Schwarzschild Metric with multiple masses

[DiamondGeezer]

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?

 

[George Jones]

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

Regards,
George

 

[robphy]

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.

 

[pervect]

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)

 

[DiamondGeezer]

Thank you for your answers. That helps a lot.