Re: [math-fun] physics thought experiment (Dave Dyer)
Also, Goucher's brownian motion solution is technically wrong in the sense brownian motion has infinite path length
Yes, it does, but you can cover that in finite *proper* time. Admittedly the heat-death of the universe would have occurred in that amount of time, but you would still be alive.
so even if you travel at C and do not mind infinite acceleration, you'll still never get anywhere.
Only from the perspective of a static observer. From your perspective, you'll cover the universe in finite time. Sincerely, Adam P. Goucher
Yes, but what if we place a limit of 20g's on the acceleration? Can you still do your random walk? Will 20g's allow you to see the heat-death of the universe? At 06:27 AM 2/2/2014, Goucher wrote:
Also, Goucher's brownian motion solution is technically wrong in the sense brownian motion has infinite path length
Yes, it does, but you can cover that in finite *proper* time. Admittedly the heat-death of the universe would have occurred in that amount of time, but you would still be alive.
so even if you travel at C and do not mind infinite acceleration, you'll still never get anywhere.
Only from the perspective of a static observer. From your perspective, you'll cover the universe in finite time.
I think even 1g will. It doesn't take long to get to 0.9999. Brent On 2/2/2014 6:35 AM, Henry Baker wrote:
Yes, but what if we place a limit of 20g's on the acceleration? Can you still do your random walk? Will 20g's allow you to see the heat-death of the universe?
At 06:27 AM 2/2/2014, Goucher wrote:
Also, Goucher's brownian motion solution is technically wrong in the sense brownian motion has infinite path length Yes, it does, but you can cover that in finite *proper* time. Admittedly the heat-death of the universe would have occurred in that amount of time, but you would still be alive.
so even if you travel at C and do not mind infinite acceleration, you'll still never get anywhere. Only from the perspective of a static observer. From your perspective, you'll cover the universe in finite time.
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Well, about a year with (Newtonian) acceleration = g, and about 5 years with proper acceleration = g. But I think the problem is turning -- Brownian motion would require far more turning than you could afford at that slow pace, and even an acceleration-limited modification would probably require much more acceleration than you could afford within your lifespan. (Right?) Charles Greathouse Analyst/Programmer Case Western Reserve University On Sun, Feb 2, 2014 at 3:02 PM, meekerdb <meekerdb@verizon.net> wrote:
I think even 1g will. It doesn't take long to get to 0.9999.
Brent
On 2/2/2014 6:35 AM, Henry Baker wrote:
Yes, but what if we place a limit of 20g's on the acceleration? Can you still do your random walk? Will 20g's allow you to see the heat-death of the universe?
At 06:27 AM 2/2/2014, Goucher wrote:
Also, Goucher's brownian motion solution is technically wrong in the
sense brownian motion has infinite path length
Yes, it does, but you can cover that in finite *proper* time. Admittedly the heat-death of the universe would have occurred in that amount of time, but you would still be alive.
so even if you travel at C and do not mind infinite acceleration, you'll
still never get anywhere.
Only from the perspective of a static observer. From your perspective, you'll cover the universe in finite time.
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How much turning would be required by Brownian motion would depend on some scale factor. But the idea is to keep your speed relative to the galaxy close to c while searching so you'd want to get up close to c and then just turn (i.e. acceleration normal to velocity). So your best search path would be relatively smooth orbits around the galaxy. Physically the problem is that you have to exhaust a enormous mass/energy, even with an exhaust velocity of c, in order to accelerate. So you have to start off with this enormous mass. I'm not sure how the beacons play into this. You can't just signal one another because the photon round-trip time is long compared to your life time. Brent Meeker On 2/2/2014 1:34 PM, Charles Greathouse wrote:
Well, about a year with (Newtonian) acceleration = g, and about 5 years with proper acceleration = g. But I think the problem is turning -- Brownian motion would require far more turning than you could afford at that slow pace, and even an acceleration-limited modification would probably require much more acceleration than you could afford within your lifespan. (Right?)
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Feb 2, 2014 at 3:02 PM, meekerdb <meekerdb@verizon.net> wrote:
I think even 1g will. It doesn't take long to get to 0.9999.
Brent
On 2/2/2014 6:35 AM, Henry Baker wrote:
Yes, but what if we place a limit of 20g's on the acceleration? Can you still do your random walk? Will 20g's allow you to see the heat-death of the universe?
At 06:27 AM 2/2/2014, Goucher wrote:
Also, Goucher's brownian motion solution is technically wrong in the
sense brownian motion has infinite path length
Yes, it does, but you can cover that in finite *proper* time. Admittedly the heat-death of the universe would have occurred in that amount of time, but you would still be alive.
so even if you travel at C and do not mind infinite acceleration, you'll
still never get anywhere.
Only from the perspective of a static observer. From your perspective, you'll cover the universe in finite time.
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There is a natural way to define Brownian motion in R^3 that starts and ends at the same point (1,0,0) and is a one-to-one arc. It's an interesting question to ask what are the probabilities that it forms various knots, like say a trefoil. --Dan
My thought is that you'd start with a straight run through the center until you picked up your buddy's beacon. At that point, and forever after, you'd have a fix on the location, speed, and heading your buddy had at some time in the past.
My thought is that you'd start with a straight run through the center until you picked up your buddy's beacon. At that point, and forever after, you'd have a fix on the location, speed, and heading your buddy had at some time in the past.
