Is the universe 13.7 Billion years old? There seems to be a contradiction

[surajt88]

The universe is said to be 13.7 Billion years (http://en.wikipedia.org/wiki/Age_of_the_universe" .
So, for the present galaxy UDFj-39546284 and the present Earth to have traveled as far as they are now, and even if we assume that the big bang occurred right between Earth and UDFj-39546284, it must have taken at least 6.6 Billion years for the objects to have been positioned as they are now (Even if they are assumed to have traveled at the speed of light). The light that left the galaxy UDFj-39546284 6.6 Billion years after the Big Bang (as we see now) should have taken another 13.2 Billion years to reach us.
So the universe must be at least 19.8 Billion years old now. Where am I wrong?

 

[Drakkith]

You havn't taken inflation and expansion of the universe into account.

 

[Dmitry67]

What do you mean by "big bang occurred right between Earth and UDFj-39546284"?

No point of space is special, there is no spatial location where Big bang had occured. It had occurred everywhere

 

[surajt88]

Thanks a lot...But both Earth and that galaxy are receding each other now. I would suppose both(considering Earth and the galaxy are two points in space) must have been very close at some point in time. How long would you suppose it took for the two points to be in the current position from being very close. The light rays that left the galaxy then took 13.2 Billion light years to reach earth.
Of course I haven't taken into account inflation and the expansion of the universe. May I know how it affects what I mentioned above.

 

[Drakkith]

When that light left that galaxy it would not have been 13.6 billion light years away, but probably MUCH closer. Something like 500 million light years perhaps. Thanks to inflation and expansion, that light has taken 13.2 billion years to reach us since we were speeding away and it had to catch up.

Edit: I think this is correct. I'm really tired due to being on shift for 12 hours now, so I can't really think that clearly at the moment. About to get off though!

 

[Bill_K]

This discussion has already gotten out of hand, so let me start over. In the first place the currently accepted age of the universe is 13.75 Gy, not 13.7.

Secondly, yes when the light from that galaxy was emitted 13.6 Gy ago, it *was* 13.6 Gy away. Light, after all, travels at the speed of light!

However, the galaxy did not take 13.6 Gy to arrive at its position. Immediately following the Big Bang was a period of cosmic inflation, in which space itself underwent a rapid exponential expansion. The expansion of space is not limited to light speed.

 

[Nabeshin]

Inflation is completely unnecessary and irrelevant to this discussion. Since inflation occurred when the universe was but a tiny quark-soup of sorts, there were certainly no galaxies around then to be emitting light. So we can safely start our analysis in a post-inflationary era where the universe is described by a normal ΛCDM model.

Ok, let's take UDFj-39546284 as an example where redshift z~10. This means that when light from this galaxy which is just now reaching us was emitted, the universe was ~11 times smaller than it is today. So what you should imagine is the two objects, namely ourselves and UDFj-39546284 much closer at the beginning of the universe, ~300Myr after the big bang. UDFj-39546284 emits some light, which begins traveling through space towards us. Since the universe is expanding (and also flat), it takes much longer than you would naively think for the light to actually reach us. You can think of, if you like, extra "space" being created between the two objects which the beam of light must now traverse. The net effect here is that when the light finally reaches us, it has traveled perhaps the 1 billion years (made this number up. Someone can calculate the actual comoving distance if they want) it might have, but over 13 billion years.

Also note that in cosmology there is some ambiguity as to what is meant by "distance". For example, there are luminosity distances, comoving distances, and angular diameter distances, none of which are the same when we are looking at distant objects.

 

[Drakkith]

So light from a galaxy that is 13 billion LY away was emitted...when?

 

[surajt88]

I seem to be getting closer to understanding this now. I'll put it in my words. Plz correct me if I'm wrong.
1. Big bang occurs.
2. After 'X' billion years, UDFj-39546284 and the point in space which is to become Earth now, are 'A' light years apart. The light which we see now, left the galaxy then.
3. Universe continues to expand.
4. 13.75 billion years after the big bang, UDFj-39546284 and the Earth , are 13.2 billion light years apart. The light reaches us now.
5. So, if we know the present rate of increase in distance between the two, we may calculate the distance between them when the light left the galaxy.

So far so good...
As far as i know, the distance is calculated based on the red shift when the light left the galaxy (i.e when they were 'A' light years apart or around 480 million years after the big bang). So 'A' must be 13.2 billion light years as we see now.
Also the time the light took to reach us cannot be determined without considering the rate of expansion of the universe.

If the universe was 13.2billion light years in radius 480 million years after the big bang, I would suppose the universe must be older and bigger than what is said now.

 

[qraal]

Hi Guys

Best place to go to help illuminate this question is Prof. Ned Wright's Cosmology Tutorial...

