The h Bar

0celo7

23:30

@ACuriousMind How do I measure the proper length of a spacelike geodesic

in a lab with human tools

ACuriousMind

@0celo7 wat

Have you got a spacelike geodesic in your lab?

0celo7

@ACuriousMind there's probably one around

there is always one

ACuriousMind

Then just hold a ruler to it.

0celo7

but how do I find it

ACuriousMind

I'm not responsible for where you left your rulers!

0celo7

23:32

sorry if this sounds silly, but there's a construction in GR called "Fermi coordinates"

and you need the length of a spacelike goedesic

but I have no clue how one would go about finding it

ACuriousMind

@0celo7 Uh, isn't a straight spatial line a spacelike geodesic (at least for Minkowski)?

Slereah

"The condition of stationarity yields new topological obstructions[22]; for example, if
(M, g) is compact, stationary and n = 3 then M is a Seifert manifold. "

wot

0celo7

@ACuriousMind curved space brah

straight spatial line

ACuriousMind

@0celo7 Well, first measure how space is curved, then compute what shape the spacelike geodesic have, then measure them

0celo7

how is straight defined

ACuriousMind

23:35

@0celo7 In my human lab with human tools, a normal human-made ruler is straight.

Slereah

Spacetime curvature in the lab is usually measured by interferometry

Just check if the light's path is the same in all direction at all points

0celo7

@ACuriousMind what about over large distances

suppose the spatial slices are curved

ACuriousMind

I seem to recall that measuring over "large" distances is a terrible mess in GR in any case

But I'm not a GR person, dunno why you ask me this

0celo7

@ACuriousMind It is

I'm the one who told you that :)

@ACuriousMind because I have a different question

Slereah

Over large distances you just integrate some spacelike curve :p

0celo7

23:38

forget spacetime

@Slereah that does not even begin to make sense physically

@ACuriousMind forget spacetime, take some general curved Riemannian manifold

suppose I'm "in" this space

and a geodesic shoots by me

what do I see

ACuriousMind

@0celo7 That doesn't make any sense

A curve is not a physical object

0celo7

will it appear perfectly straight to me?

ACuriousMind

@0celo7 If you stand on a sphere, do the meridians appear perfectly straight to you?

0celo7

@ACuriousMind suppose we fix something to the geodesic and it moves along it

Slereah

@0celo7 : You are on a manifold right now

0celo7

23:41

@ACuriousMind I have no cluie.

ACuriousMind

@0celo7 You are on earth. Earth is a sphere. How does a meridian look to you?

Slereah

Even better

Take two meridians

crossing the poles

Parallel lines do not cross each other in flat space

0celo7

@ACuriousMind never seen one

and I'm not really "on Earth" ayy lmao

2

I'm in 3-space

Slereah

@0celo7 confirmed for space alien

0celo7

what if I was 2-dimensional

ACuriousMind

23:42

Yes, but just think about what @Slereah says. They cross at the poles. You'll see them intersect.

Slereah

Also there's the whole thing about triangles having odd angle sums

You can have a triangle with 3 90° angles on a sphere

0celo7

the sphere is a bad example

let's talk about 3-space

Slereah

which one

0celo7

hmm

that is a good question

ACuriousMind

I have honestly no idea how to imagine curved 3-space.

0celo7

23:44

then how do I measure the length of a spacelike geodesic

Slereah

I mean really the same arguments we said apply to $S^3$ I think

ACuriousMind

@0celo7 Find one, roll a tape along it

I don't see the issue.

You measure the length of timelike geodesics by sending clocks along them, you measure spacelike geodesics by rolling measuring tape along them.

0celo7

@ACuriousMind you sob

this is a serious q

but how do I even find a spacelike geodesic

Slereah

Well unless you are some sort of time lord, anything you measure with a ruler will be spacelike

ACuriousMind

@0celo7 Take a tachyon and look where it moves :P

0celo7

23:46

@ACuriousMind seriously

ACuriousMind

@Slereah I think the "geodesic" part is the thing he has trouble finding, not the "spacelike".

0celo7

I'm trying to make sense of Fermi normal coordinates

they're made out to be this natural coordinate system

Slereah

Iteratively, I guess?

