Depth of gravitational well within our local Virgo supercluster?

Question

For simplicity we could imagine standing at the north pole so we do not have to care about how fast we are moving in relation to the gravitational field of the earth.

Now calculating how much faster our clock will tick if the earth suddenly "magically disappeared" is in principle a simple calculation exercise. The same thing goes for the Sun.

Now if we take things one step further, things get more difficult. Calculating how deep we are in the combined gravitational field of all the massive objects of the Milky way is not a simple task.

In order to ask a well defined question:

1. How deep in the gravitational well of all the objects of the Milky Way are we and what can be said about other locations within our local Virgo Supercluster? Are there large scale inhomogeneities so that for instance clocks are ticking slower near the center of the Virgo supercluster?

Answer 1

The size of the relative gravitational time dilation effect (when it is small), compared to a clock at infinity, is $\sim \Delta \phi/c^2$, where $\Delta \phi$ is the change in potential

You can then get an approximate order of magnitude effect for any situation you want to consider.

e.g. For the Sun's position in the Milky way, an approximate value for the gravitational potential is minus a few $10^{11}$ J/kg - leading to a time dilation effect of a few parts in $10^6$. i.e. Clocks run slower at the Sun's position in the Milky Way by a few parts in $10^6$ compared with a clock at infinity (where infinity just means outside the Milky Way potential). The size of the effect increases as one moves closer to the galactic centre, up to a maximum of a few parts in $10^5$, though if you move arbitrarily close to the central black hole, then the time dilation can become arbitrarily large.

Gravitational potential is of course additive, so if you want to compare the clock rate at the Sun's position in the Milky Way with the clock rate in some cosmic void, then you would have to add the potential at the Milky Way's position in its local cluster and supercluster. Some idea of the maximum size of this effect could come from using the gravitational potential at the centre of a uniform sphere of mass $M$ and radius $R$, which is $-3GM/2R$. The numerical factor of $3/2$ will change a little bit for different mass distributions.

The total mass and radius of the Virgo supercluster are $\sim 10^{15} M_\odot$ and $\sim 15$ Mpc respectively. Thus the potential at the centre is also a few $10^{11}$ J/kg.

Thus the additive affects of the Sun being part of the Milky Way and the Virgo supercluster add up to no more than about 1 part in $10^5$ in terms of time dilation (compared with a clock running somewhere in a "void").