The mathematics of an expanding space. (Split from Bug on a Band)

[michel123456]

If the bug is accelerating, an intelligent bug should be aware of the situation. The bug should feel a force.

[Daedalus]

If the bug is accelerating, an intelligent bug should be aware of the situation. The bug should feel a force.

 

This is getting a little off-topic, but you have to consider that the bug is travelling with a constant velocity in the space local to the bug, and therefore would not feel a force acting upon it due to the expansion of space / the rubber rope. We only observe an acceleration because space / the rubber rope is expanding everywhere.

[michel123456]

This is getting a little off-topic, but you have to consider that the bug is travelling with a constant velocity in the space local to the bug, and therefore would not feel a force acting upon it due to the expansion of space / the rubber rope. We only observe an acceleration because space / the rubber rope is expanding everywhere.

Off topic, yes, and really weird.

 

The bug would observe the observer at rest receding at an increasing velocity. And neither the observer at rest nor the bug would feel the acceleration. What kind of acceleration is that?

[Daedalus]

Off topic, yes, and really weird.

 

The bug would observe the observer at rest receding at an increasing velocity. And neither the observer at rest nor the bug would feel the acceleration. What kind of acceleration is that?

 

It is the kind of motion one finds when dealing with a space that is expanding. Such motion is non-newtonian and does not obey Newton's laws of motion. Therefore, force does not equal mass times acceleration when dealing with the acceleration observed from expanding space. However, Newton's laws are close approximations when dealing with interactions in the bug's local space. If you would like to continue this discussion, we should start a thread regarding the mathematics of an expanding space, or perhaps a moderator can accommodate us and split this thread into a new thread to allow this discussion to procede further without hijacking this one.

Edited March 21, 2013 by Daedalus

[mooeypoo]

!

Moderator Note

This was split off of "Bug on a Board" brain teaser.

[Daedalus]

Mooeypoo split the "Bug on a Band" thread so that we can discuss the mathematics / physics from the ant on a rubber rope problem here in this thread. Michel123456 and I are discussing the observed acceleration of the bug due to the expansion of space according to the mathematics of the "Bug on a Band / Ant on a Rubber Rope" problem. The function for the position of the bug / ant according to the problem is

 

[math]x(t) \ = \ \left(v_r \, t + x_1\right) \frac{v_a}{v_r} \, \text{ln} \left(\frac{v_r \, t \, + \, x_1}{x_1}\right)[/math]

 

such that the acceleration observed is the second derivative of the position function

[math]\frac{d^2x}{dt^2}\ x(t) \ = \ \frac{v_a \ v_r}{v_r \, t+x_1}[/math]

 

We can see that the acceleration approaches zero as time approaches infinity. However, the acceleration would be very different if a nonzero net force were to act upon the bug. In that case, the bug would be accelerating relative to the rope instead of just having a constant velocity.

 

The posts above this one resulted from discussing such acceleration, and why the bug or observer does not feel a force acting upon them even though each would observe the other to be accelerating. I will work the problem with the bug / ant having a constant acceleration relative to the expanding space / rubber rope and post the results.

Edited March 21, 2013 by Daedalus

[elfmotat]

Any point on the band, moving or stationary, will have an attributed coordinate acceleration:

 

[math]\frac{d^2x}{dt^2}=v_r v_a(L_0+v_r t)^{-1}[/math]

 

In the case where the point is stationary w.r.t the band (i.e. if we drew a little dot on the band with sharpie), the coordinate acceleration is therefore zero. This means that all points on the band are moving with constant velocity, so each point has its own attributed inertial reference frame. A bug not moving w.r.t the band will not be able to measure anything with his accelerometer.

 

However, as soon as the bug begins to move he will experience a measurable acceleration.

This is getting a little off-topic, but you have to consider that the bug is travelling with a constant velocity in the space local to the bug, and therefore would not feel a force acting upon it due to the expansion of space / the rubber rope. We only observe an acceleration because space / the rubber rope is expanding everywhere.

 

Each point on the band will move with constant velocity, however the bug will be accelerating w.r.t every point on the band. The "constant velocity" of the bug refers to the velocity the bug would have w.r.t the band if it were not being stretched. It's easy to see this by considering the reference frame of, for example, the middle of the band. The new coordinates are:

 

[math]x'=x-\frac{1}{2}v_r t[/math]

[math]t'=t[/math]

 

Therefore [math]\frac{d^2x'}{dt^2}=\frac{d^2x}{dt^2}[/math], so the bug is accelerating w.r.t the middle of the band in exactly the same way that the bug is accelerating w.r.t the left end of the band.

Edited March 21, 2013 by elfmotat

[michel123456]

Any point on the band, moving or stationary, will have an attributed coordinate acceleration:

 

[math]\frac{d^2x}{dt^2}=v_r v_a(L_0+v_r t)^{-1}[/math]

 

In the case where the point is stationary w.r.t the band (i.e. if we drew a little dot on the band with sharpie), the coordinate acceleration is therefore zero. This means that all points on the band are moving with constant velocity, so each point has its own attributed inertial reference frame. A bug not moving w.r.t the band will not be able to measure anything with his accelerometer.

 

However, as soon as the bug begins to move he will experience a measurable acceleration.

 

Each point on the band will move with constant velocity, however the bug will be accelerating w.r.t every point on the band. The "constant velocity" of the bug refers to the velocity the bug would have w.r.t the band if it were not being stretched. It's easy to see this by considering the reference frame of, for example, the middle of the band. The new coordinates are:

 

[math]x'=x-\frac{1}{2}v_r t[/math]

[math]t'=t[/math]

 

Therefore [math]\frac{d^2x'}{dt^2}=\frac{d^2x}{dt^2}[/math], so the bug is accelerating w.r.t the middle of the band in exactly the same way that the bug is accelerating w.r.t the left end of the band.