Cosmology Tutorial

...where you'll learn the difference between the Light Travel Time and the Co-moving radial distance. Two quite different distances, often confused in the media. Using the Cosmology Calculator, with the input of a Light Travel-Time of 13.2 Gyr, one gets a co-moving radial distance of 31.69 Glyr. Why so much further away? Because the space-time between the two points has expanded while light has been traveling. Read more to learn just how that works in relativity.

 

[Andrew Mason]

If the universe was 13.2billion light years in radius 480 million years after the big bang, I would suppose the universe must be older and bigger than what is said now.

I have yet to find a satisfactory explanation to the question you are asking.

Suppose a galaxy in a distant part of the universe traveling at .9998 x the speed of light (\gamma = 50) away from Earth (let's assume the Earth began at the time of the big bang) sent a light signal toward the Earth at a time, according to its own clocks, of BB+100m (100 million years after the Big Bang). Let's assume at the time of the Big Bang the galaxy and the Earth were in the same place in space and 100 million years later, according to the galaxy clock, they are separated by d1 = .9998c x 100m = 99.98 million light years as measured by an observer in the galaxy.

According to the galaxy clock, that light should arrive at Earth after another 99.98 million years have elapsed. By that time, the galaxy would be d2 = 100+99.98 x.9998 = 199.96 light years away from the Earth as measured in the galaxy's frame of reference.

Now that same event (sending the light signal) would be observed by an Earth observer as follows:

Time when light sent from galaxy: t = \gamma t_1' = 50 x 100m = 5 billion years (after BB).

Separation of galaxy at that time: d = \gamma d_1' = 50 x 99.98m = 4.999b (billion) light years.

Time on Earth when light arrives: t = \gamma t_2' = 50 x (199.98m) = 9.999b years.

So light from a galaxy that appears to us to be a tad less than 5 billion light years away when sent, arrives on Earth a tad less than 10 billion years after the Big Bang as measured on the earth. If light arrives from a galaxy that was 13 billion light years away when it was sent, as measured by the earth, this means that it arrives a tad less than 26 billion years after the Big Bang.

So my question is: how can the age of the universe (in Earth years) be much less than double the distance of the farthest galaxy/c ?

AM

 

[Arbitrageur]

Inflation is completely unnecessary and irrelevant to this discussion. Since inflation occurred when the universe was but a tiny quark-soup of sorts, there were certainly no galaxies around then to be emitting light. So we can safely start our analysis in a post-inflationary era where the universe is described by a normal ΛCDM model.

correct, inflation has nothing to do with it.

Hi Guys... with the input of a Light Travel-Time of 13.2 Gyr, one gets a co-moving radial distance of 31.69 Glyr. Why so much further away? Because the space-time between the two points has expanded while light has been traveling. Read more to learn just how that works in relativity.

Thanks for the good answer.

I agree, the 13.2Gyr is sometimes referred to as "lookback time", and is not the current distance which would be ~32Glyr.

I seem to be getting closer to understanding this now. I'll put it in my words. Plz correct me if I'm wrong.
4. 13.75 billion years after the big bang, UDFj-39546284 and the Earth , are 13.2 billion light years apart. The light reaches us now.

qraal is correct, the distance now would be much greater, more like ~32Glyr.

Time when light sent from galaxy: t = \gamma t_1' = 50 x 100m = 5 billion years (after BB).

If you had Earth A and Earth B 100 million light years apart both traveling in parallel paths away from the galaxy at 0.9998c and sending light beams to each other, then your math would be correct that the galaxy would observe that it would take 4.99 billion years for the light to travel that 100 million light year distance between Earth A and Earth B. But that's not the situation we have with the Earth receding from the galaxy. The effect of relativistic time dilation is to affect the frequency (or wavelength) of the light observed, not the time it takes to reach the observer since it wouldn't take more than 200 million light years from either frame of reference for the light to arrive if it left when the separation was 100 million light years, if space wasn't expanding.

But cosmologists believe that space itself is expanding, which isn't accounted for in your equations at all. The redshift we observe is largely the result of the expansion of space governed by general relativity, and also partly the result of the Doppler effect of special relativity. For a proper treatment of the effects of general relativity regarding a galaxy moving away from the Earth, see this paper:
A comparison between the Doppler and cosmological redshifts, by Maria Luiza Bedran
http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/cosmo/doppler_redshift.pdf

That paper may also shed some light on the original question as suggested by the abstract:
"We compare the Doppler effect of special relativity with the cosmological redshift of general
relativity in order to clarify the difference between them. Some basic concepts of observational cosmology, such as the definitions of distance and cosmological parameters, are also presented."

 

[Andrew Mason]

If you had Earth A and Earth B 100 million light years apart both traveling in parallel paths away from the galaxy at 0.9998c and sending light beams to each other, then your math would be correct that the galaxy would observe that it would take 4.99 billion years for the light to travel that 100 million light year distance between Earth A and Earth B.

I don't follow you there. Length contraction only applies in the direction of relative motion. There would be no length contraction perpendicular to the direction of travel.

