# B Relative speed between Galaxies

1. Jan 7, 2017

### Razola

My question is following:

Is there another Galaxy in the Universe that has a relative speed to the Milky Way of 0.999c (near of light speed)? The implication of that is that our life 'observed' by extraterrestrial beings in that Galaxy might be millions of years, an eternity! This is because of the time dilatation according to Lorenz Equations.

But there is another problem. The Universe is in expansion, so there is a kind of expansion of space itself. I read very far Galaxies are apparently separating from our Galaxy faster than light but in reality it is not due to motion but due to expansion of space itself. Then, how can we filter and remove the "apparent motion" due to space expansion from real motion through space of Galaxies.

It would be fascinating to think there is a lot of Galaxies with 0.999c relative to the Milky Way. The implication of that is that our life would be observed as an eternity for extraterrestrial beings of those Galaxies. It may be thought as a kind of heaven. We would live practically forever for those extraterrestrials. Each Galaxy would be considered as a kind of heaven with respect to the other one. I am thinking at Galaxies located in different filaments of the Universe. So, I imagined huged relative speed between Galaxies are possible. Does it happened?

2. Jan 7, 2017

### Orodruin

Staff Emeritus
Hi Razola, welcome to PF!

When you talk about Lorentz transformation (Lorenz was a different physicist), you are implicitly assuming that the laws of special relativity hold. For galaxies that are far away, this is no longer the case. In special relativity, the concept of time dilation arises partially due to the concept of relativity of simultaneity. In general relativity, it is no longer clear what you mean by "simultaneous" and there are many possible definitions (of which some are more common and intuitive than others). For this reason, it does not really make sense to talk about a far away object moving at 0.999c, because that statement assumes both a definition of a time interval as well as a definition of space intervals (which is also not unique).

You cannot. This is an arbitrary distinction that you can make only if you chose a convention for what you mean by "now" and what you mean by "motion".

Now, locally we can apply the laws of special relativity to a good approximation. If we do so and interpret all of the redshift as "motion", we end up with Hubble's law with galaxy clusters moving away. However, there is also a different possible choice of what it means to be "moving" where all galaxies are at rest (on average) and all of the redshift is due to space expanding.

3. Jan 7, 2017

### Razola

I didn't study general relativity, only the Lorentz Transformations and I think I've got it.
Why doesn't make sense to "measure" from the Earth the tick tack of a clock located in a very far Galaxy and to measure the distance as well?
I understand "measure" doesn't mean I am going to see the light coming from the moving Galaxy. It is enough to know that the tick tack of a very far Galaxy moving at 0.999c is occurring very slow for me. I don't need to really interact with the far away Galaxy and really measure but I know that the events of the universe for those two Galaxies are unique but looks different.
And I understand the Special Relativity through Lorentz Equation but I cannot understand nothing about space expansion. What does it mean "space expansion"? Is that an other thing overlaying the Relativity Theory? Than kills me. I cannot understand anymore.
And when you say and I say "very far Galaxies", well, what is 'very far' and from how far the theory should not apply anymore?
Is in General Relativity included the notion of space expanding?

Last edited: Jan 7, 2017
4. Jan 7, 2017

### Orodruin

Staff Emeritus
In relativity, space and time are all mixed up into spacetime. In GR, that spacetime is curved which makes the notion of what is time and what is space unclear unless you pick a convention. Given a convention you may be able to compute these things - but the result will depend on your chosen convention.

In SR, you will see the ticks of an object moving towards you occur faster than your ticks - this is the Doppler effect. Time dilation is what remains after you account for the fact that light from the later ticks were emitted closer to you.

You might chose a convention that determines an object's velocity solely on its redshift, but it is not the only possibility and certainly not the most intuitive one.

No. Metric expansion is predicted in GR with a few constraining assumptions.

Let me ask you: How old are you and what is your physics and math background?
There is no sharp border and it depends on how accurate you want to be.

5. Jan 8, 2017

### Razola

Thank you for your answers. The problem is I didn't have any knowledge about GR but I have some idea about what it is.
I think I understood something from you both. Is this correct?:
- SR is like the first term of the Taylor Polynomial. It is applicable for an homgeneous spacetime without accelerations, without gravity and so on. It is applicable between near Galaxies. So we will not see any near Galaxy moving faster than c from our Milky Way
- But in beteween Galaxies clusters the spacetime might not be homogeneous, there is apparent acceleration of Galaxies due to curved spacetime. Then if we measure a very far Galaxy separating from us at 3c it is because some how I considered a linear spacetime but it is curved instead. It is like making a right line through something that is curved. (correct?)
So, how is it the spacetime between Galaxies clusters? Curved? Inexistent?
Is the Universe expansion an apparent acceleration due to curved spacetime instead of homogeneous?
Is there a kind of a Gravity toward outside of the Universe?

6. Jan 9, 2017

### newjerseyrunner

Think smaller. If you have a rubber band that's one foot long and double its length in one second. So each end moved away from the other at 1/2 a foot per second right. But how much did each point move relative to space itself? Not at all. Now say the rubber band is a yard. Double its length, whoa, now the ends moved 1.5 feet per seconds, but still zero relative motion through space. Rubber band is a mile, double it in one second, the ends went apart at half a mile per second! Now make the rubber band the size of the universe, see why it's possible for objects to move away from each other way above the speed of light?

The edge of the universe aren't racing away from each other at some speed and stretching everything with it. The universe is scaleing evenly everywhere; it seems to double in size every few billion years.

To answer your question about what's in between, the universe as a whole appears flat. Spacetime on the grandest scale does not appear curved at all. It's a really big problem in cosmology, it should have broken symmetry early on and cascaded in one direction.

