/sci/ - Science & Math

No shit?

good good babby

now try to quantize 2+1 with <span class="math">\Lambda > 0[/spoiler] and shit brix

That's interesting, OP.

Tell me more interesting things.

>>4960401
Oh, perhaps if you fill the whole universe with matter or set a cosmological constant then you could have cosmology in flatland. I should see what 3-d Friedmann equations say...

>>4960433
Well, it has nothing to do with the above, but here it goes something that I find interesting too.

In the videogame Portal you can create two portals parallel to each other, in such a way that (locally) space has the topology of a cylinder. Mathematically what you're doing is identifying two sets of points, making them the same, "gluing" space. However, you must have in account time! If you enter a portal you must specify at which instant you exit the other one. It must be at the same time, because otherwise you could travel to the past going through portals one way and to the future going the other way. That accounts for newtonian mechanics, which have an absolute time.

But in special relativity things don't work properly, because the temporal location of events depends on the state of motion of the observer! So if you have an observer for which entering one portal is simultaneous to exiting the other one, there are other observers for which you exit the portal before entering in the first one! There's a fundamental incompatibility between special relativity and the cylinder topology of space. Portals are against the principle of relativity, they create a prefered class of observers!

>>4960401
>babby

self righteous asshole detected.
>mfw

>>4960290
doesn't that mean that you will still effect the manifold at the point where mass is so the whole space will collapse into a singularity if any mass is present.

>you can create two portals parallel to each other, in such a way that (locally) space has the topology of a cylinder
why would it matter that they are parallel?

The Ricci tensor doesn't have the same number of degrees of freedom as the Riemann tensor (which should be obvious, given that the Ricci tensor is just the trace part of the Riemann tensor). The traceless part of Riemann tensor (the Weyl tensor) is not in any way dependent on the Ricci tensor. <span class="math">R_{\mu \nu }=0[/spoiler] does not in any way imply that <span class="math">R^\sigma _{~\mu \nu \rho }=0[/spoiler].

A good example of when the Ricci tensor is zero but the Weyl (and therefore the Riemann) tensor is not would be free gravitational waves.

>>4960576
Who allowed you to stop? We demand more.

>>4960620
>massive bodies, even though they curve the spacetime where they are, they don't affect the spacetime surrounding them

>>4960646
You're right, topologically it doesn't matter. However, it makes calculations easier, because otherwise you'd have a metric similar to a conical surface.

>>4960677

Another curious fact. If you're in a space that's compact (for instance a 3-sphere), you can't have a sole electric charge. There must be always a couple, one positive, one negative.

Think about it. If you have, say, a positive charge, where would the electric field lines go?

>>4960290
>>4960576
>>4960646
>>4960733
>>4960979
I love you

On the other hand, in 1 spatial + 1 temporal, you can have gravitation again, because without tides you no longer need the spacetime in the vicinity of falling objects to be curved.

>>4961110
Thanks mate!

Here's another interesting fact, dedicated to you:

If you have a set A, you can always build a strictly bigger set by taking the subsets of A, which is called P(A) (parts of A). For instance, if you have three objects A={a,b,c}, then P(A) would have the following elements:
∅, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}
As you can see, there's a total of eight, which is <span class="math">2^3[/spoiler] It's easy to see that adding a fourth element d to A would duplicate the elements of P(A), because you'd have the preceding elements plus the ones which also have d. It follows that if A has cardinality <span class="math">n[/spoiler], then P(A) has cardinality <span class="math">2^n[/spoiler]. Of course, you can take the set of subsets of subsets P(P(A)) and nothing stops you from doing this again and again...

But one puzzling question arises when you consider infinite sets!

(to be continued in another post)

>>4960979
>Think about it. If you have, say, a positive charge, where would the electric field lines go?

Same place as gravitational field lines go.

Infinity.

Troll harder.

>>4961172
show me where infinity is on a sphere.

