Can two galaxies receded from one another faster than C?

[Chaos' lil bro Order]

Sorry if this is a lay question, here it goes:

Consider that on Earth we observed two galaxies, one directly above the north pole, the other directly above the south pole (aka. opposite directions). When we measured the redshifts of these galaxies we concluded that they had recessional velocities of 0.99C and 0.98C, respectively. The question then is simply this, are these two galaxies receding from one another with a combined recessional velocity of 1.97C?

If so, then consider this...
If we could put an observatory on the 0.99C galaxy, would we see the 0.98C galaxy as receding from us at 1.97C?

I'm pretty sure the answer is no, but I can't think of why its no. Am I simply thinking of the Universe's structure incorrectly?

Thanks

 

[pervect]

I don't have a formula that answers this specfic question at this time, though I'll do some more looking and as a last resort I might even try to figure it out from first principles.

It might be a good idea to move this question to the cosmology forum, as there is a good chance that SpaceTiger will be able to get you an answer more quickly. Part of the answer to this question depends on certain assumptions as to cosmological parameters.

There is however some very useful background information on your question at

http://www.astro.ucla.edu/~wright/cosmology_faq.html#FTL

however.

Can objects move away from us faster than the speed of light?

Again, this is a question that depends on which of the many distance definitions one uses. However, if we assume that the distance of an object at time t is the distance from our position at time t to the object's position at time t measured by a set of observers moving with the expansion of the Universe, and all making their observations when they see the Universe as having age t, then the velocity (change in D per change in t) can definitely be larger than the speed of light. This is not a contradiction of special relativity because this distance is not the same as the spatial distance used in SR, and the age of the Universe is not the same as the time used in SR. In the special case of the empty Universe, where one can show the model in both special relativistic and cosmological coordinates, the velocity defined by change in cosmological distance per unit cosmic time is given by v = c ln(1+z), where z is the redshift, which clearly goes to infinity as the redshift goes to infinity, and is larger than c for z > 1.718. For the critical density Universe, this velocity is given by v = 2c[1-(1+z)-0.5] which is larger than c for z > 3 .
For the concordance model based on CMB data and the acceleration of the expansion measured using supernovae, a flat Universe with OmegaM = 0.27, the velocity is greater than c for z > 1.407.

If we could multiply the redshifts together, it would be easy to answer the question from the above formula relating redshift to velocity, but I do not believe that this technique is justified for sevaral reasons - one reason being that the formula above is stated to work only for an empty universe. Presumably you'd want an answer that uses the current "concordance model" .

Note that this is an entirely different question than how velocities add together in SR, which is a much easier question. Not only is the geometry of the universe not flat, but as the FAQ above mentions, cosmologists tend to use different measures of distance (and velocity) than people in SR do. (For that matter, their defintion of simultaneity is different, too).

In case you were interested in the answer to the SR version of your question, that is given in many places, for instance

http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html"

 

[kmarinas86]

Sorry if this is a lay question, here it goes:

Consider that on Earth we observed two galaxies, one directly above the north pole, the other directly above the south pole (aka. opposite directions). When we measured the redshifts of these galaxies we concluded that they had recessional velocities of 0.99C and 0.98C, respectively. The question then is simply this, are these two galaxies receding from one another with a combined recessional velocity of 1.97C?

If so, then consider this...
If we could put an observatory on the 0.99C galaxy, would we see the 0.98C galaxy as receding from us at 1.97C?

I'm pretty sure the answer is no, but I can't think of why its no. Am I simply thinking of the Universe's structure incorrectly?

Thanks

Speed of light is measured as 299,792,458 meters / 1 second. Whose second is that? The key is that the length of the "second" varies. You might as well say that the speed of the observed galaxy is 299,792,458 meters * ? Hz, where the Hz of the observed galaxy depends on the fraction of the speed of light they are receeding i.e. $v/c$

$? Hz = 1 Hz * \frac{v}{c}\sqrt{\frac{1-\frac{v}{c}}{1+\frac{v}{c}}}$

The velocity addition formula can be derived in the following way:

Let:

$\alpha=\frac{v_1}{c}$$\beta=\frac{u}{c}$$\gamma=\frac{v_2}{c}$$f_1=f_0 \sqrt{\frac{1-\alpha}{1+\alpha}}$$f_2=f_1 \sqrt{\frac{1-\beta}{1+\beta}}$$f_2=f_0 \sqrt{\frac{1-\alpha}{1+\alpha}\frac{1-\beta}{1+\beta}}$$f_2=f_0 \sqrt{\frac{1-\gamma}{1+\gamma}}$$\sqrt{\frac{1-\gamma}{1+\gamma}}=\sqrt{\frac{1-\alpha}{1+\alpha}\frac{1-\beta}{1+\beta}}$$\left(1-\gamma\right)\left(1+\alpha\right)\left(1+\beta\right)=\left(1-\alpha\right)\left(1-\beta}\right)\left(1+\gamma\right)$$\left(1-\gamma\right)\left(1+\alpha+\beta+\alpha\beta\right)=\left(1-\alpha-\beta+\alpha\beta\right)\left(1+\gamma\right)$$1+\alpha+\beta+\alpha\beta-\gamma-\gamma\alpha-\gamma\beta-\gamma\alpha\beta=1-\alpha-\beta+\alpha\beta+\gamma-\gamma\alpha-\gamma\beta+\gamma\alpha\beta$$2\left(\alpha+\beta\right)=2\left(\gamma+\gamma\alpha\beta\right)$$\alpha+\beta=\gamma+\gamma\alpha\beta$$\frac{\alpha+\beta}{1+\alpha\beta}=\gamma$

 

[Loren Booda]

To all central observers, opposite horizons recede near the speed of light. Special relativity does not allow communication between these horizons, although we may observe them both individually.

