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Yes, you're quite correct that galaxies in a region of the universe with greater than average density would appear redshifted to observers outside that region.

In fact this is the origin of the Sachs-Wolfe effect, which is an important way that we study various properties of the universe.

Actually doing the calculation is hard because of course the universe is expanding and the rate of expansion changes with time. We need to take this into account and make sure that we exclude the cosmlogical red shift from our calculation. However almost everywhere in the universe the gravitational fields are weak enough that we can use an approximation called the weak field limit.

One of the ideas that physics students get taught early on is gravitational potential energy. You have probably been taught that for small distances it's:

$$ \text{PE} = mgh $$

though in GR we're normally interested in the potential energy per unit mass i.e. we set $m=1$ to get:

$$ \Delta \Phi = gh $$

And for large distances from a spherical body it's given by Newton's equation:

$$ \Delta \Phi = -\frac{GM}{r} $$

This also applies to galaxies though the equations get more complicated, and it also applies to groups of galaxies and galaxy clusters. In fact measuring the potential energy of galaxy clusters was how Fritz Zwicky first discovered the existence of dark matter.

The point of all this is that in the weak field limit the difference in gravitational potential between two places is directly related to the difference in the time dilation between those places by:

$$ \frac{d\tau_a}{d\tau_b} = \sqrt{1 - \frac{2\Delta \Phi}{c^2}} \tag{1} $$

So if you calculate the difference in the potential energy between the overdense region and the underdense region and feed it into equation (1) then it will give you the relative time dilation. Since in general $\Delta U$ will not be zero that means in general this is a difference in the time dilation and one galaxy will appear red shifted relative to the other. The reverse is of course also true - one galaxy will appear blue shifted relative to the other.

Lastly you ask:

Is the mass density at our position in the universe less dense than most other positions?

And the answer is that we seem to be about average. We're in the outer edges of the Virgo galaxy cluster, so we're neither in the overdense region at the centre of a cluster nor in the underdense region in a void.

Yes, you're quite correct that galaxies in a region of the universe with greater than average density would appear redshifted to observers outside that region.

In fact this is the origin of the Sachs-Wolfe effect, which is an important way that we study various properties of the universe.

Actually doing the calculation is hard because of course the universe is expanding and the rate of expansion changes with time. We need to take this into account and make sure that we exclude the cosmlogical red shift from our calculation. However almost everywhere in the universe the gravitational fields are weak enough that we can use an approximation called the weak field limit.

One of the ideas that physics students get taught early on is gravitational potential energy. You have probably been taught that for small distances it's:

$$ \text{PE} = mgh $$

though in GR we're normally interested in the potential energy per unit mass i.e. we set $m=1$ to get:

$$ \Delta \Phi = gh $$

And for large distances from a spherical body it's given by Newton's equation:

$$ \Delta \Phi = -\frac{GM}{r} $$

This also applies to galaxies though the equations get more complicated, and it also applies to groups of galaxies and galaxy clusters. In fact measuring the potential energy of galaxy clusters was how Fritz Zwicky first discovered the existence of dark matter.

The point of all this is that in the weak field limit the difference in gravitational potential between two places is directly related to the difference in the time dilation between those places by:

$$ \frac{d\tau_a}{d\tau_b} = \sqrt{1 - \frac{2\Delta \Phi}{c^2}} \tag{1} $$

So if you calculate the difference in the potential energy between the overdense region and the underdense region and feed it into equation (1) then it will give you the relative time dilation. Since in general $\Delta U$ will not be zero that means in general this is a difference in the time dilation and one galaxy will appear red shifted relative to the other. The reverse is of course also true - one galaxy will appear blue shifted relative to the other.

Lastly you ask:

Is the mass density at our position in the universe less dense than most other positions?

And the answer is that we seem to be about average. We're in the outer edges of the Virgo galaxy cluster, so we're neither in the overdense region at the centre of a cluster nor in the underdense region in a void.

Yes, you're quite correct that galaxies in a region of the universe with greater than average density would appear redshifted to observers outside that region.

In fact this is the origin of the Sachs-Wolfe effect, which is an important way that we study various properties of the universe.

Actually doing the calculation is hard because of course the universe is expanding and the rate of expansion changes with time. We need to take this into account and make sure that we exclude the cosmlogical red shift from our calculation. However almost everywhere in the universe the gravitational fields are weak enough that we can use an approximation called the weak field limit.

One of the ideas that physics students get taught early on is gravitational potential energy. You have probably been taught that for small distances it's:

$$ \text{PE} = mgh $$

though in GR we're normally interested in the potential energy per unit mass i.e. we set $m=1$ to get:

$$ \Delta \Phi = gh $$

And for large distances from a spherical body it's given by Newton's equation:

$$ \Delta \Phi = \frac{GM}{r^2} $$

This also applies to galaxies though the equations get more complicated, and it also applies to groups of galaxies and galaxy clusters. In fact measuring the potential energy of galaxy clusters was how Fritz Zwicky first discovered the existence of dark matter.

The point of all this is that in the weak field limit the difference in gravitational potential between two places is directly related to the difference in the time dilation between those places by:

$$ \frac{d\tau_a}{d\tau_b} = \sqrt{1 - \frac{2\Delta \Phi}{c^2}} \tag{1} $$

So if you calculate the difference in the potential energy between the overdense region and the underdense region and feed it into equation (1) then it will give you the relative time dilation. Since in general $\Delta U$ will not be zero that means in general this is a difference in the time dilation and one galaxy will appear red shifted relative to the other. The reverse is of course also true - one galaxy will appear blue shifted relative to the other.

Lastly you ask:

Is the mass density at our position in the universe less dense than most other positions?

And the answer is that we seem to be about average. We're in the outer edges of the Virgo galaxy cluster, so we're neither in the overdense region at the centre of a cluster nor in the underdense region in a void.

