# Schwarzschild Metric with multiple masses

1. Jan 8, 2006

### 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?

2. Jan 8, 2006

### George Jones

Staff Emeritus
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

3. Jan 8, 2006

### 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.

4. Jan 8, 2006

### pervect

Staff Emeritus
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
5. Jan 8, 2006