How does GR allow for recession speeds >c?

[rede96]

I've been studying Cosmology, albeit at a very laymen's level. I find it interesting, if not a little confusing, that galaxies are allowed under GR to recede from each other at speeds greater than c.

One of the explanations given (from Leonard Susskind's video lectures on cosmology) was because GR doesn't forbid objects at cosmological distances moving away from us with speeds > c, it just says that no information can be passed on at speeds > c and we can't see anything moving past us locally with speeds > c.

I've heard of a few other explanations such as distant galaxies aren't actually 'moving' wrt us but the space is expanding, and so it is 'ok' for the space to between them to grow >c. But from my study into Cosmology this does not seem to be the case. It's simply that everything is moving away from everything else.

So I was wondering if someone could help me understand this better and why something in the distance can move away from me at speeds > c but something locally can not.

 

[QuantumQuest]

As far as I know GR allows for such recessional velocities. Thing is, that Special Relativity is invoked and used in an improper way, so this is the source of confusion. You can take a look at this paper: http://arxiv.org/abs/astro-ph/0011070v2

 

[phinds]

. It's simply that everything is moving away from everything else.

That's a mis-statement of what is actually happening because "moving" implies proper motion, but recession is not proper motion. Things are not moving apart in the sense of proper motion, they are just getting farther apart due to the geometry of expansion. Google "Metric Expansion".

 

[PeterDonis]

I find it interesting, if not a little confusing, that galaxies are allowed under GR to recede from each other at speeds greater than c.

The "speed" in question here is not a "relative speed" in the sense of SR, which is what would be limited to less than $c$. It is just a coordinate speed; there is no limitation on coordinate speeds in GR (or, indeed, in SR if non-inertial coordinates are chosen). This does not pose a problem because coordinate speeds in general have no physical meaning; they are artifacts of a particular choice of coordinates.

One of the explanations given (from Leonard Susskind's video lectures on cosmology) was because GR doesn't forbid objects at cosmological distances moving away from us with speeds > c

This is basically saying what I said above (that the "speeds" of distant galaxies that are quoted as being greater than $c$ are just coordinate speeds and have no physical meaning) in different words.

it just says that no information can be passed on at speeds > c and we can't see anything moving past us locally with speeds > c.

This is basically saying that in order to apply the SR rule that "nothing can move at a speed greater than $c$", you have to be working in a single local inertial frame. If we construct a local inertial frame around us in which we are at rest, then we will never measure any object moving faster than $c$ within that frame. But the cosmological coordinates in which distant galaxies can have coordinate speeds greater than $c$ are certainly not local inertial frames.

 

[pervect]

I've been studying Cosmology, albeit at a very laymen's level. I find it interesting, if not a little confusing, that galaxies are allowed under GR to recede from each other at speeds greater than c.

One of the explanations given (from Leonard Susskind's video lectures on cosmology) was because GR doesn't forbid objects at cosmological distances moving away from us with speeds > c, it just says that no information can be passed on at speeds > c and we can't see anything moving past us locally with speeds > c.

I've heard of a few other explanations such as distant galaxies aren't actually 'moving' wrt us but the space is expanding, and so it is 'ok' for the space to between them to grow >c. But from my study into Cosmology this does not seem to be the case. It's simply that everything is moving away from everything else.

So I was wondering if someone could help me understand this better and why something in the distance can move away from me at speeds > c but something locally can not.

The explanation I think is the most accurate is rather abstract and not particularly satisfying. This is taken from Baez's "The Meaning of Einstein's equation", http://math.ucr.edu/home/baez/einstein/node2.html

Before stating Einstein's equation, we need a little preparation. We assume the reader is somewhat familiar with special relativity -- otherwise general relativity will be too hard. But there are some big differences between special and general relativity, which can cause immense confusion if neglected.

In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another. The reason is that in this theory, velocities are described as vectors in 4-dimensional spacetime. Switching to a different inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them point the same way.

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

At the minimum, Baez's point suggests that any explanation of "the velocity" between distant points must consider the operational details of how this velocity is observed, since there is a reasonable possibility that different methods of measuring "the velocity" might give different results.

 

[PAllen]

As an additional point, it is worth noting that SR trivially allows recession speeds >c if the same operational definition used in cosmology is applied (underscoring Peter and Pervect's points that this is not a relative velocity and there is no possible definition of relative velocity at a distance in GR).

The cosmological definition of recession speed is growth in proper distance between objects divided by cosmological coordinate time. Even in an SR inertial frame, this can approach 2c for objects moving apart from each other in that inertial frame. Further, if a non-inertial frame is used (as Peter mentioned) then in SR there is no upper bound on recession velocity per the cosmology definition. More specifically, there is a specific non-inertial frame in SR that mimics cosmloogical coordinates (it falls out of the zero density limit of FLRW solutions, which is flat space in cosmological style coordinates), for which the cosmologically defined recession rate has no upper bound - in SR.