General relativity, however, does allow points in the expanding universe to accelerate, thus enlarging the boundary and spacetime beyond where we are unable to observe light because v>c. In effect, the vast majority of spacetime regresses faster than the speed of light.

Remember that Einstein's postulate of physics equivalency in inertial frames does not necessarily apply to GR, where non-interial frames are considered.

 

[pervect]

$\frac{\alpha+\beta}{1+\alpha\beta}=\gamma$

[/quote]

This looks like it is equivalent to the SR velocity addition formula

$$v_tot = \frac{v1+v2}{1+\frac{v1 v2}{c^2}}$$

while this is correct for flat space-time (i.e. Special Relativity) it's not correct for the original problem.

 

[pervect]

I think I've dug up the correct way to proceed, though I could use a double-check by another SA who works with cosmology more than I do.

The relationship between velocity and $D_{now}$, the comoving radial distance, is NOT given by the standard SR doppler formula. (Ned Wright mentions this explicitly in his cosmology tutorial, which was my primary source for this post:)

http://www.astro.ucla.edu/~wright/cosmo_02.htm

All we really need to work this problem is Hubble's law, v = D H

From the above link:

Many Distances

With the correct interpretation of the variables, the Hubble law (v = HD) is true for all values of D, even very large ones which give v > c. But one must be careful in interpreting the distance and velocity. The distance in the Hubble law must be defined so that if A and B are two distant galaxies seen by us in the same direction, and A and B are not too far from each other, then the difference in distances from us, D(A)-D(B), is the distance A would measure to B. But this measurement must be made "now" -- so A must measure the distance to B at the same proper time since the Big Bang as we see now. Thus to determine Dnow for a distant galaxy Z we would find a chain of galaxies ABC...XYZ along the path to Z, with each element of the chain close to its neighbors, and then have each galaxy in the chain measure the distance to the next galaxy at time to since the Big Bang. The distance to Z, D(us to Z), is the sum of all these subintervals:

Dnow = D(us to Z) = D(us to A) + D(A to B) + ... D(X to Y) + D(Y to Z)

And the velocity in the Hubble law is just the change of Dnow per unit time. It is close to cz for small redshifts but deviates for large ones.

We can thus determine D1 = v1/H, and D2 = v2/H

The co-moving distances add, so that the total distance between the two galaxies is just D1+D2 = (v1+v2)/H.

We multiply the result by H, to find that the velocities add linearly!

It should be noted, again, that these are NOT the velocities as defined in Special Relativity. To quote Ned Wright from the above link.

The time and distance used in the Hubble law are not the same as the x and t used in special relativity, and this often leads to confusion. In particular, galaxies that are far enough away from us necessarily have velocities greater than the speed of light.

In particular, the "comoving radial distance" is measured along a curve that is not a geodesic, i.e. it's measured along a curve that's not the shortest distance between the two points. You can thus think of it as being larger than the distance that SR gives, which helps explain why the rate of change of this distance (which is the cosmological defintion of velocity) can be greater than 'c' - the cosmological distance scale is larger than the SR defintion, so the cosmological velocities (the rate of change of cosmological distance) are likeways larger.

 

[Chaos' lil bro Order]

In particular, the "comoving radial distance" is measured along a curve that is not a geodesic, i.e. it's measured along a curve that's not the shortest distance between the two points. You can thus think of it as being larger than the distance that SR gives, which helps explain why the rate of change of this distance (which is the cosmological defintion of velocity) can be greater than 'c' - the cosmological distance scale is larger than the SR defintion, so the cosmological velocities (the rate of change of cosmological distance) are likeways larger.




So basically, two comoving galaxies CAN recede from one another at a combined rate of over C and you say since space is curved, the distance is actually greater than the geodesic path between the two galaxies because of this curvature.

As for Hubble's Law, I thought that it predicts a maximum velocity of C. z=1000 equals something like .99999999999C, so doesn't velocity approach C as z approaches infinite?

Wouldn't: Recessional Velocity = Beta/ Hubble Constant imply that recessional velocity is constrained by C since Beta=v/C.

Since these two forumulas, the 1st cosmology, the 2nd Relativity are linked mathematically, I don't understand the comment saying:

The time and distance used in the Hubble law are not the same as the x and t used in special relativity, and this often leads to confusion. In particular, galaxies that are far enough away from us necessarily have velocities greater than the speed of light.

 

[Severian596]

So basically, two comoving galaxies CAN recede from one another at a combined rate of over C and you say since space is curved, the distance is actually greater than the geodesic path between the two galaxies because of this curvature.

It looks like you destroyed pervect's well-spoken explanation. He didn't say that two galaxies can recede from one another at a combined rate over C. He said that we on Earth can measure their apparent velocities relative to each other as being greater than C. But this is pretty meaningless...if the two galaxies measured their velocities relative to each other they would arrive at a v < c.

 

[DrChinese]

Just as a reminder in case you are not familiar with Lineweaver & Davis. There are plenty of galaxies receding from us at velocities greater than 2c, including some which are experimentally verified as such via red shift.

 

[pervect]

So basically, two comoving galaxies CAN recede from one another at a combined rate of over C and you say since space is curved, the distance is actually greater than the geodesic path between the two galaxies because of this curvature.

That's not quite what I'm saying.

What I'm saying is that there are several distance measures used in cosmology, and that one has to read the "fine print' of what cosmologists are actually measuring, to make sure that one is using the right distance for the right application.

Each of these different distance measures has an associated "velocity", the rate of change of the distance with respect to some time parameter (said time parameter being usually, but perhaps not always, our time).

Thus it is ambiguous and potentially misleading to talk about "the" velocity of a distant galaxy without being more explicit as to _which sort_ of velocity one means.

