The principle of Maximum ageing

[md65536]

On 1/22/2020 at 10:45 PM, Markus Hanke said:

To be honest, I am having a hard time thinking of a physical scenario where you’d have both maxima and minima on the same world line, with said world line still remaining a free fall geodesic.

I'm trying to figure that out, too. I'm not even sure of any examples of a free-fall world line with a minimum proper time.

I think that something like a massive ring or washer would work. If you free-fall through its center, I think that might be a minimum. If you free-fall around it, that would be a maximum (similar to if it was a point mass).

Then to make a world line with both maximum and minimum sections, have a test particle orbit a normal mass in an eccentric orbit, and add a minimum part at its apogee (a massive washer to pass through, if that works). Make it eccentric enough that each of the masses has negligible effect when the particle is near the other.

Edited January 24, 2020 by md65536

[md65536]

Actually, if I understand the description of a geodesic with minimal ageing as described in Gravitation, I don't think any exist.

It makes sense that a free-fall path over a saddle point is minimized in the sense that any spatial deviation will result in a longer path. However if a non-freefall particle follows the same spatial path, but speeds up and slows down in order to stay "nearby" the free-fall particle, they should have even lower ageing than the free-fall particle. I suspect that there are free-fall paths that do not maximize ageing, but that those paths have neither maximum nor minimum proper time among nearby (in 4D) world lines.

[Markus Hanke]

13 hours ago, md65536 said:

I'm trying to figure that out, too. I'm not even sure of any examples of a free-fall world line with a minimum proper time.

Well, massless particles do not accumulate any proper time, since they trace out null geodesics, so for them ds=0. But that’s not strictly a minimum, because all possible paths they can take are null geodesics, so this (I believe) is called an infinum, not an extremum.

To get a true minimum (saddle point) I think we would have to go to a region of spacetime in the interior of a mass-energy distribution, rather than vacuum. One can probably set up a relevant scenario there, but even then it’s not trivial.

I’d also like to mention that any confusion about maxima and minima can be avoided entirely if one considers not proper time, but rather the action (in the theoretical physics sense) of the system in question. One then applies the principle of least action (of which the principle of extremal ageing is only a special case) - the path that is taken then is always a minimum of the action, both locally and globally. 

 

[moth]

17 minutes ago, Markus Hanke said:

Well, massless particles do not accumulate any proper time, since they trace out null geodesics, so for them ds=0. But that’s not strictly a minimum, because all possible paths they can take are null geodesics, so this (I believe) is called an infinum, not an extremum.

To get a true minimum (saddle point) I think we would have to go to a region of spacetime in the interior of a mass-energy distribution, rather than vacuum. One can probably set up a relevant scenario there, but even then it’s not trivial.

I’d also like to mention that any confusion about maxima and minima can be avoided entirely if one considers not proper time, but rather the action (in the theoretical physics sense) of the system in question. One then applies the principle of least action (of which the principle of extremal ageing is only a special case) - the path that is taken then is always a minimum of the action, both locally and globally. 

 

Thanks, i was wondering about geodesics and minimizing the action earlier but didn't want to muddle things up. That's the Lagrangian? (analyzing the action).
I wonder about ds=0 for massless particles though. Is that because of the Lorentz factor or is there another reason?  

[Markus Hanke]

8 minutes ago, moth said:

That's the Lagrangian? (analyzing the action).

Yes, that’s right.

9 minutes ago, moth said:

I wonder about ds=0 for massless particles though. Is that because of the Lorentz factor or is there another reason?

I would simply say that it follows from Fermat’s principle, in free space. 
It should also be possible to formally and explicitly derive this, by starting with Maxwell’s equations in curved spacetime, and deriving the path a wave vector would follow. The result is a null geodesic. 

[moth]

I think it's because time and frequency seem so related it's hard to understand how a photon can exhibit different frequencies in the same amount of time, zero.

[Markus Hanke]

23 hours ago, moth said:

I think it's because time and frequency seem so related it's hard to understand how a photon can exhibit different frequencies in the same amount of time, zero.

Time is a purely local thing. When we talk about frequencies, then these will always be measured by an external observer, using his own clock - so it isn’t a really a problem. It would only be an issue if there was such a thing as a photon’s rest frame, but such a thing does not physically exist.

[moth]

So frequency is just an expression of a photon's energy (momentum) and not a little clock. That makes sense, but there are photons from the B.B. that are billions of years old out there, and i guess it's strange to me that in some ways it was no time at all. Probably just me being anthropocentric.

[Markus Hanke]

4 minutes ago, moth said:

So frequency is just an expression of a photon's energy (momentum) and not a little clock

Yes, you could indeed say that.

5 minutes ago, moth said:

guess it's strange to me that in some ways it was no time at all

Time (in physics) is what clocks measure; since photons do not have a rest frame associated with them, there is no physical clock that could be “attached” to a photon. In essence, there is no meaningful notion of “time passing” that could be attributed to photons. This does not, however, imply that they do not trace out ordinary world lines in spacetime, like any other particle; it’s just that they are confined to the surface of a light cone centred on any given event.

[moth]

wow sometimes i lookat things and don't even see it: ct^2-dx^2-dy^2-dz^2=0 or we're not in kansas anymore.Thanks for helping me with that.