I How is "speed of expansion" of an object defined?

[Jaime Rudas]

[Moderator's note: Spun off from a previous thread since this is really a separate topic.]

was the universe expanding at the speed of light (or faster)?

In this regard, I think it's important to note that something can't travel faster than light, but it could expand faster than the speed of light.

 

[phinds]

In this regard, I think it's important to note that something can't travel faster than light, but it could expand faster than the speed of light.

That is, SPACE (/the universe) can expand faster that c, not a material object.

 

[Jaime Rudas]

That is, SPACE (/the universe) can expand faster that c, not a material object.

I believe that, in fact, we already have the technology to build something that expands faster than the speed of light. Look:

If we take a 1 cm elastic band and, in one second, stretch it until it is 4 cm long, it will have expanded by 3 cm in one second—that is, it will have expanded at a speed of 3 cm/s.

Now, in order to increase the speed, we can build a mechanism according to the following scheme:

The blue line is the elastic band, the black dots are fixed pins, and the red dot is a pin that can move vertically.

In one hundredth of a second, the red pin moves 2 cm upward, like this:

Under these conditions, the band stretches to 4.12 cm, so it has expanded by 3.12 cm at a speed of 3.12/0.01 = 312 cm/s = 3.12 m/s.

Now, if we duplicate the mechanism, we initially have the following:

And after 0.01 seconds we see the following:

In this way, the band, which initially had a length of 2 cm, after one hundredth of a second will have a length of 2 × 4.12 = 8.24 cm, which means it has expanded by 6.24 cm at a speed of 6.24/0.01 = 624 cm/s = 6.24 m/s.

In general, if we multiply the mechanism by n, the band initially has a length of n cm and after one hundredth of a second it will have a length of n×4.12, so it will have expanded by (n×4.12)−n=3.12n centimetres at a speed of 3.12n/0.01=312n cm/s.

If we take n=100 million, we find that the band would initially be 100 million centimetres long (1,000 km) and after one hundredth of a second it would be 412 million centimetres (4,120 km), which means it expanded 3,120 km at a speed of 3,120/0.01 = 312,000 km/s.

 

[PeroK]

That is, SPACE (/the universe) can expand faster that c, not a material object.

This is not necessarily true. The diameter of an object could expand at twice the speed of light.

 

[jbriggs444]

This is not necessarily true. The diameter of an object could expand at twice the speed of light.

As @PeroK is aware, the less than 2c limit is for a material object in flat spacetime. Separation velocities (the rate at which distance between two objects is increasing or decreasing over time) are limited by this. One object can be moving less than 1c in one direction. The other object can be moving less than 1c in the opposite direction for a total just less than 2c as judged by an inertial frame in the middle.

In the more general case of a curved spacetime such as our expanding universe, there is no limit to how rapidly the distance between two material objects (or two rims on one extended object) can increase over time. The farther apart they are, the faster the distance between them can grow.

Admittedly, at some size, the two rims of an "extended object" may not be in each other's observable universe. At that point it becomes difficult to continue calling such an entity an "object".

 

[Jaime Rudas]

As @PeroK is aware, the less than 2c limit is for a material object in flat spacetime.

As I explain in post #3, an object can be made to expand its length at speeds much higher than 2c

 

[PeroK]

As I explain in post #3, an object can be made to expand its length at speeds much higher than 2c

My post related to the diameter of an object. If the length of the diameter increases at almost 2c, then the circumference of a circular object can increase at over 6c. And a circle could transform into an irrelugar shape with a much greater circumference in an almost arbitrarily short time interval.

Ultimately, however, these calculations rely on measuring the distance between adjacent particles as they change position, which becomes somewhat ambiguous. The circumference ultimately is inferred from the position of the adjacent constituent particles, rather than being a mathematically continuous line.

 

[Jaime Rudas]

My post related to the diameter of an object.

Yes, that's why my response isn't to your post, but to @jbriggs444's where he says that the limit for a material object is less than 2c

 

[jbriggs444]

Yes, that's why my response isn't to your post, but to @jbriggs444's where he says that the limit for a material object is less than 2c

I, in turn, was responding to @PeroK who had spoken of "diameter" rather than "length".

Yes, I understand your point that we can play games with the path over which a length is evaluated and find unlimited extension rates in that manner.

 

[Jaime Rudas]

I, in turn, was responding to @PeroK who had spoken of "diameter" rather than "length".

Yes, @PeroK had spoken of "diameter," but you said the limit for a "material object" was less than 2c. That's precisely why I was responding to you, not to @PeroK.

Yes, I understand your point that we can play games with the path over which a length is evaluated and find unlimited extension rates in that manner.