Nutshell: superluminal recession speed is not a distinguishing feature between SR and GR, though a long history of sloppy treatment conceals this. What is true is that the space-time geometry of a 'near flat' ultra-low density universe is completely different from the space-time geometry we observe. Specifically, in coordinates showing isotropy and homogeneity, the near SR universe would have constant cosmological time slices that have extremely hyperbolic geometry, while our universe has nearly flat spatial slices of constant cosmological time. This is only possible in a substantially curved space-time geometry, thus GR.

 

[rede96]

That's a mis-statement of what is actually happening because "moving" implies proper motion, but recession is not proper motion.

Would you mind explaining what you mean by proper motion?

 

[rede96]

As far as I know GR allows for such recessional velocities. Thing is, that Special Relativity is invoked and used in an improper way, so this is the source of confusion. You can take a look at this paper: http://arxiv.org/abs/astro-ph/0011070v2

Thanks for that, not sure I understood well enough to answer my question but liked the paper.

The "speed" in question here is not a "relative speed" in the sense of SR, which is what would be limited to less than ccc. It is just a coordinate speed; there is no limitation on coordinate speeds in GR (or, indeed, in SR if non-inertial coordinates are chosen). This does not pose a problem because coordinate speeds in general have no physical meaning; they are artifacts of a particular choice of coordinates.

At the minimum, Baez's point suggests that any explanation of "the velocity" between distant points must consider the operational details of how this velocity is observed, since there is a reasonable possibility that different methods of measuring "the velocity" might give different results.

I sort of understand that different coordinate systems would give different results and it seems defining proper velocities at large cosmological distances isn't clear. But in any case there is relative movement apart (distances increase) between two objects separated by large distances in space. And as there is no upper limit to rate these objects are allowed to separate over time, then at some point the separation rate will increase to over 2c, meaning at least on of the objects relative to the other will be moving away at speeds >c. Is that correct?

Nutshell: superluminal recession speed is not a distinguishing feature between SR and GR, though a long history of sloppy treatment conceals this. What is true is that the space-time geometry of a 'near flat' ultra-low density universe is completely different from the space-time geometry we observe. Specifically, in coordinates showing isotropy and homogeneity, the near SR universe would have constant cosmological time slices that have extremely hyperbolic geometry, while our universe has nearly flat spatial slices of constant cosmological time. This is only possible in a substantially curved space-time geometry, thus GR.

So to put this another way, does that mean anything traveling locally is in essence traveling through almost flat spacetime (As the curvature is very large) so there is a limit of c, but object far enough away 'feel' the effects of curved spacetime more and thus we can observe them moving away at speeds >c due to this curvature?

 

[phinds]

Would you mind explaining what you mean by proper motion?

Change in position as seen from a single, local, frame of reference. There IS no local frame of reference in cosmological distances that can cover recession. If the effects of recession are discounted, there IS a small proper motion of distant galaxies relative to us but it is tiny compared to recession.

 

[PAllen]

So to put this another way, does that mean anything traveling locally is in essence traveling through almost flat spacetime (As the curvature is very large) so there is a limit of c, but object far enough away 'feel' the effects of curved spacetime more and thus we can observe them moving away at speeds >c due to this curvature?

Not really. I think the key point of the sloppy history on this is failure to recognize that recession rate and relative velocity are two completely different categories, and that even locally in SR there is no c limit on recession rate (e.g. 2c in an inertial frame, higher in a non-inertial frame); and that relative velocity, the category limited to c in SR, does not exist except locally in GR (and then it is limited to c, just like SR). It's that simple, and highly unfortunate that perhaps a majority of cosmology presentations make a fundamental category error when discussing this.

 

[rede96]

Not really. I think the key point of the sloppy history on this is failure to recognize that recession rate and relative velocity are two completely different categories, and that even locally in SR there is no c limit on recession rate (e.g. 2c in an inertial frame, higher in a non-inertial frame); and that relative velocity, the category limited to c in SR, does not exist except locally in GR (and then it is limited to c, just like SR). It's that simple, and highly unfortunate that perhaps a majority of cosmology presentations make a fundamental category error when discussing this.

To be honest I am not too sure just what the difference is between relative velocity and recession velocity, other than one is a measure of how something is moving relative to me and the other is a rate of separation.

However that aside, imagine for a moment it was possible to have an infinitely long ruler. I attach a ruler of sufficient length to a rocket ship is such a way that as the rocket ship passes by me, I read off the distance per unit time from the attached ruler and work out the rocket ship is traveling away from me at 0.5c relative to me. (Taking into account length contraction etc)

Sometime in the past a similar ruler of sufficient length was attached to an object in space which is now receding away from me at some velocity due to expansion. Let's the object is around 7Gly away. Again I read off the distance per unit time of the ruler attached to the distant object and work out it is traveling also at 0.5c relative to me.

