Ant on an expanding balloon

1. Nov 16, 2011

granpa

This isnt homework but nobody is going to believe that so here it is:

1. The problem statement, all variables and given/known data
at t=1 (or any nonzero value of t)
an ant starts moving across a balloon at velocity
v=c=1 (c is the speed of light)​
the radius of the balloon as a function of time is given by
r(t)=nct (n is a constant much larger than 1)​

how far does the ant move as a function of time?

2. Relevant equations

3. The attempt at a solution
the angular velocity of the ant is
ω(t) = v/r(t) = c/nct = 1/nt​
integrating to get angular displacement I get
θ(t)=log(t)/n​
multiplying by radius to get distance I get
d(t) = r(t)*θ(t) = nct*log(t)/n = ct*log(t)​

The trouble is that this cant be right.
if n>>1 then the ant can never get all the way around the balloon
therefore the angular displacement θ must approach a limit as time goes to infinity.
But log(t) increases without limit.

I've gone over and over it
it seems too simple to be wrong but it must be.

Last edited: Nov 16, 2011
2. Nov 16, 2011

DaveC426913

Can the balloon physically expand as fast as nct? Or is that an assumption?

3. Nov 16, 2011

granpa

that is assumed

4. Nov 16, 2011

DaveC426913

I am not sure you can assume this. If the balloon's expansion is limited then your paradox might never occur.

5. Nov 16, 2011

granpa

the point is that the balloon is much expanding faster than the ant can walk
so the ant cant walk all the way around it.

there is no paradox. My math must be wrong somehow

the trouble still remains even if the radius of the balloon were as low as r(t) = ct/2π
the circumference would then be c(t) = 2πct/2π = ct

Last edited: Nov 16, 2011
6. Nov 16, 2011

cepheid

Staff Emeritus
Part of the problem seems to be the initial singularity (the fact that the balloon has 0 radius) at the beginning of the expansion. Your angular velocity blows up at t = 0 if you assume that the ant has always been moving at c "around" the surface of the balloon.

$$\theta(t) = \frac{1}{n}\int_0^t \frac{d\tau}{\tau}$$

$$= \frac{1}{n} [\log(t) - \log(0)]$$.

The logarithm of 0 is not defined. This integral is not defined.

I'm curious what happens if you stipulate that the ant has to start at rest and start moving at c only after a finite time. OR you could stipulate that the balloon has a finite radius at t = 0.

7. Nov 16, 2011

granpa

that was stipulated but I assumed that the equation would be the same.

8. Nov 16, 2011

granpa

I'm more concerned by the fact that log(infinity)=infinity

for large values of n it should by finite

9. Nov 16, 2011

granpa

log does have the nice property of equaling zero at t=1
so I originally tried to make that the point where the ant starts moving.

actually I dont see any trouble with doing that,
you just have to adjust the distance scale to match it so that c=1

Last edited: Nov 16, 2011
10. Nov 16, 2011

cepheid

Staff Emeritus
Sorry, my bad. So if we just consider things from t = 1 onward, then it is indeed true that θ(t) = log(t)/n.

So it seems like the answer is that the ant will make it all the way around in a finite time equal to t = exp(2πn). So the time required to make a full circle increases exponentially with n, but it is still finite.

Why are you so convinced that this answer cannot be right?

11. Nov 16, 2011

granpa

because some points on teh balloon will be receding from the point of origin of the ant faster than c (much faster)

consider the rate at which the circumference is increasing.

c(t) = 2πr(t)=2πnct

a point on the opposite side of the balloon will be receding at half that speed

space is opening up between the ant and that opposite point much faster than the ant can move forward.
therefore it should be receding from the ant.

now the total distance the ant travels will increase without limit
but the angle θ(t) will not.

Last edited: Nov 16, 2011
12. Nov 16, 2011

cepheid

Staff Emeritus
Your expression for the distance travelled s(t) is wrong. It's NOT true that s(t) = θ(t)r(t). Instead, we must write:

ds = r(t) dθ = r(t) (dθ/dt) dt = r(t)ω(t) dt = c dt

s(t) = ∫ds = ∫cdt = ct

So, that's progress. We found one mistake.

13. Nov 16, 2011

granpa

thats just the velocity of teh ant.

you also have to consider taht the space already traveled by the ant is expanding.

14. Nov 16, 2011

granpa

according to someone on another forum

ds/dt = r dθ/dt + θ dr/dt.

the first part is the velocity of the ant
the second part is the expansion of space.

but the second part just seems to reduce to what I already have above.

I cant believe that such a simple looking problem has not only baffled me but even you guys arent sure about it.

Last edited: Nov 16, 2011
15. Nov 16, 2011

granpa

θ(t) is teh total angular distance traveled.

it seems to me that if you know the radius then it should be trivial to figure out teh total distance

16. Nov 16, 2011

Dick

That 'correction' is dead wrong. The original analysis is correct, even it SEEMS paradoxical. s(t) is supposed to be the displacement from the original point at time t. Remember the space behind the ant is growing as well as the space in front of the ant. The ant will circle the balloon. Very slowly if n is large, but it will.

17. Nov 16, 2011

granpa

so even though a point opposite the origin of the ant is indeed receding from teh ant
it will recede slower and slower till it begins to move toward the ant?

Last edited: Nov 16, 2011
18. Nov 16, 2011

granpa

how can that be?

the rate at which a point opposite the origin of the ant is receding depends only on the amount of space between it and the ant.
The amount of space is increasing.
Therefore the rate at which it is receding should increase not decrease.

19. Nov 16, 2011

Dick

The 'c' is a red herring. This isn't a relativity problem. Even it it were there are probably regions of our universe that are receding from us faster than light. We just haven't seen them yet. And may not ever, if the universe is really in accelerated expansion.

20. Nov 16, 2011

Dick

Slow down. You worked it out correctly. You already pointed out the flaw in reasoning like that in post 13. You have to consider the expansion of the space behind the ant as contributing to his progress.