is compatible with the laws of physics. This is why Einstein, for example, describes it as a "convention" in his special relativity paper, rather than asserting it as fact. It's not fact, it's just a convenient way of labeling the diagram.
Having said all that, there is one thing we can experimentally verify (and have done so): The time it takes for light to travel along a closed inertial path of length L is L/c, as measured at the start/endpoint of the path, regardless of what the path looks like or how it is shaped. So, if there are any variations in the one-way speed of light that depend on angle, they *always* cancel each other out. As a result, if you simply assume that the one-way speed of light is a constant, then you can use that assumption to "synchronize" distant clocks by sending light pulses between them and adjusting for the speed of light delay. Although this "synchronization" is ultimately just a convention, it behaves a lot like you would expect "true" synchronization to behave (at least with respect to the clocks' reference frames, anyway). Einstein describes how this works in his paper, and others have elaborated on the details (in short, the clocks should be in inertial reference frames and not in relative motion).
Posted Feb 25, 2021 22:39 UTC (Thu)
by apoelstra (subscriber, #75205)
[Link] (2 responses)
Could you entangle two clocks?
Posted Feb 25, 2021 22:47 UTC (Thu)
by mathstuf (subscriber, #69389)
[Link] (1 responses)
Posted Feb 25, 2021 22:56 UTC (Thu)
by apoelstra (subscriber, #75205)
[Link]
[1] https://en.wikipedia.org/wiki/Twin_paradox
Axioms
This definition will appear quite natural to the reader
familiar with the invariant space-time formulation of
special relativity, as described for example in [1] or the
first chapter of [2]. In relativity, the ordering of events is
defined in terms of messages that could be sent. However,
we have taken the more pragmatic approach of only
considering messages that actually are sent. We should
be able to determine if a system performed correctly by
knowing only those events which did occur, without
knowing which events could have occurred.
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Wol
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Cheers,
Wol
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g is the gravity of earth, approximately 9.81 m/s².
The equation is F = G⋅m1⋅m2/r². In case of Earth, that's F = G⋅m⋅m_earth/r_earth², with g = G⋅m_earth/r_earth². So in that case we get F = m⋅g.
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And some comments about it: https://medium.com/cantors-paradise/richard-feynman-on-th...
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You can construct a universe with multiple time axes, with many more dimensions, and so on. The math will be internally consistent.
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Wol
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Who told you that? A six-dimensional classic (Newtonian) universe works just fine. Sure, you won't have stable orbits but apart from that it's OK.
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2. The pulse arriving at B.
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