As some of you know, there is a fundamental flat space-time metric that describes our universe without any energy or matter in it. Correct me if I am wrong, but this metric and existance of the limiting speed of signal propagation which is not dependent of the observer are the same thing. Or? Could we understand one as a consequence of the other? So, could we say that the constant $c$ is a consequence of the metric? And, could there be some other, fundamentally different flat spacetime metric which allowes other forms of motion? Eg, greater speeds, in a nutshell, a different causality? What do you think, is there some more fundamental physical reality which determines the metric?
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As some of you know, there is a fundamental flat space-time metric that describes our universe without any energy or matter in it.
It doesn't matter whether the universe has matter in it or not. The metric is locally always Minkowski.
Correct me if I am wrong, but this metric and existance of the limiting speed of signal propagation which is not dependent of the observer are the same thing.
Given any metric whose eigenvalues have the signs $+---$, we can choose coordinates such that at a particular point, the form of the metric is $\operatorname{diag}(1,-1,-1,-1)$, which is the Minkowski form and which implies a speed $c$ (equal to 1 in these units) that all observers agree on. The set of signs is called the signature of the metric. So logically, there is a universal speed that arises because the metric has a particular signature.
And, could there be some other, fundamentaly different flat spacetime metric which allowes other forms of motion?
No, the canonical form described above applies to all metrics with this signature. Any variation in $c$ can be absorbed into the change of coordinates that brings the metric into the canonical form.
What do you think, is there some more fundamental physical reality which determines the metric?
General relativity says that the metric is determined by the presence of mass-energy. What is physically observable about the metric is not $c$ or the signature, but the curvature.
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I am getting at fundamentally different metric, id est, one with different signature? – Žarko Tomičić Oct 16 '18 at 14:44
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The idea I think you're trying to get at is why a lot of people set $c=1$. As a sort of definition: as long as your pseudometric is (3,1) -- so, 3 space dimensions and 1 time -- and Riemannian, then you have an region locally looking like $\mathbb{R}^4$, and some cone of that space that has proper length of zero. The slope of that cone is then called $c$. But we could also just define our coordinate system locally to have that slope be $1$, and then you have the two coordinates on the same footing somewhat more equally.
In this sense, I guess you could say that $c$ is a consequence of the metric.
I don't think you can reasonably say that somewhere else a "different flat spacetime" could give "greater speeds". Speed is defined relative to that slope, of $c$. If you somehow rescaled spacetime so that $c$ was smaller, then all time would evolve slower, and the resulting physics would be the same.
The only way you can get different causal structure is on a larger scale (e.g. closed timelike curves), or a different local topology (at event horizons, extra time dimensions).
Alex Meiburg
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So, you could not say, there is some metric in which there is no such thing as c? Maybe? – Žarko Tomičić Oct 16 '18 at 14:46
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No, that's not really possible. Not without changing the signature of spacetime (so that there's no longer a time dimension, or longer any space dimension), or making spacetime be no-longer pseudo-Riemannian (vectors wouldn't obey anything like the Pythagorean theorem anymore.) – Alex Meiburg Oct 16 '18 at 19:19
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but that is my point, can we imagine a universe with different signature of spacetime? Like, (-,-,-,-)? – Žarko Tomičić Oct 17 '18 at 13:33
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Yes, certainly. See for instance, https://en.wikipedia.org/wiki/Anthropic_principle#Spacetime . Hopefully it's clear that (+,+,+,+) is the same as (-,-,-,-), by a simple sign flip, and similarly (-,-,-,+) is the same as (+,+,+,-). A signature with all signs the same doesn't naturally lend itself to a notion of "time evolution", and thus "dynamics". A (2,2) signature like (+,+,-,-) is also studied, and yields some interesting phenomena. – Alex Meiburg Oct 17 '18 at 20:56
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Great! Yes, I know that, that is why I didnt propose +++- :-) But nice to know it makes some sense. And since we are on the subject, could we imagine universes with higher dimensions with all kinds of metrics? Is there any sense to it, in applying to the real world, eg, QGravity? Could some phenomena we observe be a property of some other, wierd dimensions, just as the time evolution is the property of (-) part of the (-,+,+,+) metric? – Žarko Tomičić Oct 18 '18 at 16:56