Journalists do this all the time, unfortunately, which tends to lead to a confused public :-(.

Because you ARE talking about "the" velocity and "the" distance, one assumes that you have some particular model/application in mind, but it's unclear which of the many candidates for distance (or velocity) would be the closest match to your application.

Ned Wright's cosmology tutorial is a very useful reference, the Lineweaver and Davis paper that Dr Chinese posted a link to is also very useful.

I posted the link for Ned Wright's tutorial earlier, I'll repeat it here, in case you haven't at least skimmed it (there's quite a bit of material):

http://www.astro.ucla.edu/~wright/cosmolog.htm

(The main tutorial is under "enter tutorial').

Usually, we think of distance as being measured along the shortest path between two points. (Well, at least I do.)

The particular sort of distance used in the Hubble's law calculation (comoving distance) is very useful, but aside from not being the distance measured along a straight-line path (as one might intuitively think that distances should be), it is not even the only sort of distance that cosmologists use.

Among other sorts of distance in common use there are:

light travel time
angular size distance
luminosity distance

these are all conveniently computed in terms of z by a calculator program avaliable from the URL I quoted earlier.

This is not an exhaustive list of distance measures, one of my favorites, affine distance, is not even on the list. This concept of distance is closely related to light travel time distance, it could be regarded as counting the number of wavelengths of monochromatic light along the path of a lightbeam emitted by a visible object. It's probably more of theoretical interest than practical utiltiy, however.

As a conclusion to a rather long post, one can look to GR for guidance as to how to define distance, but all GR will tell you is that there isn't any unique way to determine the velocity of a distant observer when there is gravity involved (or when space-time is curved). There are some useful and general ways to proceed when one has a static metric - unfortunately, this doesn't help with cosmology, because the metric of the universe is not static, but is changing with time.

 

[Loren Booda]

The fastest speed that can be measured is the speed of the measuring particle, the photon itself.

 

[pervect]

Let me post a simple example that explains part (not all) of the more complex cosmological situation. The good news is that the analogy is relatively simple - the bad news is that it is not exact.

Suppose that you are on Earth, somewhere above the equator (say 45 degrees lattitude to be specific).

Suppose you have two laboratories with a clear line of sight between them, and that the second laboratory is at exactly the same latitude, but 18.6 miles due east of the first.

Now, since light travels at 186,000 miles per second, you expect light to take .0001 seconds to travel from one laboratory to the other.

But you observe that the light reaches the laboratory sooner than this. Is light then moving faster than light?

The answer is no. The path that you've measured along a circle of constant lattitude by moving "due east" is actually not the shortest path. The shortest path, the one that light actually takes (ignoring height for the purposes of this analogy, you'd actually need a fairly tall tower to get an 18.6 mile LOS) follows a path that's a great circle.

It is not incorrect to say that the laboratory is 18.6 miles due east, it's just that one has chosen the wrong defintion of distance for the problem at hand - by appling plane geometry concepts to a geometry that's actually spherical.

The cosmological situation is more complicated, because it is not stationary like the simple example on the Earth. But co-moving coordinates are a lot like latitude and longitude in many respects - a galaxy always has a constant value for its cosmological coordinate, even as the universe expands.

For more details, I would again urge readers to read some of the references posted earlier in the thread. The devil is in the details, as they say.

Note that the cosmological problem is harder than the problem on Earth because the universe is not static - it's expanding. distance = time * velocity does one no good if one measures the wrong distance - and in an expanding universe, because the distance is constantly changing, picking the right distance and how to time-average it is even trickier than the simple example on Earth.

 

[Chaos' lil bro Order]

Thought I made it clear that I understood the difference between a geodesic path and a curved path. Cosmology deals with the curved path as space is curved.

As for semantics, velocity = recessional velocity
distance = Light Years (of course)
z = redshift (usually Lyman series for distant objects)


From the Lineman & Davis paper, its clear that galaxies can recede at superluminal velocities, in fact any with red shifts above ~1.46 meet the requirement.

Thanks for the ref. Dr. Chinese, now I won't need to read tons of half-true posts, I love a good Review paper.

 

[Chaos' lil bro Order]

Severian, yup ok, you are just plain wrong.

Thanks for destroying your credibility.

 

[pervect]

Thought I made it clear that I understood the difference between a geodesic path and a curved path. Cosmology deals with the curved path as space is curved.

So far so good, but...

As for semantics, velocity = recessional velocity
distance = Light Years (of course)
z = redshift (usually Lyman series for distant objects)

These are the sort of statements that make me think you are NOT understanding what I'm trying to say at all :-(.

After having spent a lot of time trying to explain that there are many ways of measuring distance, you talk about velocity as "recessional velocity" and the units of distance as being light years, as though that defined what velocity it is that you're talking about.

This gives me the feeling that real communication just isn't happening.

I rather suspect that you may be trying to understand cosmology without first understanding special relativity first. If my suspicion is correct this is a big mistake - you really need to understand SR first, before you tackle cosmology. (Understanding at least a little bit about GR as well as SR would be even better.)

 

[Chaos' lil bro Order]

I think I will reserve my cosmology questions to the cosmology forum, Space Tiger seems to know everything inside out.

But thanks.

 

[da_willem]

Let me post a simple example that explains part (not all) of the more complex cosmological situation. The good news is that the analogy is relatively simple - the bad news is that it is not exact.

Suppose that you are on Earth, somewhere above the equator (say 45 degrees lattitude to be specific).

Suppose you have two laboratories with a clear line of sight between them, and that the second laboratory is at exactly the same latitude, but 18.6 miles due east of the first.

Now, since light travels at 186,000 miles per second, you expect light to take .0001 seconds to travel from one laboratory to the other.

But you observe that the light reaches the laboratory sooner than this. Is light then moving faster than light?