But that's not the situation we have with the Earth receding from the galaxy. The effect of relativistic time dilation is to affect the frequency (or wavelength) of the light observed, not the time it takes to reach the observer since it wouldn't take more than 200 million light years from either frame of reference for the light to arrive if it left when the separation was 100 million light years, if space wasn't expanding.

I'm having trouble following you there too. If that was the case, then the observers on the Earth would not measure the speed of light as c. Would that not violate the principle of relativity?

AM

 

[bcrowell]

A mistake that has been made at least once in this discussion is addressed on p. 10 of this article: http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf See the box titled "How large is the observable universe?"

The use of Lorentz gamma factors in #11 doesn't actually work, since Lorentz transformations are local things. GR doesn't have the equivalent of a global Lorentz transformation.

 

[Andrew Mason]

A mistake that has been made at least once in this discussion is addressed on p. 10 of this article: http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf See the box titled "How large is the observable universe?"

The use of Lorentz gamma factors in #11 doesn't actually work, since Lorentz transformations are local things. GR doesn't have the equivalent of a global Lorentz transformation.

This Scientific American article is a little short on detailed explanations of any of this. The following quote is particularly puzzling because it invokes a form of cosmological ether:

The solution is that special relativity applies only to “normal” velocities—motion through space. The velocity in Hubble’s law is a recession velocity caused by the expansion of space, not a motion through space.

Einstein took great pains to show that there is no such thing as motion relative to "space" - there was only motion relative to different inertial frames of reference.

The treatment of recession velocity of a galaxy as something fundamentally different than motion relative to an inertial frame of reference is never explained. Is it any wonder so many cosmologists are "confused"?

AM

 

[Andrew Mason]

Cosmological theories of superluminal expansion of space etc. are interesting but one should remain skeptical. The authors of that Scientific American article (Lineweaver and Davis) seem to think that there is a right and wrong explanation. This in itself makes be very suspicious of their claims.

In science there are only two kinds of explanations: wrong ones and possibly right ones. There are explanations that conflict with proven facts. These are wrong explanations. There are explanations those that do not yet conflict with proven facts. These are possibly right.

Lineweaver and Davis identify "wrong" explanations not because such explanations conflict with any facts. Rather they identify them as "wrong" because they conflict with other explanations that are possibly right (eg. the theory of the expansion of space and superluminal speeds).

The author of http://arxiv.org/abs/astro-ph/0601171" does not seem to agree that superluminal speeds are needed to explain cosmological red shift and other phenomena:

AM

 

[Dan Frank Kue]

Great thread! I'm still reviewing all the links and thoughts on this page...

IF all this is true, and the universe is NOT 27.4 billion light years across (say, 13.7 billion light years to the left and 13.7 billion light years to the right)

THEN, those distant, very young galaxies must actually be very close together, moving away from us, but also closer to each other...

SO the Universe isn't exactly expanding; those distant galaxies are moving (at near the speed of light) TOWARD that singularity, the Big Bang.

Galaxies apparently 26.4 billion light years apart are actually less than a billion light years apart and CONVERGING, not expanding.

This is the part where my brain starts trickling out my ears, which is nice because then it won't explode. Is there somewhere to get this cleared up?

 

[bcrowell]

This Scientific American article is a little short on detailed explanations of any of this. The following quote is particularly puzzling because it invokes a form of cosmological ether:

The solution is that special relativity applies only to “normal” velocities—motion through space. The velocity in Hubble’s law is a recession velocity caused by the expansion of space, not a motion through space.

The material you've quoted doesn't have anything to do with an aether.

Einstein took great pains to show that there is no such thing as motion relative to "space" - there was only motion relative to different inertial frames of reference.

The quote doesn't refer to motion relative to space, it refers to motion through space.

The treatment of recession velocity of a galaxy as something fundamentally different than motion relative to an inertial frame of reference is never explained. Is it any wonder so many cosmologists are "confused"?

Cosmologists are not confused. This is standard GR that's been understood since the 50's.

 

[bcrowell]

The author of http://arxiv.org/abs/astro-ph/0601171" does not seem to agree that superluminal speeds are needed to explain cosmological red shift and other phenomena:

What Davis and Lineweaver are referring to as a superluminal speed is the following. Fix a surface of simultaneity, which is defined by the FRW time coordinate t (which would be measured by observers at rest relative to the Hubble flow). Within this surface of simultaneity, construct a spacelike geodesic from galaxy A to cosmologically distant galaxy B. Find the proper length L of this geodesic. The derivative dL/dt is greater than c for many galaxies. This does not contradict SR, since SR doesn't apply on cosmological distance scales. Only locally does GR reduce to SR. The following may help.

FAQ: What does general relativity say about the relative velocities of objects that are far away from one another?

Nothing. General relativity doesn't provide a uniquely defined way of measuring the velocity of objects that are far away from one another. For example, there is no well defined value for the velocity of one galaxy relative to another at cosmological distances. You can say it's some big number, but it's equally valid to say that they're both at rest, and the space between them is expanding. Neither verbal description is preferred over the other in GR. Only local velocities are uniquely defined in GR, not global ones.