0celo7

but the fact that you have to measure a spacelike geodesic on each axis

that doesn't even make sense

Slereah

Well

Ask yourself

ACuriousMind

23:50

@0celo7 Well, again, if you know the shape of your space, you can just compute the shape of the geodesics, can't you?

Slereah

How do we know what a geodesic is in flatspace

0celo7

@ACuriousMind why would I know the shape

ACuriousMind

Because you measured it?

Slereah

it is $\sqrt{\sum x^2}$

But this was obtained after a while

0celo7

@ACuriousMind huh?

ACuriousMind

23:51

@0celo7 Dunno, if it isn't strongly enough curved so that you know how it's curved, it's kinda pointless to even do this whole stuff, isn't it?

Slereah

Obtained via the Pythagorean theorem, among other thing

0celo7

@ACuriousMind how do I know if it is curved

Slereah

in curved space, you'd have to find that too, except that will depend on your location

0celo7

this is a good question

what is the best way to figure out if spacetime is curved

geodesic deviation?

ACuriousMind

@0celo7 Just look at light?

Slereah

23:52

Triangle test I'd say

Check if the angles of a triangle add up to $180°$

0celo7

@ACuriousMind how does one look at light

@Slereah ok

how does one do that in spacetime

ACuriousMind

@0celo7 With a telescope.

Slereah

Spacetime is locally euclidian

0celo7

@ACuriousMind are you trolling me

Slereah

So you can always measure an angle like in flat space

since that's a local property

0celo7

23:53

so what you're telling me is that I will never measure spacetime curvature

ACuriousMind

@0celo7 No, recall how we tested GR in the first place!

0celo7

@ACuriousMind ah

we're on completely different pages

no need to remind me of GR

I want something completely geometric

"light rays follow null geodesics" is physics

ACuriousMind

Then do what @Slereah says and parallel transport some vector around a large triangle.

0celo7

how the heck do you do that??

Slereah

Just measure the angles jeez

0celo7

23:55

how do you parallel transport

ACuriousMind

Some variation of that Chinese device?

Or measure the angles, yes.

0celo7

@ACuriousMind lol, what?

@Slereah OF WHAT

Slereah

The corners

Of the triangle

ACuriousMind

South-pointing chariot

The south-pointing chariot (or carriage) was an ancient Chinese two-wheeled vehicle that carried a movable pointer to indicate the south, no matter how the chariot turned. Usually, the pointer took the form of a doll or figure with an outstretched arm. The chariot was supposedly used as a compass for navigation, and may also have had other purposes. There are legends of earlier south-pointing chariots, but the first reliably documented one was created by Ma Jun (c. 200–265 CE) of Cao Wei during the Three Kingdoms, about eight hundred years before the first navigational use of a magnetic compass...

0celo7

I think you physicists are too physicsy

Slereah

23:56

What is too physical about measuring an angle

Take your protractor

Put it on the triangle

0celo7

@ACuriousMind how did you remember that thing

I learned about it ages ago

ACuriousMind

> "The chariot can be used to detect straight lines or geodesics. A path on a surface the chariot travels along is a geodesic if and only if the pointer does not rotate with respect to the base of the chariot."

There you go.

0celo7

@ACuriousMind indeed, will investigate

ACuriousMind

Now you just need to figure out the 3D version of it :P

0celo7

now how does one do that for spacelike geodesics

@ACuriousMind 4D, really. and I have to figure out how to make it spacelike

@Slereah WHAT TRIANGLE

Slereah

23:58

Any triangle.

ACuriousMind

@0celo7 Hm? It already detects "spacelike" only

Slereah

Do you remember

THE MYTH

ACuriousMind

It detects geodesics on the Riemannian 2-sphere.

Slereah

Of Karl Friedrich Gauss

Some people said he tried to prove the geometry of the universe

0celo7

@ACuriousMind I need to lift it to 4D

Slereah

23:59

By measuring the angles between three mountain points

0celo7

and somehow make it distinguish spacelike and timelike

what if it goes on its merry way along a timelike geodesic

Slereah

You take some geodesic survey equipment

ACuriousMind

@Slereah Pooping on prime-numbered days?

Slereah

You go to a mountain peak

You look at the two other peaks

You measure the angle between the two

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