The answer is no. The path that you've measured along a circle of constant lattitude by moving "due east" is actually not the shortest path. The shortest path, the one that light actually takes (ignoring height for the purposes of this analogy, you'd actually need a fairly tall tower to get an 18.6 mile LOS) follows a path that's a great circle.

It is not incorrect to say that the laboratory is 18.6 miles due east, it's just that one has chosen the wrong defintion of distance for the problem at hand - by appling plane geometry concepts to a geometry that's actually spherical.

The cosmological situation is more complicated, because it is not stationary like the simple example on the Earth. But co-moving coordinates are a lot like latitude and longitude in many respects - a galaxy always has a constant value for its cosmological coordinate, even as the universe expands.

For more details, I would again urge readers to read some of the references posted earlier in the thread. The devil is in the details, as they say.

Note that the cosmological problem is harder than the problem on Earth because the universe is not static - it's expanding. distance = time * velocity does one no good if one measures the wrong distance - and in an expanding universe, because the distance is constantly changing, picking the right distance and how to time-average it is even trickier than the simple example on Earth.

Great analogy! Thanks!

 

[LnGrrrR]

Pervect,
I'm trying to work out your dang signature...I'm at an area of trial and error though, and I'm trying to fight laziness to figure it out :)

 

[Conal]

To all central observers, opposite horizons recede near the speed of light. Special relativity does not allow communication between these horizons, although we may observe them both individually.

If, for example, a spaceship is flying away from Earth at near the speed of light and communicating with us, we should percieve the signal to be traveling at the speed of light anyway. So if another spaceship is traveling in the opposite direction at near the speed of light, the signal should still be able to reach it too shouldn't it?

 

[marcus]

Sorry if this is a lay question, here it goes:

Consider that on Earth we observed two galaxies, one directly above the north pole, the other directly above the south pole (aka. opposite directions). When we measured the redshifts of these galaxies we concluded that they had recessional velocities of 0.99C and 0.98C, respectively. The question then is simply this, are these two galaxies receding from one another with a combined recessional velocity of 1.97C?

If so, then consider this...
If we could put an observatory on the 0.99C galaxy, would we see the 0.98C galaxy as receding from us at 1.97C?

I'm pretty sure the answer is no, but I can't think of why its no. Am I simply thinking of the Universe's structure incorrectly?

Thanks

The answer to the originally posted question is yes. One galaxy is receding from the other at 1.97 times the speed of light.

It is a popular mistake to think of recession speeds as limited by c. This was addressed in a Scientific American article last year. Also in the article cited by Dr Chinese in this thread.

A numerically simpler example would be the case where it is determined that galaxy A is receding from us at speed 2c and galaxy B is receding in the opposite direction at speed 3c. Then A is receding from B at speed 5c.

link to the SciAm article on popular misconceptions about the expansion of the universe is here
https://www.physicsforums.com/showpost.php?p=482299&postcount=76

It sounds like Chaos brother got discouraged and left this thread

I think I will reserve my cosmology questions to the cosmology forum, Space Tiger seems to know everything inside out.

But thanks.

Not a bad move. SpaceTiger would certainly be able to clear the matter up. (But it seems a pity if questions like this cannot be handled in the forum specifically set up to handle them.)

 

[pervect]

I would say that the cosmology forum is probably best forum for cosmology questions, a better forum than the SR/GR forum. I would agree entirely that Space Tiger is a great reference, in fact I recommended him in #2.

If you scroll back over the past discussion, Chaos got a correct answer from me in post #6, and from Dr Chineese in #9. All the answers from science advisors were pretty much right on the mark, though there were a number of posts that missed the mark from non-SA's.

Thus it is possible to get good answers about cosmology in the GR forum, but it's probably easier and quicker to get good answers about cosmology in the cosmology forum.

I actually attempted to go into a litle more depth, and provide an explanation of how cosmologists do not use a coordinate system compatible with special relativity, citing Ned Wright's cosmology FAQ

http://www.astro.ucla.edu/~wright/cosmology_faq.html#FTL

Can objects move away from us faster than the speed of light?

Again, this is a question that depends on which of the many distance definitions one uses. However, if we assume that the distance of an object at time t is the distance from our position at time t to the object's position at time t measured by a set of observers moving with the expansion of the Universe, and all making their observations when they see the Universe as having age t, then the velocity (change in D per change in t) can definitely be larger than the speed of light. This is not a contradiction of special relativity because this distance is not the same as the spatial distance used in SR, and the age of the Universe is not the same as the time used in SR. In the special case of the empty Universe, where one can show the model in both special relativistic and cosmological coordinates, the velocity defined by change in cosmological distance per unit cosmic time is given by v = c ln(1+z), where z is the redshift, which clearly goes to infinity as the redshift goes to infinity, and is larger than c for z > 1.718. For the critical density Universe, this velocity is given by v = 2c[1-(1+z)-0.5] which is larger than c for z > 3 .
For the concordance model based on CMB data and the acceleration of the expansion measured using supernovae, a flat Universe with OmegaM = 0.27, the velocity is greater than c for z > 1.407.

I didn't cut and paste this quote in the original response, unforutnately, perhaps this would have been a good idea in retrospect for the convenience of lazy readers.

These different conventions (on the measure of basic things like distance and velocity) on the part of cosmologists are the reason this thread has generated so much confusion, and the source of many of the incorrect answers given by people who are familiar with SR, but not GR or cosmology.

In any event, questions on GR&SR are certainly welcome in the SR & GR forum. We do get a tiny bit tired about questions about FTL, though.

If marcus's point was that all "FTL" questions should be sent to the SR/GR forum as a sort of "dumping ground", I suppose I have to disagree. The cosmologists have to take their fair share of the load, when the issue actually does involve cosmology. Of course the vast majority of such questions do not involve cosmology, and are best addressed with the SR velocity addition formula.

