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Comment by jiggawatts | Hacker News Reader

If you define pi based on physical circles, then it isn't even the "same pi" as the mathematical one in this universe. In fact, it changes locally from point to point. That's literally what General Relativity says: spacetime is curved!

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I don't think this works in GR the way you think it works (regardless of what 'physical circle' means), aren't you always able to change coordinates for the space to be locally flat?

(This always confuses me, can someone that knows GR shed some light on this? I would greatly appreciate it. I've been trying to teach myself GR a while back from the lectures of Frederic Schuller and Alex Flournoy that are available on youtube but it's hard without being able to ask questions)

"Locally flat" means flat only very close to the point of consideration. That is the mathematical definition, the "practical" definition is "flat only as long as you do not stray further away than some characteristic measure of distance defining your problem and accuracy requirements". So as long as your circle is very small then the " experimentally measured" pi and the math pi are the same. The moment your circle becomes big enough to pass through interesting (measurable) gravitational effects, the experimental value will be different. If anything, experimentally measuring that difference is one way to experimentally measure spacetime curvature.

That's why I asked about GR specifically, to have a common framework to work with. Without it you can say words like experimentally, practical and talking about differences between different pis whatever that means but this will only lead to misunderstandings at best and nonsense at worst

Not in accelerating frames of reference, which includes both rotating frames (such as spinning objects) and frames with high gravitational distortion.

Spinning black holes have a measured circumference that is not 2 x pi x r.

(Consequences of the accelerating frame of reference include the spaghettification in a black hole... We can't use a mathematical conversion to go to a frame of reference where the subject isn't being spaghettified).

Why not? (to the first sentence, locally is locally)

But I think I have found the source of my confusion in my original post, I was thinking about manifolds only in the topological sense and for some reason forgot that when there is a metric then we have to preserve distances. No 'mug and a donut' type of shenanigans.

As distance to center of black hole approaches zero, curvature approaches infinity. There is no "locally" one can constrain small enough to observe no curvature upon approach to a singularity... Given two particles arbitrarily close together at arbitrary velocity (< c) relative to the center of a black hole, there is a distance from the center at which they will encounter unequal acceleration.

In Riemannian geometry (i.e. GR) each point has some local invariants associated to it, a famous one being curvature. "Local" here means that it only depends on the metric in an arbitrarily small disc/ball around the point. So the answer is no, in GR you can't always change coordinates so that space is locally flat. (This is in contrast to symplectic geometry for example, where each point does look locally like the model space).

Ok, yeah. We obviously can't get rid of a curvature. I think SE answer [1] explains what I assumed incorrectly to mean 'locally flat'. I meant it so that you can make a metric look like Minkowski metric at a point p, basically existence of normal coordinates. Of course second derivatives (and thus curvature) won't vanish in the neighborhood. Case closed, thank you.

[1] https://physics.stackexchange.com/a/157019

Pi is defined over a flat space with an L2 norm. It has nothing to do with physical circles except that physical space is locally flat and has an L2 norm.

But you can define an experimentally measured pi. Define an experimental circle (the set of points experimentally measured to be at a given distance from a center) and then experimentally measure its circumference. The fact that this experimental measurement is different from the pi defined through an l2 norm in Euclidean space is certainly cool and interesting. It is basically a way to measure gravitational curvature.

Something being a circle or an oval would depend on your relative velocity, meaning the idea of a physical circle doesn't really make sense.

Not quite.

Observing an object with high relative velocity distorts the observed shape, but all observers can agree what the shape would be at zero relative velocity. For similar reasons astronomers say distant objects are redshifted rather than assuming they are the colors directly observed.

Although in the special case of a sphere, despite the length contraction, the observed behaviour is a rotation because light travel time perfectly balances length contraction: https://www.spacetimetravel.org/fussball/fussball.html

That would be very unusual (unorthodox) definition of Pi though.

Just because it's the same number doesn't mean it's the same relationship.

Any definition that relies on trig (sometimes disguised as trig expansion) assumes the existence of a flat Euclidean space - which is a convenient mathematical abstraction in real-world terms, but not required in the general case.

Most everyday physics assumes a flat space, so it's not surprising certain constants fall out of this.

There are more complex and interesting definitions which rely on relationships defined by group theory. I don't know enough about those to say if they're just more complex ways to operate in a flat space, or more complex ways to disguise simple trig expansions, or if they're completely independent, or if something else is going on.

That's not fully true.

You can get trig functions without any kind of geometry.

But wouldn't the measuring device also be curved?

I measure distances in the time it takes light to traverse them, don't you?

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