# Are distances in the Milky Way small enough to disregard GR?

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## Main Question or Discussion Point

For the purposes of distance measurement, time measurement of signals, and so on, can we approximate the distance across the Milky Way as an extended inertial reference frame? Or is that way out of bounds? (I mean obviously there are massive bodies that already prohibit that, I suspect, but say I wanted to use a light signal to measure the distance between me and a star across the galaxy, would I be justified in just assuming SR?)

If not, about how far away does the approximation of an inertial reference frame begin to wane, or is it just the mass/energy/pressure that's the problem, screwing up the metric? That is, if mass were negligible would an infinite inertial reference frame be a reasonable approximation?

Thanks.

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mfb
Mentor
The distances are large enough to ignore relativistic effects in most cases. Relativity becomes relevant if you have a lot of mass in a small space or if things move fast (the two often go together). It also becomes relevant again if you consider very large volumes (->cosmology), but that is way beyond the scale of a galaxy.
Relativistic effects are small if you are not close to massive objects within the galaxy.

I was referring specifically to the difficulty of defining distance. But I take it this is not a problem for objects within our galaxy?

Ibix
Strictly speaking you always have a more general version of the "distance in which frame?" problem familiar from SR. Always. And the underlying problem is a generalisation of the clock synchronisation issue - not only is there no unique way to synchronise clocks, sometimes there just plain isn't a way that isn't completely arbitrary. In SR, almost any sane procedure for synchronising inertial clocks gives you Einstein synchronisation. Not so in GR.

In a galaxy you can imagine scattering clocks around that "hover" at some point stationary relative to the galaxy's centre. Then you can imagine synchronising them by exchanging light pulses. As long as you draw big spheres around any black holes and keep your clocks and light pulses outside that, you aren't going to have significant synchronisation issues. So an inertial frame will do. Depending on how precise you want to be, of course - one physicist's "not significant" synchronisation issues is another's "unacceptable inaccuracy".

This is much like calling my lawn flat. It isn't, even to my eye, but it'll do for most purposes.

If you go up to cosmological distances, though, the expansion makes it difficult to define "at rest" in any large scale sense. So constructing (even notionally) a network of clocks "at rest" and interchanging light pulses becomes tricky. And the reason there is the large-scale curvature.

This is like trying to claim my whole country is flat. Galaxies are large enough that local curvature effects are like the grass on my lawn and small enough that cosmological curvature is like the curvature of the Earth. Both negligible for most purposes.

Strictly speaking you always have a more general version of the "distance in which frame?" problem familiar from SR. Always. And the underlying problem is a generalisation of the clock synchronisation issue - not only is there no unique way to synchronise clocks, sometimes there just plain isn't a way that isn't completely arbitrary. In SR, almost any sane procedure for synchronising inertial clocks gives you Einstein synchronisation. Not so in GR.

In a galaxy you can imagine scattering clocks around that "hover" at some point stationary relative to the galaxy's centre. Then you can imagine synchronising them by exchanging light pulses. As long as you draw big spheres around any black holes and keep your clocks and light pulses outside that, you aren't going to have significant synchronisation issues. So an inertial frame will do. Depending on how precise you want to be, of course - one physicist's "not significant" synchronisation issues is another's "unacceptable inaccuracy".

This is much like calling my lawn flat. It isn't, even to my eye, but it'll do for most purposes.

If you go up to cosmological distances, though, the expansion makes it difficult to define "at rest" in any large scale sense. So constructing (even notionally) a network of clocks "at rest" and interchanging light pulses becomes tricky. And the reason there is the large-scale curvature.

This is like trying to claim my whole country is flat. Galaxies are large enough that local curvature effects are like the grass on my lawn and small enough that cosmological curvature is like the curvature of the Earth. Both negligible for most purposes.
Thanks. Since we're talking about synchronisation, the same applies to time, right? I believe in cosmology the comoving coordinate system is the preferred one among astronomers, right? But over galactic distances, could I use SR to a reasonable level of accuracy for time as well as distance?

Ibix
But over galactic distances, could I use SR to a reasonable level of accuracy for time as well as distance?
Yes. So if you're writing a story about a colonisation mission to the other side of the galaxy, as long as our heroes don't get anywhere near a black hole standard SR will do.
I believe in cosmology the comoving coordinate system is the preferred one among astronomers, right?
Note that "preferred" in this case is preferred in the same sense as your local Earth's surface rest frame is preferred. It's a major factor in your daily life, but it's in no way physically special. Otherwise, yes.

pervect
Staff Emeritus
For the purposes of distance measurement, time measurement of signals, and so on, can we approximate the distance across the Milky Way as an extended inertial reference frame? Or is that way out of bounds? (I mean obviously there are massive bodies that already prohibit that, I suspect, but say I wanted to use a light signal to measure the distance between me and a star across the galaxy, would I be justified in just assuming SR?)