 

[Chaos' lil bro Order]

Thanks Marcus, best response yet!

 

[marcus]

Thanks Marcus, best response yet!

You are welcome!
There is another calculator besides Wright's that might be useful to you.
http://www.earth.uni.edu/~morgan/ajjar/Cosmology/cosmos.html

You asked a question back in post #7 about something involving z = 1000. This calculator would tell you corresponding speeds and also would tell the light travel time. So I mention z = 1000 as an illustration. But that is a CMB size redshift and the numbers are a bit stupendous. So you might also want to try redshift 1.5 or 2 or 3.

Morgan's calculator has the same basic functions as Ned Wright's -----you put in parameters like 71 for Hubble and 0.27 for matter and 0.73 for dark energy (the famous "73%") and once that is taken care of then you can put in anything for the redshift z and it will tell you the Hubble-law distance

it does NOT have some other features that Wright does, but it does have a couple of features his doesnt. Morgan's will give you the distance, say for redshift z = 1000 (essentially same as Wright's) but it will also give you the PRESENT recession SPEED of the matter that emitted the light that we are now receiving that has redshift 1000.

And also interesting it will give you the speed that the matter had when it emitted the light.In Morgan's simplified terminology "Omega" means matter density Omega_matter. So you type in 0.27----that is a common estimate (27%) comprising ordinary matter plus dark matter. Again in Morgan's terms, "Lambda" means Omega_Lambda-----the dark energy or cosmol. const. part of total.

so to use the calculator you have to type in 0.27 for Omega ("matter density") and 0.73 for Lambda ("cosmological constant") and 71 for the Hubble parameter (71 km/s per Megaparsec). then you are ready.
then if you put in z = 3 or whatever redshift you want to see distance for (or age when the light was emitted, light travel time, or present/past recession speeds)
you should get that...

If you don't want to try it, or prefer to stick with Ned Wright's distance-from-redshift it is OK my feelings won't be hurt

but just as a check, in case you do try it: you should get that for z =3 the present distance is 21.07 billion LY.

that is where you put in, for "matter density", "cosmological constant", Hubble parameter, and z the numbers 0.27, 0.73, 71, 3

have to go help with supper.

Back now. Basically the calculator gives you a simple hands-on encounter with GENERAL relativity. what the calculator is using is a certain SOLUTION TO THE EINSTEIN EQUATION associated with names like Friedman, Lemaitre, Robertson, Walker. the metric is called the FRW metric (for Friedman-Robertson-Walker).

In the example I suggested you try----with 0.27 + 0.73 = 1-----the spatial slices are FLAT
so the Hubble-law distance that we are talking about is a STRAIGHT LINE DISTANCE. It is the shortest distance between two points barring excursions into the past or the future.
(It is misleading to attribute superluminal recesssion speeds to the Hubble-law distance not being "along geodesics".)

Anyway, if you are at all interested in General Relativity the spatially flat case of the FRW metric is a simple example to play around with and you have the basics of it on Morgan's calculator.

Bear in mind that individual solutions of Gen Rel do not have to have Lorentz symmetry---they don't need to obey Special Rel (except as a local approximation). So in a particular solution of Einstein's Equation like the FRW metric HAS a idea of rest----it makes sense to say a galaxy is sitting still in some space that is receding from another galaxy----and there is an idea of SIMULTANEOUS. Whereas in Special Rel you do NOT have a notion of rest or simultaneity.

some history: Special Rel was 1905. Gen Rel was 1915.
The earlier theory is the one that has no rest, no simultaneity, and no idea of a distance increasing faster than light.

I don't know if anyone here is particularly interested in Gen Rel or this kind of introduction to it. So I will stop and see if there is any response. First one example. The microwave background comes from z=1100. So think about the MATTER that emitted the light that we are now seeing as microwaves. That matter had a certain RECESSION SPEED at the time it emitted the light.
What was the recession speed of the matter that emitted the CMB photons that we are now receiving?
Anybody?

 

[pervect]

I don't know if anyone here is particularly interested in Gen Rel or this kind of introduction to it. So I will stop and see if there is any response. First one example. The microwave background comes from z=1100. So think about the MATTER that emitted the light that we are now seeing as microwaves. That matter had a certain RECESSION SPEED at the time it emitted the light.

What was the recession speed of the matter that emitted the CMB photons that we are now receiving?
Anybody?

One first needs to define how the recession speed is measured. I gather cosmologists usually mean the rate of change of the comoving distance, as per Ned Wright - a coordinate dependent defintion.

Even given the precise coordinate dependent defintion of "velocity" that cosmologists use, the answer to the above question is highly model dependent. Plugging in z=1100 into Ned Wright's cosmological calculator and picking the model parameters of the cosmology that one believes are accurate, allows one to calculate the comoving distance and hence the "recessional velocity".

For example, for z=1100 with

H0=71, omegam=.27, and omegavac=.73 the calculator computes the comoving distance as about [correction] 14000 Mpc.

Multiply 14000 Mpc by H0, and one gets V_now 994,000 km/sec

[edit]
But with different parameters, for example H0=71, omegam=1, omegav=0 the distance is only about 8200 Mpc, significantly lower. The velocity changes correspondingly.

Hence the remarks about the model dependence of this velocity.

I should note that in cases with a static metric, I would greatly prefer to define a notion of velocity in a coordinate independent manner, rather than in the coordinate dependent way in which cosmologists measure velocity.

For instance, if one has a black hole, I would (and have)! suggest defining the idea of a "velocity relative to the black hole" of a test object X as taking an object "at rest" with respect to the black hole, and measuring the velocity between this object and the test object X when the two objects are at the same point in space-time.

