Do different points along a string travel faster than others points along the same string?
I don't think you can think of a string as made up of points since there is nothing smaller than a string. Rather a string would be in a particular vibration state at a point in time. In one of my threads (minimum time) I stated that there is no smaller unit of time than Planck time which defines one complete "vibration" in one complete string. To go to a smaller point would need a distance smaller than Planck distance and a period of time smaller than Planck time. Then again, I still don't know if my observations are valid.
They do integrations along the world-sheet, don't they? Doesn't that assume that time is a continuous quantity?
In open bosonic strings(now just preserved as a teaching theory, except for string field theory, I believe) the free ends of the string are constrained by other physics to move at the speed of light. The vibrational motion of the string, AFAIK, is not linked to its translational movement in anything I have seen, but then I haven't seen very much.
is time continuous?
I'm not familiar with the world sheet (sorry, I'm just an ignorant physician). I am assuming that what you mean by continuous is that points along a range are infinitely divisible. However, time has been always "assumed" to be continuous just as distance has always been "assumed" to be continuous. String theory has shown that distance cannot be continuous in terms of infinite indivisibility because sub-Planck distances cannot exist within the framework of the theory (a string wouldn't fit). Furthermore, continuity as defined was a problem with point particles (not strings) since they had a zero dimension. The string does not have zero dimension. In fact it has 11 dimensions and a length of a Planck length.
I don't think continuity (as defined above) is a problem in terms of integration. My understanding is that integration is a summation of quantities in a defined range. Time would still be continuous in the sense that you must define it in multiples of Planck time. In other words, an integration across any defined range of time would be expressed in multiples of Planck length. I hope this makes sense.
Doesn't that mean that because the free ends of the string move at the speed of light that there is a finite amount of time (Planck time?) in which the vibration has to occur?
Re: is time continuous?
If time and space are discrete, then there is absolutely nothing between different points in space and different points in time. But the fact that different strings interact at a distance implies some connection between them from one Plank distance to the next. There can be no mathematical relationship across boarders of discontinuity. That is what discontinuous means, no connections, no relation from one side to the other. You may instead be thinking more of discrete quantum levels in some of the quantities in physics. But these are the result of continuous differential equations.
Re: Re: is time continuous?
If you are thinking of space and time in terms of points, then your premise is correct. However, there can not be absolutely nothing between them because a "point" in space does not exist as a zero dimension without length but rather because each unit of space has one Planck length.
Think of a ruler with, say, Planck length markings. From a point particle standpoint, then I agree, between the marks there is absolutely nothing. But what if that whole space between the marks is taken up by an indivisible entity (such as a string). Then if you try to divide the space into a smaller measurement, you also divide the string, which renders the whole point of fundamentality moot.
Continuity and discontinuity are concepts in a point particle theory. Because the string has length, it takes up the whole Planck space from start to finish and there is no "in between." In the same way, Planck time as defined (the period it takes for a photon to travel a Planck distance while traveling at the speed of light) cannot be thought of in terms of a start or finish because the entire Planck time is required for one string vibration and the photon takes up the whole space.
Please tell me if I'm starting to blather. Thank you.
Re: Re: Re: is time continuous?
Still, whether there is interviening space or not between entities, you are still talking about a boarder serving as a discontinuity between one region and the next. There is no way to mathematically derive a discontinuity. Such discontinuities are merely proscribed for engineering purposes. But there is no process of derivation that results in a discontinuity. It is an illogical entity. For it would say that some entity has two different values at the same place. There must be a continuity for one thing to effect another that is some distance away.
However, there might be a minimum space-time distance between zero nodes of some function describing physical entities. This probably would serve your purposes since we could never measure anything less than the distance between such nodes.
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