Confusion on this point is at the root of many other problems in understanding GR:

Question: How can distant galaxies be moving away from us at more than the speed of light?

Answer: They don't have any well-defined velocity relative to us. The relativistic speed limit of c is a local one, not a global one, precisely because velocity isn't globally well defined.

Question: Does the edge of the observable universe occur at the place where the Hubble velocity relative to us equals c, so that the redshift approaches infinity?

Answer: No, because that velocity isn't uniquely defined. For one fairly popular definition of the velocity (based on distances measured by rulers at rest with respect to the Hubble flow), we can actually observe galaxies that are moving away from us at >c, and that always have been moving away from us at >c.[Davis 2004]

Question: A distant galaxy is moving away from us at 99% of the speed of light. That means it has a huge amount of kinetic energy, which is equivalent to a huge amount of mass. Does that mean that its gravitational attraction to our own galaxy is greatly enhanced?

Answer: No, because we could equally well describe it as being at rest relative to us. In addition, general relativity doesn't describe gravity as a force, it describes it as curvature of spacetime.

Question: How do I apply a Lorentz transformation in general relativity?

Answer: General relativity doesn't have global Lorentz transformations, and one way to see that it can't have them is that such a transformation would involve the relative velocities of distant objects. Such velocities are not uniquely defined.

Question: How much of a cosmological redshift is kinematic, and how much is gravitational?

Answer: The amount of kinematic redshift depends on the distant galaxy's velocity relative to us. That velocity isn't uniquely well defined, so you can say that the redshift is 100% kinematic, 100% gravitational, or anything in between.

Davis and Lineweaver, Publications of the Astronomical Society of Australia, 21 (2004) 97, msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf

 

[Andrew Mason]

The material you've quoted doesn't have anything to do with an aether.


The quote doesn't refer to motion relative to space, it refers to motion through space.

Perhaps then, you could explain the reference frame used to determine motion through space. ie. what is motion through space relative to if not an inertial frame of reference? An inertial frame may have motion relative to some other inertial reference frame. If you say it is moving through space but not relative to space I lose you.

Cosmologists are not confused. This is standard GR that's been understood since the 50's.

Ok. The article referred to astronomers being confused. I don't think it is GR per se that is the problem. Rather it is the physical interpretation of GR that is the issue.

AM

 

[Andrew Mason]

What Davis and Lineweaver are referring to as a superluminal speed is the following. Fix a surface of simultaneity, which is defined by the FRW time coordinate t (which would be measured by observers at rest relative to the Hubble flow).

I lose you there. If they are at rest relative to the Hubble flow does that not mean that they are at rest relative to some inertial reference frame? If so, they must be at rest relative to each other. If not, then the Hubble flow does not represent an inertial reference frame, rather a group of different inertial reference frames that are all traveling at the same speeds but in different directions. But if that is the case, then the concept of simultaneity goes out the window (metaphorically speaking). As I understand it, simultaneity has meaning only for spatially separated observers at rest relative to the same inertial reference frame.

Question: How can distant galaxies be moving away from us at more than the speed of light?

Answer: They don't have any well-defined velocity relative to us. The relativistic speed limit of c is a local one, not a global one, precisely because velocity isn't globally well defined.

If velocities are not defined then it seems to me that distance and time are not well defined for cosmological scales. If so, does it really make any sense to speak about the size or age of the universe?

AM

 

[bcrowell]

Perhaps then, you could explain the reference frame used to determine motion through space. ie. what is motion through space relative to if not an inertial frame of reference? An inertial frame may have motion relative to some other inertial reference frame. If you say it is moving through space but not relative to space I lose you.

The motion can be relative to any inertial frame of reference you choose, provided that that inertial frame, which is a local thing, is defined in the neighborhood of the object you're talking about. GR doesn't have global frames of reference.

 

[bcrowell]

I lose you there. If they are at rest relative to the Hubble flow does that not mean that they are at rest relative to some inertial reference frame?

Yes, but of course every material object is trivially at rest relative to some inertial reference frame -- a frame fixed to the object.

If so, they must be at rest relative to each other.

No, because frames in reference in GR are local, not global. If galaxies A and B are separated by a cosmological distance, then there is no frame of reference that encompasses them both.

If not, then the Hubble flow does not represent an inertial reference frame, rather a group of different inertial reference frames that are all traveling at the same speeds but in different directions.

Right.

But if that is the case, then the concept of simultaneity goes out the window (metaphorically speaking). As I understand it, simultaneity has meaning only for spatially separated observers at rest relative to the same inertial reference frame.

Simultaneity is never uniquely defined in SR or GR. That's why the L and dL/dt defined in #19 are not uniquely determined. They's just one possible way of defining the distance between the two galaxies and the rate at which their separation increases.

If velocities are not defined then it seems to me that distance and time are not well defined for cosmological scales.