Note that I would not suggest using this concept of velocity without explaining it - the very idea of the relative velocity of two objects at different points in space-time is ambiguous without further explanation. But I feel it is a good defintion, once explained, because it does not depend on the usage of any specific coordiante system.

This coordinate independence is possible because there is a unique object that is "stationary" with respect to the static metric of the black hole at any given location in space.

Unfortunately, this coordinate independent technique will not work for FRW cosmologies, because they do not have static metrics. Hence, we are stuck with the rather ugly (and potentially confusing) facts that not only can velocities as defined by cosmologists be greater than 'c', but that the point where the objects recessional velocity becomes equal to 'c' is not the same as either the event horizon or the particle horizon!

 

[marcus]

One first needs to define how the recession speed is measured. I gather cosmologists usually mean the rate of change of the comoving distance,...

Hi pervect, how nice that someone wants to try calculating!
It is after midnight here and I am going off to bed soon but before doing so I will calculate for z = 1100. Then if you do so we can compare results!

Remember that this is all done with the FRW metric which is one particular solution to the Einst. equation. Remember also that the Hubble parameter CHANGES. So the figure of 71 applies only to the present moment!

However the model is able to extrapolate back and find the Hubble that prevailed at any given time in the past.

OK, so we plug in z = 1100, the redshift of the Cosmic Microwave Background radiation.

I get that the recession speed of the emitter at the time it radiated the light which we are now picking up was 57c.

The recession speed at emission was FIFTY-SEVEN TIMES C. Is this what you get? I hope so.The present distance of the matter that emitted that light is now 45.5 billion LY and it is receding at a speed of 3.3 c. Did you get any of these numbers? If not, you may have made a mistake.
Perhaps someone else will use one of the available online calculators and check my answers.

I will compare and see what Ned Wright's gets. z=1100 is really pushing it! So I wouldn't be surprised if there were some minor discrepancy. The distance should come out within say 10 percent 45.5 billion LY but maybe not exact due to round-off error and stuff.

=====================

WOW NED WRIGHT'S GIVES VERY NEARLY THE SAME ANSWER!
It gives the distance as 45.655 billion LY
And Morgan's gave it as 45.5 billion

This is when I put in the same numbers: 71, 0.73, 0.27 and say z=1100.

GREAT! they are probably using the same formulas and very similar numerical integration routines.
Ned's is just a few more decimal places accurate.

Ned's gives the age of universe at the time the CMB light was emitted as 372,000 years. That is close to what people usually give for the time of last scattering.

The light travel time is given by the calculator to be 13.655 billion years. Almost the present age of the universe.

OK that all seems fairly consistent. Anybody trying it, prevect or others, you should get that the emitter is now some 45 billion LY away
and the light has been traveling for some 13.6 billion years
and the emitting matter, when it radiated the light, was receding at around 57c.

If you have any trouble getting these answers, give me a brief description of what you did and i will try to help.

 

[RandallB]

What was the recession speed of the matter that emitted the CMB photons that we are now receiving?

What “was” or what “is”?
Seems to me your z=1100 giving a recession speed of 57c;
“Is” relative to the emitter then very near to the BB and “us” now & the current reference frame our matter (receiver) is in now - right.
But, if we compare the recession speed to “us”, when the matter that makes us "was" also very near to the BB (before most of Hubble has kicked in) shouldn’t that ‘z’ and recession speed be much lower, near c?

The present distance of the matter that emitted that light is now 45.5 billion LY and it is receding at a speed of 3.3 c.

Also, if our relative speed now to the emitter beginning point is 57c.
It seems the “now” for that matter should also have a 57c recession from us as we emitted light back then near the BB as well.
How do we get the relative speeds between our current now, and present location of the “matter that emitted that light” down to only 3.3c?

 

[marcus]

What “was” or what “is”?
Seems to me your z=1100 giving a recession speed of 57c;
“Is” relative to the emitter then very near to the BB and “us” now & the current reference frame our matter (receiver) is in now - right.
But, if we compare the recession speed to “us”, when the matter that makes us "was" also very near to the BB (before most of Hubble has kicked in) shouldn’t that ‘z’ and recession speed be much lower, near c? Also, if our relative speed now to the emitter beginning point is 57c.
It seems the “now” for that matter should also have a 57c recession from us as we emitted light back then near the BB as well.
How do we get the relative speeds between our current now, and present location of the “matter that emitted that light” down to only 3.3c?

Randall, first off thanks for thinking about these things (which are sort of mind stretchers) I was apprehensive that virtually nobody would try out the calculation. BTW I think actually punching in the numbers is important to understanding. Are you doing that? I hope so. Dont just take words, mine or anyone's, without doing a little calculation.

Otherwise, about your comment...I don't understand what you mean by "near" the BB since it did not occur at some point in space, it occurred everywhere, so it is not a very meaningful reference point that one can be near to or far from. Maybe I just don't understand what you are saying and you should try to explain again.

I think I understand this question part of your comment

How do we get the relative speeds between our current now, and present location of the “matter that emitted that light” down to only 3.3c?

so I will try to answer.

Forgive me if I proceed very slowly and take a long time

Gen Rel is basically the einstein equation. The einstein eqn has a lot of solutions and one very popular one is the flat FRW metric. (there are pos and neg curved versions of FRW but observations show U so nearly flat that people mostly use the flat version----the spatial sections are flat).

The calculators use the FRW metric---they don't force you to use the flat case (e.g. 0.27+0.73 = 1)---they let you play around with pos and neg spatial curvature if you want---but I am sticking to the commonly-used flat case to keep it simple.
There is an absolute time parameter built into the FRW metric so words like "now" and "then" don't need quotemarks----they have a clear unambiguous meaning. This is Gen Rel (1915 theory) not Special Rel, and FRW is one particular solution. so there is a universal clock.

