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Kerr-Newman Metric Tensor

  1. Oct 12, 2013 #1
    Our galaxy is rotating and is charged therefore the choice for the metric is the Kerr-Newman Metric.

    I want to solve for the Kerr-Newman Metric Tensor.

    There are a few questions.

    1-What is the value for Q in the equation:
    ##r_Q^2=\frac{Q^2*G}{4*\pi*\epsilon_0*c^4}##
    where
    ##G=6.674E-20 \frac{km^3}{kg*s^2}##
    ##c=299792.458 \frac{km}{s}##
    ##\epsilon_0=8.8541878E-9 \frac{F}{km}##
    Charge of an Electron ##e=1.602E-19 C## if needed for Q?

    2-How do you calculate and position the Angular Momentum of the Milky Way Galaxy?


    ##J=r \times (m*v)## I know r and v are just part of a list not vectors.
    ##a=\frac{J}{M*c}##
     
    Last edited: Oct 12, 2013
  2. jcsd
  3. Oct 12, 2013 #2

    Bill_K

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    No, the Kerr-Newman metric is the metric of a charged, rotating BLACK HOLE. The Milky Way does not even remotely fit that description. There are many other solutions (an infinite number) having mass, charge and angular momentum.
     
  4. Oct 12, 2013 #3
    Could you please give me an example of a Kerr-Newman Black Hole.

    Hopefully with a Q and a J.

    Is the Milky Way Galaxy only rotating?
     
    Last edited: Oct 12, 2013
  5. Oct 12, 2013 #4

    WannabeNewton

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    The formulas you listed for ##Q## and ##J## are not even remotely correct. Just out of curiosity (based on your various posts) have you used an actual GR text before?
     
  6. Oct 12, 2013 #5
    I went thru a Undergrad GR lecture series but never read a full Grad School GR text.

    Any recommendations?
     
  7. Oct 12, 2013 #6

    fzero

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    A black hole, as a mathematical solution to the Einstein equations, describes a situation where the source (contributing to the energy-momentum tensor) is concentrated at a single point. Therefore, the only parameters are mass, charge and spin. A galaxy, or even a star, is not an example of black hole, since these are composed of spatially distributed matter, with relative motion. The state of such a system is complex and rather than just specifying mass, charge and spin, one would need to specify the constituent masses, relative locations and relative velocities as initial conditions.

    We also would not expect an exact solution, since even the solution to Einstein's equations for two point masses has not been solved exactly. In the limit where one mass is much smaller than the other, ##m\ll M##, we can arrive at approximate solutions in an expansion in powers of ##m/M##, typically by computer. For many more masses, we would not expect it to be computationally feasible to track individual constituents. Instead, one could try an approximation where we treat the source as a smoothly distributed gas of particles. This is what is done in the cosmological solutions like FRW.

    A black hole solution can be considered as an approximation to a star or galaxy in the limit where we are very far from the extended object, so that treating it as a point source is appropriate. The approximation will break down as we get nearer to the object and the internal structure becomes important.

    It is nevertheless useful to study the black hole solutions. The book Introducing Einstein's Relativity by D'Inverno is a somewhat gentle introduction that discusses charged and rotating black holes separately, i.e. Reissner-Nordstrom and Kerr, but not Kerr-Newman. Wald's General Relativity is an excellent graduate-level text that discusses the features of Kerr-Newman, but does not actually derive the solution. It is probably a worthwhile exercise to derive it, by putting together the ingredients that go into the Reissner-Nordstrom and Kerr solutions.
     
  8. Oct 13, 2013 #7

    Bill_K

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    First of all, the idea of modeling the gravitational field of the Milky Way using General Relativity is quite pointless, since the field is weak and Newtonian gravity is adequate. Nevertheless...

    Not a fair comparison. What makes the field of two point masses difficult to solve is that it is time-dependent and involves the emission of gravitational radiation. To a good approximation the field of the Milky Way is axially symmetric and time-independent, and much easier to model.

    Or a continuous mass distribution in a spatially limited region.

    As I said above, the black hole solutions are not the end of the story. There are an infinity of known exact solutions to the vacuum field equations which are time-independent and axially symmetric.

    Multipole expansions are useful in the far-field limit, and one may continue in the same spirit. A Kerr black hole has mass M, angular momentum J = Ma, quadrupole moment Ma2, and so on. All the higher moments are related, and given in terms of just the two parameters M and a.

    But one can also write down exact solutions for any prescribed set of multipole moments. Unlike the field of a black hole, the near field of these solutions contain naked singularities, but in a realistic model these would be covered by the matter distribution of the source.
     
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