"emphasis mine"

Right.

As seen from an observer at rest:

_a point A somewhere left on the band will have a constant velocity V1

_a point B in the middle of the band will have constant velocity V2

_a point C somewhere on the right of the band will have constant velocity V3

 

Where v3>v2>v1

 

So point A will observe point B receding. And so will B observe that A is receding.

In fact all points A B C will observe all other points receding.

The receding rate will be increasing with distance, so this is a good analogy to Hubble's law.

 

That is the situation when the bugs at points A B C are standing still on the rubber band.

 

And bugs A B C will feel no force. (I wonder)

 

Except that there must be some interaction between the rubber band and the feet of the bug. If the bug is made of other "stuff" and has no interaction with the band (as if there were oil between the band and the bug, or simply distance), then nothing of the above would happen.

 

And IF there is an interaction between the band and the bugs feet, then the bugs would feel a force, wouldn't they?

Edited March 22, 2013 by michel123456

[Daedalus]

Except that there must be some interaction between the rubber band and the feet of the bug. If the bug is made of other "stuff" and has no interaction with the band (as if there were oil between the band and the bug, or simply distance), then nothing of the above would happen.

 

And IF there is an interaction between the band and the bugs feet, then the bugs would feel a force, wouldn't they?

 

Think of it this way michel123456. If the expansion of space exerted a force upon the bug, then such force would be pushing the bug in all directions effectively cancelling out the effect. For instance, space is expanding behind the bug pushing it forward at the same time it is expanding in front of the bug pushing it backwards. If the the force is equal in all direction, then the effect would yield a zero net force upon the bug when dealing with its motion. Now, I'm not saying that the expansion of space exerts such forces upon the bug. Only that if it did, the effect would cancel out regarding the bug's motion. Furthermore, if a force was exerted upon the bug in all directions, then the bug should observe some sort of pressure as a result.

 

Now imagine that you are in a spaceship that is crossing a void that seperates superclusters of galaxies. You have already accelerated to your desired velocity and are coasting through the void. Although you are actually coasting along with a constant velocity in your local space, we here on Earth would observe you to be accelerating away from us and vice-versa from your FoR due to the expansion of space. Although we observe you to be accelerating, you would not experience a force because you are moving with a constant velocity through your local space.

Edited March 22, 2013 by Daedalus

[ydoaPs]

Think of it this way michel123456. If the expansion of space exerted a force upon the bug, then such force would be pushing the bug in all directions effectively cancelling out the effect. For instance, space is expanding behind the bug pushing it forward at the same time it is expanding in front of the bug pushing it backwards. If the the force is equal in all direction, then the effect would yield a zero net force upon the bug when dealing with its motion. Now, I'm not saying that the expansion of space exerts such forces upon the bug. Only that if it did, the effect would cancel out regarding the bug's motion. Furthermore, if a force was exerted upon the bug in all directions, then the bug should observe some sort of pressure as a result.

 

Now imagine that you are in a spaceship that is crossing a void that seperates superclusters of galaxies. You have already accelerated to your desired velocity and are coasting through the void. Although you are actually coasting along with a constant velocity in your local space, we here on Earth would observe you to be accelerating away from us and vice-versa from your FoR due to the expansion of space. Although we observe you to be accelerating, you would not experience a force because you are moving with a constant velocity through your local space.

While the band is obviously a metaphor for space, given the details of the problem, it cannot be modeled as such. A bug moving on a stretched band is accelerating.

[Daedalus]

While the band is obviously a metaphor for space, given the details of the problem, it cannot be modeled as such. A bug moving on a stretched band is accelerating.

 

I realize that, and I am also aware that the expansion of space cannot be modeled as such (at least not without being highly speculative and needing evidence to support such notion, which is contrary to observations). However, the topic is dealing with the bug on a band concept as it relates to a simpler form of a mathematical space that is expanding. I'm sorry I didn't clarify that, but this thread was split off from a thread in the brain teasers forum due to discussing the bug on a band concept, which is an analogy to help explain expanding space, and the effects this particular model has on motion. I naturally assumed that everyone understood that we are discussing expanding space and not the literal concept of a bug on an actual rubber band that is being stretched.

Edited March 22, 2013 by Daedalus

[michel123456]

Think of it this way michel123456. If the expansion of space exerted a force upon the bug, then such force would be pushing the bug in all directions effectively cancelling out the effect. For instance, space is expanding behind the bug pushing it forward at the same time it is expanding in front of the bug pushing it backwards. If the the force is equal in all direction, then the effect would yield a zero net force upon the bug when dealing with its motion. Now, I'm not saying that the expansion of space exerts such forces upon the bug. Only that if it did, the effect would cancel out regarding the bug's motion. Furthermore, if a force was exerted upon the bug in all directions, then the bug should observe some sort of pressure as a result.

 

Now imagine that you are in a spaceship that is crossing a void that seperates superclusters of galaxies. You have already accelerated to your desired velocity and are coasting through the void. Although you are actually coasting along with a constant velocity in your local space, we here on Earth would observe you to be accelerating away from us and vice-versa from your FoR due to the expansion of space. Although we observe you to be accelerating, you would not experience a force because you are moving with a constant velocity through your local space.

"then the bug should observe some sort of pressure as a result."

 

Exactly.

 

You said it.

 

A "pressure" coming from all directions.

 

The next step is to show that this "pressure" is function of the square of the distance.

[derek w]

The space might be expanding because there is a bug on it?

A spiders web expands because there is a spider on it,the web stretches on 1 axis.

Edited March 24, 2013 by derek w