How Are Fluid Solutions Used in Astrophysics and Cosmology?

[shounakbhatta]

I was reading wikipedia and came across the following statement:

"In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid.
In astrophysics, fluid solutions are often employed as stellar models. (It might help to think of a perfect gas as a special case of a perfect fluid.) In cosmology, fluid solutions are often used as cosmological models."

Can anyone please explain me the details.

Thanks,

-- Shounak

 

[ImaLooser]

I was reading wikipedia and came across the following statement:

"In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid.
In astrophysics, fluid solutions are often employed as stellar models. (It might help to think of a perfect gas as a special case of a perfect fluid.) In cosmology, fluid solutions are often used as cosmological models."

Can anyone please explain me the details.

Thanks,

-- Shounak

Astrophysicist are quite loose with the words gas, fluid, and plasma and use them interchangeably. They are talking about gas/plasma clouds here but using the word fluid. That's just what they do. Get used to it.

 

[shounakbhatta]

Hello,

Thanks for the reply. I was actually trying to get over a conceptual problem.

As you have told they are talking about gas/plasma clouds, then the fluid solution which is found in the Einstein equation producing the stress density fluid, which the astrophysicist use for studying stellar models, how they are related?

If you can please help me to understand the concept.

-- Shounak

 

[Chronos]

It has been the custom for over a century to model cosmological processes using the laws of fluid dynamics. Enormous clouds of gas, dust, and/or plasma are ideally suited for analysis using the Navier-Stokes equations. The relativistic version of Navier-Stokes is commonly used in cosmology to analyze these processes under high energy conditions. This paper may help shed some light on the subject - Relativistic Fluid Dynamics: Physics for Many Different Scales http://arxiv.org/abs/gr-qc/0605010

 

[shounakbhatta]

Hello,

Thank you very much for the answer. Your details as well as the details in the paper is helping me to make my concept clear. I am not too much into maths, but the relationship of relativistic fluid and general relativity is slowly clearing up in my mind. If I have any conceptual query can I ask you some questions?

Thanks,

-- Shounak

 

[Chronos]

Everyone struggles with this stuff, it is not easy.

 

[twofish-quant]

Thank you very much for the answer. Your details as well as the details in the paper is helping me to make my concept clear. I am not too much into maths, but the relationship of relativistic fluid and general relativity is slowly clearing up in my mind. If I have any conceptual query can I ask you some questions?

Getting into the math can actually make things easier. The reason that "fluids" appear all over the place is that the fluid equations say in mathematical language the following things...

1) things move around
2) mass is conserved
3) energy is conserved
4) momentum is conserved

In GR, mass, energy, and momentum is combined into one "thing" called the stress energy tensor. You start with the idea that this "thing" is conserved, and that it "curves" space, and you get GR.

 

[shounakbhatta]

Hello,

Thank you very much. This has been a fantastic reply. Special thanks to twofish-quant. Actually for me study of GR is a childhood dream that I am pursuing. For me, sometimes the maths part gets too difficult to handle, hence, I try to get over the conceptual part. As you have written, a little bit help on the conceptual part, helps me to mend the threads of though which connects me well to the mathematical part. Initially, how fluid is connected to GR and the stellar models became a bit difficult to understand. But now it is ok.

Thank you all for the wonderful help.

-- Shounak

 

[Chronos]

Linear algebra is probably the most valuable skill you can have in your toolbox for making sense out of fluid dynamics.

 

[twofish-quant]

Actually for me study of GR is a childhood dream that I am pursuing. For me, sometimes the maths part gets too difficult to handle, hence, I try to get over the conceptual part.

The math part of full GR is too difficult for most scientists to handle. What often happens is that someone who is a specialist in GR spends a few months figuring out the solution for one part of GR, and then people use that solution. Even the specialists don't completely understand GR, and there are a lot of open questions.

On the other hand, a basic understanding of GR is something that most college physics majors can have without too much trouble. One thing that makes it more difficult than it has to be is that people don't know exactly the best way to explain GR.

One irony is that most of my understanding of GR comes from knowing when to avoid GR. My background is supernova simulations. You have a limited amount of computer CPU, and if you can get a good simulation using Newtonian gravity you use that. It turns out that you can show through some basic algebra that the effects of GR aren't important for supernova.

And then there is the question of whether GR is in fact the right theory of gravity. One thing that happens is that GR is the standard comparison theory. For example, if you come up with a different theory of gravity (and there are hundreds), what you do when you write the paper is to compare your gravity theory with GR. This is useful because if your theory is very different for things within the solar system, then it's wrong.

Initially, how fluid is connected to GR and the stellar models became a bit difficult to understand. But now it is ok.

The one bit of math that you have to master is partial differential equations. This is usually a sophomore course in college for physics majors.

Imagine a pond of water. Now drop a rock in it. The rock creates a force. Where the force is high the water gets pushed away faster. Where it is low, the water doesn't move as much.

Now put numbers to all of this, and try to calculate what happens. You end up with a page full of greek letters. It takes about three months for a college sophomore or junior to understand the techniques to do the calculations.

Imagine a ball. It's round. How do you say "round" with numbers. Another three months of learning greek letters.

 

[twofish-quant]

Also some things that seem hard have easy math, and some things that seem easy have hard math.

