Tom,
One possible application, mentioned in the article, is radar beams for which the angle-of-arrival at target does not reveal the actual location of the emitter. I guess the same would hold for higher energies (weapons).
chromodynamic equivalent of the lorentz force?
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happyjack27
- Posts: 1439
- Joined: Wed Jul 14, 2010 5:27 pm
here are some considerations to make this a little bit more formal:
in 3 dimensions, if you have 2 points each traveling at fixed velocity, and they can "sense" changes in their distance...
moreover they can "sense" changes in a variational sense w/respect to hypothetical acceleration. that is, particle a knows how much its distance from particle b would change with time if they both went on their normal course, but it also knows how much this would change if in addition it accelerated in any given direction. so in this sense, in addition to its knowledge of its current distance, it's surrounded by a spherical field of scalars - the scalars representing the differential change in distance to point b if it suddenly accelerated in that direction.
now this information, in 3-dimensions, is enough to uniquely determine the orientation of 2 vectors from point a: the vector going from a to b, and the vector representing the cross-product of a and b.
if we presume that this is all the information that can be known by the point, that directly implies that all forces must act in one of these two directions. (to first-order.)
and that suffices to explain the orientation of the electromagnetic force as well as gravity.
but now what happens when we add another spatial dimension? well now we still get the vector from a to b uniquely determined by what is now our 4-sphere of scalars, but what was once simply a cross product that gave us a vector, no longer uniquely determines an orientation. it uniquely determines a _plane_, but not an orientation in that plane. we still have a full circle of possibilities.
so now we can do two things. we can leave it at that and say "fine, well, then the force represents an expansion/contraction through that plane." that is we accept _all_ answers as correct and now our "point" is really some spatial function that diffuses or anti-diffuses (contracts or expands) across the plane, in a radially symmetric fashion (at least on average).
another way we can do it is say that we simply need another moving point to uniquely determine the orientation of forces (vectors of acceleration) that can be transmitted to our point "a". now this may be overcomplete, but putting that aside for now (after all, we can always find a higher number of dimensions where it is not, we are just looking for basic principles right now). so now we have 2, arguably 3 or even 4, uniquely determined orientations at which force can act.
notice that means where before forces were just a particle-to-particle interaction, now they may (MUST!) involve an interplay of at least 3 particles. this has profound implications.
not the least of which is, where before we were talking about cross product, which if you recall the magnitude is proportional to the area of a parrellegram cut out by them, now we are talking about something more like the _volume_ cut out by three vectors, so our forces are going to act much differently with changes in distance and speed.
also, and possibly more profoundly, the geometry of harmonic motion has taken on a whole new dimension (and not just literally). there may be harmonic motion of 3 points together. or maybe as in our alternative formulation, a point and a circle, or two circles? or in higher dimensions spheres and so forth!
so the basic idea is to start with this variational idea of sensing differential changes in distance and rate of change of distance given accelerations around the n-sphere of a point's velocity, and from there determining what the possible orientations of forces are. then you just have to fit differential functions to those orientations and - assuming a point can only sense a variational infinitesimal n-sphere around it (a rather weak and reasonable assumption)) you've fully described all of the physics that are possible! (well, to first order, but second order, etc, follows from there.)
in 3 dimensions, if you have 2 points each traveling at fixed velocity, and they can "sense" changes in their distance...
moreover they can "sense" changes in a variational sense w/respect to hypothetical acceleration. that is, particle a knows how much its distance from particle b would change with time if they both went on their normal course, but it also knows how much this would change if in addition it accelerated in any given direction. so in this sense, in addition to its knowledge of its current distance, it's surrounded by a spherical field of scalars - the scalars representing the differential change in distance to point b if it suddenly accelerated in that direction.
now this information, in 3-dimensions, is enough to uniquely determine the orientation of 2 vectors from point a: the vector going from a to b, and the vector representing the cross-product of a and b.
if we presume that this is all the information that can be known by the point, that directly implies that all forces must act in one of these two directions. (to first-order.)
and that suffices to explain the orientation of the electromagnetic force as well as gravity.
but now what happens when we add another spatial dimension? well now we still get the vector from a to b uniquely determined by what is now our 4-sphere of scalars, but what was once simply a cross product that gave us a vector, no longer uniquely determines an orientation. it uniquely determines a _plane_, but not an orientation in that plane. we still have a full circle of possibilities.
so now we can do two things. we can leave it at that and say "fine, well, then the force represents an expansion/contraction through that plane." that is we accept _all_ answers as correct and now our "point" is really some spatial function that diffuses or anti-diffuses (contracts or expands) across the plane, in a radially symmetric fashion (at least on average).
another way we can do it is say that we simply need another moving point to uniquely determine the orientation of forces (vectors of acceleration) that can be transmitted to our point "a". now this may be overcomplete, but putting that aside for now (after all, we can always find a higher number of dimensions where it is not, we are just looking for basic principles right now). so now we have 2, arguably 3 or even 4, uniquely determined orientations at which force can act.
notice that means where before forces were just a particle-to-particle interaction, now they may (MUST!) involve an interplay of at least 3 particles. this has profound implications.
not the least of which is, where before we were talking about cross product, which if you recall the magnitude is proportional to the area of a parrellegram cut out by them, now we are talking about something more like the _volume_ cut out by three vectors, so our forces are going to act much differently with changes in distance and speed.
also, and possibly more profoundly, the geometry of harmonic motion has taken on a whole new dimension (and not just literally). there may be harmonic motion of 3 points together. or maybe as in our alternative formulation, a point and a circle, or two circles? or in higher dimensions spheres and so forth!
so the basic idea is to start with this variational idea of sensing differential changes in distance and rate of change of distance given accelerations around the n-sphere of a point's velocity, and from there determining what the possible orientations of forces are. then you just have to fit differential functions to those orientations and - assuming a point can only sense a variational infinitesimal n-sphere around it (a rather weak and reasonable assumption)) you've fully described all of the physics that are possible! (well, to first order, but second order, etc, follows from there.)