>>4961172
> Infinity
> on any sphere

There are infinite natural numbers. That infinity is the smallest infinity you can think about, and it's called <span class="math">\aleph_0[/spoiler] (Aleph sub zero). But now, remember what we did about considering the set of all the subsets? We can do that with the natural numbers! We can say <span class="math">P(\aleph_0)=\aleph_1[/spoiler], when now <span class="math">\aleph_1[/spoiler] is the set of all sets of natural numbers. For example, it will have elements such as {1}, {5,9001,7} {1,3,5,7...} and so on...

How many elements does <span class="math">\aleph_1[/spoiler] have? Infinite. But a bigger infinity! In a slight abuse of notation we can say it has cardinality <span class="math">2^{\aleph_0}[/spoiler]. This new infinity is bigger in the sense that it's impossible to build a bijection between it and Aleph sub zero. But nothing shall stop us from continuing!

We can take <span class="math">P(P(\aleph_0))=P(\aleph_1)=\aleph_2[/spoiler], and <span class="math">P(P(P...(P(P(\aleph_0))...)))[/spoiler] and so on and so on, building more and more different kinds of infinities...

The question now is clear.. ¿How many kinds of infinities are there? Infinite, sure, but... ¿which of those infinities?

>>4961189

<span class="math">\aleph_\infty[/spoiler]

>>4960290
its threads like this that make me realize how fucking stupid i am. and they make me study more. so thanks OP

>>4961199
Good idea!
But remember that <span class="math">\infty[/spoiler] doesn't mean anything, because there are many kinds of infinities. Would that mean <span class="math">\aleph_{\aleph_0}[/spoiler]? Or <span class="math">\aleph_{\aleph_1}[/spoiler]?

>>4961189
The majority of your post is wrong.

>>4961209

<span class="math">\aleph_0[/spoiler] and <span class="math">\aleph_1[/spoiler] are not "kinds of infinities", but rather "sets that contains infinity products". Their difference is in the density of such products, i.e., how many of them are.

>>4961209
Is there an answer or is it an unsolved problem like the "who wants to be a millionaire" thing?

>>4961215
The number of types of infinities is greater than any cardinality.

>>4961189

You're tacitly assuming CH. But even then, your post is wrong.

>>4961211
It's possible. I'm no expert in set theory. My understanding of this subject is quite "layman". If you could please point out my mistakes I'd be thankful, because that way I learn too.

>>4961225
Why?

>>4961226
How? They didn't mention the real numbers, so how could CH possibly be involved in it?

>>4961229
As this guy said >>4961226
you assumed CH. In fact, you assumed the generalized continuum hypothesis.

The statement P(\aleph_0)=\aleph_1 is independent from ZFC set theory.
>>4961231
It's a bit hard to explain, but you can think of constructing bigger and bigger ordinals in the same way as you construct bigger and bigger sets by taking unions and power sets.

>>4961234

"The set of all sets of natural numbers" is the real numbers. They claimed that <span class="math">\aleph_1[/spoiler] is that set, but that's true only given CH. Even granting that, it is not the case that <span class="math">\mathcal{P}(\aleph_0)=\aleph_1[/spoiler]. The powerset of <span class="math">\aleph_0[/spoiler] is not an ordinal. It will contain things like {1}/

>>4961234
Nvm this post, I didn't read >>4961189 carefully enough. Oops.

>>4961189
How is this correct? I'm not trying to troll but it sounds like you're saying the cardinality of the continuum is the cardinality of the power set of the rationals. But this would allow one to count the cardinality of the continuum, which is impossible. Further, it would provide an example of a cardinality inbetween aleph naught and c, which disproves the continuum hypothesis. Help

>>4961271
You could have saved yourself some trouble if you had skimmed the replies.

>>4961271
What? No. The power set of any set has a strictly larger cardinality. It's the basic diagnolization argument of Cantor.

| pow(rationals) | = | reals |

This is basic stuff.

>>4960290
Go to bed Víctor.

>>4961271

Here's a hint. What is a real number?

