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Hijack from meaning of arc length


madmac

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SFNQuestions -- post#1.

Like if i have a parabola of velocity versus time or position versus time, is there any physical meaning to the arc length of that function?

 

I can see some physical meanings, re a parabola.

Firstly it is impossible to have a parabola of vel versus time. A parabola has infinite length, but vel is limited to c.

However it is possible (even in an exclusively Einsteinian forum such as this forum) to have a need for a velocity quantity of up to 2c.

I suppose that the sum of the velocities of a number of objects can be much greater than c, but this would still be finite.

 

The torchlight-effect can give great speed, but perhaps not infinite.

But, even if infinite, it still wont give u a parabola, because a parabola here also needs an infinite time (on the time axis), in fact it needs two infinite times, one going back to the start of the measurements, & one going forward to the end.

Ignoring here that perhaps it is possible to construct a graph where the parabola is leaning such that it is asymptotic to the velocity at one end, but even here the other end will need an infinite time.

 

On this forum an infinite time is impossible in either direction. Firstly, going back in time, time started with the BigBang -- no infinity here.

And fast-forwarding a bit, there might be a time when our universe ceases to exist, in which case there will be no infinite time in that direction.

And even if time does go on forever, we will not be able to draw our parabola until time stops, in which case it wont be a parabola anyhow, it will be a part of a parabola, ie parabolic, but not a parabola.

 

A parabola of position versus time is simpler, but still impossible.

Position means distance. Unlike velocity or speed, distance is not limited by the speed of light c. But a graph of a parabola still suffers from problems re time (see above).

 

But even if u could draw a parabola describing vel or position versus time, there wont be much physical meaning to the arc length of that function. Firstly the arc length would be infinite. All such arc lengths would be infinite. Hencely the analysis might boil down to what is the meaning of infinity.

 

If we change the question to relate to parabolic curves then that might be more meaningful (re arc length of the function), but still not good.

Here i suppose that the meaning is arc length per unit time.

The total arc length depends on when or where u decided to start & finish, not much scope for any interesting meaning here.

The arc length per unit time of any small part of the parabolic curve seems to me to have no interesting ponderable meaning.

The function of the change in this small arc length will i suppose have a simple equation, which i suppose will have no interesting ponderable meaning.

 

If the parabolic curve has an axis of symmetry then i suppose that the arc lengths on one side will have mirror images on the other, no big deal. Unless the axis of symmetry isn't vertical, in which case u wont.

 

Which leaves me with only one more physical meaning that i can think of.

Arc lengths never reach the vertical, & therefore never reverse (turnabout)(return) back along the time axis.

This means that (real) time cannot physically go backwards.

I think that this means that time is mathematically a scalar not a vector.

Edited by madmac
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SFNQuestions -- post#1.

(even in an exclusively Einsteinian forum such as this forum)

 

...

 

On this forum an infinite time is impossible

!

Moderator Note

Leave your agenda out of the discussion, and try and stay on topic when answering.

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