OK, suppose I wanted to simulate this thought experiment -- perhaps first in 2D (plane of the galaxy). My first thought was to generate "time" segments, in between which all movement was inertial (special relativity). My problem is what to do at each segment end-point where forces are applied and accelerations must be accounted for. I'm willing to keep 2 (or even 3) different clocks, but I'm not sure how to do the update functions. The problem is that for the segment end-points, the accelerations are infinite, but also for a very short time. Normally this is handled by considering impulses of a finite energy, but I'm not sure what to do with the quite large effective masses of humans travelling at very near light speed. In any case, simply doing the computer simulation would finally teach me how GR really works in this instance. At 02:16 PM 2/2/2014, meekerdb wrote:
How much turning would be required by Brownian motion would depend on some scale factor. But the idea is to keep your speed relative to the galaxy close to c while searching so you'd want to get up close to c and then just turn (i.e. acceleration normal to velocity). So your best search path would be relatively smooth orbits around the galaxy. Physically the problem is that you have to exhaust a enormous mass/energy, even with an exhaust velocity of c, in order to accelerate. So you have to start off with this enormous mass.
I'm not sure how the beacons play into this. You can't just signal one another because the photon round-trip time is long compared to your life time.
Brent Meeker
On 2/2/2014 1:34 PM, Charles Greathouse wrote:
Well, about a year with (Newtonian) acceleration = g, and about 5 years with proper acceleration = g. But I think the problem is turning -- Brownian motion would require far more turning than you could afford at that slow pace, and even an acceleration-limited modification would probably require much more acceleration than you could afford within your lifespan. (Right?)
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Feb 2, 2014 at 3:02 PM, meekerdb <meekerdb@verizon.net> wrote:
I think even 1g will. It doesn't take long to get to 0.9999.
Brent
On 2/2/2014 6:35 AM, Henry Baker wrote:
Yes, but what if we place a limit of 20g's on the acceleration? Can you still do your random walk? Will 20g's allow you to see the heat-death of the universe?
At 06:27 AM 2/2/2014, Goucher wrote:
Also, Goucher's brownian motion solution is technically wrong in the sense
brownian motion has infinite path length Yes, it does, but you can cover that in finite *proper* time. Admittedly the heat-death of the universe would have occurred in that amount of time, but you would still be alive.
so even if you travel at C and do not mind infinite acceleration, you'll
still never get anywhere. Only from the perspective of a static observer. From your perspective, you'll cover the universe in finite time.
So long as you stay away from black holes I don't think you need but a bit of general relativity. In order to systematically search the (2D) galaxy, you would boost up to a speed such that you can travel in a circle who's diameter is the galaxies radius (about 5e4Ly) at a speed such that gives you 1g centripetal acceleration (for your comfort). If I've done my algebra right this is very close to c, so you will hardly age at all. Then your search path is a circle starting at the edge of the galaxy, passing thru the center (avoiding the black hole) and returning out to the edge. Because of general relativity (this is the bit you need) this orbit won't close; it will precess. But by adjusting your centripetal acceleration you can make this precession sweep out a dense set of orbits in the 2D universe. This is probably not the most efficient solution since you go near the center every orbit and so "over search" it. And of course it neglects the enormous energy required and the gravitational and other effects on the stars you pass by, or through. Brent Meeker On 2/2/2014 8:37 PM, Henry Baker wrote:
OK, suppose I wanted to simulate this thought experiment -- perhaps first in 2D (plane of the galaxy).
My first thought was to generate "time" segments, in between which all movement was inertial (special relativity).
My problem is what to do at each segment end-point where forces are applied and accelerations must be accounted for.
I'm willing to keep 2 (or even 3) different clocks, but I'm not sure how to do the update functions.
The problem is that for the segment end-points, the accelerations are infinite, but also for a very short time.
Normally this is handled by considering impulses of a finite energy, but I'm not sure what to do with the quite large effective masses of humans travelling at very near light speed.
In any case, simply doing the computer simulation would finally teach me how GR really works in this instance.
At 02:16 PM 2/2/2014, meekerdb wrote:
How much turning would be required by Brownian motion would depend on some scale factor. But the idea is to keep your speed relative to the galaxy close to c while searching so you'd want to get up close to c and then just turn (i.e. acceleration normal to velocity). So your best search path would be relatively smooth orbits around the galaxy. Physically the problem is that you have to exhaust a enormous mass/energy, even with an exhaust velocity of c, in order to accelerate. So you have to start off with this enormous mass.
I'm not sure how the beacons play into this. You can't just signal one another because the photon round-trip time is long compared to your life time.
Brent Meeker
On 2/2/2014 1:34 PM, Charles Greathouse wrote:
Well, about a year with (Newtonian) acceleration = g, and about 5 years with proper acceleration = g. But I think the problem is turning -- Brownian motion would require far more turning than you could afford at that slow pace, and even an acceleration-limited modification would probably require much more acceleration than you could afford within your lifespan. (Right?)
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Sun, Feb 2, 2014 at 3:02 PM, meekerdb <meekerdb@verizon.net> wrote:
I think even 1g will. It doesn't take long to get to 0.9999.
Brent
On 2/2/2014 6:35 AM, Henry Baker wrote:
Yes, but what if we place a limit of 20g's on the acceleration? Can you still do your random walk? Will 20g's allow you to see the heat-death of the universe?
At 06:27 AM 2/2/2014, Goucher wrote:
Also, Goucher's brownian motion solution is technically wrong in the sense > brownian motion has infinite path length Yes, it does, but you can cover that in finite *proper* time. Admittedly the heat-death of the universe would have occurred in that amount of time, but you would still be alive.
so even if you travel at C and do not mind infinite acceleration, you'll > still never get anywhere. Only from the perspective of a static observer. From your perspective, you'll cover the universe in finite time.
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participants (6)
-
Adam P. Goucher -
Charles Greathouse -
Dan Asimov -
Dave Dyer -
Henry Baker -
meekerdb