>>4961271
>But this would allow one to count the cardinality of the continuum, which is impossible.
To be clear, this is simply wrong. The power set of the rationals has a larger cardinality than the set of the rationals. The rationals are countable, and thus the power set of the rationals is not countable.

>>4961242
>>4961238
>>4961226

I think I didn't take CH, because I never compared natural numbers with real numbers... Let's just say that <span class="math">aleph_1[/spoiler] is notation for <span class="math">P(\aleph_0)[/spoiler]. And let's call C the cardinality of real numbers (which we won't use).

BTW >>4961225 is really close, or perhaps he's completely right and I'm wrong in my conclusion!

But let me finish...

Well, it turns out that when you finish to build all the possible <span class="math">\aleph_n[/spoiler], for n integer, you can do something amazing! Take the union of all of them. That's an aleph bigger than all the possible <span class="math">\aleph_n[/spoiler]!

Let's call it <span class="math">\aleph_\omega[/spoiler]. It's a huge infinity!

But nothing shall stop us here. We can build the set of all its subsets! Take <span class="math">P(\aleph_\omega)=\aleph_{\omega+1}[/spoiler]

And now <span class="math"> P(\aleph_{\omega+1})=\aleph_{\omega+2} [/spoiler]

And so on and so on...

You'll eventually have every <span class="math"> \aleph_{\omega+n}[/spoiler]

And now you can take the union of all the infinities so far, and build <span class="math">\aleph_{2\omega}[/spoiler]

You can easily see that this continues in a horrible way with <span class="math">\aleph_{3\omega}[/spoiler], <span class="math">\aleph_{n\omega}[/spoiler], <span class="math">\aleph_{\omega^2}[/spoiler], <span class="math">\aleph_{\omega^7+8\omega^4}[/spoiler] ... <span class="math">\aleph_{\omega^\omega}[/spoiler]... and there's no way to stop it.

So we arrive at the answer...

I'd lie if I told you that I understand this, but here's the answer that a friend mathematician gave to me:

"You can't talk about the cardinality of all kinds of infinities, because there is no such a thing as a set that contains all infinities". In other words, "being infinite" does not define a set. To build new sets you must always start with a previous one, and apply a condition on it. You can't just say the condition and expect that to define a set.

So elementary set theory saved the day! And now I'm exhausted

So we arrive at the answer...

I'd lie if I told you that I fully understand this, but here's the answer that a mathematician friend gave to me:

"You can't talk about the cardinality of all kinds of infinities, because there is no such a thing as a set that contains all infinities". In other words, "being infinite" does not define a set. To build new sets you must always start with a previous one, and apply a condition on it. You can't just say the condition and expect that to define a set.

So elementary set theory saved the day! And now I'm exhausted, so I think I'll follow >>4961274 's advice

>>4961295

...That's even more wrong.

>I think I didn't take CH, because I never compared natural numbers with real numbers

The set of sets of the natural numbers is the real numbers.

>Let's just say that aleph1 is notation for P(<span class="math">\aleph_0[/spoiler])

But that's wrong. Aleph 1 is the first uncountable cardinal.

>Well, it turns out that when you finish to build all the possible @n , for n integer, you can do something amazing! Take the union of all of them. That's an aleph bigger than all the possible @n !

...No. If you take the union of a finite number of infinite sets you do not increase the cardinality.

>>4961306
0/10
Troll harder.

cant you just replace all the alephs with beths and everything will be fine?

>>4961295
>BTW >>4961225 is really close, or perhaps he's completely right and I'm wrong in my conclusion!

I'm confused. Your post makes no comment about the number of possible cardinalities.

>>4961309

Consider a finite sequence of infinite ordinals, <span class="math">A_1[/spoiler] through to <span class="math">A_n[/spoiler], satisfying <span class="math">|A_i|\leq |A_{i+1}|[/spoiler]. I claim that the cardinality of <span class="math">A_n[/spoiler] is the same as the cardinality of the union of all <span class="math">A_i[/spoiler].