7. Jan 9, 2017

### newjerseyrunner

You need to think about who is obs raving because the maximum speed limit is different. The max velocity that two objects can see each other recede at is the speed of light.

However, the maximum velocity that you can OBSERVE FROM A THIRD POSITION, two objects moving away from each other is TWICE the speed of light.

If you light a match, photons are flying off of it in opposite direction at c. From your perspective, those photons move apart at 2c. From each other's perspective, the other photon might as well not even exist, it can never interact with it in any way.

In fact, because the universe is (we think) infinite, the rate velocity at which those galaxies travel away from us will also go to infinity, well just never see them or interact with them or anything.

8. Jan 10, 2017

### Orodruin

Staff Emeritus
Yes it does. It is *space* that is close to flat (defined as hypersurfaces of cosmological time).

This is not the qualifier we use when we talk about frames. Furthermore, "perspective" usually implies a rest frame, which photons do not have.

What is a rate velocity? The rate of expansion is decreasing (the Hubble parameter) and the speed of separation depends on your definition of space.

9. Jan 10, 2017

### pervect

Staff Emeritus
Going back to the original question, there are as Odoruin has mentioned a lot of technical issues with the idea of relative velocity between galaxies, it's largely a matter of convention.

The good news is that I can say that the most distant object I've seen mentioned has a redshift factor (z-factor) of 11.09, which is the current record holder for "most distant object". The z-factor is a commonly used measure in the professional litearture which can be converted either into distance or recessional velocity. The z-factor is what's really measureed, the "distances" and "velocities" are probably not terribly physically significant but are calculated from the z-factor.

But I'm not quite sure of the accepted way of converting this z-number into a velocity, to answer the original question. I've seen some links, but - I'm a bit suspicious of them. And I can't even say for sure if there's a single standard or multiple standards for this conversion process, really.

I do think it would provide some perspective to give the recessional velocity of the most distant known object - but I don't have that info, just the z-factor.

10. Jan 10, 2017

### Staff: Mentor

There isn't, because there isn't a single standard meaning to "velocity" for spatially separated objects in a curved spacetime; there can't be, because there is no invariant that could be used to support such a meaning. (Not to mention the additional issue of time delay--that furthest object at z=11.09 is being seen as it was billions of years ago, when the observable universe was much smaller than it is now and was expanding faster.)

11. Jan 10, 2017

### Orodruin

Staff Emeritus
You can always smash it into the Doppler formula. This will give you the velocity corresponding to the gamma factor you obtain when you take the inner product between your 4-velocity and the 4-velocity of the object parallel transported along the light-like geodesic that the light took to reach you. This is an invariant, but you might ponder about the physical meaning of this velocity for a very long time ...

12. Jan 10, 2017

### pervect

Staff Emeritus
Many of the sources I've been reviewing indicate that "recession velocity" is usually - or at least very often - calculated via Hubble's law from the distance. WIth the caveat that one must use the correct formulation of distance. For some specific examples, we have Ned Wright's cosmology tutorial, http://www.astro.ucla.edu/~wright/cosmo_01.htm, we have for instance a plot of recession velocity vs distance.

We also have an illuminating remark about "Many Distances" in part 2, http://www.astro.ucla.edu/~wright/cosmo_02.htm.

I've seen similar remarks in Lineweaver & Davis paper "Expanding Confusion", so this point of view isn't just something unique to Ned Wright - though I haven't attempted to dig up a more complete set of references that talk about this.

I actually agree with the point that the velocity is not really well defined, because it ultimately winds up being conventional, depending on human convetional choices such as coordinate choices. Baez for instance makes this point in the introduction of "The Meaning of Einstein's Equation"

[quote = Baez]
In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another.

...

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.
[/quote]

So I'm not quite sure what the best answer is. I believe that cosmologists do tend to use a more-or-less standard set of conventions regarding distances (and velocities) when they report their findings in the popular literature. I believe it's based on Hubble's law and the associated definition of distance, that also defines a "recession velocity". I believe that this is done even though this choice is conventional, and that the cosmologists convention has a confusing side effect with velocities greater than "c", which is not compatible with special relativity and causes a lot of angst to lay readers.

13. Jan 10, 2017

### Staff: Mentor

Yes. This "velocity" is a coordinate speed in FRW coordinates. It does not have a direct physical interpretation (that should be obvious since it can be faster than light). It's a convenient number for cosmologists to use, and that's about all. (IMO even the convenience aspect can be overstated: the main convenience seems to be that cosmologists can quote velocities instead of redshifts, but I don't see that that makes much difference in their actual models, it mainly serves to confuse lay people who are trying to understand what cosmologists are saying.)

This is my understanding as well. See above.

14. Jan 11, 2017

### Mordred

The FrW coordinates is a commoving coordinates system. In many ways I find trying to use recessive velocity more trouble than its worth lol. Gets far too confusing to the average person.

Trying to use the cosmological redshift formula for velocity simply doesn't work. Its really intended to handle the volume change. However the form one commonly comes across is only accurate up to the Hubble horizon. Which is where your recessive velocities start to exceed c.

Just noticed Orodruin mentions this above

Took me awhile to find this formula for redshifts beyond Hubble horizon thought I might share it

$$z=\frac {H_0l^2}{c+\frac {1}{2}(1+q_0)H^2_0l^2/c^2+O (H^3_0l^3)}$$

Part of the reason for the complexity of this formula is that the scale factor a (t) isn't linear
Another good article is the following

http://arxiv.org/abs/astro-ph/?9905116"Distance measures in cosmology" David W. Hogg

Last edited: Jan 12, 2017