Yes, you're quite correct that galaxies in a region of the universe with greater than average density would appear redshifted to observers outside that region.

In fact this is the origin of the Sachs-Wolfe effect, which is an important way that we study various properties of the universe.

Actually doing the calculation is hard because of course the universe is expanding and the rate of expansion changes with time. We need to take this into account and make sure that we exclude the cosmlogical red shift from our calculation. However almost everywhere in the universe the gravitational fields are weak enough that we can use an approximation called the weak field limit.

One of the ideas that physics students get taught early on is gravitational potential energy. You have probably been taught that for small distances it's:

$$ \text{PE} = mgh $$

though in GR we're normally interested in the potential energy per unit mass i.e. we set $m=1$ to get:

$$ \Delta \Phi = gh $$

And for large distances from a spherical body it's given by Newton's equation:

$$ \Delta \Phi = -\frac{GM}{r} $$

This also applies to galaxies though the equations get more complicated, and it also applies to groups of galaxies and galaxy clusters. In fact measuring the potential energy of galaxy clusters was how Fritz Zwicky first discovered the existence of dark matter.

The point of all this is that in the weak field limit the difference in gravitational potential between two places is directly related to the difference in the time dilation between those places by:

$$ \frac{d\tau_a}{d\tau_b} = \sqrt{1 - \frac{2\Delta \Phi}{c^2}} \tag{1} $$

So if you calculate the difference in the potential energy between the overdense region and the underdense region and feed it into equation (1) then it will give you the relative time dilation. Since in general $\Delta U$ will not be zero that means in general this is a difference in the time dilation and one galaxy will appear red shifted relative to the other. The reverse is of course also true - one galaxy will appear blue shifted relative to the other.

Lastly you ask:

Is the mass density at our position in the universe less dense than most other positions?

And the answer is that we seem to be about average. We're in the outer edges of the Virgo galaxy cluster, so we're neither in the overdense region at the centre of a cluster nor in the underdense region in a void.

Yes, you're quite correct that galaxies in a region of the universe with greater than average density would appear redshifted to observers outside that region.

In fact this is the origin of the Sachs-Wolfe effect, which is an important way that we study various properties of the universe.

Actually doing the calculation is hard because of course the universe is expanding and the rate of expansion changes with time. We need to take this into account and make sure that we exclude the cosmlogical red shift from our calculation. However almost everywhere in the universe the gravitational fields are weak enough that we can use an approximation called the weak field limit.

One of the ideas that physics students get taught early on is gravitational potential energy. You get taught that for small distance it's:

$$ \Delta U = mgh $$

and for large distances from a spherical body it's given by Newton's equation:

$$ \Delta U = \frac{GM}{r^2} $$

This also applies to galaxies though the equations get more complicated, and it also applies to groups of galaxies and galaxy clusters. In fact measuring the potential energy of galaxy clusters was how Fritz Zwicky first discovered the existence of dark matter.

The point of all this is that in the weak field limit the difference in gravitational potential between two places is directly related to the difference in the time dilation between those places by:

$$ \frac{d\tau_a}{d\tau_b} = \sqrt{1 - \frac{2\Delta U}{c^2}} \tag{1} $$

So if you calculate the difference in the potential energy between the overdense region and the underdense region and feed it into equation (1) then it will give you the relative time dilation. Since in general $\Delta U$ will not be zero that means in general this is a difference in the time dilation and one galaxy will appear red shifted relative to the other. The reverse is of course also true - one galaxy will appear blue shifted relative to the other.

Lastly you ask:

Is the mass density at our position in the universe less dense than most other positions?

And the answer is that we seem to be about average. We're in the outer edges of the Virgo galaxy cluster, so we're neither in the overdense region at the centre of a cluster nor in the underdense region in a void.

Yes, you're quite correct that galaxies in a region of the universe with greater than average density would appear redshifted to observers outside that region.

In fact this is the origin of the Sachs-Wolfe effect, which is an important way that we study various properties of the universe.

Actually doing the calculation is hard because of course the universe is expanding and the rate of expansion changes with time. We need to take this into account and make sure that we exclude the cosmlogical red shift from our calculation. However almost everywhere in the universe the gravitational fields are weak enough that we can use an approximation called the weak field limit.

One of the ideas that physics students get taught early on is gravitational potential energy. You have probably been taught that for small distances it's:

$$ \text{PE} = mgh $$

though in GR we're normally interested in the potential energy per unit mass i.e. we set $m=1$ to get:

$$ \Delta \Phi = gh $$

And for large distances from a spherical body it's given by Newton's equation:

$$ \Delta \Phi = \frac{GM}{r^2} $$

This also applies to galaxies though the equations get more complicated, and it also applies to groups of galaxies and galaxy clusters. In fact measuring the potential energy of galaxy clusters was how Fritz Zwicky first discovered the existence of dark matter.

The point of all this is that in the weak field limit the difference in gravitational potential between two places is directly related to the difference in the time dilation between those places by:

$$ \frac{d\tau_a}{d\tau_b} = \sqrt{1 - \frac{2\Delta \Phi}{c^2}} \tag{1} $$

So if you calculate the difference in the potential energy between the overdense region and the underdense region and feed it into equation (1) then it will give you the relative time dilation. Since in general $\Delta U$ will not be zero that means in general this is a difference in the time dilation and one galaxy will appear red shifted relative to the other. The reverse is of course also true - one galaxy will appear blue shifted relative to the other.

Lastly you ask:

Is the mass density at our position in the universe less dense than most other positions?

And the answer is that we seem to be about average. We're in the outer edges of the Virgo galaxy cluster, so we're neither in the overdense region at the centre of a cluster nor in the underdense region in a void.