It's the same situation as in SR. They can be defined, but the definitions are not unique. In SR, the length of a meter stick can be defined, but the definition depends on your state of motion relative to the meter stick.

If so, does it really make any sense to speak about the size or age of the universe?

When people refer to these things, they're either implicitly or explicitly assuming some specific definition, such as the definitions of L and t given in #19.

-Ben

 

[Arbitrageur]

I don't follow you there. Length contraction only applies in the direction of relative motion. There would be no length contraction perpendicular to the direction of travel.

You're right about length contraction. But I didn't say there would be length contraction, I said there would be time dilation. So the length doesn't contract, it just takes 50 times longer to go the same length, from the perspective of the observer outside that reference frame.

I'm having trouble following you there too. If that was the case, then the observers on the Earth would not measure the speed of light as c. Would that not violate the principle of relativity?

They would still measure the speed of light as c. Why wouldn't they?

 

[Andrew Mason]

You're right about length contraction. But I didn't say there would be length contraction, I said there would be time dilation. So the length doesn't contract, it just takes 50 times longer to go the same length, from the perspective of the observer outside that reference frame.

They would still measure the speed of light as c. Why wouldn't they?

Perhaps I misunderstood what you were saying. If it took 5 billion years for light to travel the distance between A and B in the galaxy frame then by definition, the separation would have to be 5 billion light years in the galaxy frame, would it not? So I thought you were saying there was length contraction making it appear to be 100 million lyr.

Light beams at right angles to the direction of motion tend to be rather tricky things. What happens, I think, is that the light is not seen as being sent at right angles but at a sharp forward angle (\alpha) such that \sin\alpha = 1/\gamma.

AM

 

[bcrowell]

If it took 5 billion years for light to travel the distance between A and B in the galaxy frame then by definition, the separation would have to be 5 billion light years in the galaxy frame, would it not?

GR doesn't have global frames of reference, only local ones. See #19.

 

[Arbitrageur]

Perhaps I misunderstood what you were saying. If it took 5 billion years for light to travel the distance between A and B in the galaxy frame then by definition, the separation would have to be 5 billion light years in the galaxy frame, would it not?

The galaxy frame sees the two receding clocks going slowly. The speed of light it will see in the receding reference frame is determined by those receding clocks. So the light is still traveling at the speed of light by traveling 100 million light years in 100 million years according to the receding clocks which is 5 billion years according to the galaxy clock. The distance is still 100 million light years in either reference frame because as you said we aren't measuring distance in the direction of motion.

100 million light years in one reference frame is the same as 5 billion years in the other reference frame (I'm using the math example you provided with 50x time dilation). So if you're on one of the two Earths traveling parallel 100 million light years apart it only takes one hundred million years for light to reach the other earth.

This animation illustrates the concept: http://en.wikipedia.org/wiki/File:Time_dilation02.gif

Note the passage of time in the two reference frames represented by the clocks at the top. It illustrates a time dilation of 2:1 rather than 50:1, but it's the same concept. The thing in motion takes twice as long to travel when the clock moves half as fast, but you have to pay attention to which reference frame you're in. Time passes normally to you inside your frame of reference, no matter what reference frame you're in. You have to observe the motion from a different reference frame to see the time dilation, and that's why the math didn't work in your original post, because you didn't really have a different reference frame. The animation shows a different scenario where your 50:1 math will work because the light is traveling between two different points in the same moving reference frame outside your reference frame.

 

[bcrowell]

If you had Earth A and Earth B 100 million light years apart both traveling in parallel paths away from the galaxy at 0.9998c and sending light beams to each other, then your math would be correct that the galaxy would observe that it would take 4.99 billion years for the light to travel that 100 million light year distance between Earth A and Earth B.

No, Andrew Mason's calculation would still be incorrect in that case. The planets are separated by cosmological distances, so SR is not a useful approximation. Therefore you can't use Lorentz transformations to calculate anything meaningful.

-Ben

 

[Arbitrageur]

No, Andrew Mason's calculation would still be incorrect in that case. The planets are separated by cosmological distances, so SR is not a useful approximation.

Thanks. So roughly what would be the largest distance over which SR would be a useful approximation? any idea?

 

[bcrowell]

Thanks. So roughly what would be the largest distance over which SR would be a useful approximation? any idea?

This is essentially what the Riemann tensor measures. The flatness of spacetime in SR can be defined by the fact that parallel-transporting a vector around a closed loop doesn't change the vector. When we observe that it does change, the Riemann tensor measures how much it changes. The fractional change in the vector is essentially the Riemann tensor multiplied by the area A enclosed by the loop. (I'm leaving out the tensor indices and oversimplifying, but that's the basic physical idea.) The Riemann tensor sets a length scale L (the Riemann tensor varies as 1/L^2) such that the fractional change in a vector under parallel transport around a closed loop is on the order of A/L^2. For a cosmological solution, L is basically the inverse of the Hubble constant. So basically if you want to estimate the size of the fractional error when you apply SR over a distance d, the error goes like d^2H^2, where H is the Hubble constant. In Andrew Mason's example, d was 5 billion light years at a time when 1/H was 14 billion light years. That means that dH was of order unity, so the fractional error in applying SR was of order unity.