As far as I understand it that is a perfectly good way to measure relative velocity as long as it were possible to have an infinite meter stick. However even if not, I know that at some point a distant object will be moving away from me at speeds greater than 2c due to accelerated expansion. Which means one of us must be traveling relative to other at speeds >c And from what I understand this is allowed in GR as the object isn't in an inertial frame of reference. Is that correct?

 

[bcrowell]

PAllen's #10 gets it right.

This may also be helpful: http://physics.stackexchange.com/questions/13388/at-what-speed-does-our-universe-expand

That's a mis-statement of what is actually happening because "moving" implies proper motion, but recession is not proper motion. Things are not moving apart in the sense of proper motion, they are just getting farther apart due to the geometry of expansion. Google "Metric Expansion".

You seem to be implying that there is a distinction between two phenomena, recession due to metric expansion and recession that is proper motion. They are not two phenomena. They are the same phenomenon, just described using different words.

 

[phinds]

PAllen's #10 gets it right.You seem to be implying that there is a distinction between two phenomena, recession due to metric expansion and recession that is proper motion. They are not two phenomena. They are the same phenomenon, just described using different words.

My understanding is that recession is not proper motion. How is that wrong?

 

[bcrowell]

My understanding is that recession is not proper motion. How is that wrong?

It's possible that I'm misunderstanding you, so you might want to state how you're defining your terms. But based on my interpretation of your words, the reason I think you're wrong is the reason I gave in #12.

You may find these papers helpful:

Francis et al., "Expanding Space: the Root of all Evil?," http://arxiv.org/abs/0707.0380v1

E.F. Bunn and D.W. Hogg, "The kinematic origin of the cosmological redshift," American Journal of Physics, Vol. 77, No. 8, pp. 694, August 2009, http://arxiv.org/abs/0808.1081v2

Bunn and Hogg, for example, argue for a certain interpretation, but they make it pretty clear that it's a pedagogical/philosophical preference, not something empirically verifiable.

 

[phinds]

But your objection there was that I seemed to think there were two kinds of recession, but that had never been my belief. I see recession and proper motion as distinct concepts, with proper motion being local and recession being on cosmological scales. If I define a frame of reference in which I am at rest, a person walking past me has proper motion in that frame of reference but a galaxy 10billion light years away is not IN that frame in the same sense as is the person walking past me and the galaxy has no proper motion due to recession.

It is my further understanding that there actually IS a small proper motion of remote galaxies that is independent of recession and that for really far away galaxies is trivial in magnitude compared to the recession velocity.

 

[PAllen]

But your objection there was that I seemed to think there were two kinds of recession, but that had never been my belief. I see recession and proper motion as distinct concepts, with proper motion being local and recession being on cosmological scales. If I define a frame of reference in which I am at rest, a person walking past me has proper motion in that frame of reference but a galaxy 10billion light years away is not IN that frame in the same sense as is the person walking past me and the galaxy has no proper motion due to recession.

It is my further understanding that there actually IS a small proper motion of remote galaxies that is independent of recession and that for really far away galaxies is trivial in magnitude compared to the recession velocity.

There is no way to take an arbitrary world line (of a galaxy) and factor its 4-velocity into proper motion and recession rate in any invariant sense. Any such factorization is a feature of the coordinates chosen. Any notion of recession rate at all is a feature of coordinates chosen. In GR, per se, there is just 4-velocity (as a covariant quantity); and two 4-velocities can only be meaningfully compared if they are near each other.

 

[PeterDonis]

it seems defining proper velocities at large cosmological distances isn't clear

It's not that it isn't clear; it's that it isn't well-defined. There is no such thing as "proper velocities" (meaning velocities that work like relative velocities do in SR) at large cosmological distances; the concept simply does not apply.

But in any case there is relative movement apart (distances increase) between two objects separated by large distances in space.

In the sense of coordinate distances, yes. But nobody measures these distances directly. They are calculated according to a particular model using a particular coordinate chart.

does that mean anything traveling locally is in essence traveling through almost flat spacetime (As the curvature is very large) so there is a limit of c, but object far enough away 'feel' the effects of curved spacetime more and thus we can observe them moving away at speeds >c due to this curvature?

Sort of. It is true that the reason that the concept of "relative velocity" in the SR sense does not apply at cosmological distances is spacetime curvature; this is just a special case of the general fact that the concept of "relative velocity" in the SR sense does not apply in any situation where spacetime curvature is significant, which means any situation that is not confined to a single local inertial frame. Note that "local" means local in time as well as space--the expansion of the universe over time is itself a manifestation of spacetime curvature, so we can't define a "relative velocity" between an object on Earth now and an object on Earth a billion years ago any more than we can define a "relative velocity" between an object on Earth now and an object in a galaxy a billion light years away.

But I'm not sure I would describe the effects of spacetime curvature as "objects can now move away from us faster than c", because that invites the very confusion that has been giving you trouble. I would say that, whenever the effects of spacetime curvature are significant, "speed" is not a concept that has a well-defined meaning. You can calculate numbers that look like speeds, but thinking of them as speeds just causes confusion.