No, that's not the case. I haven't proposed anything like "we can play games with the path over which a length is evaluated". What I proposed is that lengths be measured in the traditional way: with a ruler.

 

[jbriggs444]

Yes, @PeroK had spoken of "diameter," but you said the limit for a "material object" was less than 2c. That's precisely why I was responding to you, not to @PeroK.

The limit for the rate of increase of the diameter of a material object is what I had in mind.

No, that's not the case. I haven't proposed anything like "we can play games with the path over which a length is evaluated". What I proposed is that lengths be measured in the traditional way: with a ruler.

On a route that runs along the object rather than on a direct path.

 

[PeterDonis]

an object can be made to expand its length at speeds much higher than 2c

Not in any sense of "expand its length" that is physically meaningful.

In your scenario, no atom of the elastic band moves faster than 200 cm/s, the speed at which the red pins, and therefore the atoms co-located with them, move upward. The atoms at the black pins don't move at all. The atoms in between the red and black pins move at some intermediate speed that depends on their location.

Those are the only physically meaningful speeds in the scenario. Your calculation of "speed of length expansion" is not physically meaningful. (Note that the same is true of the "recession speeds" that can be greater than $c$ in FRW spacetime.)

 

[Jaime Rudas]

[Moderator's note: Some off topic content has been deleted.]

On a route that runs along the object rather than on a direct path.

Yes, on a path that runs along the object, which is the traditional way of measuring the length of objects. For example, this is how @PeroK appears to measure the circumference of a circular object in post #7.

 

[Jaime Rudas]

Those are the only physically meaningful speeds in the scenario. Your calculation of "speed of length expansion" is not physically meaningful.

And how do you determine whether something is physically meaningful or not?
Where can we find references to this?

 

[PeterDonis]

how do you determine whether something is physically meaningful or not?

By whether I can attach any physical meaning to it. I can't.

If you think the "speed of expansion" you defined in your scenario is physically meaningful, what do you think its physical meaning is?

 

[PeterDonis]

this is how @PeroK appears to measure the circumference of a circular object in post #7.

But @PeroK is not claiming that the "speed of increase" of the circumference is physically meaningful. He's saying it's like the "recession speeds" in FRW spacetime.

 

[Jaime Rudas]

Those are the only physically meaningful speeds in the scenario.

What physical meaning can you attach to the speed of the red pins?

 

[PeterDonis]

What physical meaning can you attach to the speed of the red pins?

The speed at which the red pins are physically moving. They move 2 cm in one hundredth of a second, or 200 cm/sec.

Nothing is physically moving at the "speed of expansion" you defined.

 

[Jaime Rudas]

If you think the "speed of expansion" you defined in your scenario is physically meaningful, what do you think its physical meaning is?

The speed at which the length of the rubber band physically expands. It expands 3,120 km in one hundredth of a second, or at 312,000 km/s.

And, as I mentioned from the beginning, this isn't a speed of movement, but rather a speed of expansion or increase.

 

[martinbn]

The expansion rate of the universe is around 70km/sMpc and the speed of light is about 300000km/s. The unitis are different, so how can you even compare them and ask which one is bigger?

 

[Ibix]

The expansion rate of the universe is around 70km/sMpc and the speed of light is about 300000km/s. The unitis are different, so how can you even compare them and ask which one is bigger?

This spun off a thread by someone who seemed to be thinking of the universe as a spherical region of expanding matter surrounded by vacuum, which might be said to have an expansion speed (of the frontier at least) and is how this discussion started.

And, as I mentioned from the beginning, this isn't a speed of movement, but rather a speed of expansion or increase.

I would prefer to call this a rate of change rather than a speed, but I accept the terminology probably isn't universally agreed. In any case, I think it's another example of things like the speed of a laser pointer dot or the speed of the crossing point of the blades of scissors - quantities that have units of speed but, since no physical object is travelling with that speed, are not limited to $c$.

 

[Jaime Rudas]

I would prefer to call this a rate of change rather than a speed,

I think it would be a rate, if I had said it grows 312 times per second, but since I expressed it in absolute terms, I think it is correct to call it the speed of expansion (or increase).

 

[Ibix]

I think it would be a rate, if I had said it grows 312 times per second, but since I expressed it in absolute terms, I think it is correct to call it the speed of expansion (or increase).

It's a rate of change of length. I'd prefer not to use "speed" here simply to avoid arguing over why it's a speed that can exceed $c$.

Edit: of course, this is simply a choice of terminology, similar to whether you want to call $\gamma m_0$ "mass". The choice makes no difference to the physics, nor even the maths.