As an example where SR *is* a good approximation, the Andromeda galaxy lies at a distance of d=2.5 million light years, so d^2H^2 is on the order of 10^-7. Therefore SR is an excellent approximation over those distances. We can define a frame of reference that encompasses both our galaxy and Andromeda. We can define multiple frames of this type, and transform among them using Lorentz transformations. We can unambiguously determine how much of the Doppler shift of light from Andromeda is kinematic (almost all of it) and how much is gravitational (almost none of it). If light takes 2.5 million years to get to us from Andromeda, it's correct to multiply by c and get Andromeda's distance from us. We wouldn't be able to do any of these things on a distance scale of billions of light years.

 

[marcus]

Thanks. So roughly what would be the largest distance over which SR would be a useful approximation? any idea?

Ben Crowell might have a different idea but I'd say that what's a "useful" approximation depends a lot on how accurate you want to be and what kind of calculation you have in mind.

Just to pull a number out of the hat, I'm thinking of an application where 140 million light years is too far for comfort. That is the instantaneous distance that is increasing at 1 percent of the speed of light.

By instantaneous distance I mean what you would measure by ordinary means (radar ranging, yardsticks, tape measure) if you could stop the expansion process at this moment. That is the measure of distance that Hubble law tells us the instaneous rate of increase of.

SR is flat non-expanding geometry, so it is mainly good where gravity is not too strong (so curvature can be neglected) and where the expansion of distance is so small or slow that it can be neglected.
=====================
Oh! I see Ben already answered the question! I didn't need to post. But I will leave this up anyway because it illustrates a different way of responding to the question "what size distance can't you apply SR to usefully?"

 

[bcrowell]

Ben Crowell might have a different idea but I'd say that what's a "useful" approximation depends a lot on how accurate you want to be and what kind of calculation you have in mind.

I would definitely agree with this. I picked what the Riemann tensor measures (failure of parallelism) as my measure of how good or bad the approximation was, and that led to d^2H^2 as an order-of-magnitude estimate of the level of fractional error. But I'm sure it would be easy to come up with other examples, perhaps equally well motivated physically, where the fractional error would be on the order of (dH)^p, where p is 1 rather than 2, or something like that. OTOH, I think these things really do have to depend only on the quantity dH, since H is the only characteristic scale in the metric, and dH is the only unitless quantity you can construct from d and H.

BTW, the OP of this thread is a good example of how these error estimates work:

The universe is said to be 13.7 Billion years (check). [...Reasons based on SR...] So the universe must be at least 19.8 Billion years old now.

The OP tried using x=ct to determine the time required for a ray of light to travel a distance x. This is only valid in SR, and the error that results from applying it here is 44.5%, i.e., the fractional error is of order unity. This is exactly what we expect, since dH is of order unity.

 

[Jamders]

I think big bang was not a bang after all. for now may said. Observing our planet and Jupiter or Saturn, they are basically big masses with big gravitational forces, A solid center mass and an atmosphere where storms are visible that resemble galaxies. and if you'r in Florida can u see or feel a Typhon in Australia? Can u see it? All this math confusion seems similar to the ancient Greeks confusion thinking the planet was flat. This Universe is only a big storm that exist in the Elemental Gravity Mass Energy system wish is similar to his sons the palnets. even they hold flat formations when the distance reach some limits

 

[Jamders]

correct, inflation has nothing to do with it.

Thanks for the good answer.

I agree, the 13.2Gyr is sometimes referred to as "lookback time", and is not the current distance which would be ~32Glyr.

qraal is correct, the distance now would be much greater, more like ~32Glyr.



If you had Earth A and Earth B 100 million light years apart both traveling in parallel paths away from the galaxy at 0.9998c and sending light beams to each other, then your math would be correct that the galaxy would observe that it would take 4.99 billion years for the light to travel that 100 million light year distance between Earth A and Earth B. But that's not the situation we have with the Earth receding from the galaxy. The effect of relativistic time dilation is to affect the frequency (or wavelength) of the light observed, not the time it takes to reach the observer since it wouldn't take more than 200 million light years from either frame of reference for the light to arrive if it left when the separation was 100 million light years, if space wasn't expanding.

But cosmologists believe that space itself is expanding, which isn't accounted for in your equations at all. The redshift we observe is largely the result of the expansion of space governed by general relativity, and also partly the result of the Doppler effect of special relativity. For a proper treatment of the effects of general relativity regarding a galaxy moving away from the Earth, see this paper:
A comparison between the Doppler and cosmological redshifts, by Maria Luiza Bedran
http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/cosmo/doppler_redshift.pdf

That paper may also shed some light on the original question as suggested by the abstract:
"We compare the Doppler effect of special relativity with the cosmological redshift of general
relativity in order to clarify the difference between them. Some basic concepts of observational cosmology, such as the definitions of distance and cosmological parameters, are also presented."

the first stars should be at this present time at the edge or almost of the universe I m observing light that start traveling 13.5by ago. I was born 40 years ago right? but my location? what if that star is my neighbor which is let's said 5 million years apart?