If I understand your question. You ask HOW COULD THE RECESSION SPEED of that matter be 57 c back when it emitted the light and NOW BE ONLY 3.3 c?

The answer is that gravity slowed the expansion down. the universe was denser back then, and matter (both dark and baryonic) was more concentrated so it played a dominant role. It slowed expansion.

Now, by contrast, the U has expanded so much that the same amount of matter is all thinned out in a larger vol. So the cosmo constant or dark energy has become dominant, and the expansion is slowly accelerating. But for MOST of that long 13 billion years matter was concentrated enough to overwhelm that acceleration effect and the expansion was slowing. There are some nice hornshape pictures that show this----maybe I can find a link---the scale parameter plotted as a function of time, at first convex (slowing) and then inflecting to concave (speeding up)

For now, Randall, please let me know if you have actually used the Morgan calculator. If you have, then we could do another example with a less extreme redshift-----like say z=10-----then we wouldn't be straining the accuracy of the calculator and might be able to learn more.

 

[marcus]

Here is another example, a less extreme redshift

Same numbers (0.27, 0.73, 71) but now z=10

age then (when light emitted) 0.48 billion years
age now (when light received) 13.66 billion years

distance from our galaxy then 2.86 billion LY
distance from our galaxy now 31.5 billion LY

emitter's recession speed away from our galaxy then 3.94 c
recession speed now 2.28

================

this is a better example because the numerical accuracy of Morgan's calculator can handle it better and the round-off is not such a problem.
and you get the same qualitative effects

1. recession was faster in the past, but it slowed down from around 4c down to near 2c
basically this is because the Hubble parameter has declined*.

2. the light travel time is 13.18 billion years, a substantial part of the age of the universe. (to get it you just subtract the age then from age now: 13.66 - 0.48)

3. the distance (which is a straightline shortest-distance-type distance in this simple spatial flat case) is much bigger now----around 31 versus 3 billion LY----because the universe expanded a lot during those 13 some billion years.

The calculator link is
http://www.earth.uni.edu/~morgan/ajjar/Cosmology/cosmos.html

Anybody have questions about this?

* in this example you can see from the calculator that the Hubble parameter declined from 1347 to the present value of 71

 

[RandallB]

Marcus,
Couldn’t find the (Ned) Wright calculator –
SO, just sticking with the Morgan calculator I see several different measurements points and I’m not clear which ones are used.

I like your use of the CBR as everything everywhere “saw” the universe become transparent at the same time as the “fog” lifted; take it as z=1100.

Let me define the points of reference:
“US-THEN” Where and when ‘our’ matter was while send our CBR emissions when the fog lifted.

“US –NOW” Some 13 Billion Yrs later and for the first time seeing light from an ‘OBJECT’ sending CBR from back then.

“OBJECT-THEN” Where and when the light we see at some sky point from which its CBR originated from – some 13 Billion Light Yrs from us. The first visual notice we are given that something is there.

“OBJECT-NOW” A location much future away we will need to wait a VERY long time to see any light being generated at that location “now”. In fact, it may take turning the expansion around to a contraction to ever actually see this light being generated 'now'.

I think I had been misreading Morgan - expecting to give the speeds between us as a receiver which would be “US-NOW” and the actual point of sending the CBR we see which would be “OBJECT-THEN”.

But maybe I should read:
the RED numbers as the speeds and distance between “US-THEN” and “OBJECT-THEN”. (To far apart to know about each other and will not see till we see each others CBR at a time defined as “NOW” for both “us” and “object”.)

And the Black numbers for between “US –NOW” and “OBJECT-NOW”.
(much to far apart to expect any kind of contact except with each others ancient history)

So for a z=1100 do we have a relative speed between “US –NOW” and “OBJECT-THEN” that would be more directly related to the z of 1100 we see from the CBR? That is relative to the time place and speed of the object that generated the CBR light we observe.

I guess I’m seeing a slowing down of the Hubble expansion in the calculator results I didn’t expect. It makes me suspect I’m not reading ( or understanding) the results correctly.

 

[pervect]

<rubbing eyes> OK, I get 14,000 Mpc, not 12,000 Mpc, when I enter the numbers into Ned Wright's calculator today. Which is 45 GLy. This is with the default $\Lambda$-CDM values of .27 and .73 entered into Ned Wright's calculator and z=1100.

This corresponds to V_now of 994,000 km/sec, just by multiplying by Hubble's constant (the current value).

To find the velocity "at emission", though, we need the generalized Friedmann equation.

Unfortunately I can't quite figure out what $\Omega_{ro}$ is supposed to be, numerically in

http://www.astro.ucla.edu/~wright/Distances_details.gif

eq 4. I assume that it represents the effect of radiation. It's a bit puzzling why this isn't an input parameter into the model in the first place.

My text (MTW) covers the Friedmann equation with totally different notation, unfortunately, so it's not a big help.

[add]I see that Morgan's calculator spits V @ emission out for us automatically. I guess I'm concerned that the radiation effects may not be being modeled, because we may not have good information on them - i.e. we may be implicitly assuming that $\Omega_{ro}$ = 0 when we use either Morgan's or Ned Wright's calculator.

[add^2] This is probably not important for z=10, but may be important for z=1100.

 

[marcus]

... I assume that it represents the effect of radiation. It's a bit puzzling why this isn't an input parameter into the model in the first place.
...

I think you are right. I think it is approximately zero as fraction of total energy density and therefore usually ignored or lumped in with matter.

