I want to understand the universe

[twofish-quant]

If someone could come up with actual solutions to those 10 partial differential equations would it be a big deal?

Solutions are known for specific cases. It's being able to understand the dynamics for the general case that we don't know about.

Or does the numerical solutions give enough detail that actually analytically solving them wouldn't give us any new information?

There's a lot of work with numerical solutions, but numerical relativity is a hard and difficult field itself. Setting up those differential equations so that you can solve them on a computer is a very non-trivial thing to do.

This is one of those things that *seems* easy until you actually do it. You think to yourself, all I have are ten equations, how hard can it be to numerically solve them, and then you try, it you run into the same problems that everyone else does.

 

[twofish-quant]

As an outsider to this discussion, I'm wondering if someone can tell me why string theory does have so much activity.

Much activity is relative. There are probably at most about a few dozen people working on it. The thing about string theory is that it gets a lot of press, because you are trying to figure out the theory of everything. People that work on string theory can talk a lot about "God" whereas people that work on ocean physics usually don't.

Perhaps it is experimentally unfounded, though I would hope that there are people who found the mathematics of an established, successful (experimentally, and theoretically pretty sound) theory of physics to strongly suggest certain string theory approaches.

Yes. In the late-1960's and early-1970's, there were some huge breakthoughs in particle physics by using symmetry principles to come up with things like electroweak unification. From the standpoint of the early-1980's, it wasn't totally crazy to try to look for deep symmetries that would unify everything.

Something else to bear in mind is that it takes about seven years to get a Ph.D., and maybe another decade or so to get a permanent position, so if it takes about twenty years to figure out that you are on the wrong path, that's not an unreasonable amount of time.

 

[xdrgnh]

That's very interesting. If I make it to Grad school and beyond I plan on doing research in GR and QFT. If there is no proof that those 10 general PDE in GR have no solutions then I don't see a reason why I should got give it a shot one day. If the solutions are known for specific cases what is preventing them to be known for the general case? Are the equations chaotic in nature?

 

[Nabeshin]

Much activity is relative. There are probably at most about a few dozen people working on it. The thing about string theory is that it gets a lot of press, because you are trying to figure out the theory of everything. People that work on string theory can talk a lot about "God" whereas people that work on ocean physics usually don't.

One reason string theory is still described as the most active field is that there were a lot of hires around 2000, compared to theoretical physics as a whole. In some years, more then half of the new high energy hires were string theory.

See: http://www.physics.utoronto.ca/~poppitz/Jobs94-08.pdf for the distribution, and note what it's been doing the past few years.

 

[Feodalherren]

What's F=ma? Adding, subtracting, multiplying, and dividing. Almost all the math in your general physics class, you've learned in 5th grade. That's just one equation, but applied in different ways.

Do you call doing different ways of adding, subtracting, multiplying and dividing "hard math"?

That's called arithmetic. Math itself is based on proofs.

Not really. I don't know where you did your basic physics but it was much more than that for us. You can plug plenty of stuff into that equation, especially when you are supposed to show why something happens mathematically.
At any rate, that's besides the point. My argument was that a physicist needs to be good at math like a novelist needs to be a good linguist.

 

[Nabeshin]

At any rate, that's besides the point. My argument was that a physicist needs to be good at math like a novelist needs to be a good linguist.

I have to agree with nano, this is just a result of you not having done much mathematics at all. Mathematics is not really at all about the symbol manipulation physicists do. That's not to say that it's easy, but it's not mathematically interesting. The heart of mathematics lays more in the abstract proof, generation of structure, discovery of relationships, etc. See for example http://en.wikipedia.org/wiki/Abstract_algebra , and note that (aside from its motivation) this isn't really about solving equations at all! I advise you to click around some links and see some more 'mathy' mathematics, and realize that it's nothing at all like what's done in a physics class!

 

[Feodalherren]

I have to agree with nano, this is just a result of you not having done much mathematics at all. Mathematics is not really at all about the symbol manipulation physicists do. That's not to say that it's easy, but it's not mathematically interesting. The heart of mathematics lays more in the abstract proof, generation of structure, discovery of relationships, etc. See for example http://en.wikipedia.org/wiki/Abstract_algebra , and note that (aside from its motivation) this isn't really about solving equations at all! I advise you to click around some links and see some more 'mathy' mathematics, and realize that it's nothing at all like what's done in a physics class!