 

[PAllen]

Well, the invariant way to define expansion is using the kinematic decomposition of a congruence representing the elements of the body. leading to the expansion tensor. Interpreted any any one point in standard units it observes the light speed limit. What defines an expanding universe in an invariant sense is whether you can construct a global expanding (per this tensor) congruence. Ultimately, this is a statement about the shape of spacetime, independent of any slicing choice. Note, that you can almost always find a local expanding congruence, but only for special spacetimes, including the non-degenerate FLRW spacetimes, can you find a global expanding congruence (given, of course, by the Hubble flow).

As for distance growth between widely separated points, this almost entirely a matter of convention, determined by how you foliate spacetime and define a time parameter. A key example is that using the Milne foliation of flat spacetime of SR (which is, in fact, a degenerate solution of the the FLRW metric), you get a distance growth proportional to distance, with no upper bound, just as in realistic cosmologies. Yet, there is no curvature and no dynamics to the spacetime. Further, even though the redshift is looks cosmological with the Milne foliation, using a standard foliation, it is exactly pure relative motion Doppler of SR.

 

[PAllen]

In SR, an even simpler example of unbounded distance growth rate is celerity:

To describe an interstellar trip, I take flat foliation determined by e.g. the galactic rest frame, but use a time parameter of the proper time of a rocket traveling rapidly relative to the galactic center. Then, the distance traveled divided by rocket time is unbounded, and also the rate of separtion of two stars may be 'many times c'.

[added] The relevance to @Jaime Rudas example is that is using, however you want to dodge, a very specific foliation (the extremely non-geodesic measuring path), combined with a time parameter related to a standard global inertial foliation.

 

[Jaime Rudas]

Well, the invariant way to define expansion is using the kinematic decomposition of a congruence representing the elements of the body. leading to the expansion tensor. Interpreted any any one point in standard units it observes the light speed limit.

I completely agree. In the example I propose in post #3, nothing travels faster than the light, so this limit is observed. However, this does not prevent the length of the elastic band from expanding at a speed greater than the speed of light, because the limit refers to the speed of travel, not the speed at which a given magnitude increases in size.

 

[PeterDonis]

this does not prevent the length of the elastic band from expanding at a speed greater than the speed of light, because the limit refers to the speed of travel, not the speed at which a given magnitude increases in size.

In other words, you agree that "the length of the elastic band...expanding at a speed greater than the speed of light", as you're defining those word, is not physically meaningful--because the speed of anything physically meaningful can't be greater than the speed of light. It's just "the speed at which a given magnitude increases in size", without any implication that anything is actually moving at that speed.

 

[PeterDonis]

The speed at which the length of the rubber band physically expands.

But nothing is physically moving at that speed. "Length" is not a physical thing. It's just a "magnitude" (to use your terminology).

 

[Jaime Rudas]

The relevance to @Jaime Rudas example is that is using, however you want to dodge, a very specific foliation (the extremely non-geodesic measuring path), combined with a time parameter related to a standard global inertial foliation.

I had understood that, when a problem involved very small masses and travel velocities, its solution using the theory of relativity was asymptotically close to the solution using Newtonian mechanics and Euclidean geometry. The problem I posed involves negligible masses and travel velocities well below the speed of light, so I considered that the solution would be close to the correct one if solved exclusively using Euclidean geometry.

 

[A.T.]

"Length" is not a physical thing.

I don't see why the length of a physical object should be less physical than it's position. Therefore I don’t get why 'speed' in the sense of 'length change per time' should be less physically meaningful than speed in the sense of '(magnitude of) position change per time'.

Whether one should use the same term 'speed' for both concepts is another question. But it's simply about two different meanings that need to be kept appart. No need to get into philosophical arguments about what is more physicaly meaningful.

 

[PeterDonis]

I don’t get why 'speed' in the sense of 'length change per time' should be less physically meaningful than speed in the sense of '(magnitude of) position change per time'.

As I've said, in the OP's scenario, nothing actually moves at the "speed of length expansion".

And @PAllen points out in posts #24 and #25 that the "distance growth" the OP defines is a convention. In relativity, such things are standardly considered to be not physically meaningful. Physical meaning is in the invariants.

 

[A.T.]

In relativity, such things are standardly considered to be not physically meaningful. Physical meaning is in the invariants.

But speed in the usual sense of '(magnitude of) position change per time' is not an invariant either, so based on the above it's not more physically meaningful than 'length change per time'.

 

[PeterDonis]

speed in the usual sense of '(magnitude of) position change per time' is not an invariant either

No, but the inner product of two 4-velocity vectors is an invariant, and that is what gives you the relative speed of two objects, which is the meaningful quantity in relativity (and the one that can't exceed $c$).

There is no such invariant corresponding to the "speed of length expansion".

 

[Jaime Rudas]

No, but the inner product of two 4-velocity vectors is an invariant, and that is what gives you the relative speed of two objects, which is the meaningful quantity in relativity (and the one that can't exceed $c$).

