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.)