/sci/ - Science & Math

>>16153992
>everyone
>4chan schizos
They hate what they don't understand.

General relativity is based. It's probably just an approximation of an underlying emergent phenomenon, which we have yet to figure out, but it's still extremely useful.

Let me guess, you've never solved a nontrivial Einstein field equation

>>16154046
What's the matter? Not too nontrivial for you?

I'm okay with GR but all that shit with curvature tensors and all their indexes is tedious af.

>>16153992
Just crackpots really and in like 99% of cases they are fighting with something they don't even understand.

>>16153992
awwww how cute, a bottom trying to understand GR. GR is too difficult for you, you should only be thinking about how to look pretty for your top

>>16153992
>i love general relativity why does everyone seem to hate it?
Bro can you help a brother out?
How do you measure distances in curved 4D spacetime, between two points along some curve, including along the geodesics (the shortest path)?

I think the hate is towards Einstein rather than theory itself. This is perfectly valid as Einstein himself had ripped off Lorentz and Minkowski while formulating STR. Similarly GTR was a ripoff of Ricci, Riemann and Hilbert. Nobel committee were aware of that but because they're a nepotism club, they decided to give him a Nobel prize anyway on photoelectric effect lmao which is a ripoff of Planck.
>>16154442
Not really, in most cases the indices are cyclic or they appear both in contra and covariant form and can be removed (Einstein summation convention)
>>16155099
By using the metric tensor of your manifold. Remember in 2d Euclidean space the L2 distance is
[eqn]||x||_2 = \sqrt{\sum_{i=1}^{n} x_i^2}[/eqn]
That's because the metric is [[1,0],[0,1]]. Or do you mean something like geodesic equation and parallel transport?

You love relativity, so I, relative to you, appear to hate relativity

>>16154043
>geometry can be emergent
Nope

>>16153992
It's disproven by quasars, so there's that. Also it predicts stationary charges to radiate in gravitational fields, which is also not observed.

>>16154652
Says the bottom bitch fag with a completely wrecked asshole.

It's very exciting to go through classical mechanics and then see where it fails, and how this (even to this day) undermines the layperson's intuitive notion of what time is. Then you realize, it's just another model with its own flaws as well, and end up a sadder yet wiser man.

>>16155315
Einstein didn’t even like maths, you really think he’d read all those mathematicians? LMAO

>>16155537
That paradox was resolved by Fritz Rohrlich in 1965: Maxwell's equations hold only within an inertial frame.

>>16155602
And how exactly does a stationary election know whether it's in an inertial frame?

>>16155623
How exactly does a point knows whether it's in Polar or Cartesian coordinates?

>>16155627
Nice deflection, I'll answer my own question: It can't. The equivalence principle is violated, therefore GR is disproven.

>>16155315
>Remember in 2d Euclidean space the L2 distance is
Doesnt this use the minkowski metric in GR?
>>16155315
>Or do you mean something like geodesic equation and parallel transport?
Curves in general, including the geodesic
I was also wondering if theres some way to embedd 4D space inside some higher dimensional space, to better visualize the curvature. I know curvature has been treated as an intrinsic property since Gauss started doing that but should be possible to do an embedding, i think theres some theorem about it

>>16153992
I guarantee there is no one on this board who legitimately understands GR.

>>16153992
Why the fuck are you back you disgusting faggot?

>>16155315
bump so i can get an answer
your distance formula works with minkowski metric?

>>16155825
>each image is only posted by one person
are you smart enough for this board?

>>16156698
Of course it does. In 4d Minkwoski space, the metric is usually written as [[-1, 0,0,0],...] where ... Is the Euclidian metric for 3d space and is a diagonal metric with 1 in diagonal. Then, your distance is -dx0^2 + dx1^2 +dx2^2 +dx3^2. This is what we call the spacetime interval if x0 is time and rest 3 are space dimensions

>>16156783
For that image yes, dumb faggot

>>16156971
retard
there are many bottomposters

>>16156859
So how would one calculate a distance in curved spacetime? Just using that metric?
Can it come out negative? Whats the interpretation of a geodesic when there are negative distances?

>>16157766
Your distance travelled through spacetime is always the same - one second per second. It's simply a question of how much of your path through spacetime is going through the "space" direction versus the "time" direction as your relative motion and the curvature of spacetime changes.

>>16157883
>Your distance travelled through spacetime is always the same - one second per second.
Cool but how do you calculate the distance (in 4D) in between any two points?
I say this because i want to understand what a geodesic is. If im to talk about a short path, i want to know how to calculate the length of any path
How is it done in practice, could you start by doing some parametric curve for the integration?

