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

bcrowell

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.

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

bcrowell

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

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.

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

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 travelling 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 seperated observers at rest relative to the same inertial reference frame.

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

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

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

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.

Right.

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.

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.

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

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?

Andrew Mason

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 ([itex]\alpha[/itex]) such that [itex]\sin\alpha = 1/\gamma[/itex].

AM

bcrowell

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

Arbitrageur

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

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

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

bcrowell

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

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.
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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

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 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

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