There is no such invariant corresponding to the "speed of length expansion".

So the speed at which a rail expands, or the speed at which a disease spreads, or the speed at which a country's population grows, or the speed at which, on average, children grow in adolescence, have no physical meaning?

 

[PeterDonis]

the speed at which a rail expands

Defined how?

the speed at which a disease spreads

Defined how?

the speed at which a country's population grows

Defined how?

the speed at which, on average, children grow in adolescence

Defined how?

For the first and last of these, I can think of a definition that would make the "speed" in question physically meaningful, in that some physical object could actually be thought of as moving at that speed. But those definitions just highlight how they're different from the definition you're using in your scenario in this thread.

For the second item, of course there's an obvious physical limit involved, since diseases spread by some sort of physical thing being transmitted, like a germ, and the speed those things can travel is limited. But again, that just highlights how different this is from your scenario in this thread.

For the third item, I can't think of a physically meaningful definition at all. But you're welcome to try to suggest one.

 

[Jaime Rudas]

Defined how?

The speed at which a rail expands, defined as the increase in its length per unit of time.

The speed at which a disease spreads, defined as the increase in the area where infected people live per unit of time.

The speed at which a country's population grows, defined as the increase in the number of people living in that country per unit of time.

The speed at which, on average, children grow during adolescence, defined as the average increase in height of children during adolescence.

 

[Jaime Rudas]

For the first and last of these, I can think of a definition that would make the "speed" in question physically meaningful, in that some physical object could actually be thought of as moving at that speed.

So, a speed only has physical meaning if there is a physical object that can move at that speed?

 

[PeterDonis]

The speed at which a rail expands, defined as the increase in its length per unit of time.

Is anything actually moving at this speed? For example, would an atom at one end of the rail be moving at this speed?

The speed at which a disease spreads, defined as the increase in the area where infected people live per unit of time.

But this has nothing to do with the speed at which germs move.

The speed at which a country's population grows, defined as the increase in the number of people living in that country per unit of time.

This isn't even a speed, since the number of people is not a distance.

The speed at which, on average, children grow during adolescence, defined as the average increase in height of children during adolescence.

Would any actual part of the child be moving at this speed?

 

[PeterDonis]

So, a speed only has physical meaning if there is a physical object that can move at that speed?

Since we're talking about physics, that certainly seems like a reasonable way to look at it.

 

[Jaime Rudas]

So, a speed only has physical meaning if there is a physical object that can move at that speed?

Since we're talking about physics, that certainly seems like a reasonable way to look at it.

Do you have any reference for this?

I ask because I wouldn't have doubted that the speed at which the temperature of a liquid increases or the speed at which two chemical compounds react would have a clear physical meaning.

 

[PAllen]

Do you have any reference for this?

I ask because I wouldn't have doubted that the speed at which the temperature of a liquid increases or the speed at which two chemical compounds react would have a clear physical meaning.

Actually, in scientific literature, I have only seen those called rates, not speeds.

 

[PeterDonis]

Do you have any reference for this?

For what? I said it seems like a reasonable way to look at it. Earlier I gave the definition of relative speed as an invariant (which you can find in any relativity textbook), and also referred to the view that physical meaning is contained entirely in invariants (which you can also find in many relativity textbooks). I'm not sure what other kind of reference you're looking for.

I ask because I wouldn't have doubted that the speed at which the temperature of a liquid increases or the speed at which two chemical compounds react would have a clear physical meaning.

Neither of those are speeds since their numerators are not distances.

 

[PeterDonis]

Actually, in scientific literature, I have only seen those called rates, not speeds.

Which would make sense since neither temperature nor concentration of a chemical compound are distances, so their rates of change with respect to time are not speeds.

 

[Jaime Rudas]

Neither of those are speeds since their numerators are not distances.

The word 'speed' doesn't just mean "how fast something moves," it also means "how fast something happens."

 

[PeterDonis]

The word 'speed' doesn't just mean "how fast something moves," it also means "how fast something happens."

You defined speed as the change of a length with time. That's the definition we're using in this thread. Any other definition is irrelevant and off topic for this discussion.

 

[Jaime Rudas]

The word 'speed' doesn't just mean "how fast something moves," it also means "how fast something happens."

You defined speed as the change of a length with time. That's the definition we're using in this thread. Any other definition is irrelevant and off topic for this discussion.

Yes, I defined 'speed of expansion' as the change of a length with time, regardless of whether or not something is moving at that speed.

 

[PeterDonis]

Yes, I defined 'speed of expansion' as the change of a length with time, regardless of whether or not something is moving at that speed.

Yes, which means this...

it also means "how fast something happens."

...is irrelevant for this thread.