# Are there contradictions with Special relativity in the expansion of the Universe?

1. Feb 9, 2006

### mitchellmckain

I have read that the expansion of the universe is not a relative velocity and therefore not restricted to the limitation of the speed of light. I read that this creates a horizon which we cannot see beyond due the fact that the portions of the universe are moving away from us at a speed greater than the speed of light.

I accepted this when I first read it, although I was surprised, but now, thinking about it more, I am seeing logical contradictions and beginning to suspect that this was misinformation.

I believe that simply an accelerating expansion would create a horizon even though there is no relative velocity, due to the expansion, which exceeds the speed of light. So I don't think the existence of a horizon is sufficient to support the idea that any relative velocity due to the expansion can exceed the speed of light.

My difficulty is that I cannot see any logical difference between a relative velocity due to the expansion of the universe and a relative veocity discussed in special relativity. I imagine a space ship leaving a galaxy towards another at relativistic speeds and wonder how to calculate the relative speed towards this target galaxy if it is receding from the starting galaxy, especially if this recession can accelerate to something beyond the speed of light. In fact the whole idea of an expansion which is exempt from special relativity implies to me some sort of priviledged refrence frame which distinguishes the relative velocity due to expansion from a relative velocity which is not due to expansion.

So I hope you guys which are more educated in general relativity that I am can clear this up for me. And if you can do this then maybe you can address the further topic of the superluminal expansion of the galaxy in the cosmological theory of inflation.

2. Feb 9, 2006

### Mike2

There is the velocity of objects moving relative to the space they are in. And there is the velocity percieved by observers quite a distance away. The galaxies are not moving much with respect to (wrt) the space they are in or wrt to nearby galaxies. But because there is so much expanding space between us the them, we preceive them as moving away from us near the speed of light.

That said, I see your point: there is no preferred space which would enable us to say they are not moving fast wrt their local space. But since we do not observe galaxies whose stars have greatly increased mass due to relativistic speed influencing each other, this indicates that they are not moving very fast. Or is this just a frame dragging effect?

3. Feb 9, 2006

Staff Emeritus
First, if there is a horizon between the home planet of the spaceship and the target galaxy, how did the spaceshp acquire the target?

Secondly what a spaceship sees is that during its journey (many centuries, at sublight speeds, relative to the target) the space between the ship and the target has expanded and the target is farther away than it was in the calculated progress when the ship started. But the rate of expansion between two points is proportional to the volume of space included, so as the ship lessens the distance, the expansion between it and the galaxy is less. I don't know how to work this out in detail; I am sure there is a true compoutation for it.

One thing to remember is that expansion is a general relativistic effect, so calculations in special realtivity may not apply. spacetime is curved in general ralativity, and special relativity only holds locally (as the tangent space to the spacetime manifold at a point).

4. Feb 9, 2006

### Nucleonics

The examples you provided are not taken as contradictions because, according to the LCDM (Lamda-Cold Dark Matter) concordance model, the motion is not in any observer’s inertial frame, every observer will measure the speed of light locally to be 'c', and, unfortunately for NASA fans, none of the observers can ever overtake a beam of light.

5. Feb 9, 2006

### Nucleonics

ditto what selfAdjoint said

6. Feb 9, 2006

### pervect

Staff Emeritus
Relative velocity in GR turns out to be a tricky concept - as it is in SR, when one of the objects is accelerating.

You want to know the relative velocity of an object "now". In order to detremine "now", you have to use some defintion of simultaneity. But one of the lessons SR tells us is that simultaneity is relative. If the velocity is changing with time due to acceleration, the concept of relative velocity becomes ill-defined.

This shows up in situations as mundane as finding one's velocity relative to a black hole. Some purists argue that the concept (velocity relative to a black hole) has no meaning, but I think this is too restrictive. It is possible to determine if an observer is stationary with respect to the black hole (by looking for an unchanging metric). Given that there is only one "stationary" worldline at any specific point in space-time, one can use this worldline as a reference to measure one's velocity.

I should add that sometimes I find the purists were right and I was wrong - but this is how I think about the situation at the current time.

[add]I should probably add that there is no problem measuring relative velocity when two objects are at the same point in space-time even when one is accelerating, the simultaneity issues do not arise.

In any event I would encourage you to think about the broader question about how you measure the velocity of a distant obsever in the presence of gravitational fields.

Last edited: Feb 9, 2006
7. Feb 10, 2006

### mitchellmckain

Well anyway the consensus opinion here seems to be that what I read was not mis-information. I cannot say I am perfectly happy since I would have liked a neat simple anwer to my dilemna (like confirmation that what I read was nonsense) but I accept that there are good reasons (which some of you provided) why Special Relativity is not applicable.

It does bring up another interesting question for me. I have read and studied introductory texts on GR (Schutz and Wald, I also have Hawkings "Large Scale structure of space time"), and I don't remember reading anything that fulfills the expectation that GR is a generalization of SR to include acceleration. Maybe some of you can point me in the right direction to somewhere that this topic is mentioned. In other words, where would I look if I wanted to calculate something like the nonlocal SR-like effects (lorentz contraction, time dilation, or velocity addition) in curved space such as when the expansion of space is a factor, or maybe some discussion of accelerating reference frames, or anything that might look like a generalization of SR.

