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Schwarzschild Metric with multiple masses

  1. Jan 8, 2006 #1
    I know that it's possible to calculate the rate at which time flows when in the gravitational field of a single spherical mass.

    But how do you calculate the rate when there are two masses or more? How do they add together?
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  3. Jan 8, 2006 #2

    George Jones

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    In order to calculate this, you need to know the metric that is the solution to Einstein's equation for this particular physical situation.

  4. Jan 8, 2006 #3


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    Since Einstein's Equations are nonlinear PDEs, one may not be able to superpose solutions exactly. Some approximate model will probably have to be used.
  5. Jan 8, 2006 #4


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    Exact solutions are very difficult. The PPN approximation, which is valid only in weak fields/low velocities gives the metric coefficient g_00 = 1-2U (in geometric units), where U is the Newtonian potential energy. This means that the time dilation factor is sqrt(1-2U), which is equal to 1-U in the region where the approximation is valid (U << 1).

    Note that gravity in the solar system can be considered to be "weak field".

    By "time dilation factor" I mean a number less than 1, i.e. a time dilation factor of .5 means that a clock at that location run half - fast (though such a large time dilation factor would be outside the region where the PPN approximation worked well).

    U is the [correction] negative of the newtonian potential energy / unit mass (U is always positive, the energy is always negative) which is dimensionless when c=1 (i.e when one is using geometric units). U must be zero at infinity.

    So roughly speaking, if one is at a distance r1 from mass m1 and a distance r2 from mass m2 in a weak field

    U = m1/r1 + m2/r2 (in geometric units). (Note the sign correction).

    In non geometric units, one would write the time dilation factor (defined in the same way) as

    1 - G*m1/(r1*c^2) - G*m2/(r2*c^2)
    Last edited: Jan 8, 2006
  6. Jan 8, 2006 #5
    Thank you for your answers. That helps a lot.
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