 

[Arbitrageur]

As an example where SR *is* a good approximation, the Andromeda galaxy lies at a distance of d=2.5 million light years, so d^2H^2 is on the order of 10^-7. Therefore SR is an excellent approximation over those distances.

Thanks to you and Marcus for the explanations and clarification with the example, because without it I would have also thought the distance to Andromeda might also be considered cosmological per your earlier reply. You gave me enough to research so I can figure out how to do the calculations (I hope) on what the errors are in the SR approximations.

Thanks very much for teaching me something regarding the limitations of SR approximations!

 

[Andrew Mason]

So, I take it that on cosmological distance scales the unit of distance as measured by an observer on earth, the light year, is NOT the distance that light takes one year to travel as measured by an observer on earth. So what is it and how is it measured?

AM

 

[marcus]

So, I take it that on cosmological distance scales the unit of distance as measured by an observer on earth, the light year, is NOT the distance that light takes one year to travel as measured by an observer on earth. So what is it and how is it measured?

AM

That's right, a lightyear is the distance light would travel in a non-expanding universe in one year. In our galaxy (and local group of galaxies) the effect of expansion is essentially non-existent. There is an alternative unit of distance called "parsec" defined using parallax angle and a parsec is 3.26 lightyears. You could define a lightyear as 1/3.26 parsecs.

In cosmology typically what we measure is redshift. Distances and times are determined from redshift using a model.

One kind of distance measure is instantaneous "freeze-frame" distance which is what you would measure (say by how long light would take) if you could stop the expansion process right now.

A time measure that has a complicated relation to distance is the "lookback time" or "light travel time" that it is also possible to calculate from the redshift, using the model.

I'd say anyone interested in cosmology would be well-advised to learn to think in terms of the redshift. And get used to using the available calculators that convert redshifts to distances, and also give travel times.

If you google "cosmo calculator" and put in a redshift like z = 0.5 it will tell you the instantaneous distance now (labeled comoving in the readout) and also what the instantaneous distance WAS back when the light was emitted and started out on its way to us. That is labeled "angular size distance" in the readout.

These will be given both in parsecs and in lightyears (which are 1/3.26 of a parsec). The calculator will also give the light travel time in years. Not terribly useful as a measure of distance but nevertheless nice to know.

 

[twofish-quant]

Here is a useful paper:

http://arxiv.org/PS_cache/astro-ph/pdf/9905/9905116v4.pdf

The problem is here is defining what we mean by a "distance". What exactly is a distance, anyway. What do I mean when I say that television set is four meters away? Well, I do an experiment, and the answer comes back four meters. The interesting thing is that I can do different experiments and the answer comes back "four meters".

The complication with cosmology, is that if you do different experiments that come back with the same answer when you measure "short" distances, you end up coming back with different numbers. So when you are talking about something being "far" away, you have do define what you mean by distance and there are ten of so different definitions for "distance". The paper that I referenced lists then and tells you how to convert between different types of "distance."

 

[bcrowell]

So, I take it that on cosmological distance scales the unit of distance as measured by an observer on earth, the light year, is NOT the distance that light takes one year to travel as measured by an observer on earth. So what is it and how is it measured?

General relativity doesn't provide any uniquely defined way of measuring distances, nor does it associate a particular distance-measuring procedure with a particular observer's state of motion, as do Galilean relativity and special relativity.

 

[Tanelorn]

Ben, does this mean that there is a point where we no longer use distance, and that distance no longer has meaning? If so under what circumstances does this happen and what is the real world physical source for this?

 

[bcrowell]

Ben, does this mean that there is a point where we no longer use distance, and that distance no longer has meaning?

You can define measures of distance, but they're not uniquely defined. Someone else can define a different one, and it can be equally valid. In cosmology, the most convenient measure of distance is usually proper distance, which is defined as what you would get with a chain of rulers, extending along a spacelike geodesic, each of them at rest relative to the Hubble flow.

If so under what circumstances does this happen and what is the real world physical source for this?

Under the circumstances described in #30.

 

[marcus]

... In cosmology, the most convenient measure of distance is usually proper distance, which is defined as what you would get with a chain of rulers, extending along a spacelike geodesic, each of them at rest relative to the Hubble flow...

I agree. Nice concise description of what is commonly called "proper distance". I often call it instantaneous or "freeze-frame" distance because it is what you would get if you could freeze the expansion proces (the Hubble flow) at a particular moment in time.

And then, having frozen it, so that distances would not be changing while you measured, use any familiar means like rulers or radar.