8. Feb 10, 2006

### robphy

Your expectation should be that "GR is a generalization of SR to include gravitation"... using the modern/professional usage of GR and SR, where SR is GR with zero curvature on R4. (Historically, it has been said that GR is needed to handle accelerations or accelerating frames. With modern usage, SR is perfectly fine handling accelerations and accelerating frames, as long as the spacetime curvature is zero. However, one will probably have to go beyond the elementary techniques taught in introductions to SR. [Note that Newtonian Physics handles accelerations and accelerating frames pretty well.])

I try to follow up with some references to the literature on non-inertial frames.

9. Feb 10, 2006

### chronon

In special relativity the proper time of someone moving with respect to you will not agree with yours. Although neither frame is the 'correct' one, in order to do calculations it is necessary to choose a particular frame. In general relativity there is more freedom in choosing coordinates, and cosmologists generally choose one in which the time coordinate corresponds to the proper time of matter taking part in the expansion (comoving matter). This does not agree with special relativistic coordinates, and this difference is reponsible for most of the confusion in the subject. Thus in SR nothing can travel faster than c, but in comoving coordinates galaxies can have superluminal recession velocities. Note that these are perfectly visible - a cosmological event horizon is due to acceleration of expansion, which is a separate matter.

10. Feb 10, 2006

### pervect

Staff Emeritus
The best treatment of accelerating reference frames that I've seen is in "Gravitation" by MTW. This would be a good place to start, IMO, if you can handle the notation. Basically you need to know that a 4-vector is written as $v^a$, and that the length of a 4-vector v is written as $v^a v_a$ and is equal to $g_{ab} v^a v^a$.

Some confusion might arise here in that basis vectors really are vectors even though they are written as $e_0$. The text does explain that in component notation this really means $(e_0)^a$, i.e. "the subscript tells which vector, not which component", but it's still easy to get confused by this :-(.

Other than the possibilities of difficulty with the unfamaliar notation, the treatment of accelerating reference frames in MTW is very good (and is referred to by the sci.physics.faq as well, so I'm not the only one who thinks their treatment is especially good).

As far as time dilation goes, to calculate the proper time along a world line, you integrate the proper time differential, i.e. $\tau = \int d\tau$. You simply calculate dtau with a different metric in GR than you do in SR.

This won't necessarily answe all of your questions as to how to calculate simple things in GR, but should get you started.

Last edited: Feb 10, 2006
11. Feb 10, 2006

### Son Goku

12. Feb 11, 2006

### mitchellmckain

Thanks everyone. Mostly reminders of what I know already or ought to know already, but it is appreciated all the same. The MTW book is expensive for me right now but I am very tempted and will think about it.

13. Feb 11, 2006

### robphy

accelerated observers

concerning "accelerated observers"
possibly useful references... on my "to read someday" list:

Marzke-Wheeler coordinates for accelerated observers in special relativity
Authors: Massimo Pauri, Michele Vallisneri
http://arxiv.org/abs/gr-qc/0006095

M. Vallisneri, Relativity and Acceleration, thesis, doctorate in physics (University of Parma, Italy, 2000), 96 pp.
http://www.vallis.org/publications/pub_tesidott.html
http://www.vallis.org/publications/tesidott.pdf

Simultaneity, radar 4-coordinates and the 3+1 point of view about accelerated observers in special relativity
Authors: D. Alba (Univ. Firenze), L. Lusanna (INFN Firenze)
http://arxiv.org/abs/gr-qc/0311058

Generalized Radar 4-Coordinates and Equal-Time Cauchy Surfaces for Arbitrary Accelerated Observers
Authors: David Alba (Firenze Univ.), Luca Lusanna (INFN, Firenze)
http://arxiv.org/abs/gr-qc/0501090

What is the reference frame of an accelerated observer?
Authors: K.-P. Marzlin
Comments: 5 pages, 2 figures, published in Phys. Lett. A 215, 1 (1996)
http://arxiv.org/abs/gr-qc/9409061

A new mathematical formulation of accelerated observers in general relativity. I. [Journal of Mathematical Physics 23, 96 (1982)].
D. G. Retzloff, B. DeFacio, P. W. Dennis

On the physical meaning of Fermi coordinates
Authors: Karl-Peter Marzlin
Journal-ref: Gen.Rel.Grav. 26 (1994) 619
http://arxiv.org/abs/gr-qc/9402010

14. Feb 11, 2006

### pervect

Staff Emeritus
Especially you only want the one chapter, it might be worthwhile getting it via inter-library loan instead of buying it. It seems to have become extremely pricely.

You could probably get by with photocopying just the one chapter on accelerated observers, if thats your main interest, esp. since you have other books on GR available.

In many respects, MTW is a bit of a dionsaur, but because of its size, it tends to discuss things at length that other textbooks omit, making it a very valuable reference. (On the flip side of being a valuable reference is the fact that it's not always easy to find the relevant material in the book, the index is a bit weak, and its very large).