I guess I'm concerned that the radiation effects may not be being modeled, because we may not have good information on them - i.e. we may be implicitly assuming that $\Omega_{ro}$ = 0 when we use either Morgan's or Ned Wright's calculator

in the usual percentage breakdown dark energy or the cosm. const is 73% and matter is 27%
and radiation is a fraction of a percent, so it is "in the noise". One is not expressing the 27% accurately enough for something as slight as density of radiant energy to register. So instead of writing down "~0%", which would seem a bit pedantic, one just ignores it.

However if you are especially interested I have seen detailed energy breakdowns of the contents of the universe with estimates carried out to several decimal places. I might be able to find a link.

A possibly interesting detail is that almost all the radiation energy (which is already small on a per unit volume basis) is in the CMB!
Compared with the CMB the rest of the radiant energy-----like starlight for example----is negligible. Note that the temperature of space is the temperature of the CMB radiation (the starlight contributes almost nothing to the temperature). And Planck's cavity radiation density formula can provide the estimate---using the measured 2.75 Kelvin as input.

Let me know if you are especially interested in the average density of EM radiation in space and I will take time to look it up or do a back-of-the-envelope.

but for the moment, just call it zero. After all baryonic matter is only around 4%!

 

[pervect]

I think you are right. I think it is approximately zero as fraction of total energy density and therefore usually ignored or lumped in with matter.
in the usual percentage breakdown dark energy or the cosm. const is 73% and matter is 27%
and radiation is a fraction of a percent, so it is "in the noise". One is not expressing the 27% accurately enough for something as slight as density of radiant energy to register. So instead of writing down "~0%", which would seem a bit pedantic, one just ignores it.

However if you are especially interested I have seen detailed energy breakdowns of the contents of the universe with estimates carried out to several decimal places. I might be able to find a link.

Well, I'm mainly interested in how far back we can go with the online calculators. Mostly because it was in your example, but it also seems like a reasonable thing to want to know about the results (the realm of their validity).

The effect of radiation becomes more important the earlier one goes.

If we look at the generalized Friedmann eq, Ned Wright version, we find that

$$\dot{a} = H_0 \sqrt{\frac{\Omega_m}{a} + \frac{\Omega_r}{a^2} + \Omega_v a^2 + (1-\Omega_t)}$$

http://www.astro.ucla.edu/~wright/Distances_details.gif, eq 4

a is defined such that it is now unity, $H_0$ is the current value of the Hubble constant, and the various $\Omega$ represent the proportion of energy in mass, radiation, and vacuum energy.

Early in the history of the universe, a was much smaller. Since a=1/z+1, for z=1000, a was about .001. We can see that this amplifies the importance of the $\Omega_r$ term by a factor of 1000.

Thus in order for the above equation to be accurate to this era, we must have $\Omega_r << .001 \Omega_m$.

 

[marcus]

The effect of radiation becomes more important the earlier one goes.
...

good observation!
That is quite true. In fact in the very early universe radiation is dominant.

so perhaps you do not care what modest fraction of a percent it is now

Instead, perhaps you would like to know the epoch, or the redshift, at which radiation ceases to be dominant and matter takes over?

my guess is that it is well before 300,000 years old but that is merely a guess. It would take some looking up. Would you like to do that?

My sense is you are very capable of searching out that kind of information, so if you would like to track it down, please do! (I will be interested to know the result.)

 

[SpaceTiger]

The epoch of matter-radiation equality is around z~10000 and $\frac{\Omega_r}{\Omega_m}\sim 10^{-5}$. The radiation content of the universe is presently dominated by the CMB, so the previous relation can be derived by considering the energy density of a blackbody (at 2.7 K):

$$u=aT^4$$

and comparing it to the energy equivalent of the critical density:

$$\rho_c=\frac{3H_0^2}{8\pi G}$$

 

[marcus]

Excellent! Thanks SpaceTiger. See also:

A possibly interesting detail is that almost all the radiation energy (which is already small on a per unit volume basis) is in the CMB!
Compared with the CMB the rest of the radiant energy-----like starlight for example----is negligible. Note that the temperature of space is the temperature of the CMB radiation (the starlight contributes almost nothing to the temperature). And Planck's cavity radiation density formula can provide the estimate---using the measured 2.75 Kelvin as input.

Let me know if you are especially interested in the average density of EM radiation in space and I will take time to look it up or do a back-of-the-envelope.

so maybe we should use the radiation density formula----involving the fourth power of temp----that Nick has written for us, and do a back-of-envelope calculation?

If I remember correctly the critical density (which one assumes is THE REAL density in a flat model) is around one joule per cubic kilometer. Nick provides us with a formula which we could use to check that.
the rest should be easy. would anyone like to try?

=========================

EDIT: I am delighted with how the thread is going and will reply to your next post here, pervect, so as not to add a superfluous post. I don't think you really need to worry about radiation in what you are doing since it is not close enough to z = 10,000. but if you happen to want the constant "a" in the fourthpower energy density law it is

the same as the stefanboltzmann constant but multiplied by 4/c
so stefanboltzmann is 5.67E-8 W/sq meter per K to the fourth
and the "a" in what Nick wrote is
7.56E-16 Joule/cubic meter per K to the fourth

What that means for example is that if you have a cubic kilometer of space at temp 100 Kelvin then the thermal radiation in that cubic kilometer is 75.6 Joules
but if the temp is only 1 Kelvin then the radiation in the cubic kilometer is only 75.6 E-8 Joules
which is to say a bit under a microJoule

the critical density of the universe, at present, is around one joule per cubic km
I calculated it one time and seem to remember something like 0.86 but anyway something like 1.
You can calculate it from the Hubble parameter using the formula Nick mentioned.
And the critical density corresponds pretty well with the actual real density (the galaxies the dark energy etc. plus a little light)
And so in an average cubic km most of that Joule is made up of dark energy and dark matter and a little baryonic.

things seem to be OK and it is my bedtime so buenas noches.