Einstein's field equations

Discussion in 'Physics & Math' started by tashja, Jan 5, 2011.

  1. tashja Registered Senior Member

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    Hi!

    I've been reading popular books about Einstein, relativity, etc, but even though they are for non-scientists, I don't understand what 'field equations' are. They say that some 'solutions' are black holes, while others give the possibility of something else. I just don't understand this. How can an equation have many solutions? How many solutions have been discovered? Are there many more to be discovered? I just cannot grasp this, please help me understand.
     
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  3. kurros Registered Senior Member

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    Field equations are equations which describe the structure and dynamics of fields. Einstein's describe the gravitational field. It is a bit confusing perhaps because the Einstein field equations describe gravity in terms of the shape of spacetime, so there isn't actually a field embedded in some space like everyone used to think, and like there is for say the electromagnetic field. You could say that the name is due to a relic of terminology from the days when gravity was considered a field. Some people still prefer this point of view though so take your pick.

    The Einstein field equations tell you what shape space you end up with if you supply them with a distribution of matter and energy. I.e. start with some matter, turn the crank (solve the equations) and you end up with mathematical objects which tell you the shape of spacetime. Since there are infinitely many distributions of matter that you can start with there are infinitely many solutions. Some of these are black holes, as you mention.

    To give you a simple example of an equation that has multiple solutions, consider
    \(0=(x-1)(x-2)\)
    Solving for \(x\), there are two solutions, \(x=1\) and \(x=2\).

    I guess you are not familiar with calculus, but consider that Newtons laws describe the dynamics of every day objects. Take a ball and throw it. Newtons laws describe the balls trajectory based on the launch angle and velocity, and the forces acting on it (i.e. gravity

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    ). Since you can throw the ball will infinitely many different angles and velocities there are infinitely many possible trajectories, and these are all solutions of the ball's equations of motion, i.e. Newtons laws. Lots of solutions are very similar to each other though so often people might describe a whole class of solutions that differ only by tweaking some parameters as the same solution, or a more general kind of solution if you like.
     
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  5. tashja Registered Senior Member

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    I'm sorry, Kurros, I don't understand :-(
     
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  7. tashja Registered Senior Member

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    Ok, what I don't understand is why, after writing those equations, he didn't find the solutions for them? Why did other physicists had to come along and finish them? Like for example, Schwartzschild or Godel. You would think that him been the person that wrote the equations, he would be the best person to find the solutions, no?
     
  8. Lady Historica Banned Banned

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    Mathmatics can be thought of the human brain child in which we evolve along with our use for the principals. Now did anchient astronomers know how to calculate how fast a star was moving from its point in the sky. Maybe/ maybe not, but they had all the simple principals necessary for the task. Which brings us to why other mathmaticians are left to solve problems. In some instances it comes down to an "if its not broken then don't fix it" and other expansions could be thought of as seemless integrations of one field of math into another. Often giving a more simple understanding.
     
  9. arfa brane call me arf Valued Senior Member

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    He did. But
    The "trick" is to find meaningful solutions; this is something a lot of people are still working on. His theories didn't get to be mainstream until the late 1950's. They were considered interesting up until then, but too difficult to work with (or understand). Physicists were still waking up to light being bent around the sun's gravitational field.

    Einstein understood that his were only a part of a larger group of theories, which at the time he had no concrete idea of. And today there are a lot more ideas--lots of research has been done in the meantime; his theories have been developed, extended and there is a recognition that, whatever gravity is, a new theory that subsumes Eintein's will have to be in terms of quantum theories (perhaps again, new ones).

    Einstein's ideas have been pretty successful, nonetheless, despite the incomplete picture.
     
  10. tashja Registered Senior Member

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    715
    Is there like a computer out there looking for 'meaningful' solutions to this equations 24/7 in some lab?
     
  11. arfa brane call me arf Valued Senior Member

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    I wouldn't be in the least surprised.

    Actually, computer simulations and models are quite a rigorous way to test ideas.
     
  12. kurros Registered Senior Member

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    Short answer: it is really really hard to do. They are just really nasty equations. Oh sure they are beautifully simply from an aesthetic point of view but they are a bitch to solve. Computers can solve them approximately for some useful scenarios but you learn a lot more about the structure of the equations and how they work by finding "analytical" solutions, i.e. exact ones like the Schwarzschild solution. Even onces you have found these spacetimes though there is still a lot more work to be done in analysing the solutions and seeing what kinds of interesting things happen in them.