If not, about how far away does the approximation of an inertial reference frame begin to wane, or is it just the mass/energy/pressure that's the problem, screwing up the metric? That is, if mass were negligible would an infinite inertial reference frame be a reasonable approximation?

Thanks.
What sort of accuracy are you looking for, and what effects are you worried about? Are you concerned with local effects due to local massive objects, or just the expansion of the universe? As far as local objects go, the black hole at the center of the galaxy seems the most likely to cause problems. Some hint of the specific case you're interested in could be helpful to make sure you get a good answer.

What sort of accuracy are you looking for, and what effects are you worried about? Are you concerned with local effects due to local massive objects, or just the expansion of the universe? As far as local objects go, the black hole at the center of the galaxy seems the most likely to cause problems. Some hint of the specific case you're interested in could be helpful to make sure you get a good answer.
Mostly expansion of the universe and the idea that my approximate inertial reference frame does not extend indefinitely, although what that truly means I'm still beginning to learn (I'm assuming along the way the very way that distance is calculated will change due to spacetime curvature, rendering my neat little inertial coordinate chart useless). For what I'm considering, I have no plans of looking at light that goes near the center of the galaxy.

I'm thinking on this scale that the expansion of the universe is a non-issue, but I want to be sure of this.

Just to use light signals to measure distance to objects, or to determine a minimum age of an object based upon light signals coming from it (using standard syncing convention, of course). As in, presumably, since I can see the object, and if I know its distance, I can calculate the minimum age it has to be. Or if I bounce a light signal onto a distant object, I can find out how far away it is (although I'm assuming a different method than radar is used for that on that scale).

mfb
Mentor
I'm thinking on this scale that the expansion of the universe is a non-issue
It is a non-issue because it is not even there within the galaxy.
As in, presumably, since I can see the object, and if I know its distance, I can calculate the minimum age it has to be.
Astronomers usually refer to "now" as to the time where we see things here. As an example, the prominent supernova "in 1987" had its light arriving here in 1987. This is purely convention, of course.
Or if I bounce a light signal onto a distant object, I can find out how far away it is (although I'm assuming a different method than radar is used for that on that scale).
Radar works in the inner solar system and with the big outer planets, beyond that the reflected intensity is way too low. There is no signal we could send that would get reflected enough to see this reflection.

pervect
Staff Emeritus
Mostly expansion of the universe and the idea that my approximate inertial reference frame does not extend indefinitely, although what that truly means I'm still beginning to learn (I'm assuming along the way the very way that distance is calculated will change due to spacetime curvature, rendering my neat little inertial coordinate chart useless). For what I'm considering, I have no plans of looking at light that goes near the center of the galaxy.

I'm thinking on this scale that the expansion of the universe is a non-issue, but I want to be sure of this.

Just to use light signals to measure distance to objects, or to determine a minimum age of an object based upon light signals coming from it (using standard syncing convention, of course). As in, presumably, since I can see the object, and if I know its distance, I can calculate the minimum age it has to be. Or if I bounce a light signal onto a distant object, I can find out how far away it is (although I'm assuming a different method than radar is used for that on that scale).
Minimum age since the big bang?

To be really confident, alas, I think you need to do a GR analysis and show if it does or doesn't matter, and by how much. To be really confident, you probably have to do the analysis yourself, though you can gain some confidence by having someone you trust do the analysis.

Possibly relevant to the question of aging is this interesting thread on another forum: https://physics.stackexchange.com/questions/161453/why-isnt-the-center-of-the-galaxy-younger-than-the-outer-parts/161482#161482

Basically, the idea is that we can use linearized gravity to get a handle on the GR effects, and that what we need is the Newtonian potential $\phi$ of the galaxy.

But this is about aging, more than distance. Mathematically, we know how to calculate the length of a curve. The problem of distance, as I see it at least, it to specify what curve in space-time we want to calculate the length of. For instance, do we want to use cosmological coordinates and curves of constant cosmological time, do we want to find a space-time geodesic and find the length of said geodesic, or do we want to use "radar" methods and calculate the proper time for a radar pulse to reach the destination and get back to the source?