Algebra is just one field of mathematics though? As far as I've been told, math is about finding patterns. And physics is about patterns in the natural world.

I will find out soon enough as I progress toward my degree.

 

[chiro]

Algebra is just one field of mathematics though? As far as I've been told, math is about finding patterns. And physics is about patterns in the natural world.

I will find out soon enough as I progress toward my degree.

It's easier to talk about math in terms of representation, constraints, and transformations. These three things underly all of mathematics including analysis, algebra, logic, probability, topology and so on.

Finding patterns can be seen in terms of the above things: patterns are discernable by the representation used to describe something. The more compact a representation is, the easier it will be to discern a pattern.

You also have to remember that you can decompose something in many ways, and a decomposition is a transformation. Each decomposition will tell you something specific to the context of that decomposition.

By taking a large system and reducing it to descriptions of lower descriptive complexity, you are finding common patterns. Scientists and mathematicians talk about beauty being simple, and this is one way of understanding that statement.

 

[Nabeshin]

That's very interesting. If I make it to Grad school and beyond I plan on doing research in GR and QFT. If there is no proof that those 10 general PDE in GR have no solutions then I don't see a reason why I should got give it a shot one day. If the solutions are known for specific cases what is preventing them to be known for the general case? Are the equations chaotic in nature?

The known solutions apply in situations of ridiculously simplifying symmetry. For example, the FLRW solution has a universe which is completely homogeneous and isotropic. The black hole solutions are completely axisymmetric, as is the TOV star. Now, to try to move a step up to a solution containing TWO of these objects, no analytical solution exists. While it can still be symmetric, the complication of the second object simply hasn't been overcome (which is not to say people aren't trying...).

If you don't understand what's standing in the way of a 'general' solution to the einstein equations, consider the following:
1) We still have no general solution for the Navier-Stokes equation, which describes simple fluid flow. Indeed, existence and smoothness haven't even been shown!
2) Write out the Einstein equations in terms of the metric tensor. This should help you understand the magnitude of the undertaking.
3) MOST nonlinear differential equations don't have general solutions anyways. Furthermore, the EE are very complicated nonlinear DE, so to hope for a generic solution describing every possible spacetime which is possible is folly.

 

[xdrgnh]

Well I took an intro to DE class and I hated the fact that it was a cook book class and that is why I remember nothing from that class. What I did get from it is the flaw of using purely mathematical/theoretical methods to solve DE. Instead I prefer to use physics and intuition to guess the solution to the DE.This usually works in physics when math techniques becomes to complicated to use. Maybe it's our lack of intuitively understanding these equations that is preventing us to find a solution to them. For example the equation for simple harmonic oscillator can be solved without now how to solve 2nd order DE and characteristic equations because have a intuitive feel of the motion.

 

[Feodalherren]

It's easier to talk about math in terms of representation, constraints, and transformations. These three things underly all of mathematics including analysis, algebra, logic, probability, topology and so on.

Finding patterns can be seen in terms of the above things: patterns are discernable by the representation used to describe something. The more compact a representation is, the easier it will be to discern a pattern.

You also have to remember that you can decompose something in many ways, and a decomposition is a transformation. Each decomposition will tell you something specific to the context of that decomposition.

By taking a large system and reducing it to descriptions of lower descriptive complexity, you are finding common patterns. Scientists and mathematicians talk about beauty being simple, and this is one way of understanding that statement.

I suppose that does make sense. I must admit, the notion that physics and maths are as different as you claim sounds alien to me. Don't get me wrong, I believe you, but it's hard for me to grasp. So far there seems to be little difference.

 

[chiro]

I suppose that does make sense. I must admit, the notion that physics and maths are as different as you claim sounds alien to me. Don't get me wrong, I believe you, but it's hard for me to grasp. So far there seems to be little difference.

It's more to do with the focus itself of the two disciplines than them being different.

Mathematicians focus on different things than physicists do and as a result of this, the context is different.

Mathematicians like generality, physicists and other scientists like specificity. Mathematicians focus on situations corresponding to any reality, physicists focus on this one (look up Platonic viewpoint for more information).

It's not just that mathematicians care more about proofs or formality that makes them different: it's just the focus which determines how one particular person looks at the world.

It's the kind of the thing where if you got five random people to say what they thought something was without any of them having any kind of serious exposure (and the five were from completely different backgrounds in stark contrast to each other), you would see things in all of the candidates that would be also in contrast in all likelihood.