For example, the entire universe. It turns out that if you make a key assumption, that the equations for the whole universe is an "easy" equation. The assumption is that on the average, all parts of the universe are the same. Once you assume that (and it looks to be a good assumption), then the calculation to do the whole universe is about five lines. The reason it's easy is that you are calculating just one number and how it changes with time.

Now light a match. Blow out the fire and look at the smoke. That's a painfully difficult equation to solve.

 

[twofish-quant]

Also some things that seem hard have easy math, and some things that seem easy have hard math.

For example, the entire universe. It turns out that if you make a key assumption, that the equations for the whole universe is an "easy" equation. The assumption is that on the average, all parts of the universe are the same. Once you assume that (and it looks to be a good assumption), then the calculation to do the whole universe is about five lines. The reason it's easy is that you are calculating just one number and how it changes with time.

Now light a match. Blow out the fire and look at the smoke. That's a painfully difficult equation to solve.

 

[shounakbhatta]

Hello,

Thank you very much twofish-quant. You have explained some very complex things, so simply that...Well, if somebody gets one like u who can explain conceptual things so easily, it wud be great. Well, yes, I am going over linear algebra and vector operations. Soon wud be going deep with calculus. Actually, while dealing with calculus like I want to understand the basic crux, u can call it philosophical implications. Like differentiation...What actually it is?

Well, I might bother u with certain concepts in GR. I am on with tensor calculus right now and slowly going thru it. In GR I find certain concepts, for which first I need to understand what actually it is, but hardly get anyone to explain.

But ur above explanations seem to ease out might path, especially areas in gravity and that there are other theories.

Thank u so much. Will be bothering u in future.

Thanks a lot.

-- Shounak

 

[codex34]

So regarding fluids such as plasma/gas/dust clouds, are the 'particulates' generally considered to be perfect? (to make the maths easier).

I've been finding I can't get solutions to certain things without using imperfect non-uniform dust, by using spherically symmetric dust I run into gridlock, as the forces acting on the dust directly oppose each other.

 

[twofish-quant]

You have explained some very complex things, so simply that...

Well there is a reason for this. In the past, I've had a lot of situations when I didn't understand something, and I always promise myself that if I figure out something, that I'll remember the explanation so that I can explain it to a younger me.

Actually, while dealing with calculus like I want to understand the basic crux, u can call it philosophical implications. Like differentiation...What actually it is?

Well, asking that question will send you down a path of very deep philosophy and controversy. You could spend an entire lifetime with that question.

That's an interesting path, but not the one that I've gone on. My philosophy regards this sort of math as "mental tools". I don't think too deeply about deep mathematical philosophy, not because it's unimportant, but because I got sucked up by other mysteries.

Well, I might bother u with certain concepts in GR. I am on with tensor calculus right now and slowly going thru it. In GR I find certain concepts, for which first I need to understand what actually it is, but hardly get anyone to explain.

Yup. Been there myself. What I ended up doing was that when I finally figured out an explanation that made sense, I just made a note of it.

One thing is that mathematicians like to make things as abstract as possible. Which is great for doing mathematics. However, I think in terms of physical objects. So when I learn tensor calculus, it's easier for me to start with a sailboat.

 

[twofish-quant]

So regarding fluids such as plasma/gas/dust clouds, are the 'particulates' generally considered to be perfect? (to make the maths easier).

What happens is that in some situations, the details of the interactions between the particles doesn't matter much. It's like an ideal gas. If you have a gas that is very hot or very low density, things behave very much like an ideal gas. When things start getting dense, then the details of the interactions matter.

There's no "general." Part of a physics argument is to start up with "in this situation, we can assume that this does/doesn't matter."

I've been finding I can't get solutions to certain things without using imperfect non-uniform dust, by using spherically symmetric dust I run into gridlock, as the forces acting on the dust directly oppose each other.

Part of the physics is to know when you can use what approximation.

 

[Chronos]

So regarding fluids such as plasma/gas/dust clouds, are the 'particulates' generally considered to be perfect? (to make the maths easier).

I've been finding I can't get solutions to certain things without using imperfect non-uniform dust, by using spherically symmetric dust I run into gridlock, as the forces acting on the dust directly oppose each other.

An ideal gas is one in which all collisions between atoms, or molecules, are perfectly elastic. Of course, there is no such thing in the real universe, but, it is a very close approximation for the kind of environments that exist in space. re: http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html

 

[codex34]

Thanks for the answers guys.

Part of the physics is to know when you can use what approximation.

...and what assumptions to use/ignore.

Would a rough approximation to shape be to use a random vector, generated at each interval for each point, to represent any deflection caused by surface interactions?
The argument being that the data set was random to begin with and any actual calculation of real shape would be essentially arbitrary.

 

[shounakbhatta]

Yes, that is what I have been doing twofish. Last night, I was working on fluid dynamics, Newtonian fluid, perfect fluid and how the Newtonian stress-energy tensor through calculation became a 2nd.order tensor. After that the fluid solution, how it can be related to the centre of the stars...Got stuck up... In the dawn felt like my eyes are tiring, went to sleep...

Well, yes, taking a bit of notes, really helps.

 

[shounakbhatta]

Then again I sat down in the morning studying Descartes on the Discourses and Method, well it relates to each other, maths, physics and philosophy...