One direction is obvious. For the other direction, observe that <span class="math">|A_i|\leq |A_{n}|[/spoiler] for all i. As such let <span class="math">f_i[/spoiler] be a one to one function from <span class="math">A_i[/spoiler] to <span class="math">A_n[/spoiler].

We define an injection <span class="math">f[/spoiler] from <span class="math">A_n[/spoiler] to the union of all <span class="math">A_i[/spoiler] as follows. For x = ny+i (extending the notions of remainder in a fitting manner to the infinite ordinals), let <span class="math">f(x)=f_i(y)[/spoiler]. It is easy to verify that f is indeed an injection.

6/10, made me reply.

>>4961669
He was taking a union of an infinite amount of ordinals. Your post is invalid.

>>4960401
Isn't that how you get Chern-Simons on an AdS background

>>4961688

>Well, it turns out that when you finish to build all the possible @n , for n integer, you can do something amazing! Take the union of all of them. That's an aleph bigger than all the possible @n !

He was taking the union of all alephs up to a given integer. That's a finite number of alephs.

>>4961704
No, he was not.
>Get all @n FOR ALL n
>Take the union of ALL OF THEM

>>4961690
Do you have any good references for learning string theory? I'm very curious about that AdS/CFT duality everybody talks about.

>>4963021
do you know any or do you want to start from scratch?

From scratch. I know it's something that will take years, but It's better if I start as soon as possible

>>4963053
whats your background knowledge? do you know QFT and GR?

>>4963063
Nothing. The last physics course I took was in highschool. Majoring in gender studies now.

>>4963064
Hahahaha

>>4963063
I know a little GR, but no QFT. I suppose I should master these topics before attempting strings...

>>4963064
ok, thats a long way to go then. you will need the following:
Lagrangian mechanics
Hamiltonian mechanics
electrodynamics
special relativity
classical field theory
introduction to quantum mechanics
relativistic wave mechanics
general relativity
quantum field theory

the math you will need is
calculus
linear algebra
differential equations
abstract algebra
group theory
differential geometry

then you can start with string theory

i most likely missed some things. others can add.

best thing to do is look for a textbook on each of those subjects and start from the top of each list.

>>4963068
yes you should know GR (actually, just the math, so just deferential geometry). QFT is a must,. work through something like An Introduction To Quantum Field Theory and then you can go to A First Course in String Theory by B.Zwiebach.

>>4963073
Mmmkay

Currently I'm studying classical field theory from this webpage http://academic.reed.edu/physics/courses/Physics411/html/index.html
(In the last lecture there is a brief discussion on the Nambu-Goto action) What I lack most is quantum mechanics stuff, I think I'll read Feynman's lectures to get some general ideas before taking a textbook.

>>4963075
Zwiebach? I'll look it up.

Thanks, man

>>4963073
Thanx. Do you have any textbook that covers all of them?

>>4963087
Zwiebach only introduces QM in chapter 12 (of 23) so you should be able to start right away.

>>4963089
unfortunately, no.

>>4963089
For a (very) soft introduction one could try to read through "Nakahara" Geometry topology and physics. He basically goes through a lot of topics, none too deep, but at least one gets an idea of what one needs. The book has many chapters on differential geometry and shows a glimps of strings in the end. Again, it's not a book about one subject, but it's an overview and I'd recommend it.

>>4963073
All the compex stuff is missing (unless you count operator product expansions and functional analysis under "calculus" :)

>>4961165
this is not "me" btw., I was confused for a moment...

@the set theory people, what do you do, academia wise?

>>4963111
>this is not "me" btw

Yes, it was. You made a retarded post and got called out. Don't deny it.

>>4963114
as far as I can see, there wasn't even a response to that post.
Apart from that, I can savely say that I wouldn't deny it.

>>4960646
this was me yesterday.

>>4963120
Get a tripcode.

Can somebody give me a list things to know about differential geometry to get into GR

>>4960290
What about photons?