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Absolutely, I mean String Theory is actually on a larger metric: it's (10,1), or ten spatial dimensions. The reason it doesn't become 'boring' in the way that the Anthropic principle wiki article suggests is that the extra dimensions have finite length, and so forces can't 'diffuse' out too far. There are also serious studies of theories with 3 spatial and 2 time dimensions, to understand how that would be different from our world. Finally, in condensed matter, people often study lower dimensions: for instance, graphene, being flat (2 spatial dimensions), is modelled as (-,-,+). – Alex Meiburg Oct 18 '18 at 18:12
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hmh, what does it mean, extra dimensions have finite lenght? What is dimension anyway? – Žarko Tomičić Oct 18 '18 at 18:14
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That's getting pretty out of scope for this, but it's about the large-scale structure of the universe -- the topology -- and not the local structure. The metric signature is a local property, and the point is people study these other metrics in contexts like graphene or string theory. – Alex Meiburg Oct 18 '18 at 21:44
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so, could we say that cylinder has one finite dimension? and a sphere all of them? – Žarko Tomičić Oct 18 '18 at 22:52
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Yes, exactly. And a plane (or paraboloid, or hyperboloid) have two infinite dimensions. – Alex Meiburg Oct 18 '18 at 23:47
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You seem to ask whether the laws of nature are governed by gravity, or vice versa. Spacetime existed at the big bang while one cannot speak of the big bang being a consequence of something else, since the big bang is the initial moment in time of our universe. So did gravity exist in its current form since the big bang, or were the fundamental constants of the universe determined shortly thereafter?
As for the cosmological constant: perhaps the nonzero vacuum energy density of spacetime acts as a limiting factor. We know for example that light travels slower in water than in air. Spacetime being a medium, might it have been "easier" to travel through with a smaller cosmological constant? This would make the cosmological constant more fundamental than the speed of light. In turn, the cosmological constant isn't even constant according to the theory of cosmic inflation.
To my knowledge it's unclear what the laws of physics were in the crazy energy regime of the big bang.
Geert VS
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1Spacetime being a medium, might it have been "easier" to travel through with a smaller cosmological constant? This would make the cosmological constant more fundamental than the speed of light. No, this is wrong. Spacetime is always locally Minkowski, regardless of the presence of a cosmological constant. – Oct 15 '18 at 21:02
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So did gravity exist in its current form since the big bang, or were the fundamental constants of the universe determined shortly thereafter? When people talk about this kind of theory in which the fundamental constants arise in a contingent way, $c$ is not the kind of fundamental constant they're talking about. The value of $c$ is always 1 if you choose units in which it's 1. Its SI value is an accident of the construction of the SI. See https://physics.stackexchange.com/questions/3644/the-origin-of-the-value-of-speed-of-light-in-vacuum – Oct 15 '18 at 21:04
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I think I understand now. Say humanity used distance units 300 million times as large as the meter is now, while still using the second for time. We simply get Δx ≤ Δt, and we can change the question into: why can't we move faster through time? But it wouldn't result in faster travel, the physics would remain the same as Alex Meiburg wrote in his answer. – Geert VS Oct 16 '18 at 23:35
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The reason why 300 million meters feels intuitively like a special distance is just because we humans have evolved in a certain way, not because of the basis vectors of spacetime. – Geert VS Oct 16 '18 at 23:44
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As some of you know, there is a fundamental flat space-time metric that describes our universe without any energy or matter in it.
Yes, the FRW-universe "without any energy or matter", often called the "empty universe", is mathematically equivalent to Minkowski spacetime by coordinate transformation which is globally flat. This means that special relativity holds throughout Minkowski space.
So, could we say that the constant c is a consequence of the metric?
No, the metric measures - sloppy said - lengths and angles - Whereas the lightspeed c is a universal physical constant which has the same value measured locally in flat or curved spacetime independent of the metric.
What do you think, is there some more fundamental physical reality which determines the metric?
There are speculations that gravity could be an emergent phenomenon based on microscopic degrees of freedom. But if so that wouldn't affect general relativity and also not the FRW-metric.
timm
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if the metric is such that it leaves maximum speed of signal propagation unchanged under boosts etc than it is connected to that speed somehow...constancy of the speed of light, as oberved, IS something that gave us insight into this metric.. – Žarko Tomičić Oct 16 '18 at 17:57