    Here is an analogy. Say you were designing a device to transmit some radio waves. To see what radio waves you end up with you stick your input arrangement of electrical currents into Maxwells equations and solve them to find the electromagnetic field that those currents create. It doesn't do you much good to just keep sticking in random arrangements of electric currents and seeing the interesting fields they create, you need a bit more purpose guiding you, i.e. maybe you see a transmitter on top of a building and want to see what kind of field it emits, so you take that as your input.

    Similarly with the Einstein equations you might start with something you are physically interested in, say the Earth's gravitational field. So, you stick in a big sphere of matter with some density distribution you imagine the earth has and solve the equations to find out how spacetime looks around the Earth.
    It is a little more complicated than that but that is something like how it goes.
     
  13. tashja Registered Senior Member

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    Do you know of any recent physicist that has found a new solution for these equations?
     
  14. kurros Registered Senior Member

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    793
    Not off the top of my head, but it is an active area of research. New exact solutions don't come along all that often I think (although a quick search found me this paper: New exact solutions of the Einstein vacuum equations for rotating axially symmetric fields - Ahmad T Ali 2009 Phys. Scr. 79 035006 )
    On the other hand there are lots of people doing numerical calculations for general relativity all the time, trying to figure out how stars, neutron stars, blacks hole etc form, what happens when black holes collide, how the jets from quasars are generated, all sorts of things regarding the evolution of the universe etc.

    I don't do anything in that area though so I have only a superficial understanding of what those researchers are really doing with their time

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    .
     
  15. AlphaNumeric Fully ionized Registered Senior Member

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    A particular equation for particular boundary conditions might (and often does) have only one solution but the same equation with different boundary conditions has different solutions.

    A similarly complicated but more familiar system to GR is fluid mechanics. The Navier-Stokes equations govern fluid motions, be it water in a pipe, exhaust in a rocket, blood through your body, all the same equation but with different boundary conditions. Water in a pipe is in a cylinder with smooth walls, rockets have nozzles and are extremely hot, your blood vessels can bend and flex and your heart beats to push the fluid (unlike say water pipes which don't have a 'heart beat' to them). For each system you have to solve the same sort of equation but the systems define different boundary conditions, which can vastly alter the results.

    Solving them on a computer can be done and is big business but its still very very difficult. To model aircraft you have to split up the air around the wings into billions of different regions and then work out how they all interact with one another, which you have to do millions of times to get how they change in time. This is super-computer territory, just to do one aircraft for one simulation.

    GR systems are worst. Each system has a different distribution of mass and energy, which affects space curvature, which in turn affects the mass and energy.

    New solutions in particular cases of GR are found all the time but the issue is how relevant they are. For instance, any space with vanishing Ricci tensor is a solution to the Einstein field equations so every Calabi-Yau is a valid GR space and there's infinitely many. The issue is finding the solution for a distribution of mass and energy we know is relevant to physics. We know the space-times for a single star or planet but we don't know for 2 stars orbiting one another, because they each alter the other's space curvature. Even NASA supercomputer simulations have limited success in this stuff.
     
  16. tashja Registered Senior Member

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    If a physicist is devoted to this area, can he/she be called a 'relativist theorist?' and can you cite a modern example of one? Also, is there a solution in Einstein's equations that give out, for example, spiral or elliptical galaxies? like for example, after solving so-and-so equation, the space-time and matter in that region of the universe will form an elliptical or an spiral galaxy. I will also like to see, if you can, what the solution for space-time around the Earth looks like in its mathematical form.
     
  17. temur man of no words Registered Senior Member

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    Called "relativist", as in "numerical relativist", or "mathematical relativist". If you at a math department working on GR you most likely are a geometric analyst, a mathematical physicist, or a PDE-person.

    For galaxies and stuff you would not need full GR, I think some kind of approximation would suffice. But you need to do better than Newton because at least the vast size of galaxies will make the instantaneous interaction approximation useless. The Earth's gravitational field can be approximated by an exterior Kerr solution. Recent missions such as Gravity Probe did measurements of the Earth's gravitational field to test against the theory, which is a perturbation (due to Earth's irregular shape etc) of the Kerr solution.