To amplify something Ben said, one reason it is so convenient is that it is the distance which is used in formulating basic features of the standard model, like the Hubble law v=Hd. This is a statement about proper distances and their instantaneous rate of increase.

 

[marcus]

Here is a useful paper:

http://arxiv.org/PS_cache/astro-ph/pdf/9905/9905116v4.pdf
...

That paper by David Hogg is a total classic! Somebody should have it tattooed in their signature. It gets referred to here regularly over the years. In fact you are pointing out the importance of operational definitions in science. A term like "distance" has no meaning in the abstract. It only means something if you have some procedure in mind for measuring the distance. We do not know or predict Nature. We only know and predict events observations measurements...

Tanelorn asks "does the idea of distance become useless?" and the answer is no it remain useful. Very useful! We just have to be more aware of the processes used to evaluate it---its operational meanings.

Google "wright calculator" and find the distances corresponding to some redshift and think about the fact that the "angular size distance" which it gives is actually the same as the proper or instantaneous distance at the moment that the light was emitted and started on its way to us. Think about what angular size distance means, how you would determine it (how far away a meterstick looks, by what angle it subtends) and why it is the same as the proper distance back then. I still find the equivalence intriguing

 

[Tanelorn]

Marcus, Ben, So proper distance is the real distance between the objects, the one we measure in everyday situations with a ruler. The speed of the objects and the expansion of space are all frozen for the measurement and we have an extremely long ruler that can be instantaneously moved into position between the objects?

 

[yheyman]

13.7 billion years is the distance to the Hubble sphere... the age of the Universe need to be recomputed.

If the recession speed exceeds the speed of light, the photon would never reach the observer, this is why there exists a horizon of the visible Universe (the Hubble sphere), beyond which light would never reach us. Historically the age of the Universe was computed from the loockback time between a redshift zero and infinity, which yields 1/Ho. Note that this measure gives the lookback time to the Hubble sphere because the redshift must converge towards infinity at the horizon of the visible Universe. Here is a reference showing the calculations with a De Sitter Universe (http://www.jrank.org/space/pages/2440/look-back-time.html). Another reference where the age of the Universe is computed with the look-back time between a redshift of zero and infinity: http://www.mpifr-bonn.mpg.de/staff/hvoss/DiplWeb/DiplWebap1.html . See A.36 et A.37.

Using another approach we can show that an apparently steady Hubble coefficient in the light travel distance framework is equivalent to a time-varying Hubble coefficient in the Euclidean framework of order two (i.e. Universe expanding at a steady acceleration pace). This approach gives an age of the Universe of about 20-25 billion years. This figure is compatible with the age of the Universe obtained from the datation of old stars. According to Chaboyer (1995) who analysed metal-rich and metal-poor globular clusters, the absolute age of the oldest globular clusters are found to lie in the range 11-21 Gyr. Bolte et al. (1995) estimated the age of the M92 globular cluster to be 15.8 Gyr. Th/Eu dating yields stellar ages of up to 18.9 Gyr (Truran et al., 2001). A paper describing this appoach is available online: http://fr.calameo.com/books/00014533338c183febd92

 

[mitrasoumya]

How is the universe exactly 13.7 billion years old, in absence of absolute time?
Distribution of mass across the universe is not even. Therefore, passage of time should vary according to gravity. Which means at places time will pass at a higher pace or a lower pace than in respect of other places. Then, how is the entire Universe exactly 13.7 billion years old?

 

[Chronos]

The source of cmb photons was a mere ~46 million light years distant [in comoving coordinates] when the photons we now detect were emitted 13.7 billion years ago. Hellfire has a very useful cosmo calculator for this sort of thing. Cosmology is a fickle mistress.

 

[Mark M]

How is the universe exactly 13.7 billion years old, in absence of absolute time?
Distribution of mass across the universe is not even. Therefore, passage of time should vary according to gravity. Which means at places time will pass at a higher pace or a lower pace than in respect of other places. Then, how is the entire Universe exactly 13.7 billion years old?

Cosmological time is the time measured by an observer at rest with respect to the CMB.

 

[Lino]

... SR is flat non-expanding geometry, so it is mainly good where gravity is not too strong (so curvature can be neglected) and where the expansion of distance is so small or slow that it can be neglected...

... As an example where SR *is* a good approximation, the Andromeda galaxy lies at a distance of d=2.5 million light years, so d^2H^2 is on the order of 10^-7. ...

Marcus / Ben, I understand the basics of what your are saying, re the impact of curvature, but I would have thought that the impact was greater (and therefore more significant) at shorter distances! What I mean is: over shorter distances (Milkyway to Andromeda) objects are gravitationally bound (implying greater curvature), while over larger distances, like those mentioned in earlier posts, expansion rules (implying that the average curvature would be closer to (an average of) zero). I appreciate that I am fundementally miss-stating something - could you give me a pointer as to where I am going wrong please?


Regards,


Noel.