[jeffreyH (OP)]

Can we determine, with respect to the centre of our galaxy, what the time dilation is on the surface of the earth. This would mean determining a value for the mass of the galaxy with the dark matter element included. Then calculating the potential that the earth is at. I was considering this with respect to external galaxies. From our perspective do we view the universe as running faster than it actually is due to our time being time dilated. This would skew our measurements of expansion.

[Kryptid]

The calculated escape velocity of the Milky Way around the area of the Solar System is 537 (+59/-43) kilometers per second according to this study: https://arxiv.org/abs/1309.4293v1. You could probably calculate local gravitational field strength from that value and then time dilation from that, right?

[evan_au]

Can we determine, with respect to the centre of our galaxy, what the time dilation is on the surface of the earth.

Not really.
The center of our galaxy is believed to harbour a supermassive black hole with a mass around 3 million times the mass of the Sun.
- At the event horizon of this black hole, the time dilation of the Earth would be zero.
- Inside the event horizon of this black hole, the question almost loses its meaning, since spacetime is so twisted in there, and you cannot communicate the result back to the Earth anyway.

Normally the reference location for time dilation would be away from any local masses, such as outside our local galaxy cluster. Kryptid implied this when he talked about "escape velocity of the Milky Way".

[Bored chemist]

I'm fairly sure that the answer to this "What is the mean time dilation at the surface of the earth" is zero.

because the time reference standard is set  at sea level.
"Scale interval is the SI second at mean sea level"
from
http://www.npl.co.uk/upload/pdf/20081203_t-f_whibberley_1.pdf

[jeffreyH (OP)]

Well I just didn't get my point across then.

[jeffreyH (OP)]

I have just modified the first post replacing the word universe with galaxy. Taking the mass of the universe and dark matter into account would be wrong.

[Bored chemist]

Most of the universe is too far away to influence the rate of passage of time.

The clocks on the GPS satellites would drift by 38 microseconds a day if they were in the earth's gravitational field, rather than in free fall. (Strictly, part of that difference- around 7 microseconds- is due to speed, and some- 45 microseconds- to gravity)
That's a pretty small effect (about half a part per billion), and it tells you how much difference the earth's gravity makes.

Near the Earth's surface the only significant  source of gravity is the Earth itself.
There's a small contribution from the sun + moon which gives rise to the tides. Technically, that must affect the speed of the clocks but I don't know if anyone has measured the effect.
Everything else in the universe  is too far away to make a measurable difference.

[Colin2B]

Most of the universe is too far away to influence the rate of passage of time.

@jeffreyH what @Bored chemist says is equivalent of taking the std gp calculation removing test particle to ∞. You could start with that as a working assumption which would give you a magnitude to work with.

[jeffreyH (OP)]

I'm going to have to think about this a bit more. I am taking in all the points made in the replies.

[Janus]

Can we determine, with respect to the centre of our galaxy, what the time dilation is on the surface of the earth. This would mean determining a value for the mass of the galaxy with the dark matter element included. Then calculating the potential that the earth is at. I was considering this with respect to external galaxies. From our perspective do we view the universe as running faster than it actually is due to our time being time dilated. This would skew our measurements of expansion.

For your intended purpose ( how it effects observations of distant galaxies), you don't want the Earth's gravitational potential with respect to the our galaxy's center but with respect to some far distant observer.   For that, you can get a rough estimate from our distance from the center of the galaxy and the mass of that part of the galaxy closer to the center than we are.  This comes out to a time dilation of 1 part in 624,000 due to gravitational potential.   However, this would be mitigated some by the fact that the Sun has a orbital velocity with respect to the Galaxy's center,  which accounts for about 1 part in 3.5 million. ( if we are viewing some distant point in the universe, we would see its clock run fast due to the relative gravitational potential, but run slow due to our  relative velocity difference.
So how would this effect our observation of the Universe?  Consider our nearest large neighbor, the Andromeda galaxy. We measure its radial velocity as being 301 ±1 km/sec.  Which means we can only measure it to ab accuracy of 1 part in 301.    In other words, the excepted error bar in our observation far exceed any correction required to account for time dilation.

[jeffreyH (OP)]

Thank you @Janus. More food for thought.