>>16157890
Line path integrals. Start with something simple in 2D.
A = (0,0)
B = (1,1)
Along the path [math]y = x[/math]

[math]ds^2 = dx^2 + dy^2 = dx^2 + \left(\frac{dy}{dx}\right)^2\;dx^2[/math]
[math]ds = \sqrt{\left(1+ y'(x)^2\right)}\;dx = \sqrt{\left(1+1\right)}\;dx = \sqrt{2}\;dx[/math]
[math]s = \int^{1}_{0} \sqrt{2}\;dx = \sqrt{2}[/math]

Along the path [math]y = x^2[/math]

[math]ds = \sqrt{\left(1+ y'(x)^2\right)}\;dx = \sqrt{\left(1+4x^2\right)}\;dx[/math]
[math]s = \int^{1}_{0} \sqrt{\left(1+4x^2\right)}\;dx = \left(2 \sqrt{5}+\sinh^{-1}\left(2\right)\right)/4 \approx 1.479[/math]

Where it gets messy in GR is that you're dealing with three spatial and one temporal coordinate, all of which are coupled. You need to solve the Einstein Field Equations to determine what ds is in the first place (which depends on your coordinate system and the distribution of matter and energy in space, and requires solving a system of ten differential equations simultaneously; 64 if you include solving for all the Levi-Civita symbols). Once you have ds, you still need to find a scheme for parameterizing your path, and then solve for the integral of ds, which can only rarely be accomplished analytically.

The reason you hear about things like the Minkowski metric (flat Cartesian space), the Schwarzchild metric (point mass in a vacuum in spherical coordinates), the Kerr metric (including rotation), and a handful of other metrics is that those are the few we have actual nice analytic models for. And even for these models, only a few trivial, highly symmetric paths have analytic solutions.

>>16157927
Oh but you ruined it by not using the metric. See thats the notorious part.
The minkowski metric. Its easy in 2D with euclidean metrics.

>>16157927
>You need to solve the Einstein Field Equations
How about you just dedine what the curvature is and eschew the einstein equation? Take it as a math problem.
Ill make it easier for you. Do it for flat space.
Distance between any two points, it can be zero or negative becuse of the metric, isnt it?

You seem to be a söyboy who likes little mathematic challenges but you lack the ability to criticize anything. The relativistic transformations will give light too much notion with cost of deforming the universe.

>>16157890
Look this for derivation of geodesic distance
https://wikipedia.org/wiki/Geodesics_in_general_relativity
The Christoffel symbol in the geodesic equation contains the metric g

>>16154652
Disgusting.

>>16158120
plz do an example in flat space, could be in 2D but one has to be time

>>16153992
I just dislike theoretical physics as a discipline in general. Don't get me wrong, there's some interesting mathematics and modeling those people do, but they tend to have problems seeing the forest for the trees.

Theoretical physics is an inferential modeling discipline in which we try to map the immaterial and platonic fields of analysis and algebra to the material. They've achieved some great things with these models, but many in physics lose sight that they are in fact just models.

There's nothing actually material to suggest (as an example) Hamilton's principle of least action is actually something intrinsic to the state evolution of real material bodies. It's a modeling convention that we then develop other conventions on top of. Just don't forget the golden rule, "All models are wrong, but some models are useful."

>>16158474
>There's nothing actually material to suggest (as an example) Hamilton's principle of least action is actually something intrinsic to the state evolution of real material bodies
Hamilton's principle is just a restatement of F = ma. If you accept the former as something intrinsic to the dynamics of systems, then the latter is also necessarily fundamental.

>>16158561
> Hamilton's principle is just a restatement of F = ma.

That is absolutely not true. You can derive a result equivalent to F = ma from Hamilton's principle, but it's a bit deeper than that. Hamilton's principle states that the true state evolution between any two time instances MUST be the result of a stationary point for the functional (a.k.a. only the "least costly" path is possible).

This is something that is an apriori assertion and is absolutely not to be taken at face value for realizable systems (meaning in meat space rather than on paper) where there are always stochastic uncertainties and process uncertainties regardless of what system you are trying to model. There's no such thing as a frictionless system, there's no such thing as a truly linear system with true superposition, it's a good approximation but it's a modeling approximation. It isn't "source code" it's a map we've written in an effort to describe the territory.

>>16153992
>everyone
You mean larping retards on /sci/? You know this is a joke board right? A zoo?

>>16158638
a mongolian basket weaving asylum

>>16158644
I think it's more nuclear than that.