    In general there are very powerful techniques that give you detailed information about the behavior of GR in generic regimes, meaning that you study a whole lot (much more than exact solutions with a few parameters) of solutions together and say precise statements about their behavior. Perhaps the "simplest" kind in this category is perturbation analysis where you study solutions close to a family of exact solutions, or close to having a nice symmetry. If you want to study things globally, meaning that you want near-full understanding of the Einstein field equations, there are a very few rigorous results, and a lot of conjectures, and moving towards proving these conjectures is at the forefront of mathematical relativity research today. some of the successes are the Penrose-Hawking theorems on singularity formation, the Christodoulou-Klainerman proof of the nonlinear stability of Minkowski spacetime, and Christodoulou's proof of black hole formation out of gravitational waves in vacuum. One long term goal is the so-called final state conjecture, which says that if you start with an arbitrary state of gravitational field in vacuum then after black hole forming, radiating gravitational waves, and doing what it supposed to do, in the end the universe will consist of a few Kerr black holes moving with constant velocity infinitely far from each other.
     
  18. Guest254 Valued Senior Member

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    Excellent post.
     
  19. tashja Registered Senior Member

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    715
    Super! Thanks you guys!
     
  20. D H Some other guy Valued Senior Member

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    A similar thing happens in fluid flow. It's called turbulence.

    Regarding which is hairier: The Clay Math Institute made making "substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations" one of their million dollar millennial questions. To win the prize you only have to address a simplified form of the Navier-Stokes equation, dealing with incompressible fluids only. You don't have to solve the problem, either. You just need to show a smooth solution always exists for all smooth, divergent-free initial conditions or show that some smooth, divergent-free initial condition has no solution.
     
  21. tashja Registered Senior Member

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    There's a chapter in the book I'm currently reading that deals with the Higgs and how it is suppose to give mass to particles and stuff, but it doesn't go into specifics on how it does it. The book says that the Higgs was present shortly after the BB, and that it gave mass to particles. Question: Is the Higgs constantly sustaining the mass of particles like everyday, or it gave particles their mass at the beginning of the universe and now it's doing nothing?
     
  22. AlphaNumeric Fully ionized Registered Senior Member

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    6,702
    It is presently the bane of my existence. That and Mathematica's dislike of being told what to do.

    I feel I should clarify a bit. Both GR and the N-S equations have the whole "The solution affects the conditions which affects the solution" back reaction issue which makes them extremely complex but the reason I said GR was that I qualitatively think of a GR system as any 'normal' system plus the issue of altering space-time. For instance, you could embed a fluid into space-time and then you have the double issue, solving the N-S equations in a background altered by the fluid. The metric would then depend on the flow, which depends on the metric etc. Putting QFT into a gravitational background is extremely difficult when compared to QFT or GR separately and they are already tough as hell. Having got and read bits of the books by Bachelor and Lamb recently its clear that fluid mechanics was to the 19th century what GR was to the 20th century and with many analogous problems.

    As for the N-S equations themselves, I have enough issue numerically solving inviscid incompressible low Reynolds number (insert additional simplifications here) systems. I actually find string theory more accessible, there's so many radically different parts to it, with so many avenues of research open that you can find a bit to suit your mathematical or conceptual preferences. I'm in no doubt there's plenty of different things related to fluid mechanics mathematical physics can do but that's just the knee jerk impression I get.

    By 'present shortly after the BB' it means that very early on in the universe's life there was so much energy packed into such a small space that such particles could appear easily via the \(E=mc^{2}\) relation. After the universe expanded a bit (and thus cooled) there wasn't enough energy in a small enough region flying about to make Higg bosons easily.

    In the same way that the photon couples to electrons mean the electrons are 'charged' the Higgs coupling to particles means they have mass, ie the electron couples to the Higgs also. If the photon coupling turns off then the electron becomes neutral and if the Higgs coupling turns off then the electron would become massless, hence the coupling is 'on' all the time.

    The coupling is on all the time despite no Higgs particles floating about for us to see naturally because the couplings act via virtual particles, which can flitter in and out of existence without needing to put the energy in directly, thanks to things like the uncertainty principle.
     
  23. tashja Registered Senior Member

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    715
    Got it!

    The book also mentions magnetic monopoles, so a few questions: To which family of particles does the magnetic monopole belong to? What would be its mass? Could magnetic monopoles come together and form some kind of visible matter?
     

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