A programmer might look at system in terms of algorithms, structure, design, flow and so on. An artist might use more visual or dynamic interpretations to understand something. A teacher may use cognitive knowledge and understanding gained through teaching experience.

They all have a completely different focus.

One great thing though, is that we are moving from isolated disciplines to an interdisciplinary approach to learning. It used to be that someone would study one or two main areas, but now people are starting to mix areas that were segregated previously together.

This is resulting in forms of thinking that only polymaths could do, but because it is becoming a lot more widespread, and because of the availibility of both raw and processed information (like say the internet or big university libraries), this is becoming a common thing.

 

[Jorriss]

I suppose that does make sense. I must admit, the notion that physics and maths are as different as you claim sounds alien to me. Don't get me wrong, I believe you, but it's hard for me to grasp. So far there seems to be little difference.

I'm a bit surprised. I wonder what an intro course could of been like to think physics is just math.

 

[twofish-quant]

If the solutions are known for specific cases what is preventing them to be known for the general case?

The equations are non-linear. That means that you can have a solution for scenario A. A solution for scenario B, but when you mix the two, you end up with behavior that is completely different from the two scenarios. In other words, because the equations are non-linear, you can't break up the equations.

This isn't a problem just with GR equations. You are going to run into non-linear PDE's all over the place in physics.

Are the equations chaotic in nature?

Under some situations. Yes.

Also there is the issue of what "chaos" means in GR...

http://arxiv.org/pdf/gr-qc/9612017v1.pdf

 

[twofish-quant]

Instead I prefer to use physics and intuition to guess the solution to the DE.This usually works in physics when math techniques becomes to complicated to use.

That works some times, but the trouble comes in if you are in a situation in which you have no real physical intuition. For example, since I don't run into black holes in daily life, I don't have any physical intuition as to how black holes behave. Same thing with electrons. If I try to use intuition to figure out how electrons behave based on things that I see in my daily life, I'll get it wrong.

Also something that people often do is to take the equations, and try to "create" physical intuition. I've stared at the equations for stellar evolution and the equations for cosmology long enough so that I have a "gut feeling" for how those equations behave.

Maybe it's our lack of intuitively understanding these equations that is preventing us to find a solution to them.

Sure. If we had some black holes nearby we could play with them, but we don't.

 

[nucl34rgg]

The best advice I can give is to make sure you know the math really well before attempting some of the harder topics. Otherwise it is just that much more confusing.

 

[amiras]

The best advice I can give is to make sure you know the math really well before attempting some of the harder topics. Otherwise it is just that much more confusing.

What topics in physics are considered "harder" ?

 

[chill_factor]

Not really. I don't know where you did your basic physics but it was much more than that for us. You can plug plenty of stuff into that equation, especially when you are supposed to show why something happens mathematically.
At any rate, that's besides the point. My argument was that a physicist needs to be good at math like a novelist needs to be a good linguist.

I think the first question is irrelevant but if you must know I did it at a University of California campus.

No, it is NOT much more than that. There is absolutely no math in basic physics that you haven't seen before unless you are behind on math. What you said about being able to plug many things in is the science being hard. Executing the calculation itself (what you think is "math") is not hard in basic physics, in my opinion.

Now, it might be hard in upper division classes, but that's not math, that's the arithmetic being hard. 5.598866*e^0.74795 is analytically very hard to solve; try it without a calculator. But that's arithmetic, not math. Math is about proofs and logic.

Even upper division quantum mechanics has only basic linear algebra and basic calculus as absolutely necessary to solve problems, the rest you should be able to pick up in the class itself. There may be arithmetic manipulations that are hard, but that's not math. Math is about proofs and logic.

Also, your understanding of physics and math is mostly around fundamental theoretical physics, but that's not what 99% of physicists do. Outside of fundamental theoretical physics, such as in applied physics, they have nothing in common. Just as an novelist doesn't care about the theory of linguistics, syntax and patterns in languages...

How much math is in this physics research article? http://arxiv.org/pdf/1207.0895.pdf

 

[chill_factor]

The best advice I can give is to make sure you know the math really well before attempting some of the harder topics. Otherwise it is just that much more confusing.

I think you have this spot on:

You don't need to know too much advanced math for most courses in physics.

But you must MASTER basic math, and if you have mastered basic math, everything else will fall into place.